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+DEEP BIOLOGICAL PATHWAY INFORMED PATHOLOGY-
+GENOMIC MULTIMODAL SURVIVAL PREDICTION
+Lin Qiu1, Aminollah Khormali2, Kai Liu1
+1 Division of Research and Early Development, Genentech
+{qiul13, liuk3}@gene.com
+2 Department of Electrical and Computer Engineering, University of Central Florida
+aminollah.khormali@gmail.com
+ABSTRACT
+The integration of multi-modal data, such as pathological images and genomic data,
+is essential for understanding cancer heterogeneity and complexity for personalized
+treatments, as well as for enhancing survival predictions. Despite the progress made
+in integrating pathology and genomic data, most existing methods cannot mine the
+complex inter-modality relations thoroughly. Additionally, identifying explainable
+features from these models that govern preclinical discovery and clinical prediction
+is crucial for cancer diagnosis, prognosis, and therapeutic response studies. We
+propose PONET- a novel biological pathway informed pathology-genomic deep
+model that integrates pathological images and genomic data not only to improve
+survival prediction but also to identify genes and pathways that cause different
+survival rates in patients. Empirical results on six of The Cancer Genome Atlas
+(TCGA) datasets show that our proposed method achieves superior predictive
+performance and reveals meaningful biological interpretations. The proposed
+method establishes insight into how to train biologically informed deep networks on
+multimodal biomedical data which will have general applicability for understanding
+diseases and predicting response and resistance to treatment.
+1
+INTRODUCTION
+Manual examination of hematoxylin and eosin (H&E)-stained slides of tumor tissue by pathologists
+is currently the state-of-the-art for cancer diagnosis (Chan, 2014). The recent advancements in deep
+learning for digital pathology have enabled the use of whole-slide images (WSIs) for computational
+image analysis tasks, such as cellular segmentation (Pan et al., 2017; Hou et al., 2020), tissue
+classification and characterization (Hou et al., 2016; Hekler et al., 2019; Iizuka et al., 2020). While
+H&E slides are important and sufficient to establish a profound diagnosis, genomics data can provide
+a deep molecular characterization of the tumor, potentially offering the chance for prognostic and
+predictive biomarker discovery.
+Cancer prognosis via survival outcome prediction is a standard method used for biomarker discovery,
+stratification of patients into distinct treatment groups, and therapeutic response prediction (Cheng
+et al., 2017; Ning et al., 2020). WSIs exhibit enormous heterogeneity and most approaches adopt a
+two-stage multiple instance learning-based (MIL) approach for the representation learning of WSIs.
+Firstly, instance-level feature representations are extracted from image patches in the WSI, and then
+global aggregation schemes are applied to the bag of instances to obtain a WSI-level representation
+for subsequent modeling purpose (Hou et al., 2016; Courtiol et al., 2019; Wulczyn et al., 2020; Lu
+et al., 2021). Therefore, multimodal survival prediction faces an additional challenge due to the
+large data heterogeneity gap between WSIs and genomics, and many existing approaches use simple
+multimodal fusion mechanisms for feature integration, which prevents mining important multimodal
+interactions (Mobadersany et al., 2018; Chen et al., 2022b;a).
+The incorporation of biological pathway databases in a model takes advantage of leveraging prior
+biological knowledge so that potential prognostic factors of well-known biological functionality can
+be identified (Hao et al., 2018). Moreover, encoding biological pathway information into the neural
+1
+arXiv:2301.02383v1  [q-bio.QM]  6 Jan 2023
+
+Figure 1: Overview of PONET model.
+networks achieved superior predictive performance compared with established models (Elmarakeby
+et al., 2021).
+Based on the current challenges in multimodal fusion of pathology and genomics and the potential
+prognostic interpretation to link pathways and clinical outcomes in pathway-based analysis, we
+propose a novel biological pathway-informed pathology-genomic deep model, PONET, that uses H&E
+WSIs and genomic profile features for survival prediction. The proposed method contains four major
+contributions: 1) PONET formulates a biological pathway-informed deep hierarchical multimodal
+integration framework for pathological images and genomic data; 2) PONET captures diverse and
+comprehensive modality-specific and cross-modality relations among different data sources based
+on the factorized bilinear model and graph fusion network; 3) PONET reveals meaningful model
+interpretations on both genes and pathways for potential biomarker and therapeutic target discovery;
+PONET also shows spatial visualization of the top genes/pathways which has enormous potential
+for novel and prognostic morphological determinants; 4) We evaluate PONET on six public TCGA
+datasets which showed superior survival prediction comparing to state-of-the-art methods. Fig. 1
+shows our model framework.
+2
+RELATED WORK
+Multimodal Fusion. Earlier works on multimodal fusion focus on early fusion and late fusion. Early
+fusion approaches fuse features by simple concatenation which cannot fully explore intra-modality
+dynamics (Wöllmer et al., 2013; Poria et al., 2016; Zadeh et al., 2016). In contrast, late fusion
+fuses different modalities by weighted averaging which fails to model cross-modal interactions
+(Nojavanasghari et al., 2016; Kampman et al., 2018). The exploitation of relations within each
+modality has been successfully introduced in cancer prognosis via bilinear model (Wang et al., 2021b)
+and graph-based model (Subramanian et al., 2021). Adversarial Representation Graph Fusion (ARGF)
+(Mai et al., 2020) interprets multimodal fusion as a hierarchical interaction learning procedure where
+firstly bimodal interactions are generated based on unimodal dynamics, and then trimodal dynamics
+are generated based on bimodal and unimodal dynamics. We propose a new hierarchical fusion
+framework with modality-specific and cross-modality attentional factorized bilinear modules to
+mine the comprehensive modality interactions. Our proposed hierarchical fusion framework is
+different from ARGF in the following ways: 1) We take the sum of the weighted modality-specific
+representation as the unimodal representation instead of calculating the weighted average of the
+modality-specific representation in ARGF; 2) For higher level’s fusion, ARGF takes the original
+embeddings of each modality as input while we use the weighted modality-specific representations;
+3) We argue that ARGF takes redundant information during their trimodal dynamics.
+Multimodal Survival Analysis. There have been exciting attempts on multimodal fusion of pathol-
+ogy and genomic data for cancer survival prediction (Mobadersany et al., 2018; Cheerla & Gevaert,
+2019; Wang et al., 2020). However, these multimodal fusion based methods fail to model the interac-
+tion between each subset of multiple modalities explicitly. Kronecker product considers pairwise
+interactions of two input feature vectors by producing a high-dimensional feature of quadratic ex-
+pansion (Zadeh et al., 2017), and showed its superiority in cancer survival prediction (Wang et al.,
+2021b; Chen et al., 2022b;a). Despite promising results, using Kronecker product in multimodal
+2
+
+Unimodal
+ Gene layer  Pathway layer Hidden layer
+MFB
+>C
+Atten
+um
+hm
+hm
+hm
+↑
+Bimodal
+himz
+Gene
+hg
+Gene
+MFB
+α
+Multimodal
+Atten
+zuuy
+Wm1m2
+Representation
+hm1m2
+r
+hm
+Cox
+Pathway
+Pathology
+Trimodal
+↓
+hp
+hmi
+fe
+MFB
+Atten
+Wm1m2m3
+ hm1m2m3
+CNV + MUT
+hc
+Spatial Pathway
+Data embedding
+Hierarchical multimodal fusion
+Interpretationfusion may introduce a large number of parameters that may lead to high computational cost and risk
+of overfitting (Kim et al., 2017; Liu et al., 2021), thus limiting its applicability and improvement in
+performance. To overcome this drawback, hierarchical factorized bilinear fusion for cancer survival
+prediction (HFBSurv) (Li et al., 2022) uses factorized bilinear model to fuse genomic and image
+features, dramatically reducing computational complexity. PONET differs from HFBSurv in two
+ways: 1) PONET’s multimodal framework has three levels of hierarchical fusion module includ-
+ing unimodal, bimodal, and trimodal fusion while HFBSurv only considers within-modality and
+cross-modality fusion which we argue it is not adequate for mining the comprehensive interactions;
+2) PONET leverages biological pathway informed network for better prediction and meaningful
+interpretation purposes.
+Pathway-associated Sparse Neural Network. The pathway-based analysis is an approach that a
+number of studies have investigated to improve both predictive performance and biological inter-
+pretability (Jin et al., 2014; Cirillo et al., 2017; Hao et al., 2018; Elmarakeby et al., 2021). Moreover,
+pathway-based approaches have shown more reproducible analysis results than gene expression data
+analysis alone (Li et al., 2015; Mallavarapu et al., 2017). These pathway-based deep neural networks
+can only model genomic data which severely inhibits their applicability in current biomedical re-
+search. Additionally, the existing pathway-associated sparse neural network structures are limited
+for disease mechanism investigation: there is only one pathway layer in PASNet (Hao et al., 2018)
+which contains limited biological prior information to deep dive into the hierarchical pathway and
+biological process relationships; P-NET (Elmarakeby et al., 2021) calculates the final prediction by
+taking the average of all the gene and pathway layers’ outputs, and this will bias the learning process
+because it will put more weights for some layers’ outputs while underestimating the others.
+3
+METHODOLOGY
+3.1
+PROBLEM FORMULATION AND NOTATIONS
+The model architecture of PONET is presented in Fig. 1, where three modalities are included as input:
+gene expression g ∈ Rdg, pathological image p ∈ Rdp, and copy number (CNV) + mutation (MUT)
+CNV + MUT ∈ Rdc, with dp being the dimensionality of p and so on. We define a hierarchical
+factorized bilinear fusion model for PONET. We build a sparse biological pathway-informed embed-
+ding network for gene expression. A fully connected (FC) embedding layer for both preprocessed
+pathological image feature (fp) and the copy number + mutation (fc) to map feature into similar
+embedding space for alleviating the statistical property differences between modalities, the three
+network architecture details are in Appendix C.1. We label the three modality embeddings as hm,
+m ∈ {g, p, c}, the superscript/subscript u, b, and t represents unimodal fusion, biomodal fusion and
+trimodal fusion. After that, the embeddings of each modality are first used as input for unimodal fusion
+to generate the modality-specific representation hu
+m = ωmˆhm, ωm represent the modality-specific im-
+portance, the feature vector of the unimodal fusion is the sum of all modality-specific representations
+hu = �
+m hu
+m. In the bimodal fusion, modality-specific representations from the output of unimodal
+fusion are fused to yield cross-modality representations hb
+m1m2 = ωm1m2ˆhm1m2, m1, m2 ∈ {p, c, g}
+and m1 ̸= m2, ωm1m2 represents the corresponding cross-modality importance. Similarly, the feature
+vector of bimodal fusion is calculated as hb = �
+m1,m2 hb
+m1m2. We propose to build a trimodal
+fusion to take each cross-modality representation from the output of bimodal fusion to mine the
+interactions. Similarly to the bimodal fusion architecture, the trimodal fusion feature vector will
+be ht = �
+m1,m2,m3 ωm1m2m3ˆhm1m2m3, m1, m2, m3 ∈ {p, c, g} and m1 ̸= m2 ̸= m3, ωm1m2m3
+represents the corresponding trimodal importance. Finally, PONET concatenates hu, hb, ht to obtain
+the final comprehensive multimodal representation and pass it to the Cox proportional hazards model
+(Cox, 1972; Cheerla & Gevaert, 2019) for survival prediction. In the following sections we will
+describe our hierarchical factorized bilinear fusion framework, l, o, s represents the dimensionality
+of hm, zm, ˆhm1m2.
+3.2
+SPARSE NETWORK
+We design the sparse gene-pathway network consisting of one gene layer followed by three pathway
+layers. A patient sample of e gene expressions is formed as a column vector, which is denoted by
+X = [x1, x2, ..., xe], each node represents one gene. The gene layer is restricted to have connections
+3
+
+73,703 x 50,706 px
+224 x 224 px, mpp: 0.5
+Whole Slide Image
+WSI patching
+Image Augmentation
+𝑔!!(#)
+𝑞!!(#)
+𝑔!"(#)
+𝑓!!(#)
+𝑓!"(#)
+𝑦"
+'𝑦#
+𝑧"
+̂𝑧#
+𝑝"
+𝐿(𝑝, 𝑧)
+Visual representation learning using SSL ViT
+Student Network
+Teacher Network
+Patch features
+𝑣
+𝑢
+Figure 2: Overall framework of the visual representation extraction using pre-trained self-supervised
+vision transformer.
+reflecting the gene-pathway relationships curated by the Reactome pathway dataset (Fabregat et al.,
+2020). The connections are encoded by a binary matrix M ∈ Ra×e, where a is the number of
+pathways and e is the number of genes, an element of M, mij, is set to one if gene j belongs to
+pathway i. The connections that do not exist in the Reactome pathway dataset will be zero-out. For
+the following pathway-pathway layers, a similar scheme is applied to control the connection between
+consecutive layers to reflect the parent-child hierarchical relationships that exist in the Reactome
+dataset. The output of each layer is calculated as
+y = f[(M ∗ W)T X + ϵ]
+(1)
+where f is the activation function, M represents the binary matrix, W is the weights matrix, X is the
+input matrix, ϵ is the bias vector, and ∗ is the Hadamard product. We use tanh for the activation of
+each node. We allow the information flow from the biological prior informed network starting from
+the first gene layer to the last pathway layer, and we label the last layer output embeddings of the
+sparse network for gene expression as hg.
+3.3
+UNIMODAL FUSION
+Bilinear models (Tenenbaum & Freeman, 2000) provide richer representations than linear models.
+Given two feature vectors in different modalities, e.g., the visual features x ∈ Rm×1 for an image and
+the genomic features y ∈ Rn×1 for a genomic profile, the bilinear model uses a quadratic expansion
+of linear transformation considering every pair of features:
+zi = xT Wiy
+(2)
+where Wi ∈ Rm×n is a projection matrix, zi ∈ R is the output of the bilinear model. Bilinear
+models introduce a large number of parameters which potentially lead to high computational cost
+and overfitting risk. To address these issues, Yu et al. (2017) develop the Multi-modal Factorized
+Bilinear pooling (MFB) method, which enjoys the dual benefits of compact output features and robust
+expressive capacity.
+Inspired by the MFB (Yu et al., 2017) and its application in pathology and genomic multimodal
+learning (Li et al., 2022), we propose unimodal fusion to capture modality-specific representations
+and quantify their importance. The unimodal fusion takes the embedding of each modality hm as
+input and factorizes the projection matrix Wi in Eq. (2) as two low-rank matrices:
+zi
+=
+hT
+mWihm =
+k�
+d=1
+hT
+mum,dvT
+m,dhm
+=
+1T (U T
+m,ihm ◦ V T
+m,ihm), m ∈ {p, c, g}
+(3)
+we get the output feature zm:
+zm = SumPooling
+�
+˜U T
+mhm◦ ˜V T
+mhm, k
+�
+, m ∈ {p, c, g}
+(4)
+where k is the latent dimensionality of the factorized matrices. SumPooling (x, k) function performs
+sum pooling over x by using a 1-D non-overlapped window with the size k, ˜Um ∈ Rl×ko and
+˜Vm ∈ Rl×ko are 2-D matrices reshaped from Um and Vm, Um =[Um,1, . . . , Um,h] ∈ Rl×k×o and
+Vm = [Vm,1, . . . , Vm,h] ∈ Rl×k×o. Each modality-specific representation ˆhm ∈ Rl+o is obtained
+as:
+ˆhm = hm©zm, m ∈ {p, c, g}
+(5)
+4
+
+where © denotes vector concatenation. We also introduce a modality attention network Atten ∈
+Rl+o → R1 to determine the weight for each modality-specific representation to quantify its impor-
+tance:
+ωm = Atten(ˆhm; ΘAtten), m ∈ {p, c, g}
+(6)
+where ωm is the weight of modality m. In practice, Atten consists of a sigmoid activated dense layer
+parameterized by ΘAtten. Therefore, the output of each modality in unimodal fusion, hu
+m, is denoted
+as ωmˆhm ∈ Rl+o, m ∈ {p, c, g}. Accordingly, the output of unimodal fusion, hu, is the sum of each
+weighted modality-specific representation ωmˆhm, m ∈ {p, c, g} which is different from ARGF (Mai
+et al., 2020) that used the weighted average of different modalities as the unimodal fusion output.
+3.4
+BIMODAL AND TRIMODAL FUSION
+Bimodal fusion aims to fuse diverse information of different modalities and quantify different
+importance for them. After receiving the modality-specific representations hu
+m from the unimodal
+fusion, we can generate the cross-modality representation ˆhm1m2 ∈ Rs similar to Eq. (4) :
+ˆhm1,m2 = Sum Pooling
+�
+˜U T
+m1hu
+m1◦ ˜V T
+m2hu
+m2, k
+�
+,
+m1, m2 ∈ {p, c, g}, m1 ̸= m2
+(7)
+where
+˜U T
+m1 ∈ R(l+o)×ks and ˜V T
+m2 ∈ R(l+o)×ks are 2-D matrices reshaped from Um1 and Vm2
+and Um1 = [Um1,1, . . . , Um1,s] ∈ R(l+o)×k×s and Vm2 = [Vm2,1, . . . , Vm2,s] ∈ R(l+o)×k×s. We
+leverage a bimodal attention network (Mai et al., 2020) to identify the importance of the cross-
+modality representation. The similarity Sm1m2 ∈ R1 of hu
+m1 and hu
+m2 is first estimated as follows:
+Sm1,m2 =
+l+o
+�
+i=1
+�
+eωm1 hu
+m1,i
+�l+o
+j=1 eωm1hu
+m1,j
+� �
+eωm2hu
+m2,i
+�l+o
+j=1 eωm2hu
+m2,j
+�
+(8)
+where the computed similarity is in the range of 0 to 1. Then, the cross-modality importance ωm1m2
+is obtained by:
+ωm1m2 =
+eˆωmimj
+�
+mi̸=mj eˆωmimj , ˆωm1m2 = ωm1 + ωm2
+Sm1m2 + S0
+(9)
+where S0 represents a pre-defined term controlling the relative contribution of similarity and modality-
+specific importance, and here is set to 0.5. Therefore, the output of bimodal fusion, hb, is the sum of
+each weighted cross-modality representation ωm1m2ˆhm1m2, m1, m2 ∈ {p, c, g} and m1 ̸= m2.
+In trimodal fusion, each bimodal fusion output is fused with the unimodal fusion output that does
+not contribute to the formation of the bimodal fusion. The output for each corresponding trimodal
+representation is ˆhm1m2m3. In addition, trimodal attention was applied to identify the importance of
+each trimodal representation, ωm1m2m3. The output of the trimodal fusion, ht, is the sum of each
+weighted trimodal representation ωm1m2m3ˆhm1m2m3, m1, m2, m3 ∈ {p, c, g} and m1 ̸= m2 ̸= m3.
+3.5
+SURVIVAL LOSS FUNCTION
+We train the model through the Cox partial likelihood loss (Cheerla & Gevaert, 2019) with l1
+regularization for survival prediction, which is defined as:
+ℓ(Θ) = −
+�
+i:Ei=1
+�
+�ˆhΘ (xi) − log
+�
+j:Ti>Tj
+exp
+�
+ˆhΘ (xj)
+�
+�
+� + λ (∥Θ∥1)
+(10)
+where the values Ei, Ti and xi for each patient represent the survival status, the survival time, and the
+feature, respectively. Ei = 1 means event while Ei = 0 represents censor. ˆhΘ is the neural network
+model trained for predicting the risk of survival, Θ is the neural network model parameters, and λ is
+a regularization hyperparameter to avoid overfitting.
+5
+
+4
+EXPERIMENTS
+4.1
+EXPERIMENTAL SETUP
+Datasets. To validate our proposed method, we used six cancer datasets from The Cancer Genome
+Atlas (TCGA), a public cancer data consortium that contains matched diagnostic WSIs and genomic
+data with labeled survival times and censorship statuses. The genomic profile features (mutation
+status, copy number variation, RNA-Seq expression) are preprocessed by Porpoise 1 (Chen et al.,
+2022b). For this study, we used the following cancer types: Bladder Urothelial Carcinoma (BLCA)
+(n = 437), Kidney Renal Clear Cell Carcinoma (KIRC) (n = 350), Kidney Renal Papillary Cell
+Carcinoma (KIRP) (n = 284), Lung Adenocarcinoma (LUAD) (n = 515), Lung Squamous Cell
+Carcinoma (LUSC) (n = 484), Pancreatic adenocarcinoma (PAAD) (n = 180). We downloaded the
+same diagnostic WSIs from the TCGA website 2 that were used in Porpoise study to match the
+paired genomic features and survival times. The feature alignment table for all the cancer types
+is in Appendix A. For each WSI, automated segmentation of tissue was performed. Following
+segmentation, image patches of size 224 × 224 were extracted without overlap at the 20 X equivalent
+pyramid level from all tissue regions identified while excluding the white background and selecting
+only patches with at least 50% tissue regions. Subsequently, a visual representation of those patches
+is extracted with a vision transformer (Wang et al., 2021a) pre-trained on the TCGA dataset through
+a self-supervised constructive learning approach, such that each patch is represented as a 1 × 2048
+vector. Fig. 2 shows the framework for the visual representation extraction by vision transformer
+(VIT). Survival outcome information is available at the patient level, we aggregated the patch-level
+feature into slide level feature representations based on an attention-based method (Lu et al., 2021;
+Ilse et al., 2018).
+Baselines. Using the same 5-fold cross-validation splits for evaluating PONET, we implemented and
+evaluated six state-of-the-art methods for survival outcome prediction. Additionally, we included
+three variations of PONET: a) PONET-O represents only genomic data, and pathway architecture
+for the gene expression are included in the model; b) PONET-OH represents only genomic and
+pathological image data but without pathway architecture in the model; c) PONET is our full model.
+For all methods, we use the same VIT feature extraction pipeline for WSIs, as well as identical
+training hyperparameters and loss functions for supervision. Training details and the parameters
+tuning can be found in Appendix C.2.
+CoxPH (Cox, 1972) represents the standard Cox proportional hazard models.
+DeepSurv (Katzman et al., 2018) is the deep neural network version of the CoxPH model.
+Pathomic Fusion (Chen et al., 2022a) as a pioneered deep learning-based framework for predicting
+survival outcomes by fusing pathology and genomic multimodal data, in which Kronecker product is
+taken to model pairwise feature interactions across modalities.
+GPDBN (Wang et al., 2021b) adopts Kronecker product to model inter-modality and intra-modality
+relations between pathology and genomic data for cancer prognosis prediction.
+HFBSurv (Li et al., 2022) extended GPDBN using the factorized bilinear model to fuse genomic
+and pathology features in a within-modality and cross-modalities hierarchical fusion.
+Porpoise (Chen et al., 2022b) applied the discrete survival model and Kronecker product to fuse
+pathology and genomic data for survival prediction (Zadeh & Schmid, 2020).
+Evaluation. For each cancer dataset, we used the cross-validated concordance index (C-Index)
+(Appendix B.1) (Harrell et al., 1982) to measure the predictive performance of correctly ranking the
+predicted patient risk scores with respect to overall survival.
+4.2
+RESULTS
+Comparison with Baselines. In combing pathology image, genomics, and pathway network via
+PONET, our approach outperforms CoxPH models, unimodal networks, and previous deep learning-
+based approaches on pathology-genomic-based survival outcome prediction (Table 1). The results
+show that deep learning-based approaches generally perform better than the CoxPH model. PONET
+achieves superior C-index values in all six cancer types. All versions of PONET outperform Pathomic
+1https://github.com/mahmoodlab/PORPOISE
+2https://www.cancer.gov/about-nci/organization/ccg/research/structural-genomics/tcga
+6
+
+Table 1: C-Index (mean ± standard deviation) of PONET and ablation experiments in TCGA survival
+prediction. The top two performers are highlighted in bold.
+Model
+TCGA-BLCA
+TCGA-KIRC
+TCGA-KIRP
+TCGA-LUAD
+TCGA-LUSC
+TCGA-PAAD
+CoxPH (Age + Gender) (Cox, 1972)
+0.525 ± 0.130
+0.550 ± 0.070
+0.544 ± 0.050
+0.531 ± 0.082
+0.532 ± 0.094
+0.539 ± 0.092
+DeepSurv (Kampman et al., 2018)
+0.580 ± 0.062
+0.620 ± 0.043
+0.560± 0.063
+0.534 ±0.077
+0.541 ± 0.066
+0.544 ± 0.076
+GPDBN (Wang et al., 2021b)
+0.612 ± 0.042
+0.647 ± 0.073
+0.669 ± 0.109
+0.565 ± 0.057
+0.545 ± 0.063
+0.571 ± 0.060
+HFBSurv (Li et al., 2022)
+0.622 ± 0.043
+0.667 ± 0.053
+0.769 ± 0.109
+0.581 ± 0.017
+0.548 ± 0.049
+0.591 ± 0.052
+Pathomic Fusion (Chen et al., 2022a)
+0.586 ± 0.062
+0.598 ± 0.060
+0.577 ± 0.032
+0.543 ± 0.065
+0.523 ±0.045
+0.545 ± 0.064
+Porpoise (Chen et al., 2022b)
+0.617 ± 0.048
+0.711 ± 0.051
+0.811 ± 0.089
+0.586 ±0.056
+0.527 ± 0.043
+0.591 ± 0.064
+PONET-O (ours)
+0.596 ± 0.056
+0.664 ± 0.056
+0.761 ± 0.093
+0.623 ±0.062
+0.538 ± 0.037
+0.598 ± 0.027
+PONET-OH (ours)
+0.625 ± 0.063
+0.695 ± 0.043
+0.776 ± 0.123
+0.618 ± 0.049
+0.553 ± 0.049
+0.591 ± 0.050
+PONET (ours)
+0.643 ± 0.037
+0.726 ± 0.056
+0.829 ± 0.054
+0.646 ±0.047
+0.567 ± 0.066
+0.639 ± 0.080
+Table 2: Evaluation of PONET on different fusion methods and pathway designs by C-index (mean
+± standard deviation). The best performer is highlighted in bold.
+Methods
+TCGA-BLCA
+TCGA-KIRP
+TCGA-LUAD
+TCGA-LUSC
+TCGA-PAAD
+Single fusion
+Simple concatenation
+0.585 ± 0.045
+0.652 ± 0.049
+0.554 ± 0.065
+0.525 ± 0.066
+0.568 ± 0.075
+Element-wise addition
+0.592 ± 0.047
+0.655 ± 0.055
+0.587 ± 0.065
+0.522 ± 0.046
+0.588 ± 0.055
+Tensor fusion (Zadeh et al., 2017)
+0.605 ± 0.046
+0.775 ± 0.053
+0.595 ± 0.060
+0.545 ± 0.045
+0.592 ± 0.061
+Hierarchical fusion
+Unimodal
+0.596 ± 0.035
+0.783 ± 0.063
+0.611 ± 0.056
+0.553 ± 0.073
+0.595 ± 0.053
+Bimodal
+0.602 ± 0.062
+0.789 ± 0.053
+0.601 ± 0.056
+0.552 ± 0.051
+0.598 ± 0.083
+ARGF (Mai et al., 2020)
+0.597 ± 0.054
+0.792 ± 0.043
+0.614 ± 0.051
+0.556 ± 0.063
+0.602 ± 0.065
+Unimodal + Bimodal
+0.614 ± 0.052
+0.803 ± 0.061
+0.631 ± 0.044
+0.578 ± 0.058
+0.615 ± 0.057
+Pathway design
+PASNet (Hao et al., 2018)
+0.606 ± 0.045
+0.793 ± 0.051
+0.621 ± 0.061
+0.551 ± 0.069
+0.625 ± 0.057
+P-NET (Elmarakeby et al., 2021)
+0.622 ± 0.047
+0.802 ± 0.071
+0.625 ± 0.045
+0.562 ± 0.054
+0.627 ± 0.073
+PONET
+0.643 ± 0.037
+0.829 ± 0.054
+0.641 ± 0.046
+0.567 ± 0.066
+0.639 ± 0.070
+Fusion by a big margin. Pathomic Fusion uses Kronecker product to fuse the two modalities, and
+that’s also the reason why other advanced fusion methods, like GPDBN and HFBSurv, achieve better
+performance. Also, we argue that Pathomic Fusion extracts the region of interest of pathology image
+for feature extraction might limit the understanding of the tumor microenvironment of the whole slide.
+HFBSurv shows better performance than GPDBN and Pathomic Fusion which is consistent with
+their findings, and these results further demonstrate that the hierarchical factorized bilinear model
+can better mine the rich complementary information among different modalities compared to the
+Kronecker product. Porpoise performs similarly with PONET on TCGA-KIRC and TCGA-KIRP and
+outperformed HFBSurv in these two studies, this probably is due to Porpoise partitioned the survival
+time into different non-overlapping bins and parameterized it as a discrete survival model (Zadeh
+& Schmid, 2020) which works better for these two cancer types. In other cases, Porpoise performs
+similarly to HFBSurv. Note: the results of Porpoise are from their paper (Chen et al., 2022b).
+Additionally, we can see that PONET consistently outperforms PONET-O and PONET-OH indi-
+cating the effectiveness of the biological pathway-informed neural network and the contribution of
+pathological image for the overall survival prediction.
+Ablation Studies. To assess whether the impact of hierarchical factorized bilinear fusion strategy
+is indeed effective, we compare PONET with four single-fusion methods: 1) Simple concatenation:
+concatenate each modality embeddings; 2) Element-wise addition: element-wise addition from each
+modality embeddings; 3) Tensor fusion (Zadeh et al., 2017): Kronecker product from each modality
+embeddings. Table 2 shows the C-index values of different methods. We can see that PONET
+achieves the best performance and shows remarkable improvement over single-fusion methods on
+different cancer type datasets. For example, PONET outperforms the Simple concatenation by
+8.4% (TCGA-BLCA), 27% (TCGA-KIRP), 15% (TCGA-LUAD), 8.0% (TCGA-LUSC), and 11.4%
+(TCGA-PAAD), etc.
+Furthermore, we adopted five different configurations of PONET to evaluate each hierarchical
+component of the proposed method: 1) Unimodal: unimodal fusion output as the final feature
+representation; 2) Bimodal: bimodal fusion output as the final feature representation; 3) Unimodal
++ Bimodal: hierarchical (include both unimodal and bimodal feature representation) fusion; 4)
+ARGF: ARGF (Mai et al., 2020) fusion strategy; 5) PONET: our proposed hierarchical strategy by
+incorporating unimodal, bimodal, and trimodal fusion output. As shown in Table 2, Unimodal +
+Bimodal performs better than Unimodal and Bimodal which demonstrates that Unimodal + Bimodal
+can capture the relations within each modality and across modalities. ARGF performs worse than
+Unimodal + Bimodal and far worse than PONET across all the cancer types. PONET outperforms
+7
+
+Figure 3: Inspecting and interpreting PONET on TCGA-KIRP. a: Sankey diagram visualization
+of inner layers of PONET shows the estimated relative importance of different nodes in each layer.
+Nodes in the first layer represent genes; the next layers represent pathways; and the final layer
+represents the model outcome. Different layers are linked by weights. Nodes with darker colors are
+more important, while transparent nodes represent the residual importance of undisplayed nodes
+in each layer, H1 presents the gene layer, and H2-H4 represent pathway layers; b: Co-attention
+visualization of top 4 ranked pathways in one case of TCGA-KIRP.
+Unimodal + Bimodal in 4 out of 5 cancer types indicating that three layers of hierarchical fusion can
+mine the comprehensive interactions among different modalities.
+To evaluate our sparse gene-pathway network design, we compare PONET with PASNet (Hao et al.,
+2018) and P-NET (Elmarakeby et al., 2021) pathway architecture, PASNet performs the worst due to
+the fact that it only has one pathway layer in the network, and thus limited prior information was used
+to predict the outcome. PONET constantly outperforms P-NET across all the cancer types, which
+demonstrates that averaging all the intermediate layers’ output for the final prediction cannot fully
+capture the prior information flow among the hierarchical biological structures.
+Model Interpretation. We discuss the model interpretation results for cancer type TCGA-KIRP
+here and the results for other cancer types are included in the Appendix C.3. To understand the
+interactions between different genes, pathways, and biological processes that contributed to the
+predictive performance and to study the paths of impact from the input to the outcome, we visualized
+the whole structure of PONET with the fully interpretable layers after training (Fig. 3 a). To evaluate
+the relative importance of specific genes contributing to the model prediction, we inspected the genes
+layer and used the Integrated Gradients attribution (Sundararajan et al., 2017) method to obtain
+the total importance score of genes, and the modified ranking algorithm details are included in the
+Appendix B.3. Highly ranked genes included KRAS, PSMB6, RAC1, and CTNNB1 which are known
+kidney cancer drivers previously (Yang et al., 2017; Shan et al., 2017; Al-Obaidy et al., 2020; Guo
+et al., 2022). GBN2, a member of the guanine nucleotide-binding proteins family, has been reported
+that the decrease of its expression reduced tumor cell proliferation (Zhang et al., 2019). A recent study
+identified a strong dependency on BCL2L1, which encodes the BCL-XL anti-apoptotic protein, in a
+subset of kidney cancer cells (Grubb et al., 2022). This biological interpretability revealed established
+and novel molecular features contributing to kidney cancer. In addition, PONET selected a hierarchy
+of pathways relevant to the model prediction, including downregulation of TGF-β receptor signaling,
+regulation of PTEN stability and activity, the NLRP1 inflammasome, and noncanonical activation of
+NOTCH3 by PSEN1, PSMB6, and BCL2L1. TGF-β signaling is increasingly recognized as a key
+driver in cancer, and in progressive cancer tissues TGF-β promotes tumor formation, and its increased
+expression often correlates with cancer malignancy (Han et al., 2018). Noncanonical activation of
+NOTCH3 was reported to limit tumor angiogenesis and plays a vital role in kidney disease (Lin et al.,
+2017).
+8
+
+a
+H1
+H2
+H3
+H4
+KRAS
+Downregulation of TGF-beta receptor signaling
+TGF-beta receptor signaling activates SMADs
+Neurodegenerative Diseases
+PSMB6
+Regulation of PTEN stability and activity
+NOTCH3 Activation and Transmission of Signal
+Cellular Senescence
+RAC1
+Calmodulin induced events
+Semaphorin interactions
+Signal amplification
+CTNNB1
+The NLRP1 inflammasome
+Downstream signaling events of B Cell Receptor
+Signaling by FGFR
+BCL2L1
+Noncanonical activation of NOTCH3
+M Phase
+LeadingStrand Synthesis
+GNB2
+Synthesis of PIPs in the nucleus
+Biosynthesis of the N-glycan precursor
+ER to Golgi Anterograde Transport
+outcome
+PSEN1
+Regulation of PTEN gene transcription
+Biosynthesis of DHA-derived SPMs
+Signaling by TGF-beta Receptor Complex in Cancer
+MTMR4
+RPA3
+Viral mRNA Translation
+Export of Viral Ribonucleoproteins from Nucleus
+Interferon Signaling
+HDAC3
+Complex I biogenesis
+ZBP1(DAI) mediated induction of type I IFNs
+Transportofvitaminsandnucleosides
+residual 
+Formation of tubulin folding intermediates by CCT/TriC Amine Oxidase reactions
+Transportofbilesalts,organicacids,andmetalions
+residual
+residual
+residual
+b
+TCGA-Q2-A5QZ
+Downregulation of TGF-beta
+Regulation of PTEN
+Survival Month: 14.06
+receptor signaling
+ stability and activity
+Calmodulin induced events
+The NLRP1 inflammasome
+High Attn
+Low AttnFigure 4: Kaplan-Meier analysis of patient stratification of low and high risk patients via four
+variations of PONET on TCGA-KIRP. Low and high risks are defined by the median 50% percentile
+of hazard predictions via each model prediction. Log-rank test was used to test for statistical
+significance in survival distributions between low and high risk patients.
+To further inspect the pathway spatial association with the WSI slide we adopted the co-attention
+survival method MCAT (Chen et al., 2021) between WSIs and genomic features on the top
+pathways of the second layer, visualized as a WSI-level attention heatmap for each pathway
+genomic embedding in Fig.
+3 b (algorithm details are included in the Appendix B.4).
+We
+used the gene list from the top 4 pathways as the genomic features and trained MCAT on the
+TCGA-KIRP dataset for survival prediction. Overall, we observe that high attention in different
+pathways showed different spatial pattern associations with the slide. This heatmap can reflect
+genotype-phenotype relationships in cancer pathology. The high attention regions (red) of dif-
+ferent pathways in the heatmap have positive associations with the predicted death risk while
+the low attention regions (blue) have negative associations with the predicted risk. By further
+checking the cell types in high attention patches we can gain insights of prognostic morpho-
+logical determinants and have a better understanding of the complex tumor microenvironment.
+Table 3: Comparison of model complexity
+Methods
+Number of Parameters
+FLOPS
+Pathomic Fusion
+175M
+168G
+GPDBN
+82M
+91G
+HFBSurv
+0.3M
+0.5G
+PONET
+2.8M
+3.1G
+Patient Stratification.
+In visualizing
+the Kaplan-Meier survival curves of pre-
+dicted high risk and low risk patient
+populations, we plot four variations of
+PONET in Fig. 4. PONET-ARGF rep-
+resents the model that we use the hier-
+archical fusion strategy of ARGF in our
+pathway-informed PONET model. From
+the results, PONET enables easy sepa-
+ration of patients into low and high risk
+groups with remarkably better stratifica-
+tion (P-Value = 6.60e-7) in comparison to the others.
+Complexity Comparison. We compared PONET with Pathomic Fusion, GPDBN, and HFBSurv
+since both Pathomic Fusion and GPDBN are based on Kronecker product to fuse different modalities
+while GPDBN and HFBSurv modeled inter-modality and intra-modality relations which have similar
+consideration to our method. As illustrated in Table 3, PONET has 2.8M (M = Million) trainable
+parameters, which is approximately 1.6%, 3.4%, and 900% of the number of parameters of Pathomic
+Fusion, GPDBN, and HFBSurv. To assess the time complexity of PONET and the competitive
+methods, we calculate each method’s floating-point operations per second (FLOPS) in testing. The
+results in Table 3 show that PONET needs 3.1G during testing, compared with 168G, 91G, and 0.5G
+in Pathomic Fusion, GPDBN, and HFBSurv. The main reason for fewer trainable parameters and the
+number of FLOPS lies in that PONET and HFBSurv perform multimodal fusion using the factorized
+bilinear model, and can significantly reduce the computational complexity and meanwhile obtain
+more favorable performance. PONET has one additional trimodal fusion which explains why it has
+more trainable parameters than HFBSurv.
+5
+CONCLUSION
+In this study, we pioneer propose a novel biological pathway-informed hierarchical multimodal
+fusion model that integrates pathology image and genomic profile data for cancer prognosis. In
+comparison to previous works, PONET deeply mines the interaction from multimodal data by
+conducting unimodal, bimodal and trimodal fusion step by step. Empirically, PONET demonstrates
+9
+
+PONET-O
+PONET-OH
+PONET-ARGF
+PONET
+1.0 -
+1.0
+1.0
+P-Value =1.90e-3
+P-Value =7.49e-4
+P-Value =4.27e-5
+ P-Value =6.60e-7
+0.9
+0.9
+0.9
+0.9
+0.8
+0.8
+0.8
+0.8
+0.7
+0.7
+Proportion s
+0.7
+0.7
+0.6
+0.6
+0.6
+0.6
+0.5
+0.5
+0.5
+0.5
+Low risk
+0.4
+Low risk
+0.4
+Low risk
+Low risk
+High risk
+0.4
+High risk
+High risk
+High risk
+0.3
+0
+25
+50
+75
+100
+125
+0
+25
+50
+75
+100
+125
+0
+25
+50
+75
+100
+125
+0
+25
+50
+75
+100
+125
+Time (months)the effectiveness of the model architecture and the pathway-informed network for superior predictive
+performance. Specifically, PONET provides insight on how to train biologically informed deep
+networks on multimodal biomedical data for biological discovery in clinic genomic contexts which
+will be useful for other problems in medicine that seek to combine heterogeneous data streams for
+understanding diseases and predicting response and resistance to treatment.
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+13
+
+Table 4: TCGA Data Feature Alignment Summary
+WSI
+CNV
+MUT
+RNA
+WSI+CNV+MUT
+WSI+MUT+RNA
+ALL
+Cancer Type
+BLCA
+454
+443
+452
+450
+441
+448
+437
+KIRC
+517
+509
+357
+514
+352
+355
+350
+KIRP
+294
+291
+286
+293
+284
+285
+284
+LUAD
+528
+522
+523
+522
+519
+519
+515
+LUSC
+505
+502
+489
+503
+486
+487
+484
+PAAD
+208
+201
+187
+195
+187
+180
+180
+A
+DATA
+Table 3 in Appendix A shows the number of patients with matched different data modalities: WSI
+(Whole slide image), CNV (Copy number), MUT (Mutation), RNA (RNA-Seq gene expression). For
+each TCGA dataset and each patient we have preprocessed data dimensions dg ∈ R1×2000 (RNA),
+dc ∈ R1×227 (CNV + MUT), and dp ∈ R1×32 (WSI) which will be used for our multimodal fusion.
+B
+METHODS
+B.1
+C-INDEX
+We use concordance-index (C-index) (Harrell et al., 1982) to measure the performance of survival
+models. It evaluates the model by measuring the concordance of the ranking of predicted harzards
+with the true survival time of patients. The range of the C-index is [0, 1], and larger values indicate
+better performance with a random guess leading to a C-index of 0.5.
+B.2
+WSI REPRESENTATION LEARNING
+It has been shown that the WSI visual representations extracted by self-supervised learning methods
+on histopathological images are more accurate and transferable than the supervised baseline models
+on domain-irrelevant datasets such as ImageNet. In this work, a pre-trained Vision Transformer (ViT)
+model (Wang et al., 2021a) that is trained on a large histopathological image dataset has been utilized
+for tile feature extraction. The model is composed of two main neural networks that learn from each
+other, i.e., student and teacher networks. Parameters of the teacher model θt are updated using the
+student network with parameter θs using the update rule represented in Eq. (11).
+θt ← τθt + (1 − τ)θs
+(11)
+Two different views of a given input H&E image x, uniformly selected from the training set I, are
+generated using random augmentations, i.e., u, v. Then, student and teacher models generate two
+different visual representations according to u and v as y1 = f θs (u) and ˆy2 = f θt (v), respectively.
+Finally, the generated visual representations are transformed into latent space using linear projection as
+p1 = gθs �
+gθs (y1)
+�
+and ˆz2 = gθt (ˆy2) for student and teacher networks, respectively. Similarly, feed-
+ing v and u to student and teacher networks leads to y2 = f θs (v) , ˆy1 = f θt (u) , p2 = gθs �
+gθs (y2)
+�
+and ˆz1 = gθt ( ˆy1). Finally, the symmetric objective function Lloss is optimized through minimizing
+the ℓ2 − norm distance between student and teacher as Eq. (12)
+Lloss = 1
+2L (p1, ˆz2) + 1
+2L (p2, ˆz1)
+(12)
+where L(p, z) = −
+p
+∥p∥2 ·
+z
+∥z∥2 and ∥ · ∥2 represents ℓ2 − norm.
+14
+
+B.3
+SPARSE NETWORK FEATURE INTERPRETATION
+We use the Integrated Gradients attribution algorithm to rank the features in all layers. Inspired by
+PNET (Elmarakeby et al., 2021), to reduce the bias introduced by over-annotation of certain nodes
+(nodes that are members of too many pathways), we adjusted the Integrated Gradients scores using a
+graph informed function f that considers the connectivity of each node. The importance score of
+each node i, Cl
+i is divided by the node degree dl
+i if the node degree is larger than the mean of node
+degrees plus 5σ where σ is the standard deviation of node degrees.
+dl
+i = fan − inl
+i + fan − outl
+i
+adjusted Cl
+i = f(x) =
+� Cl
+i
+dl
+i ,
+dl
+i > µ + 5σ
+Cl
+i,
+otherwise
+B.4
+CO-ATTENTION BASED PATHWAY VISUALIZATION
+After we got the ranking of top genes and pathways, we adopted the co-attention survival model
+(MCAT) (Chen et al., 2021) to show the spatial visualization of genomic features. We trained MACT
+on all our TCGA datasets, and MACT learns how WSI patches attend to genes when predicting
+patient survival. We define each WSI patch representation and pathway genomic features as Hbag
+and Gbag. The genomic features are the gene list values from the top pathways of each TCGA dataset.
+The model uses Gbag ∈ RN×dg to guide the feature aggregation of Hbag ∈ RN×dp into a clustered
+set of gene-guided visual concepts �Hbag ∈ RN×dp , dg and dp represents the dimension for the
+pathway (number of genes involved in the pathway) and patch. Through the following mapping:
+CoAttnG→H(G, H) = softmax
+�
+QK⊤
+�
+dp
+�
+= softmax
+�
+WqGH⊤W⊤
+s
+�
+dp
+�
+WvH → Acoattn WvH → �H
+where Wq, Ws, Wv ∈ Rdp×dp are trainable weight matrices multiplied to the queries Gbag and
+key-value pair (Hbag , Hbag ), and Acoattn ∈ RN×M is the co-attention matrix for computing the
+weighted average of Hbag . Here, M represents the number of patches in one slide, and N represents
+the number of pathways (We trained the top four pathways, so N = 4 in our study).
+C
+EXPERIMENTS
+C.1
+NETWORK ARCHITECTURE
+Sparse network for gene: The final gene expression embedding is hg ∈ R1×50.
+Pathology network: The slide level image feature representation is passed through an image embed-
+ding layer and encodes the embedding as hp ∈ R1×50.
+CNV + MUT network: Similarly as the pathology network, the patient level CNV + MUT feature
+representation is passed through an FC embedding layer and encodes the embedding as hc ∈ R1×50.
+C.2
+EXPERIMENTAL DETAILS
+PONET. The latent dimensionality of the factorized matrices k is a very important tuning parameter.
+We tune k = [3, 5, 10, 20, 30, 50] based on the testing C-index value (Appendix Fig. 5) and the loss
+of training and testing plot (Appendix Fig. 6) for each dataset. We choose k to maximize the C-index
+value and also it should have stable convergence in both training and testing loss. For example, we
+choose k = 10 in TCGA-KIRP for the optimized results. We can see that in Appendix Fig. 5 the
+testing loss is quite volatile when k is less than 10. Similarly, we choose k = [20, 10, 20, 20, 10] for
+TCGA-BLCA, TCGA-KIRC, TCGA-LUAD, TCGA-LUSC, and TCGA-PAAD, respectively.
+15
+
+Figure 5: C-Index value under K = 3, 5, 10, 20, 30, 50 for TCGA-KIRP. The mean value and standard
+deviation for 5-fold cross-validation are plotted.
+The learning rate and the regularization hyperparameter λ for the Cox partial likelihood loss are
+also tunable parameters. The model is trained with Adam optimizer. For each training/testing pair,
+we first empirically preset the learning rate to 1.2e-4 as a starting point for a grid search during
+training, the optimal learning rate is determined through the 5-fold cross-validation on the training
+set, C-index was used for the performance metric. After that, the model is trained on all the training
+sets and evaluated on the testing set. We use 2e-3 through the experiments for λ. The batch size is
+set to 16, and the epoch is 100. During the training process, we carefully observe the training and
+testing loss for convergence (Figure 4 in Appendix C.2). The server used for experiments is NVIDIA
+GeForce RTX 2080Ti GPU.
+CoxPH. We only include the age and gender for the survival prediction. Using CoxPHFitter from
+lifelines 3.
+DeepSurv 4. We concatenate preprocessed pathological image features, gene expression, and copy
+number + mutant data in a vector to train the DeepSurv model. L2 reg = 10.0, dropout = 0.4, hidden
+layers sizes = [25, 25], learning rate = 1e-05, learning rate decay = 0.001, momentum = 0.9.
+Pathomic Fusion 5. We use the pathomicSurv model which takes our preprocessed image feature,
+gene expression, and copy number + mutation as model input. k = 20, Learning rate is 2e-3, weight
+decay is 4e-4. The batch size is 16, and the epoch is 100. Drop out rate is 0.25.
+GPDBN 6. The learning rate is 2e-3, the batch size is 16, the weight decay is 1e-6, the dropout rate is
+0.3, and the epoch is 100.
+HFBSurv 7. The learning rate is set to 1e-3, the batch size is 16, λ = 3e-3, weight decay is 1e-6, and
+the epoch is 100.
+3https://github.com/CamDavidsonPilon/lifelines
+4https://github.com/czifan/DeepSurv.pytorch
+5https://github.com/mahmoodlab/PathomicFusion
+6https://github.com/isfj/GPDBN
+7https://github.com/Liruiqing-ustc/HFBSurv
+16
+
+0.8
+0.6
+C-Index
+0.4
+0.2
+0.0
+5
+10
+20
+30
+50
+3
+KFigure 6: Train and test loss for TCGA-KIRP under K = 3, 5, 10, 20, 50 for 5-fold cross-validation.
+17
+
+Fold 1
+Fold 2
+Fold 3
+Fold 4
+Fold 5
+Train
+K=3
+Test
+K=5
+K= 10
+K= 20
+1.06
+K=50
+ 0.65
+0.60
+0.0
+0.50C.3
+ADDITIONAL RESULTS
+Figure 7: Inspecting and interpreting PONET on TCGA-BLCA. Sankey diagram visualization of
+the inner layers of PONET shows the estimated relative importance of different nodes in each layer.
+Nodes in the first layer represent genes; the next layers represent pathways; and the final layer
+represents the model outcome. Different layers are linked by weights. Nodes with darker colors are
+more important, while transparent nodes represent the residual importance of undisplayed nodes in
+each layer.
+Figure 8: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-BLCA.
+18
+
+Gene
+Pathways
+GNB1
+PI5P,PP2Aand IER3RegulatePI3K/AKTSignaling
+Toll Like Receptor 10 (TLR10) Cascade
+Cell-extracellular matrix interactions
+PPP2R5E
+SHC-related events triggered by IGF1R
+Cell death signalling via NRAGE, NRIF and NADE
+RNA Polymerase II Transcription Elongation
+KRAS
+MAP2K and MAPK activation
+Interferon gamma signaling
+rRNA processing in the mitochondrion
+Calmodulin induced events
+Mitotic Telophase/Cytokinesis
+Regulation of Hypoxia-inducible Factor (HIF) by oxygen
+PSMA7
+Activation of G protein gated Potassium channels
+FBXW7 Mutants and NOTCH1 in Cancer
+mRNA Splicing
+KPNA2
+outcome
+YWHAB
+Activation of NF-kappaB in B cells
+Golgi-to-ER retrograde transport
+TCR signaling
+GSK3B
+Gap junction degradation
+Signaling by FGFR1 in disease
+Mitotic Spindle Checkpoint
+HSP90AB1
+Phosphate bond hydrolysis by NUDT proteins
+Biosynthesis of DPA-derived SPMs
+ESR-mediated signaling
+TBK1
+NEP/NS2 Interacts with the Cellular Export Machinery
+TCF transactivating complex
+Fatty acid metabolism
+PIK3CA
+p53-IndependentDNADamageResponse
+Interleukin-17 signaling
+Effects of PIP2 hydrolysis
+residual
+residual
+residual
+residualTCGA-4Z-AA7Y
+PI5P, PP2A and IER3
+SHC-related events
+PI3K/AKT Signaling
+MAP2K and MAPK activation
+Calmodulin induced events
+Survival Month: 50
+triggered by IGF-1R
+High Attn
+TCGA-UY-A78N
+Survival Month: 86.76
+Low AttnFigure 9: Inspecting and interpreting PONET on TCGA-KIRC. Sankey diagram visualization of
+the inner layers of PONET shows the estimated relative importance of different nodes in each layer.
+Nodes in the first layer represent genes; the next layers represent pathways; and the final layer
+represents the model outcome. Different layers are linked by weights. Nodes with darker colors are
+more important, while transparent nodes represent the residual importance of undisplayed nodes in
+each layer.
+Figure 10: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-KIRC.
+19
+
+TCGA-A3-3313
+Downregulation of
+MAP2K and MAPK activation
+Activation of the
+P53-Independent DNA
+ERBB2:ERBB3 signaling
+Survival Month: 24.15
+pre-replicative complex
+damage response
+High Attn
+TCGA-A3-3320
+Survival Month: 49.54
+Low AttnGene
+Pathways
+TFDP2
+Glucagon-type ligand receptors
+Class B/2 (Secretin family receptors)
+GPCR ligand binding
+MAPK3
+Downregulation of ERBB2:ERBB3 signaling
+G1/STransition
+G1/S DNA Damage Checkpoints
+PTPN11
+MAP2K and MAPK activation
+p53-lndependent G1/S DNA damage checkpoint
+Mitotic G1-G1/S phases
+ADCY5
+Activation of the pre-replicative complex
+RAF/MAP kinase cascade
+Defects in vitamin and cofactor metabolism
+PSMC2
+p53-Independent DNADamage Response
+CLEC7A (Dectin-1) signaling
+MAPK1/MAPK3 signaling
+MTRR
+outcome
+PLCB1
+Processing of DNA double-strand break ends
+Downregulation of ERBB2 signaling
+C-type lectin receptors (CLRs)
+PSMD11
+CLEC7A (Dectin-1) induces NFAT activation
+Defects in cobalamin (B12) metabolism
+HIV Infection
+PSMF1
+RHO GTPases Activate NADPH Oxidases
+HDR or Single Strand Annealing
+Signaling by ERBB2
+IL6ST
+SHC-related events triggered by IGF1R
+G2/M Transition
+Fatty acid metabolism
+Activation of NOXA and translocation to mitochondria
+Activation of BH3-only proteins
+Mitotic G2-G2/M phases
+residual
+residual
+residual
+residualFigure 11: Inspecting and interpreting PONET on TCGA-LUAD. Sankey diagram visualization
+of the inner layers of PONET shows the estimated relative importance of different nodes in each
+layer. Nodes in the first layer represent genes; the next layers represent pathways; and the final layer
+represents the model outcome. Different layers are linked by weights. Nodes with darker colors are
+more important, while transparent nodes represent the residual importance of undisplayed nodes in
+each layer.
+Figure 12: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-LUAD.
+20
+
+Gene
+Pathways
+CCT3
+Processive synthesis on the lagging strand
+Lagging Strand Synthesis
+Class I MHC pathwiay
+PSEN1
+HDR through Homologous Recombination (HRR)
+Leading Strand Synthesis
+Signaling by EGFR in Cancer
+Phosphate bond hydrolysis by NUDT proteins
+Antigen processing-Cross presentation
+Intrinsic Pathway for Apoptosis
+EGFR
+Chk1/Chk2(Cds1) mediated inactivation of Cyclin B:Cdk1 complex
+RAF/MAP kinase cascade
+Nucleobase catabolism
+PSMD2
+Polymerase switching
+RHO GTPase Effectors
+Homology Directed Repait
+outcome
+CCT6A
+Purine catabolism
+G2/M Checkpoints
+ER-Phagosome pathway
+PTGES3
+HDR or Single Strand Annealing
+Signaling by Rho GTPases
+Golgi Cisternae Pericentriolar Stack Reorganization
+NUDT1
+G2/M DNA damage checkpoint
+MAPK1/MAPK3 signaling
+Downregulation of ERBB2:ERBB3 signaling
+YWHAZ
+Cap-dependent translation
+GPCR downstream signalling
+Regulation of RAS by GAPs
+RUNX1
+Signaling by Overexpressed Wild-Type EGFR in Cancer
+Glycosaminoglycan metabolism
+Inhibition of Signaling by Overexpressed EGFR
+AKT2 residual
+residual
+residual
+residualTCGA-55-8621
+Processive synthesis on
+HDR through
+Phosphaste bond hydrolysis
+Chk1/Chk2(Cds1) mediated
+Survival Month: 16.92
+the lagging strand
+homologous recombination
+by NUDT proteins
+inactivation of Cyclin B:Cdk1 complex
+High Attn
+TCGA-78-7153
+Survival Month: 119.42
+LowAttnFigure 13: Inspecting and interpreting PONET on TCGA-LUSC. Sankey diagram visualization
+of the inner layers of PONET shows the estimated relative importance of different nodes in each
+layer. Nodes in the first layer represent genes; the next layers represent pathways; and the final layer
+represents the model outcome. Different layers are linked by weights. Nodes with darker colors are
+more important, while transparent nodes represent the residual importance of undisplayed nodes in
+each layer.
+Figure 14: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-LUSC.
+21
+
+Gene
+Pathways
+DLG1
+Calmodulin induced events
+TCF transactivating complex
+Cell-extracellular matrix interactions
+UBA52
+CD28 dependent PI3K/Akt signaling
+Pentose phosphate pathway disease
+Neurotransmitterclearance
+PPP2R5E
+PI5P, PP2A and IER3 Regulate PI3K/AKT Signaling
+Signaling by NTRK3 (TRKC)
+rRNA processing in the mitochondrion
+PSMC5
+NrCAM interactions
+Interferon gamma signaling
+TCR signaling
+RAC1
+Constitutive Signaling by NOTCH1
+Glutathione conjugation
+Hedgehog 'on' state
+outcome
+CREB1
+MAP2K and MAPK activation
+Toll Like Receptor 10 (TLR10) Cascade
+Base-Excision Repair, AP Site Formation
+ADAM17
+Negative regulation of MAPK pathway
+Defects in biotin (Btn) metabolism
+Hedgehog 'off state
+PAK2
+Cleavage of the damaged purine
+SUMO E3 ligases
+Regulation of Hypoxia-inducible Factor (HIF) by oxygen
+PSMC2
+AXIN missense mutants destabilize the destruction complex
+RAF-independent MAPK1/3 activation
+Signaling by NOTCH4
+NCOA1
+SHC-related events triggered by IGF1R
+Golgi-to-ER retrograde transport
+Nucleobase biosynthesis
+residual
+residual
+residual
+residualTCGA-18-3414
+CD28 dependent
+PI5P PP2A and IER3
+Calmodulin induced events
+ PI3K/Akt signaling
+regulate PI3K/AKT signaling
+NrCAM interactions
+Survival Month: 23.52
+High Attn
+TCGA-33-4538
+Survival Month: 97.86
+Low AttnFigure 15: Inspecting and interpreting PONET on TCGA-PAAD. Sankey diagram visualization
+of the inner layers of PONET shows the estimated relative importance of different nodes in each
+layer. Nodes in the first layer represent genes; the next layers represent pathways; and the final layer
+represents the model outcome. Different layers are linked by weights. Nodes with darker colors are
+more important, while transparent nodes represent the residual importance of undisplayed nodes in
+each layer.
+Figure 16: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-PAAD.
+22
+
+Gene
+Pathways
+PTGES3
+ SMAD4 MH2 Domain Mutants in Cancer
+Interleukin-6 family signaling
+Mitotic G1-G1/S phases
+C1QC
+SMAD2/3 MH2Domain Mutants in Cancer
+Signaling by FGFR3
+Class I MHC pathway
+PSMD3
+Synthesis of Prostaglandins and Thromboxanes
+AXIN mutants destabilize the destruction complex, activating WNT signalir
+Triglyceride metabolism
+CABIN1
+Regulation by c-FLIP
+Influenza Viral RNA Transcription and Replication 
+RIPK1-mediated regulated necrosis
+NRAS
+C1QB
+Formation of Senescence-Associated Heterochromatin Foci (SAHF)
+Thyroxine biosynthesis
+Reversal of alkylation damage by DNA dioxygenases
+outcome
+EIF3E
+Coenzyme A biosynthesis
+Fusion and Uncoating of the Infuenza Virion
+PIP3 activates AKT signaling
+MGAT4B
+PPCS
+RUNX3 regulates BCL2L11 (BIM) transcription
+EPH-Ephrin signaling
+Metabolism of cofactors
+YWHAG
+FGFR1 mutant receptor activation
+Signaling by NOTCH1 HD Domain Mutants in Cancer
+Platelet Aggregation (Plug Formation)
+MET activates RAP1 and RAC1
+Resolution of AP sites via the single-nucleotide replacement pathway
+GPCR downstream signalling
+Chk1/Chk2(Cds1) mediated inactivation of Cyclin B:Cdk1 complex
+Regulation of innate immune responses to cytosolic DNA
+RNA Polymerase I Promoter Clearance
+residual
+residual
+residual
+residualTCGA-2J-AABO
+Formation of senescence
+Survival Month: 14.45
+Regulation by c-FLIP
+associated heterochromatin foci
+Coenzyme A biosynthesis
+MET activates RAP1 and RAC1
+High Attn
+TCGA-3A-A9IH
+Survival Month: 33.54
+Low Attn

39FIT4oBgHgl3EQf6SuL/content/tmp_files/2301.11393v1.pdf.txt

@@ -0,0 +1,2080 @@
+The S-diagnostic—an a posteriori error
+assessment for single-reference coupled-cluster
+methods
+Fabian M. Faulstich,∗,† H˚akon E. Kristiansen,‡ Mihaly A. Csirik,‡ Simen Kvaal,‡
+Thomas Bondo Pedersen,‡ and Andre Laestadius¶,‡
+†Department of Mathematics, University of California, Berkeley
+‡Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University
+of Oslo, Norway
+¶Department of Computer Science, Oslo Metropolitan University, Norway
+E-mail: f.m.faulstich@berkeley.edu
+Abstract
+We propose a novel a posteriori error assessment for the single-reference coupled-
+cluster (SRCC) method called the S-diagnostic. We provide a derivation of the S-
+diagnostic that is rooted in the mathematical analysis of different SRCC variants. We
+numerically scrutinized the S-diagnostic, testing its performance for (1) geometry op-
+timizations, (2) electronic correlation simulations of systems with varying numerical
+difficulty, and (3) the square-planar copper complexes [CuCl4]2−, [Cu(NH3)4]2+, and
+[Cu(H2O)4]2+. Throughout the numerical investigations, the S-diagnostic is compared
+to other SRCC diagnostic procedures, that is, the T1, D1, and D2 diagnostics as well as
+different indices of multi-determinantal and multi-reference character in coupled-cluster
+theory. Our numerical investigations show that the S-diagnostic outperforms the T1,
+1
+arXiv:2301.11393v1  [physics.chem-ph]  26 Jan 2023
+
+D1, and D2 diagnostics and is comparable to the indices of multi-determinantal and
+multi-reference character in coupled-cluster theory in their individual fields of applica-
+bility. The experiments investigating the performance of the S-diagnostic for geometry
+optimizations using SRCC reveal that the S-diagnostic correlates well with different
+error measures at a high level of statistical relevance. The experiments investigating
+the performance of the S-diagnostic for electronic correlation simulations show that the
+S-diagnostic correctly predicts strong multi-reference regimes. The S-diagnostic more-
+over correctly detects the successful SRCC computations for [CuCl4]2−, [Cu(NH3)4]2+,
+and [Cu(H2O)4]2+, which have been known to be misdiagnosed by T1 and D1 diagnos-
+tics in the past. This shows that the S-diagnostic is a promising candidate for an a
+posteriori diagnostic for SRCC calculations.
+1
+Introduction
+While the underlying mathematical theory of the quantum many-body problem is, on a fun-
+damental level, well described, the governing equation, namely, the many-body Schr¨odinger
+equation, remains numerically intractable for a large number of particles. In fact, the many-
+body Schr¨odinger equation poses one of today’s hardest numerical challenges, mainly due
+to the exponential growth in computational complexity with the number of electrons. Over
+the past century, numerous numerical approximation techniques of various levels of cost
+and accuracy have been developed in order to overcome this curse of dimensionality. Ar-
+guably, the most successful approaches are based on coupled-cluster (CC) theory1, which
+defines a cost-efficient hierarchy of increasingly accurate methods, including the so-called gold
+standard of quantum chemistry—the coupled-cluster singles-and-doubles with perturbative
+triples (CCSD(T))2 model.
+Despite the great success of CC theory, its reliability is not yet fully quantifiable. More
+precisely, aside from a few heuristically derived results, there exists no universally reliable
+diagnostic that indicates if the computational result is to be trusted.
+This shortcoming
+2
+
+is most apparent in the regime of transition metal compounds and molecular bond break-
+ing/making processes, systems dominated by strong nondynamic electron-correlation effects,
+where several methods based on CC theory tend to fail along with all other numerically
+tractable approaches.
+Therefore, a posteriori error diagnostics are urgently needed in the field.
+Until very
+recently, the diagnostic approaches available were limited to the so-called T1 (also called
+τ1)3,4, D1, and D2 diagnostic5,6. Despite clear numerical evidence that diagnostics based
+on the single excitation amplitudes, such as the T1 and D1 diagnostics, do not provide
+reliable indicators7, they are commonly used due to the lack of alternatives.
+Recently,
+an alternative set of multi-reference indices was introduced which provided a number of a
+posteriori diagnostic tools8 christened the indices of multi-determinantal and multi-reference
+character in coupled-cluster theory. These tools are highly descriptive and able to determine
+different molecular scenarios in which CC theory may fail.
+We provide an alternative error diagnostic that is based on assumptions employed in the
+mathematical analysis CC theory. More precisely, our diagnostic is derived from the math-
+ematical analysis of CC theory that provides sufficient conditions for a locally unique and
+quasi-optimal solution to the CC working equations. Central to our derivation is the strong
+monotonicity property, as introduced by Schneider9, which is eponymous for our S-diagnostic.
+Compared to the recently suggested nine indices that describe the multi-determinantal and
+multi-reference character in coupled-cluster theory8, the S-diagnostic is a diagnostic tech-
+nique that can be applied to multi-determinantal and multi-reference scenarios alike. We
+complement our theoretical derivation of the S-diagnostic with numerical simulations scruti-
+nizing its validity for different geometry optimizations, and electronic correlation computa-
+tions for systems of varying numerical difficulty for single reference coupled-cluster methods.
+The rest of the article is structured as follows. We begin with a brief review of CC theory,
+followed by a short summary of the mathematical results derived in previous works which
+lay the mathematical foundation for the proposed S-diagnostics. Then, we derive the main
+3
+
+result, i.e., the S-diagnostic which is subsequently numerically scrutinized.
+2
+Theory
+2.1
+Brief overview of coupled-cluster theory
+In CC theory the wave function is parametrized by the exponential |ψ⟩ = e ˆT|φ0⟩. Here, |φ0⟩
+is the reference determinant defining the occupied spin orbitals, and ˆT = �
+µ tµ ˆXµ = �
+k ˆTk
+is a cluster operator, where ˆTk excites k = 1, . . . , N electrons—k is the excitation rank of a
+given ˆTk—from the occupied spin orbitals into the virtual spin-orbitals. All possible excited
+determinants can be expressed as |µ⟩ = ˆXµ|φ0⟩ for some multi-index µ labeling occupied and
+virtual spin-orbitals. The governing equations determining amplitudes (tµ), and therewith
+also the CC energy ECC(t), are given by fCC(t) = 0, where
+�
+�
+�
+�
+�
+ECC(t) = ⟨φ0|e− ˆT ˆHe
+ˆT|φ0⟩
+(fCC(t))µ = ⟨µ|e− ˆT ˆHe
+ˆT|φ0⟩.
+(1)
+More compactly, Eq. (1) can be expressed using the CC Lagrangian10,11
+L(t, z) = ECC(t) +
+�
+µ
+zµ(fCC(t))µ = ⟨φ0|(ˆI + ˆZ†)e− ˆT ˆHe
+ˆT|φ0⟩,
+(2)
+where (zµ) are the Lagrange multipliers which are the dual variables corresponding to (tµ). In
+the extended CC theory12–14 (ECC), which will be used to introduce additional information
+to our S-diagnostic, the Lagrangian is replaced with the more general energy expression
+EECC(t, λ) = ⟨φ0|e
+ˆΛ†e− ˆT ˆHe
+ˆT|φ0⟩.
+(3)
+4
+
+Consequently, through the substitution eˆΛ = ˆI + ˆZ, we have EECC(t, λ) = L(t, z).
+The
+stationarity condition can then be formulated as FECC = 0, where
+FECC = (∂ΛEECC, ∂TEECC)
+(4)
+is the so-called flipped gradient15. The partial derivatives with respect to the amplitudes in
+Eq. (4) are given by
+∂λµEECC = ⟨µ|e
+ˆΛ†e− ˆT ˆHe
+ˆT|φ0⟩,
+∂tµEECC = ⟨φ0|e
+ˆΛ†[e− ˆT ˆHe
+ˆT, ˆXµ]|φ0⟩.
+(5)
+Since the number of determinants, and therewith the size of the system’s governing
+equations, suffer in general from the curse of dimensionality (i.e., it grows exponentially fast
+with the number of electrons), restrictions are necessary to ensure the system’s numerical
+tractability. In practice this is achieved by restricting excitations to excited determinants
+that correspond to a preselected index set—this is referred to as truncation. Such excitation
+hierarchies are commonly denoted as singles (S), doubles (D), etc. We emphasize that the
+CC working equations, as a system of polynomial equations, typically have a large number
+of roots, and the corresponding landscape of said roots is highly non-trivial16. Consequently,
+different limit processes have to be considered separately and carefully studied. More pre-
+cisely, the convergence of the CC roots with respect to the basis set discretization, i.e.,
+convergence towards the complete basis set limit, is a fundamentally different limit process
+from the convergence with respect to the coupled-cluster truncations. Hence, it is important
+to note that the convergence of the numerical root finding procedure for the truncated stan-
+dard (or extended) CC equations does not by itself imply convergence of the roots to the
+corresponding exact roots. In other words, whether the discrete roots converge to the exact
+roots cannot simply be assumed to be true in general.
+Before proceeding further with the derivation of the S-diagnostic, we wish to provide
+the reader with a more precise description of the underlying mathematical conventions in
+5
+
+coupled-cluster theory. We first emphasize the distinction between the cluster amplitudes
+and the corresponding wave function. Although related, these objects live in different spaces
+which we shall elaborate on subsequently. First, the wave function object |ψ⟩ = e ˆT|φ0⟩ lives
+in the N-particle Hilbert space of square-integrable functions, i.e., L2 = {ψ :
+�
+|ψ|2 < +∞},
+with finite kinetic energy.1 We remind the reader of the notation for the L2-inner product
+⟨ψ′|ψ⟩, and its induced norm ∥ψ∥2
+L2 = ⟨ψ|ψ⟩.
+Second, operators that act on the wave
+function, e.g., the Hamiltonian or excitation operators. In this case, we can introduce a
+norm expression for the operator inherited from the function space it is defined on. For
+example, let O be an operator defined on L2 then we define the L2 operator norm
+∥O∥L2 = sup{∥OΨ∥L2 : ∥Ψ∥L2 = 1 and Ψ ∈ L2}.
+(6)
+Note that this reduces to the conventional matrix norm in the finite dimensional case. Third,
+the CC amplitudes (tµ) live in the Hilbert space of finite square summable sequences denoted
+the ℓ2-space. This space is equipped with the ℓ2-inner product17, i.e., let x = (xµ) and
+y = (yµ) be two finite sequences, the ℓ2-inner product is defined as
+⟨x, y⟩ℓ2 =
+�
+µ
+xµyµ,
+which induces the norm ∥x∥2
+ℓ2 = ⟨x, x⟩ℓ2. Henceforth, we shall denote the full amplitude space
+by V, and the truncated amplitude space, e.g., the space only containing single and double
+amplitudes, by V(d); note that we use “d” in this section to distinguish objects that are subject
+to imposed truncations. We moreover follow the mathematically convenient convention that
+1Mathematically, assuming finite kinetic energy is important for the well-posedness of the Schr¨odinger
+equation. In a “weak” formulation this is given by (here for simplicity leaving out spin degrees of freedom)
+�
+R3N |∇ψ(r1, . . . , rN)|2dr1 . . . drN < +∞.
+In the mathematical literature this can be summarized by ψ ∈ H1 (Sobolev space)17. This extra constraint
+of finite kinetic energy is moreover important for the “continuous” (i.e., infinite dimensional) formulation of
+coupled-cluster18.
+6
+
+uses a generic constant C, independent of the main variables under consideration, for the
+different estimations performed subsequently.
+Having laid down the basic definitions, we now recall a result that gives insight into the
+root convergence of CC theory which can be established using a basic existence result of
+nonlinear analysis9,15,18–20. To state this result, we need two more definitions.
+First, local strong monotonicity. Let t, t′, t∗ be cluster amplitudes with ˆT, ˆT ′ and ˆT∗
+denoting the corresponding cluster operators. Set
+∆(t, t′) = ⟨fCC(t) − fCC(t′), t − t′⟩ℓ2,
+(7)
+and furthermore ∆ ˆT = ˆT − ˆT ′. Then the CC function fCC is said to be locally strongly
+monotone at t∗ if for some r > 0, γ > 0 and all t, t′ within the distance r of t∗
+∆(t, t′) ≥ γ∥t − t′∥2
+ℓ2.
+(8)
+Second, local Lipschitz continuity. The function fCC is said to be locally Lipschitz con-
+tinuous at t∗ with Lipschitz constant L > 0 if
+∥fCC(t) − fCC(t′)∥ℓ2 ≤ L∥t − t′∥ℓ2
+(9)
+for any t, t′ in a ball around t∗. Note that in the finite-dimensional case, fCC is indeed locally
+Lipschitz since it is continuously differentiable.
+With these definitions at hand, we can recall the following result9,19:
+Let fCC(t∗) = 0 and assume that fCC is locally strongly monotone with constant γ > 0 at
+t∗. Furthermore, let V(d) ⊂ V be a truncated amplitude space with Pd being the orthogonal
+projector onto V(d) and fd a discretization of fCC, i.e., fd = PdfCC. Then, the following
+holds:
+1. t∗ is locally unique, i.e., |ψ∗⟩ = eT∗|φ0⟩ is the only solution within a sufficiently small
+7
+
+ball.
+2. There exists a sufficiently large d0, such that for any d > d0, there exists t(d)
+∗
+∈ V(d)
+such that fd(t(d)
+∗ ) = 0. This root is unique in a ball centered at t∗ (for some radius r)
+and we have quasi-optimality of the discrete solution t(d)
+∗
+i.e.
+∥t(d)
+∗
+− t∗∥ℓ2 ≤ L
+γ dist(t∗, V(d)),
+(10)
+where dist(v, V(d)) is the distance from v to V(d) measured using the norm of V, and L
+is the Lipschitz constant of fCC at t∗.
+3. For d > d0, the discrete equations fd(t(d)
+∗ ) = 0 have locally unique solutions, and in
+addition to the error estimate (10), we have the quadratic energy error bound
+|ECC(t(d)
+∗ ) − E0| ≤ C1∥t∗ − t(d)
+∗ ∥2
+ℓ2 + C2∥t∗ − t(d)
+∗ ∥ℓ2∥z∗ − z(d)
+∗ ∥ℓ2,
+(11)
+where E0 is the ground state energy and z∗ and z(d)
+∗
+are the Lagrange multiplier of the
+exact and truncated equations, respectively. The constants C1, C2 > 0 arise in general
+from particular continuity considerations18,19 which shall not be further characterized
+here.
+We emphasize that the result in Ref. 18 is more elaborate since it is concerned with an
+infinite dimensional amplitude space. Here, we implicitly assume a finite-dimensional am-
+plitude space which allows us to present the result in the simpler but equivalent ℓ2-topology.
+This result ensures that the CC method is convergent as the truncated cluster amplitude
+space V(d) approaches the untruncated limit and that the energy converges quadratically.
+Note also that the above results hold for conventional single-reference CC theory but can be
+formulated for the extended CC theory as well with some slight modifications (see Ref. 15).
+8
+
+2.2
+Strong Monotonicity Property
+The local strong monotonicity at a root of the CC equations is the mathematical basis of what
+we deem as a reliable solution obtained from a truncated CC calculation since this implies
+a unique solution of fd = 0 for sufficiently good approximate V(d) as well as a quadratic
+convergence in the energy. Moreover, it follows that the Jacobian of both fCC and fd are
+non-degenerate at such a solution. In order to derive the S-diagnostic, we start with a brief
+review of the proof presented in the literature15,18,20 while making some slight improvements.
+We subsequently establish Eq. (8) up to second order in ∥t−t′∥ℓ2 under certain assumptions.
+To that end, we define
+∆2(t∗; t, t′) = ⟨∆ ˆTφ0|e− ˆT∗( ˆH − E0)e
+ˆT∗|∆ ˆTφ0⟩.
+(12)
+Now, suppose that fCC(t∗) = 0, then by Taylor expansion we find
+∆(t, t′) = ∆2(t∗; t, t′) + O((∆t)3).
+(13)
+For the proof, we refer the reader to Ref. 19. We emphasize that the core idea of the proof
+is a Taylor expansion of e ˆT and e ˆT ′ around ˆT∗, which does not require t∗ itself to be small,
+rather, the assumption is that we are within a certain neighborhood of t∗.
+By Eq. (13), if ∆2(t∗; t, t′) ≥ γ′∥t − t′∥2
+ℓ2 with γ′ > 0 for t, t′ within distance r′ from t∗,
+then it is possible to find r > 0 such that Eq. (8) is true for γ ∈ (0, γ′] for t, t′ at distance at
+most r ≤ r′) from t∗. Consequently, we wish to establish
+∆2(t∗; t, t′) ≥ γ′∥t − t′∥2
+ℓ2
+(14)
+for some γ′ = γ′(t∗) > 0.
+We subsequently assume that the ground state of ˆH exists and is non-degenerate, and
+that ˆH admits a spectral gap γ∗ > 0 between the ground-state energy E0 and the rest of the
+9
+
+spectrum of ˆH, i.e.,
+γ∗ = inf
+�
+⟨ψ| ˆH − E0|ψ⟩
+⟨ψ|ψ⟩
+: |ψ⟩ ⊥ |ψ∗⟩
+�
+> 0.
+(15)
+Moreover, we assume that the reference |φ0⟩ is such that it is not orthogonal to the ground-
+state wave function.
+With these assumptions, we can establish an improved version of
+Lemma 11 in Ref. 15 and Lemma 3.5 in Ref. 19: If t∗ solves fCC(t∗) = 0 then for |ψ⟩ ⊥ |φ0⟩
+⟨ψ| ˆH − E0|ψ⟩ ≥ γeff
+∗ ∥ψ∥2
+L2,
+(16)
+where
+γeff
+∗
+=
+γ∗
+∥eT∗φ0∥2
+L2
+.
+(17)
+For the sake of clarity, we here display the used L2-norm. Equation 16 can be obtained as
+follows: Let P∗ be the projection onto the solution |ψ∗⟩, then
+⟨ψ|( ˆH − E0)ψ⟩ = ⟨ψ − P∗(ψ)| ˆH − E0|ψ − P∗(ψ)⟩
+≥ γ∗∥ψ − P∗(ψ)∥2
+L2
+= ∥ψ∥2
+L2 − 2Re⟨ψ|P∗(ψ)⟩ + ∥P∗(ψ)∥2
+L2
+= ∥ψ∥2
+L2 − |⟨ψ|ψ∗⟩|2
+∥ψ∗∥2
+L2
+= ∥ψ∥2
+L2 − |⟨ψ|(eT∗ − I)φ0⟩|2
+∥ψ∗∥2
+L2
+.
+(18)
+We next note that
+|⟨ψ|(eT∗ − I)φ0⟩|2
+∥ψ∗∥2
+L2
+≤ ∥ψ∥2
+L2 ∥(eT∗ − I)φ0∥2
+L2
+∥ψ∗∥2
+L2
+= ∥ψ∥2
+L2
+�
+1 −
+1
+∥ψ∗∥2
+L2
+�
+,
+which inserted in Eq. (18) yields the desired result.
+10
+
+With the inequality (16) at hand, we can establish the inequality
+∆2(t∗; t, t′) = ⟨∆ ˆTφ0|e− ˆT∗( ˆH − E0)e
+ˆT∗|∆ ˆTφ0⟩
+≥ γeff
+∗ ∥∆ ˆTφ0∥2
+L2 − CGCC(T∗)∥∆ ˆTφ0∥2
+H1,
+(19)
+where C is a constant that depends on the Hamiltonian ˆH and
+GCC(T∗) = ∥e
+ˆT∗ − I∥L2 + ∥e− ˆT †
+∗ − I∥L2∥e
+ˆT∗∥L2.
+(20)
+Equation (19) follows from the definition of ∆2 and that
+∆2 = ⟨∆ ˆTφ0| ˆH − E0|∆ ˆTφ0⟩ + ⟨∆ ˆTφ0| ˆH − E0|(e
+ˆT∗ − I)∆ ˆTφ0⟩
++ ⟨(e− ˆT †
+∗ − I)∆ ˆTφ0| ˆH − E0|e
+ˆT∗∆ ˆTφ0⟩,
+then, using that ˆH is a bounded operator in the energy norm and the estimate in Eq. (16),
+we obtain the desired result in Eq. (19).
+3
+The S-Diagnostic
+Given the reformulation of the strong monotonicity property in Eq. (19), we consider a
+computation to be successful if the results fulfill Eq. (19). In order to derive an a posterioi
+diagnostic, we reformulate this inequality in a way that yields a function that indicates
+a reliable computation. To ensure the tractability of the said function we introduce the
+following approximations, which will yield diagnostic functions of different flavors, later
+referred to as S1, S2, and S3, respectively.
+11
+
+Approximation (i)
+A first-order Taylor approximation of e ˆT∗ and the trivial operator
+norm inequality 2 yields
+∥e
+ˆT∗φ0∥2
+L2 ≈ 1 + ∥ ˆT∗∥2
+L2.
+(21)
+Approximation (ii)
+For GCC we use (i) and make the approximation (linearization)
+GCC(T) ≈ 2∥ ˆT∥L2.
+(22)
+Approximation (iii)
+As outlined in Ref. 20, we can moreover estimate
+(1 + ∥ ˆZ∗∥2
+L2)1/2 ≈ (1 + ∥ ˆT∗∥2
+L2)−1/2.
+(23)
+This approximation follows by equating the bra and ket wave functions (in the bivariational
+formulation) e− ˆT †
+∗(ˆI + ˆZ∗)|φ0⟩ = ∥e ˆT∗φ0∥−2
+L2 e ˆT∗|φ0⟩ with eˆΛ∗ = ˆI + ˆZ∗ and approximating
+e− ˆT †
+∗(ˆI + ˆZ∗)|φ0⟩ ≈ (ˆI + ˆZ∗)|φ0⟩.
+(24)
+With these approximations at hand, we can derive three variants of the S-diagnostic that
+we shall investigate subsequently.
+3.1
+The S1-diagnostic
+Starting from Eq. (19), we first note that we are considering the finite-dimensional case, and
+therefore there exists a constant C > 0 such that
+∆2(t∗; t, t′) ≥
+�
+γeff
+∗ − CGCC( ˆT∗)
+�
+∥∆ ˆTφ0∥2
+L2
+(25)
+2
+∥ ˆT∗φ0∥L2 ≤ ∥ ˆT∗∥L2∥φ0∥L2 = ∥ ˆT∗∥L2
+12
+
+holds. Next, we employ Approximation (ii) in the definition of GCC( ˆT∗), and combine Ap-
+proximation (i) with the definition of γeff
+∗
+in Eq. (17), i.e.,
+γeff
+∗
+≈
+γ∗
+1 + ∥ ˆT∗∥2
+L2
+.
+(26)
+This yields
+γeff
+∗ − CGCC( ˆT∗) ≈
+γ∗
+1 + ∥ ˆT∗∥2
+L2
+− 2C∥ ˆT∗∥L2.
+(27)
+Requiring that this expression is positive, we obtain the success condition
+1
+2 > C
+γ∗
+(1 + ∥ ˆT∗∥2
+L2)∥ ˆT∗∥L2.
+(28)
+3.2
+The S2-diagnostic
+By applying Approximation (iii) to Eq. (28), we obtain a success condition that involves the
+Lagrange multipliers, namely,
+1
+2 > C
+γ∗
+∥ ˆT∗∥2
+L2
+(1 + ∥ ˆZ∗∥2
+L2)
+.
+(29)
+3.3
+The S3-diagnostic
+To obtain a diagnostic that includes the Lagrangian multipliers without making use of Ap-
+proximation (iii), we shall follow the argument on strong monotonicity of the extended CC
+function FECC defined above. Note that although we use the extended CC formalism in this
+section (i.e., where the Lagrange multipliers are treated as a second set of cluster amplitudes),
+the derived diagnostic is for the conventional single reference CC method. Subsequently, we
+assume that truncations of ˆT and ˆΛ are at the same rank, i.e., the truncated scheme follows
+as described above for V(d) but takes the double form V(d) × V(d) and with Pd being the
+orthogonal projector onto Vd × Vd. Note that this aligns with practical implementations of
+the CC Lagrangian. For brevity, let ˆU = ( ˆT, ˆΛ), ˆU∗ = ( ˆT∗, ˆΛ∗) and ˆU (d)
+∗
+= ( ˆT (d)
+∗ , ˆΛ(d)
+∗ ) and
+furthermore, set Fd to be the Galerkin discretization of FECC, i.e., Fd( ˆU (d)) = PdFECC( ˆU (d)).
+13
+
+In Ref. 15 strong monotonicity of FECC was established under certain assumptions, and
+recently generalized to a class of extended CC theories21. We, therefore, refer the reader
+to these references for the full proof, here we shall only address those parts relevant to our
+diagnostics.
+Similarly to the CC case, local strong monotonicity of FECC holds if
+∆ECC := ⟨FECC(u) − FECC(u′), u − u′⟩ ≥ γ∥u − u′∥2
+(30)
+for some positive constant γ. Note that we here extended the notation such that u carries
+both the primal-, and dual variables. Furthermore, we let ∆ECC up to second order in ∥u−u′∥
+be denoted ∆ECC
+2
+and similarly to Eq. (19) we have
+∆ECC
+2
+(u∗; u, u′) ≥ γeff
+∗ ∥∆ ˆUφ0∥2
+L2 − CGECC( ˆU∗)∥∆ ˆUφ0∥2
+H1,
+(31)
+where
+GECC( ˆU) = GECC( ˆT, ˆΛ)
+= ∥e− ˆT †e
+ˆΛ∥L2∥e
+ˆT − I∥L2 + ∥e− ˆT †e
+ˆΛ − I∥L2 + K∥φ0∥H1∥e− ˆT †∥L2∥e
+ˆT∥L2∥e
+ˆΛ − I∥L2.
+for some positive constant K
+Starting from Eq. (31), we note again that since we are considering finite-dimensional
+Hilbert spaces, there exists a constant C > 0 such that
+∆ECC
+2
+(u∗; u, u′) ≥
+�
+γeff
+∗ − CGECC( ˆU∗)
+�
+∥∆ ˆUφ0∥2
+L2.
+(32)
+We next employ a variation of Approximation (iii): For GECC we make the substitution
+eˆΛ = ˆI + ˆZ and approximate with a low-order Taylor expansion
+˜GECC( ˆT, ˆZ) := GECC( ˆT, ˆΛ( ˆZ)) ≈ C(∥ ˆT∥L2 + ∥ ˆZ∥L2).
+(33)
+14
+
+Hence, we arrive at the approximation (and we remind the reader that C is used as a generic
+constant)
+γeff
+∗ − CGECC( ˆU∗) ≈
+γ∗
+1 + ∥ ˆT∗∥2
+L2
+− C(∥ ˆT∗∥L2 + ∥ ˆZ∗∥L2).
+(34)
+Requiring that this expression is positive, we find the condition
+1 > C
+γ∗
+�
+(1 + ∥ ˆT∗∥2
+L2)(∥ ˆT∗∥L2 + ∥ ˆZ∗∥L2)
+�
+≈ C
+γ∗
+�
+(1 + ∥ ˆT∗∥2
+L2)∥ ˆT∗∥L2 +
+∥ ˆZ∗∥L2
+1 + ∥ ˆZ∗∥L2
+�
+. (35)
+3.4
+Approximation of operator norms using singular values
+The above-derived success conditions Eqs. (28), (29) and (35) can be directly implemented,
+however, the quantities involved will depend on the system size. This can be illustrated
+by simply placing copies of a molecular system at a distance such that they are at least
+numerically non-interacting. In that case, the reliability of the overall CC calculation is
+determined by the CC calculations of a single copy, yet, the operator norm of the cluster
+operator ∥ ˆT∥L2 will scale with the system’s size.
+To remedy this serious difficulty, we consider an alternative interpretation of the clus-
+ter operators22: The CCSD method yields a set of single amplitudes (ta
+i ) forming a ma-
+trix in Rnocc×nvirt and a set of double amplitudes (tab
+ij ) forming a fourth-order tensor in
+Rnocc×nocc×nvirt×nvirt. As outlined in Ref. 22, in order to capture the pair correlation we re-
+shape the fourth-order tensor that describes the double amplitudes as a matrix in Rn2
+occ×n2
+virt,
+an operation that is also known as “matricization”. In order to include pair correlations
+captured by the single amplitudes, we can moreover extend (tab
+ij ) to also include products of
+single amplitudes which yields MT ∈ Rn2
+occ×n2
+virt with matrix elements
+[MT]ij,ab = tab
+ij + (ta
+i tb
+j − tb
+ita
+j).
+(36)
+15
+
+The singular value decomposition then yields
+MT = UTΣTV ⊤
+T ,
+(37)
+where UT, VT are real orthogonal matrix and ΣT is diagonal. We will subsequently use the
+spectral norm, i.e., the largest singular value, here denoted as σ(MT) to approximate the
+operator norm, i.e.,
+∥ ˆT∥L2 ≈ σ(MT) =: σ(t)
+(38)
+and similarly for the dual variable z. Incorporating this into the success conditions Eqs. (28),
+(29) and (35) yields the S-diagnostic functions used in this article
+S1(t) := 1
+γ∗
+(1 + σ(t)2)σ(t),
+(39a)
+S2(t, z) := 1
+γ∗
+σ(t)
+1 + σ(z)2,
+(39b)
+S3(t, z) := 1
+γ∗
+�
+(1 + σ(t)2)σ(t) +
+σ(z)
+1 + σ(z)2
+�
+.
+(39c)
+For computed cluster amplitudes (t) and Lagrange multipliers (z), the above functions
+will yield an S-diagnostic value.
+In the following numerical investigations, we will first
+investigate the statistical correlation between the computed S-diagnostic value and different
+measures of error. Second, we will investigate a quantitative bound for the S-diagnostic value
+beyond which the computations may not be reliable and further benchmark computations
+with more profound error classifications are advised.
+4
+Numerical simulations
+In this section, we numerically scrutinize the proposed S-diagnostic procedures derived in
+the previous sections. All simulations are performed using the Python-based Simulations
+of Chemistry Framework (PySCF)23–25.
+First, we perform geometry optimizations on a
+16
+
+medium-sized set of molecules comprising all molecules that were investigated in Refs. 3,5,6
+to test the T1, D1, and D2 diagnostic, respectively. With this data at hand, we can propose
+an initial set of values, beyond which our diagnostic suggests interpreting the computational
+results with caution and if possible benchmarking with additional methods that allow for
+a more profound error classification. Second, we target small model systems whose multi-
+reference character can be controlled by simple geometric changes. Third, we numerically
+investigate transition metal complexes that have been shown to be misdiagnosed by the T1
+and D1 diagnostics7.
+4.1
+Correlation in Geometry Optimization
+In order to quantify the correlation between the S-diagnostics and the error of the CC
+method, we numerically investigate the Spearman correlation26 between the error of in sil-
+ico geometry optimizations and the corresponding value of the S-diagnostics. We perform
+geometry optimizations for 34 small to medium-sized molecules that were previously studied
+in relation to CC error classifications3,5,6, see Table 1.
+Table 1: Molecules which are used in the geometry optimization presented here.
+H2N2
+HOF
+C2H2
+ClOH
+H2S
+O3
+FNO
+ClNO
+C2
+C3
+CO
+HNO
+HNC
+HOF
+Cl2O
+P2
+N2H2
+HCN
+CH2NH
+N2
+C2H4
+F2
+HOCl
+Cl2
+HF
+CH4
+H2O
+SiH4
+NH3
+HCl
+CO2
+BeO
+H2CO
+CH2
+The calculations are performed using the CC method with singles and doubles (CCSD)
+using the cc-pVDZ basis set provided by PySCF; the geometry optimization is performed
+using the interface to PyBerny27. The numerically obtained results are compared with exper-
+imentally measured geometries of the considered systems in their gas phases extracted from
+the Computational Chemistry Comparison and Benchmark Data Base (CCCBDB)28. Since
+the computed atomic positions cannot be directly compared, we introduce the bond-length
+matrix that describes the pairwise distance between the atoms in the molecular compound.
+17
+
+This bond-length matrix can be directly compared with the bond-length matrix provided by
+CCCBDB if we label and order the atoms of the corresponding system accordingly. We in-
+vestigate the correlation between the S-diagnostics and three possible error characterizations
+obtained from the absolute difference of the bond-length matrices denoted D(diff):
+i) The maximal absolute error (∆r(max)
+abs
+): the maximal absolute deviation of the numeri-
+cally obtained bond-length matrix to the experimentally obtained bond-length matrix,
+i.e.,
+∆r(max)
+abs
+= max
+i,j D(diff)
+ij
+ii) The averaged absolute error (∆r(ave)
+abs ): the averaged absolute deviation of the numeri-
+cally obtained bond-length matrix to the experimentally obtained bond-length matrix,
+i.e.,
+∆r(ave)
+abs
+=
+�
+i,j D(diff)
+i,j
+Natoms
+iii) The averaged relative error (∆r(ave)
+rel
+): the averaged relative deviation of the numerically
+obtained bond-length matrix to the experimentally obtained bond-length matrix, i.e.,
+∆r(ave)
+rel
+=
+�
+i,j D(diff)
+i,j
+Natoms maxi,j D(diff)
+ij
+Computing the Spearman correlation between the errors listed above and the proposed S-
+diagnostics, we find that all suggested S-diagnostics correlate well with all the error measures
+suggested, i.e., we consistently find correlations of rsp > 0.5 with p < 0.0008, see Table 2. The
+largest correlation is observed between the maximal absolute error (∆r(max)
+abs
+) and S2 and S3
+where we find a correlation of rsp = 0.58476 with p = 0.00018. For comparison, we compute
+the Spearman correlation for the previously suggested T1, D1, and D2 diagnostic in Table 2.
+We find that T1, and D1, are uncorrelated to all the errors that we investigate here, i.e.,
+rsp < 0.3 with p > 0.1. The D2 diagnostic6 shows a correlation with the averaged absolute
+error (∆r(ave)
+abs ) and the averaged relative error (∆r(ave)
+rel
+), where we find a correlation of rsp =
+18
+
+0.36886 with p = 0.026847 and rsp = 0.35496 with p = 0.033646, respectively. We moreover
+compare the S-diagnostics with the recently suggested indices of multi-determinantal and
+multi-reference character in CC theory8. We find that similar to the S-diagnostics, the EEN
+index8 correlates well with the maximal absolute error (∆r(max)
+abs
+); we observe a correlation
+of rsp = 0.53572 with p = 0.000759.
+Directly comparing the Spearman correlation of the S-diagnostics with the T1, D1, and
+D2 diagnostic, we see that the S-diagnostics have a significantly higher correlation than the
+heuristically motivated diagnostics T1, D1 and D2 diagnostics while exhibiting a higher level
+of stochastic significance. Comparing the Spearman correlation of the S-diagnostics with the
+indices of multi-determinantal and multi-reference character in CC theory, we find that the
+S-diagnostic and EEN show similar correlation with the maximal absolute error (∆r(max)
+abs
+)
+with a comparable level of stochastic significance.
+Table 2: Spearman correlation between the S-diagnostic computed form CCSD amplitudes
+and different errors in geometry optimization. The pair-entries show the rank correlation
+and the corresponding p-value, i.e., (rsp, p).
+∆r(max)
+abs
+∆r(ave)
+abs
+∆r(ave)
+rel
+S1
+(0.57910, 0.000215)
+(0.57761, 0.000225)
+(0.53668, 0.000740)
+S2
+(0.58476, 0.000180)
+(0.58584, 0.000174)
+(0.54543, 0.000581)
+S3
+(0.58476, 0.000180)
+(0.58584, 0.000174)
+(0.54543, 0.000581)
+T1
+(0.03025, 0.863034)
+(0.00489, 0.977416)
+(0.02265, 0.895674)
+D1
+(0.27675, 0.107522)
+(-0.00541, 0.975040)
+(-0.02034, 0.906294)
+D2
+(0.16974, 0.329625)
+(0.36886, 0.026847)
+(0.35496, 0.033646)
+EEN
+(0.53572, 0.000759)
+(0.42059, 0.010643)
+(0.33694, 0.044488)
+In order to obtain an approximate trusted region suggested by the S-diagnostics, we
+require a descriptive function that maps the value obtained from the S-diagnostic to the
+error in geometry. Since the Spearman correlation describes a monotone relation between
+the quantities, we may not assume that this relation is linear. Unfortunately, the Spearman
+correlation does not indicate the type of relation that connects the two measured quanti-
+ties. We, therefore, perform a piecewise linear fit to the data obtained in this simulation,
+see Fig. 1. We here allow for four segments which are optimized to reach the best approx-
+19
+
+imation by means of a piecewise linear and monotone function. We emphasize that larger
+numbers of segments yield similar approximations, see Fig. 1b. Performing this piecewise
+linear fit, we observe that the function is constant on some segments. Based on the data dis-
+tribution, we conclude that this constant behavior is artificial and caused by the test set not
+being sufficiently versatile. In particular, no quantitative conclusions can be drawn from the
+piecewise linear fit function for values S3 > 1. Therefore, from the geometry optimizations
+performed here, we can merely conjecture to raise a concern about the validity of CC calcu-
+lations performed for values of the S-diagnostics v(3)
+crit ≥ 1. Based on the piecewise linear fit,
+S3 = 1 corresponds to an error larger than 0.035 a0. A larger statistical investigation with
+a larger variety of molecules and basis set discretizations is delegated to future works. We
+emphasize that this first estimation of vcrit is particularly pessimistic since the data set is not
+versatile enough to give a precise estimation of vcrit. Indeed, in the subsequently performed
+simulations, we show a more refined estimation of vcrit that reveals v(2)
+crit = 1.9 and v(3)
+crit = 1.8,
+for S2, and S3, respectively.
+(a)
+(b)
+Figure 1:
+The maximal error in geometry optimization as a function of the S2 value. (a)
+The orange line corresponds to a piecewise linear fit to the data using four segments for
+the piecewise linear function. (b) Piecewise linear fits to the data with a varying number of
+segments.
+Aside from CC-based simulations, we can also perform MP2 simulations, and use the
+obtained doubles amplitudes to compute the S-diagnostics.
+We find that the proposed
+20
+
+0.150
+4-seg.
+0.125
+0.100
+Max. diff.
+0.075
+0.050
+0.025
+0.000
+0.5
+1.0
+1.5
+2.0
+S3 value0.150
+3-seg.
+0.125
+4-seg.
+5-seg.
+0.100
+Max. diff.
+6-seg.
+0.075
+0.050
+0.025
+0.000
+0.5
+1.0
+1.5
+2.0
+S3 valueS-diagnostics correlate similarly well with MP2 based calculations as it does for CCSD,
+see Table 3
+Table 3: Spearman correlation between S-diagnostics computed from MP2 doubles ampli-
+tudes and different errors in geometry optimization.
+∆r(max)
+abs
+∆r(ave)
+abs
+∆r(ave)
+rel
+S1
+(0.55992, 0.000384)
+(0.54569, 0.000577)
+(0.49781, 0.002006)
+S2
+(0.56687, 0.000313)
+(0.54801, 0.000541)
+(0.49858, 0.001968)
+S3
+(0.55992, 0.000384)
+(0.54569, 0.000577)
+(0.49781, 0.002006)
+4.2
+Model Systems
+In this section we investigate the use of the proposed S-diagnostics for four model systems
+whose multi-reference character can be controlled by simple geometric change: (1) twisting
+ethylene, (2) the C2v insertion pathway for BeH2 (Be · · · H2)29, (3) the H4 model (transition
+from square to linear geometry)30 (4) the H4 model (symmetrically disturbed on a circle);
+the computations are performed in cc-pVTZ basis.
+4.2.1
+Twisting ethylene
+We begin by numerically investigating the proposed S-diagnostics for ethylene twisted around
+the carbon–carbon bond, see Fig. 2.
+Θ
+H
+C
+H
+H
+C
+H
+H
+C
+H
+C
+H
+H
+Figure 2: Depiction of the ethylene (C2H4) model with twist angle Θ.
+At a twist angle of 90°, this system shows a strong multi-reference character. This can
+be seen as follows: At the equilibrium geometry, i.e., in a planar geometry, the two carbon
+p orbitals are perpendicular to the molecular plane form bonding π and anti-bonding π∗
+orbitals. In this geometry, the ground state doubly occupies the π-orbital. As we twist around
+21
+
+the carbon–carbon bond, the overlap between the two p orbitals decreases and becomes zero
+at 90°. Therefore, at 90° the π and π∗ orbitals become degenerate and the π-bond is broken.
+This (quasi) degeneracy can also be observed numerically by computing the HOMO-LUMO
+gap as a function of the twist angle, see Fig. 3a. Computing the corresponding ground state
+energy as a function of the twist angle, we observe the characteristic energy cusp at exactly
+90°, see Fig. 3b.
+(a)
+(b)
+Figure 3:
+(a) HOMO-LUMO gap of C2H4 as a function of the twist angle (b) RHF and
+RCCSD energies of C2H4 as a function of the twist angle
+Due to the quasi degeneracy around 90°, we compare the S-diagnostic with the MRI
+index suggested in Ref. 8. We clearly see the indication of the quasi degeneracy in the MRI
+index, see Fig. 4b. The S-diagnostic also indicates the problematic region around 90°. By
+numerical comparison, we find that a cut-off value of v(2)
+crit = 1.9 and v(3)
+crit = 1.8 for S2 and
+S3, respectively, indicates the same region of quasi degeneracy as the MRI index.
+4.2.2
+C2v insertion pathway for BeH2
+Next we shall investigate the C2v insertion pathway for BeH2 (Be · · · H2)29. The model
+represents an insertion of the Be atom into the H2 molecule. The transformation coordinate
+connects the non-interacting subsystems (Be + H2) with the linear equilibrium state (H-Be-
+H), see Fig. 5
+22
+
+HOMO-LUMO gap
+0.50
+0.45
+0.40
+0.35
+0.30
+0
+1
+2
+3
+Twist angle/ RadianGround state energy
+-78.0
+RHF
+-78.2
+CCSD
+-78.4
+0
+1
+2
+3
+Twist angle/ Radian(a)
+(b)
+Figure 4:
+(a) The proposed S-diagnostics of C2H4 as a function of the twist angle, the
+dotted green and red horizontal lines correspond to v(2)
+crit = 1.9 and v(3)
+crit = 1.8, respectively.
+(b) The previously suggested MRI of C2H4 as a function of the twist angle
+H
+Be
+H
+H
+Be
+H
+Figure 5: Depiction of the C2v insertion pathway for BeH2.
+23
+
+3.0
+S1
+S2
+2.5
+S3
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Twist angle/ Radian1.0
+0.5
+0.0
+-0.5
+MRI
+-1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Twist angle/ RadianWe here follow the insertion pathway outlined in Ref. 29 and denote the position of
+the beryllium atom by X-position, where X-position equal to zero corresponds to the linear
+equilibrium state and X-position equal to five corresponds to the non-interacting subsystems.
+The transition state of this chemical transformation has a pronounced multi-reference char-
+acter. Another distinguishing feature of this model system is a change in the character of the
+dominating determinant in the wave function along the potential energy surface. There are
+two leading determinants in the wave function, each of which dominates in a certain region
+of the potential energy surface while both are quasi-degenerate around the transition-state
+geometry. This leads yields to discontinuities as can be seen in Figs. 6a and 6b
+(a)
+(b)
+Figure 6:
+(a) HOMO-LUMO gap as a function of the X-position (b) RHF and RCCSD
+energies as a function of the X-position.
+Due to the quasi-degeneracy that appears along the transition path, we again compare
+the proposed S-diagnostics with the MRI index suggested in Ref. 8. We clearly see the
+indication of the quasi degeneracy in the MRI index, see Fig. 7b. The region indicated by
+MRI< −0.99 corresponds to x ∈ [2.6, 3.05]. The S-diagnostic also indicates a region where
+the CC computations are potentially unreliable. It is worth mentioning that choosing the
+critical values similar to the previous example, i.e., v(2)
+crit = 1.9 and v(3)
+crit = 1.8, the predicted
+region corresponds to x ∈ [2.5, 4.5] and x ∈ [2.5, 4.25], respectively. In order to reproduce
+the same region of quasi-degeneracy as indicated by the MRI index, the critical values have
+24
+
+Ground state energy
+-15.0
+RHF
+Hartree
+CCSD
+-15.2
+一15.4
+Energyl
+-15.6
+-15.8
+0
+1
+2
+3
+4
+5
+X-position/ aoHOMO-LUMO gap
+0.5
+0.4
+0.3
+0.2
+0
+2
+3
+4
+5
+1
+X-position/ aoto be adjusted to v(2)
+crit = 3.8 and v(3)
+crit = 3.5, respectively.
+(a)
+(b)
+Figure 7: (a) shows the S-diagnostics, the dotted green, and red horizontal lines correspond
+to v(2)
+crit = 1.9 and v(3)
+crit = 1.8, respectively. (b) shows the previously suggested MRI
+4.2.3
+H4 model (transition from square to linear geometry)
+Next, we shall investigate the proposed S-diagnostics applied to the H4 model. The H4
+model is a standard transition model that allows steering the quasi-degeneracy using a single
+parameter, namely, the transition angle α where α = 0 corresponds to a square geometry
+and α = π/2 corresponds to a linear geometry. Following Ref.30, we set a = 2.0 (a.u.),
+see Fig. 8.
+a
+a
+a
+a
+a
+α
+α
+a
+a
+a
+a
+Figure 8: Depiction of the H4 model undergoing the transition from a square geometry to
+linear geometry model by the angle α.
+We see that as the transition angle α tends to zero, the HOMO-LUMO gap closes and
+the system shows signs of (quasi-) degeneracy, see Fig. 9a
+25
+
+So
+S
+S2
+101
+100
+0
+2
+3
+4
+5
+X-position/ ao1.0
+0.5
+0.0
+-0.5
+MRI
+-1.0
+0
+1
+2
+3
+4
+5
+X-position/ ao(a)
+(b)
+Figure 9: (a) HOMO-LUMO gap of H4 as a function of the transition angle (b) RHF, CCSD
+and FCI energies of H4 as a function of the transition angle
+Due to the quasi degeneracy near α = 0, we again compare the proposed S-diagnostics
+with the MRI index. We clearly see the indication of the quasi degeneracy in the MRI index,
+see Fig. 10b. The S-diagnostic also indicates the problematic region near zero transition
+angle.
+A cut-off value of v(2)
+crit = 1.9 and v(3)
+crit = 1.8 results in S2 and S3, respectively,
+indicating the same region of quasi degeneracy as the MRI index.
+(a)
+(b)
+Figure 10:
+(a) The S-diagnostics of H4 as a function of the transition angle, the dotted
+green, and red horizontal lines correspond to v(2)
+crit = 1.9 and v(3)
+crit = 1.8, respectively. (b) The
+previously suggested MRI of H4 as a function of the transition angle.
+For this small model Hamiltonian, it is moreover feasible to perform computations at the
+26
+
+HOMO-LUMO gap
+0.50
+0.45
+0.40
+0.35
+0.30
+0.0
+0.5
+1.0
+1.5
+Angle/ radianGround state energy
+RHF
+-2.0
+CCSD
+FCI
+2.1
+-2.2
+0.0
+0.5
+1.0
+Angle/ radian6
+So
+S1
+5
+S2
+4
+3
+2
+1
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+Angle/ radian1.0
+0.5
+0.0
+-0.5
+MRI
+-1.0
+0.00
+0.25
+0.50 0.75
+¥1.00
+1.25
+1.50
+Angle/ radianFCI level of theory, see Fig. 12. This comparison yields a quantitative comparison of error
+and S-diagnostic.
+Figure 11
+Figure 12: The energy error of CCSD compared to the FCI reference energy using semi-log
+scales. The area left of the vertical solid (black), dashed (green), and dotted-dashed (red)
+lines correspond to the regions where the MRI, S2, and S3 diagnostic indicate a potential
+failure of CCSD, respectively.
+4.2.4
+H4 model (symmetrically disturbed on a circle)
+Another variant of the H4 model that is commonly employed to evaluate CC methods consists
+of four hydrogen atoms symmetrically distributed on a circle of radius R = 1.738 ˚A31.
+For small or large angles, the system resembles two H2 molecules that are reasonably well
+separated, but as the angle passes through 90, the four atoms form a square yielding a
+degenerate ground state. The exact energy is smooth as a function of the angle, but at the
+RHF level, we observe a cusp at 90, similar to the rotation of the carbon-carbon bond in
+ethylene. We follow the system’s geometry configuration outlined in Ref.32, see Fig. 13.
+We see that as the transition angle Θ tends to π/2 radians (90°), the HOMO-LUMO gap
+closes and the system shows signs of (quasi) degeneracy, see Fig. 14a
+Due to the quasi degeneracy near Θ = π/2 (90°), we again compare the proposed S-
+diagnostics with the MRI index. We clearly see the indication of the quasi degeneracy in
+27
+
+Ground state energy error
+EcCSD - FFCI
+X
+10-3
+10-3
+3
+ X
+2
+× 10-3
+0.0
+0.5
+1.0
+Angle/ radianΘ
+Θ
+Figure 13: Depiction of the H4 model undergoing a symmetric disturbance on a circle modeled
+by the angle Θ.
+(a)
+(b)
+Figure 14:
+(a) HOMO-LUMO gap of H4 as a function of the transition angle (b) RHF,
+RCCSD energies of H4 as a function of the transition angle.
+28
+
+HOMO-LUMO gap
+0.5
+0.4
+0.3
+0.2
+1.0
+1.5
+2.0
+Angle/ RadianGround state energy
+RHF
+.8
+CCSD
+FCI
+-1.9
+-2.0
+-2.1
+-2.2
+1.0
+1.5
+2.0
+Angle/ Radianthe MRI index, see Fig. 15b. The S-diagnostic also indicates the problematic region near
+zero transition angle. A cut-off value of v(2)
+crit = 1.9 and v(3)
+crit = 1.8 results in S2 and S3,
+respectively, indicating the same region of quasi degeneracy as the MRI index.
+(a)
+(b)
+Figure 15:
+(a) The S-diagnostics of H4 as a function of the transition angle, the dotted
+green, and red horizontal lines correspond to v(2)
+crit = 1.9 and v(3)
+crit = 1.8, respectively. (b) The
+previously suggested MRI of H4 as a function of the transition angle.
+For this small model Hamiltonian, it is moreover feasible to perform computations at
+the FCI level of theory, see Fig. 16. This comparison reveals the variational collapse of the
+CCSD energy, see Fig. 16a, and moreover yields a quantitative comparison of error and S-
+diagnostic. The trusted region suggested by the S-diagnostic corresponds to a CCSD energy
+error smaller than 2 · 10−4 a.u. which is below the chemical accuracy threshold.
+Since the simulations performed in the previous section suggest that the previously used
+T1, D1, and D2 diagnostics are uncorrelated, or merely weakly correlated, we do not report
+their performance here. The computations showing the performance of the T1, D1, and D2
+diagnostics can be found in the Appendix, see Figs. 26 to 29
+4.3
+Transition metal complexes
+In this section we investigate three square-planar copper complexes [CuCl4]2−, [Cu(NH3)4]2+,
+and [Cu(H2O)4]2+. Transition metal complexes are in general considered to be strongly corre-
+29
+
+S1
+15
+S2
+S3
+10
+5
+0
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+Angle/ RadianMRI
+0.5
+0.0
+-0.5
+-1.0
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+Angle/ Radian(a)
+(b)
+Figure 16: (a) The energy error of CCSD compared to the FCI reference energy. Note that
+in the region of 1.3-1.8 radians the CCSD energy is lower than the FCI reference energy,
+which indicates the variational collapse of the CCSD energy in this region. (b) The absolute
+value of the energy error of CCSD compared to the FCI reference energy using semi-log
+scales. The area between the vertical solid (black), dashed (green), and dotted-dashed (red)
+lines correspond to the regions where the MRI, S2, and S3 diagnostic indicate a potential
+failure of CCSD, respectively.
+lated systems and complete active space self-consistent field (CASSCF) theory is commonly
+applied, with multi-reference perturbation or truncated CI corrections for dynamic correla-
+tion. However, as shown in Ref. 7, the single reference CC method performs very well despite
+the large D1 diagnostic value. We use these systems to scrutinize the proposed S-diagnostics
+for larger systems that are known to be misleadingly diagnosed by the D1 diagnostics.
+Similar to Ref. 7, we perform the simulation of [CuCl4]2−, [Cu(NH3)4]2+, and [Cu(H2O)4]2+
+in 6-31G basis using UHF and ROHF as reference states. Also, He, Ne, and Ar cores were
+frozen in the nitrogen, chlorine, and copper atoms, respectively, resulting in 41 electrons
+in 50, 66, and 74 orbitals for the [CuCl4]2−, [Cu(H2O)4]2+, and [Cu(NH3)4]2+ molecules,
+respectively. We list the ground state energies obtained at the mean-field level of theory and
+the corresponding CCSD results in Table 4; we moreover list the HOMO-LUMO gap which
+enters in the S-diagnostics.
+The results in Table 4 show that UHF and ROHF calculations predict similar energy
+values. Moreover, using the UHF, or ROHF reference state results in similar CCSD energy
+30
+
+0.000
+Iartree
+-0.002
+-0.004
+Energyl
+-0.006
+-0.008
+EcOSD -EFCI
+1.0
+1.5
+2.0
+Angle/ Radian10-2.
+10
+IEcCSD －EFCll
+10-5
+1.0
+1.5
+2.0
+Angle/ RadianTable 4: Energies values and HOMO-LUMO gap obtained with UHF, ROHF, and UCCSD
+calculations given the reference state from UHF and ROHF, respectively.
+UHF
+γUHF
+UCCSD
+RHOF
+γROHF
+UCCSD
+[CuCl4]2−
+-3476.764
+0.453
+-3477.119
+-3476.763
+0.146
+-3477.119
+[Cu(NH3)4]2+
+-1862.977
+0.564
+-1863.663
+-1862.976
+0.351
+-1863.663
+[Cu(H2O)4]2+
+-1942.225
+0.677
+-1942.914
+-1942.224
+0.340
+-1942.914
+values. It is worth noticing that ROHF yields a generally smaller HOMO-LUMO gap. Since
+the performed CCSD calculations differ in their reference, we can compute the S-diagnostics
+for both sets of calculations. The results obtained from a UHF and ROHF reference are
+listed in Table 5 and in Table 6, respectively.
+Table 5: S-diagnostics obtained for the three square-planar copper complexes [CuCl4]2−,
+[Cu(NH3)4]2+, and [Cu(H2O)4]2+ in spin unrestricted formulation with UHF reference.
+S1
+S2
+S3
+T1
+D1
+D2
+[CuCl4]2−
+0.208
+0.409
+0.406
+0.019
+0.158
+0.110
+[Cu(NH3)4]2+
+0.203
+0.403
+0.398
+0.014
+0.130
+0.121
+[Cu(H2O)4]2+
+0.155
+0.308
+0.305
+0.011
+0.072
+0.116
+We see that all S-diagnostic variants suggest that the CCSD calculations were successful,
+and do not require additional numerical confirmation. This is opposed to the D1 diagnostics,
+which aligns with the results reported in Ref. 7.
+Table 6: S-diagnostics obtained for the three square-planar copper complexes [CuCl4]2−,
+[Cu(NH3)4]2+, and [Cu(H2O)4]2+ in spin unrestricted formulation with ROHF reference.
+S0
+S1
+S2
+T1
+D1
+D2
+[CuCl4]2−
+0.645
+1.285
+1.27
+0.020
+0.167
+0.110
+[Cu(NH3)4]2+
+0.326
+0.646
+0.638
+0.015
+0.139
+0.121
+[Cu(H2O)4]2+
+0.309
+0.614
+0.607
+0.011
+0.077
+0.116
+Similar to the results in Table 5, we see that all variants of the S-diagnostic suggest that
+the CCSD calculations were successful. However, it is worth noticing that the S-diagnostic
+values have increased compared to the values reported in Table 5.
+31
+
+5
+Conclusion
+In this article, we proposed three a posteriori diagnostics for single-reference CC calcula-
+tions which we called S-diagnostics, due to their origin in the strong monotonicity analysis.
+Contrary to previously suggested CC diagnostics, the S-diagnostics are motivated by math-
+ematical principles that have been used to analyze CC methods of different flavors in the
+past9,15,18,19,33.
+We performed a set of geometry optimizations for small to medium-sized molecules in
+order to reveal the correlation between the S-diagnostics and the error in geometry from
+CCSD calculations. The test set comprised all molecules that were used in previous articles
+concerning CC diagnostics3–6. Our investigations revealed that the S-diagnostics correlate
+well and with large statistical relevance with different errors in geometry. This yields a first
+estimate of the critical values for the S-diagnostics beyond which the computational results
+should be confirmed using further and more careful numerical investigations. The observed
+correlation between the S-diagnostics and the different errors in geometry are comparable
+to the recently suggested EEN index8. A heuristic test revealed that the S-diagnostics also
+correlate well and with large statistical relevance with the error in geometry at the MP2 level
+of theory. This suggests that the S-diagnostics can also be used as an a posteriori diagnostic
+for MP2 calculations. Our numerical simulations moreover showed that diagnostics based on
+single excitation cluster amplitudes, i.e., D1 and T1, are uncorrelated to errors in geometry
+optimization.
+Following we investigated the S-diagnostics for transition state models that undergo a
+transition from a region in which CC calculations are reliable to a regime where the CC cal-
+culations require further numerical investigations—in this case, due to (quasi-) degeneracy of
+the ground state. The S-diagnostic detects the corresponding regions of (quasi-) degeneracy
+well. In fact, its performance is comparable to the recently suggested MRI indicator—an a
+posteriori indicator for multi-reference character8.
+The last set of numerical simulations targeted transition metal complexes which have
+32
+
+recently been carefully benchmarked7. The previously performed benchmark calculations
+revealed that diagnostics based on single excitation amplitudes severely misdiagnose the
+performance of CCSD for these transition metal complexes. Our computations confirm this,
+and moreover, show that the S-diagnostic correctly confirms the accuracy of the CCSD
+results outlined in Ref. 7.
+These carefully performed numerical investigations suggest that the S-diagnostic is a
+promising candidate for an a posteriori diagnostic for single-reference CC and MP2 calcu-
+lations. To further confirm this, benchmarks on a larger set of molecules will be performed
+in the future. Moreover, since the mathematical analysis of the single-reference CC method
+generalizes to periodic systems as well, we believe that the S-diagnostic can moreover be
+applied to simulations of solids at the CC and MP2 level of theory.
+Throughout our numerical investigations, we observe a subpar performance of the T1 and
+D1 diagnostics. This suggests that those diagnostics should once and for all be removed as
+a posteriori diagnostic tools for single-reference CC calculations.
+Acknowledgement
+This work was partially supported by the Air Force Office of Scientific Research under the
+award number FA9550-18-1-0095 and by the Simons Targeted Grants in Mathematics and
+Physical Sciences on Moir´e Materials Magic (F.M.F.), by the Peder Sather Grant Program
+(A.L., M.A.C., F.M.F.,), and by the Research Council of Norway (A.L., M.A.C.) through
+Project No.
+287906 (CCerror) and its Centres of Excellence scheme (Hylleraas Centre)
+Project No. 262695.
+Some of the calculations were performed on resources provided by
+Sigma2 - the National Infrastructure for High Performance Computing and Data Storage in
+Norway (Project No. NN4654K). We also want to thank Prof. Lin Lin, Prof. Trygve Helgaker,
+Prof. Anna Krylov, Dr. Pavel Pokhilko, Dr. Tanner P. Culpitt, Dr. Laurens Peters, and Dr.
+Tilmann Bodenstein for fruitful discussions.
+33
+
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+37
+
+6
+Appendix
+6.1
+Correlation in Geometry Optimization
+Since we can correlate three S-diagnostic variants with three error measures, we can in
+principle perform the piecewise linear fit that is presented in Section on Correlation in
+Geometry Optimization for nine different scenarios. We here present the piecewise linear fits
+which were not addressed in the above article.
+6.1.1
+S1-diagnostic Correlations
+(a)
+(b)
+Figure 17: The averaged relative error in geometry optimization as a function of the S1 value.
+(a) The orange line corresponds to a piecewise linear fit to the data using four segments for
+the piecewise linear function. (b) Piecewise linear fits to the data with a varying number of
+segments.
+38
+
+4-seg.
+0.03
+Ave. rel. diff.
+0.02
+0.01
+0.2
+0.4
+0.6
+0.8
+1.0
+Si value0.03
+Ave. rel. diff.
+0.02
+3-seg.
+4-seg.
+0.01
+5-seg.
+6-seg.
+0.2
+0.4
+0.6
+0.8
+1.0
+Si value(a)
+(b)
+Figure 18:
+The maximal absolute error in geometry optimization as a function of the S1
+value. (a) The orange line corresponds to a piecewise linear fit to the data using four segments
+for the piecewise linear function. (b) Piecewise linear fits to the data with a varying number
+of segments.
+(a)
+(b)
+Figure 19:
+The averaged absolute error in geometry optimization as a function of the S1
+value. (a) The orange line corresponds to a piecewise linear fit to the data using four segments
+for the piecewise linear function. (b) Piecewise linear fits to the data with a varying number
+of segments.
+39
+
+0.150
+4-seg.
+0.125
+0.100
+Max. diff.
+0.075
+0.050
+.
+0.025
+★+
+0.000
+0.2
+0.4
+0.6
+0.8
+1.0
+Si value0.150
+3-seg.
+0.125
+4-seg.
+5-seg.
+0.100
+Max. diff.
+6-seg.
+0.075
+0.050
+.
+0.025
+★★
+0.000
+0.2
+0.4
+0.6
+0.8
+1.0
+Si value0.06
+★
+0.05
+Ave. abs. diff.
+0.04
+0.03
+0.02
+0.01
+4-seg.
+0.00
+0.2
+0.4
+0.6
+0.8
+1.0
+Si value0.06
+★
+0.05
+Ave. abs. diff.
+0.04
+0.03
+3-seg.
+0.02
+4-seg.
+5-seg.
+0.01
+6-seg.
+0.00
+0.2
+0.4
+0.6
+0.8
+1.0
+Si value6.1.2
+S2-diagnostic Correlations
+(a)
+(b)
+Figure 20: The averaged relative error in geometry optimization as a function of the S2 value.
+(a) The orange line corresponds to a piecewise linear fit to the data using four segments for
+the piecewise linear function. (b) Piecewise linear fits to the data with a varying number of
+segments.
+40
+
+4-seg.
+0.03
+Ave. rel. diff.
+0.02
+0.01
+0.5
+1.0
+1.5
+2.0
+S2 value0.03
+Ave. rel. diff.
+0.02
+3-seg.
+4-seg.
+0.01
+5-seg.
+6-seg.
+0.5
+1.0
+1.5
+2.0
+S2 value(a)
+(b)
+Figure 21:
+The maximal absolute error in geometry optimization as a function of the S2
+value. (a) The orange line corresponds to a piecewise linear fit to the data using four segments
+for the piecewise linear function. (b) Piecewise linear fits to the data with a varying number
+of segments.
+(a)
+(b)
+Figure 22:
+The averaged absolute error in geometry optimization as a function of the S2
+value. (a) The orange line corresponds to a piecewise linear fit to the data using four segments
+for the piecewise linear function. (b) Piecewise linear fits to the data with a varying number
+of segments.
+41
+
+0.150
+4-seg.
+0.125
+0.100
+Max. diff.
+0.075
+0.050
+0.025
+0.000
+0.5
+1.0
+1.5
+2.0
+S2 value0.150
+3-seg.
+0.125
+4-seg.
+5-seg.
+0.100
+Max. diff.
+6-seg.
+0.075
+0.050
+0.025
+☆
+0.000
+0.5
+1.0
+1.5
+2.0
+S2 value0.06
+★
+0.05
+Ave. abs. diff.
+0.04
+0.03
+0.02
+0.01
+4-seg.
+0.00
+0.5
+1.0
+1.5
+2.0
+S2 value0.06
+★
+0.05
+Ave. abs. diff.
+0.04
+0.03
+3-seg.
+0.02
+4-seg.
+5-seg.
+0.01
+6-seg.
+0.00
+0.5
+1.0
+1.5
+2.0
+S2 value6.1.3
+S2-diagnostic Correlations
+(a)
+(b)
+Figure 23: The averaged relative error in geometry optimization as a function of the S3 value.
+(a) The orange line corresponds to a piecewise linear fit to the data using four segments for
+the piecewise linear function. (b) Piecewise linear fits to the data with a varying number of
+segments.
+42
+
+4-seg.
+0.03
+Ave. rel. diff.
+0.02
+0.01
+0.5
+1.0
+1.5
+2.0
+S3 value7
+0.03
+Ave. rel. diff.
+0.02
+3-seg.
+4-seg.
+0.01
+5-seg.
+6-seg.
+0.5
+1.0
+1.5
+2.0
+S3 value(a)
+(b)
+Figure 24:
+The maximal absolute error in geometry optimization as a function of the S3
+value. (a) The orange line corresponds to a piecewise linear fit to the data using four segments
+for the piecewise linear function. (b) Piecewise linear fits to the data with a varying number
+of segments.
+(a)
+(b)
+Figure 25:
+The averaged absolute error in geometry optimization as a function of the S3
+value. (a) The orange line corresponds to a piecewise linear fit to the data using four segments
+for the piecewise linear function. (b) Piecewise linear fits to the data with a varying number
+of segments.
+43
+
+0.150
+4-seg.
+0.125
+0.100
+Max. diff.
+0.075
+0.050
+0.025
+0.000
+0.5
+1.0
+1.5
+2.0
+S3 value0.150
+3-seg.
+0.125
+4-seg.
+5-seg.
+0.100
+Max. diff.
+6-seg.
+0.075
+0.050
+0.025
+0.000
+0.5
+1.0
+1.5
+2.0
+S3 value0.06
+★
+0.05
+Ave. abs. diff.
+0.04
+0.03
+0.02
+0.01
+4-seg.
+0.00
+0.5
+1.0
+1.5
+2.0
+S3 value0.06
+★
+0.05
+Ave. abs. diff.
+0.04
+0.03
+3-seg.
+0.02
+4-seg.
+5-seg.
+0.01
+6-seg.
+0.00
+0.5
+1.0
+1.5
+2.0
+S3 value6.2
+Transition State Models
+Here we shall compare the performance of the S-diagnostics and the previously used T1, D1,
+and D2 diagnostics.
+(a)
+(b)
+Figure 26:
+(a) shows the S-diagnostics (b) shows the previously suggested T1, D1 and D2
+diagnostics
+(a)
+(b)
+Figure 27:
+(a) shows the S-diagnostics (b) shows the previously suggested T1, D1 and D2
+diagnostics
+44
+
+So
+S
+S2
+101
+100
+0
+2
+3
+4
+5
+X-position/ ao10-1
+T1
+D1
+D2
+10-2
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Twist angle/ Radian100
+T1
+D1
+D2
+10-1
+10-2
+0
+2
+3
+4
+5
+1
+X-position/ ao3.0
+S1
+S2
+2.5
+S3
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Twist angle/ Radian(a)
+(b)
+Figure 28:
+(a) shows the S-diagnostics (b) shows the previously suggested T1, D1 and D2
+diagnostics
+(a)
+(b)
+Figure 29:
+(a) shows the S-diagnostics (b) shows the previously suggested T1, D1 and D2
+diagnostics
+45
+
+6
+So
+S1
+5
+S2
+4
+3
+2
+1
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+Angle/ radianS1
+15
+S2
+S3
+10
+5
+0
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+Angle/ RadianT1
+D1
+D:
+10-
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+Angle/ radian100
+T1
+D1
+D2
+10-1
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+Angle/ RadianGraphical TOC Entry
+Running CCSD ...
+S-Diagnostic
+46
+

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+Strongly correlated physics in organic open-shell quantum systems
+G. Gandus,1, ∗ D. Passerone,2 R. Stadler,3 M. Luisier,1 and A. Valli3, 4, †
+1Integrated Systems Laboratory, ETH Z¨urich, Gloriastrasse 35, 8092 Z¨urich, Switzerland
+2Empa, Swiss Federal Laboratories for Materials Science and Technology,
+¨Uberlandstrasse 129, CH-8600, D¨ubendorf, Switzerland
+3Institute for Theoretical Physics, Vienna University of Technology,
+Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria
+4Department of Theoretical Physics, Institute of Physics,
+Budapest University of Technology and Economics, M¨uegyetem rkp. 3., H-1111 Budapest, Hungary
+Strongly correlated physics arises due to electron-electron scattering within partially-filled or-
+bitals, and in this perspective, organic molecules in open-shell configuration are good candidates to
+exhibit many-body effects. With a focus on neutral organic radicals with a molecular orbital host-
+ing a single unpaired electron (SOMO) we investigate many-body effects on electron transport in a
+single-molecule junction setup. Within a combination of density functional theory and many-body
+techniques, we perform numerical simulations for an effective model for which all the parameters,
+including the Coulomb tensor, are derived ab-initio. We demonstrate that the SOMO resonance is
+prone towards splitting, and identify a giant electronic scattering rate as the driving many-body
+mechanism, akin to a Mott metal-to-insulator transition. The nature of the splitting, and thus of
+the resulting gap, as well as the spatial distribution of the SOMO and its coupling to the electrodes,
+have dramatic effects on the transport properties of the junction. We argue that the phenomenon
+and the underlying microscopic mechanism are general, and apply to a wide family of open-shell
+molecular systems.
+I.
+INTRODUCTION
+Strongly correlated electronic physics arises in par-
+tially occupied orbitals in the presence of competing en-
+ergy scales.
+Due to the Coulomb repulsion, electrons
+display a collective behavior, leading to the breakdown
+of the single-particle picture and the emergence of com-
+plex quantum phenomena.
+Electronic correlations are
+also enhanced due to spatial confinement effects in low-
+dimensional and nanoscopic systems. While in solid-state
+physics the concept of a “strongly-correlated metal” is
+well-established, its analog for molecules is not obvious.
+In chemistry, the majority of stable organic molecules
+have closed-shell electronic configurations, and electrons
+are paired in delocalized molecular orbitals (MOs) that
+are either completely filled or empty. The energy differ-
+ence between the frontier MOs, i.e., the highest occupied
+(HOMO) and the lowest unoccupied (LUMO) orbitals
+defines the spectral gap. In particular, π-conjugated sys-
+tems display a wide HOMO-LUMO gap (∆ ∼ eV) which
+is controlled by the overlap of neighboring pz orbitals. A
+molecular system in an open-shell configuration (radical)
+is characterized by unpaired valence electrons residing
+in non-bonding singly-occupied MOs (SOMOs) found at
+intermediate energies between HOMO and LUMO. Rad-
+icals can form by breaking bonds or by adding/removing
+electrons (e.g., in photoinduced processes) and are inter-
+mediate products of chemical reactions.
+While open-shell configurations are typically associ-
+ated with high chemical reactivity, there exist also species
+∗ gandusgui@gmail.com
+† valli.angelo@ttk.bme.hu
+of relatively stable radicals, which possess interesting
+electronic, magnetic, and optical functionalities that are
+relevant to technological applications ranging from next-
+generation spintronics to quantum information [1–3].
+Tremendous advances in the synthesis and character-
+ization of organic radicals triggered recent experimen-
+tal studies with organic species that are stable enough
+to be trapped in break-junctions [4, 5] or investigated
+with scanning tunneling spectroscopy [6–9], which fu-
+eled a revival of interest in the molecular Kondo ef-
+fect [4, 6–12].
+There is a growing experimental and
+theoretical effort to unravel how many-body effects can
+dramatically influence electronic and transport proper-
+ties in light of technological applications.
+In the con-
+text of molecular electronics, noteworthy organic radicals
+include triphenylmethyl [4, 5, 12], Blatter radical [13],
+polyacetylene [14, 15], benzyl [16, 17], together with the
+whole family of polycyclic hydrocarbons with non-Kekul`e
+structure [7, 18–20]. Molecular organic frameworks with
+transition-metal centers (e.g., iron-porphyrin) are also
+typically open-shell, and have been recently suggested
+as molecular transistors [21, 22].
+From the theoretical point of view, in wide-gap semi-
+conductors, the electron-electron scattering rate is low
+due to the lack of electronic states at the Fermi energy.
+The accuracy of ab-inito prediction of the gap is a long-
+standing issue [23], and numerical simulations for insula-
+tors [24, 25] and molecules [25–32] predict a many-body
+renormalization of the spectral gap. However, these ef-
+fects do not change qualitatively the transport proper-
+ties. In open-shell configurations instead, it can be ex-
+pected that electron-electron scattering within the par-
+tially filled SOMO and many-body effects have a promi-
+nent role.
+arXiv:2301.00282v1  [cond-mat.str-el]  31 Dec 2022
+
+2
+In computational quantum chemistry,
+it is well-
+established that open-shell molecular configurations re-
+quire careful treatment (see, e.g., [33] for an overview)
+but the accuracy of quantum chemical methods comes
+at a high numerical cost. Hence, we recently witnessed
+significant advances in developing alternative simulation
+schemes, that are suitable to describe complex devices
+relevant to molecular electronics [11, 34, 35]. In the en-
+deavor to achieve predictive power and allow for a quan-
+titative comparison with experiments, a suitable method
+should be high-throughput — i.e., scalable and automa-
+tized as much as possible, and able to describe a real-
+istic chemical environment and many-body correlations
+within an ab-initio framework. This would allow a coop-
+erative effort between theory and experiments, and pave
+the path to future breakthroughs for next-generation
+quantum technologies.
+II.
+SCOPE OF THIS WORK
+The scope of this work is to investigate the emergence
+of strongly correlated electron physics in the electronic
+and transport properties of single-molecule junctions.
+To this end, we have developed a comprehensive nu-
+merical workflow that combines density functional theory
+(DFT) with quantum field theoretical methods, and it is
+able to address the complexity of a realistic chemical en-
+vironment as well as electronic correlation effects beyond
+the single-particle picture within an ab-initio framework.
+With both aspects taken into account, we are able to un-
+ravel the origin of many-body transport effects in single-
+molecule junctions.
+The art of combining ab-initio and many-body com-
+putational schemes lies in a transformation from non-
+orthogonal atomic orbitals (AOs) to recently introduced
+local orbitals (LOs) [36]. The LOs are by construction
+orthogonal within the same atom and localized in space.
+They take over the symmetries of the original AOs, while
+inheriting the information of the environment. This al-
+lows to represent the electronic wavefunction in a region
+of the spectrum close to the Fermi energy with a mini-
+mal set of orbitals, making them an ideal basis for many-
+body calculations. So far, LOs have been employed in the
+context of DFT [36]. In what follows, we also evaluate
+the Coulomb integrals that describe the electron-electron
+repulsion in the LO basis, and thus map to the origi-
+nal Hamiltonian onto an effective many-body problem,
+which we can feasibly solve with appropriate numerical
+methods. This recipe is particularly suitable to address
+strong correlation effects in the transport properties of
+molecular junctions.
+In terms of applications, we focus on molecular break-
+junctions in which the central molecule bridging the elec-
+trodes is in an open-shell configuration, which are strong
+candidates to manifest many-body effects. Specifically,
+we select a linear and a cyclic molecular bridge, i.e., a
+polyene radical, and a benzene molecule substituted with
+a methylene (CH2) radical group. While both molecules
+are π-radicals with one electron in the SOMO, we show
+that many-body effects bring out profound differences.
+We identify the fingerprint of strong electronic correla-
+tions in the splitting of the SOMO resonance. The details
+of the splitting and the spatial distribution of the SOMO
+on the molecular backbone have dramatic consequences
+on the transport properties of the junction.
+Finally, we demonstrate that such a splitting cannot
+be obtained with less sophisticated techniques, such as
+many-body perturbation theory. We argue that this phe-
+nomenon and the underlying microscopic mechanism are
+general, and apply to a wide family of open-shell molec-
+ular systems.
+III.
+METHODS
+A.
+Local orbitals and low-energy models
+The LOs method [36] is a transformation-based ap-
+proach that aims at retrieving hydrogen-like orbitals for
+atoms in molecules and solids. By construction, LOs are
+locally orthogonal on each atom. The starting point is a
+DFT calculation in an AOs basis set. The Hilbert space
+H is then spanned by a finite set of non-orthogonal or-
+bitals {|i⟩}, i.e., with a overlap matrix ⟨i|j⟩ = (S)ij ̸= δij
+for |i⟩ , |j⟩ ∈ H. A set of LOs {|m⟩} ∈ M ⊆ H can be
+obtained for any atom α in subspace M by a subdiago-
+nalization of the corresponding Hamiltonian sub-block
+Hα |m⟩ = ϵmSα |m⟩
+(1)
+The LOs are then linear combinations of AOs and are
+by definition orthogonal on each atom. This allows for a
+more natural physical interpretation of the LOs as atomic
+orbitals [36].
+In order to obtain an ab-initio effective
+model, we formally separate the Hilbert space into an ac-
+tive space (A) and an environment (E). The active space
+consists of a subset of LOs {|a⟩} = A ⊆ M which are ex-
+pected to describe the relevant physics close to the Fermi
+energy, and at the same time can be efficiently treated
+within quantum many-body techniques.
+Insytead, the
+environment consists of all the remaining LOs and AOs,
+i.e., {|e⟩} ∈ E ≡ H \A. Embedding the active space into
+the environment ensures that the effective model pre-
+serves all information of the original single-particle DFT
+Hamiltonian [36]. Finally, it is convenient to perform a
+L¨owdin orthogonalization [37] of the LO {|a⟩} states and
+redefine the A subspace in terms of this new orthonormal
+basis set with elements
+��a⊥�
+=
+�
+a
+(S−1/2)aa⊥ |a⟩ .
+(2)
+Since the overlap between LOs on different atoms is typi-
+cally low, i.e., (S)ij ≪ 1, the L¨owdin orthonormalization
+of the active space results only in a weak deformation of
+the original LOs, which preserves their atomic-like sym-
+metry.
+
+3
+In practice, the LO low-energy model is constructed
+embedding the active subspace into the environment
+through a downfolding procedure [38, 39]. Taking into
+account the non-orthogonality between the A and E sub-
+spaces [34], we write the Green’s function projected onto
+the A subspace as
+GA(z) = S−1
+A SAHGH(z)SHAS−1
+A ,
+(3)
+where z = E + iη is a complex energy with an infinites-
+imal shift η → 0+. GH denotes the Green’s function of
+the full Hilbert space, and SAH the overlap matrix be-
+tween orbitals
+��a⊥�
+∈ A and orbitals |i⟩ ∈ H, while the
+overlap SA between the
+��a⊥�
+states is, by construction,
+the identity matrix and will be omitted in what follows
+for notational simplicity. The effect of the environment
+on the A subspace is described by the hybridization func-
+tion
+∆A(z) = g−1
+A (z) − GA(z)−1,
+(4)
+where
+gA =
+�
+z − HA
+�−1
+(5)
+is Green’s function of the isolated A subspace. Rewriting
+GA in terms of ∆A and using the definition of gA yields
+GA(z) =
+�
+z − HA − ∆A(z)
+�−1.
+(6)
+Then, GA can be seen as the resolvent of an effective
+A subspace renormalized by the environment through a
+dynamical hybridization. The Green’s function describes
+the physics of the whole system, projected onto a sub-
+space.
+For a single-particle Hamiltonian, the partition above
+is arbitrary, and the procedure remains valid indepen-
+dently of the subset of LOs included in the active space.
+In the context of π-conjugated organic molecules, the
+projection onto a single pz LO per C atom (and pos-
+sibly other species such as N or S) is usually sufficient
+to achieve a faithful representation of the frontier MOs,
+and hence suitable to describe the physics close to the
+Fermi energy [36]. The possibility of considering a re-
+stricted subset of LOs in the effective model is of pivotal
+importance in view of performing computationally-heavy
+many-body simulations.
+B.
+cRPA and ab-initio Coulomb parameters
+In order to derive the electronic interaction parameters
+in the A subspace beyond the semi-local density approx-
+imations, we employ the constrained Random Phase Ap-
+proximation (cRPA) [34, 40, 41]. Within the cRPA, we
+select a region R ⊃ A where the formation of electron-
+hole pairs is expected to screen the Coulomb interaction
+between the A electrons. Because of the strong local na-
+ture of the LOs, it is sufficient that R comprises the A
+subspace and few atoms nearby. Defining GR to be the
+Green’s function projected onto the R subspace in anal-
+ogy with Eq. (3), the screened Coulomb interaction at
+the RPA level is given by
+WR =
+�
+I − VRPR
+�−1VR,
+(7)
+where VR is the bare Coulomb interaction
+(VR)ij,kl =
+�
+dr
+�
+dr′ψi (r)ψ∗
+j (r)
+e2
+|r − r′|ψ∗
+k(r′)ψl (r′),
+(8)
+being ψi(r) the orbitals in the R region, and PR is the
+static component of the polarizability
+(PR)ij,kl = −2i
+� dz′
+2π Gik(−z′)Glj(z′).
+(9)
+The projection of WR onto the A subspace then yields
+the static screened interaction WA. Since we aim at per-
+forming many-body simulations of the effective model,
+we need to partially unscreen the Coulomb parameters,
+eliminating from WA the screening channels arising from
+A-A transitions included in PR, which will be treated at
+a more sophisticated level of theory. This can be done
+according to the following prescription
+UA = WA
+�
+I + PAWA
+�−1,
+(10)
+using the polarization PA of the A electrons obtained
+from GA similarly to Eq. (9). The matrix elements in
+UA can therefore be regarded as the effective (partially
+screened) Coulomb parameters.
+C.
+Solutions of the low-energy models
+The Green’s function of Eq. (6), together with the in-
+teractions parameters of Eq. (10), define a low-energy
+model which can be solved with many-body techniques.
+Here, we propose two somewhat complementary strate-
+gies, i.e., exact diagonalization (ED) and the dynamical
+mean-field theory (DMFT) [42] as implemented within its
+real-space generalization (R-DMFT) for inhomogeneous
+systems [43–47].
+1.
+Exact diagonalization
+The ED technique requires a Hamiltonian formulation
+of the effective model.
+If the states of the active and
+embedding subspaces are energetically well-separated, it
+is possible to neglect the dynamical character of the hy-
+bridization function and construct an effective Hamilto-
+nian as
+Heff
+A = HA + ∆A(z = 0).
+(11)
+
+4
+Including the screened Coulomb interaction, the model
+Hamiltonian then reads
+H =
+�
+ij,σ
+�
+Heff
+A − Hdc
+A
+�
+ijc†
+iσcjσ
++ 1
+2
+�
+ijkl,σσ′
+�
+UA
+�
+ij,klc†
+jσc†
+kσ′clσ′ciσ,
+(12)
+where c(†)
+iσ denote the annihilation (creation) operator of
+an electron at LO i with spin σ, and the double-counting
+correction Hdc
+A accounts for the interaction already in-
+cluded at the mean-field level by DFT (see Sec. III D).
+The diagonalization of this Hamiltonian yields the many-
+body spectrum (eigenstates and eigenvalues) which can
+be used to construct the Green’s function GED
+A
+through
+its Lehmann representation [48]. The many-body self-
+energy is obtained from the Dyson equation
+ΣED
+A (z) = z − Heff
+A −
+�
+GED
+A (z)
+�−1,
+(13)
+and it describes both local Σii and non-local Σi̸=j elec-
+tronic correlations in the LO basis. An obvious advan-
+tage of ED over, e.g., quantum Monte Carlo [49], is that it
+provides direct access to retarded self-energy and Green’s
+function, and hence the electron transmission function,
+without the need to perform an analytic continuation
+numerically, which is an intrinsically ill-defined prob-
+lem [50]. Note that within ED, we obtain a many-body
+self-energy which is, by construction, spin-independent,
+i.e., Σσ
+ij = Σ¯σ
+ij since Heff
+A follows from a restricted DFT
+calculation.
+2.
+Real-space DMFT
+The idea behind R-DMFT consists of mapping a many-
+body problem onto a set of auxiliary Anderson impurity
+models (AIMs) —one for each atom α— described by the
+projected Green’s function [44–46]
+gσ
+α(z) = (Gσ
+A(z))α .
+(14)
+The solution of AIM α (see details below) yields a local
+many-body self-energy Σσ
+α(z), so that the self-energy of
+the A subspace is block diagonal in the atomic subspaces
+Σσ
+A(z) = diag(
+�
+Σσ
+α(z) | α ∈ A
+�
+).
+(15)
+The set of auxiliary AIMs are coupled by the Dyson equa-
+tion
+Gσ
+A(z) =
+�
+z+µ−(HA−Hdc
+A )−∆A(z)−Σσ
+A(z)
+�−1, (16)
+where the Green’s function Gσ
+A includes the many-body
+self-energy and the double-counting correction, and the
+chemical potential µ is determined to preserve the DFT
+occupation of the A subspace. Finally, Eqs. (14-16) are
+iterated self-consistently starting with an initial guess
+(typically Σσ
+A = 0) until convergence.
+More in detail, in AIM α the impurity electrons inter-
+act through a screened local Coulomb repulsion projected
+onto atom α, i.e., Uα = (UA)ij,kl | i, j, k, l ∈ α [51].
+Moreover, the impurity is embedded in a self-consistent
+bath of non-interacting electrons, which describes the rest
+of the electronic system, encoded in the hybridization
+function
+∆σ
+α(z) = z +µ−(Hα −Hdc
+α )−
+�
+gσ
+α(z)
+�−1 −Σσ
+α(z). (17)
+Also within R-DMFT, it is convenient to use ED to
+solve the AIMs to have direct access to retarded func-
+tions. This requires to discretize the hybridization func-
+tion with a finite number of bath orbitals, described by
+orbital energies ϵσ
+m and hopping parameters to the impu-
+rity tσ
+mi. The hybridization parameters together with the
+local Coulomb blocks Uα, define the AIM Hamiltonian
+HAIM =
+�
+ij,σ
+�
+Hα − Hdc
+α
+�
+ijc†
+iσcjσ − µ
+�
+iσ
+c†
+iσciσ
++
+�
+m,σ
+ϵσ
+ma†
+mσamσ +
+�
+mi,σ
+tσ
+mi(a†
+mσciσ + c†
+iσamσ)
++ 1
+2
+�
+ijkl,σσ′
+�
+Uα
+�
+ij,klc†
+jσc†
+kσ′clσ′ciσ,
+(18)
+where c(†)
+iσ and a(†)
+mσ denote the annihilation (creation)
+operator of an electron at LO i with spin σ, or at bath
+orbital m with spin σ, respectively. Once the many-body
+spectrum of the AIM is known, the local self-energy is
+evaluated in terms of the local Green’s function Gσ
+α as
+Σσ
+α(z) =
+�
+gσ
+α(z)
+�−1 −
+�
+Gσ
+α(z)
+�−1.
+(19)
+At convergence, we define the R-DMFT self-energy as
+Σσ,R−DMFT
+A
+(z) = Σσ
+A(z) − Hdc
+A − µ,
+(20)
+so that it contains all shifts related to the density matrix.
+In terms of approximations, R-DMFT takes into ac-
+count local electronic correlations (Σii), neglecting non-
+local correlations (i.e., Σij = 0), but some degree of
+non-locality is retained as Σii ̸= Σjj, and the AIMs
+are coupled through the self-consistent Dyson equation.
+Therefore, R-DMFT is suitable to treat intrinsically in-
+homogeneous systems [26, 46, 47, 52–54].
+Moreover,
+R-DMFT is considerably lighter in terms of computa-
+tional complexity with respect to the direct ED of the
+original many-body problem and can treat systems with
+hundreds of atoms in the active space, inaccessible to
+ED [26, 44, 46]. Finally, besides the restricted solution
+Σσ
+A = Σ¯σ
+A, within R-DMFT we also have the freedom
+of breaking the spin degeneracy, and describe magnetic
+solutions [28, 30, 31, 44, 55].
+D.
+Double-counting correction
+The double-counting (DC) correction Hdc
+A
+aims at
+eliminating the correlations in the A subspace included
+
+5
+at a mean-field level by DFT, which are instead to be
+included in a more sophisticated level of theory within
+the many-body simulations. Unfortunately, an analyt-
+ical expression of the correlation effects accounted for
+within DFT is unknown, and therefore several approxi-
+mations [47, 56–58] have been developed in the context of
+DFT+DMFT [59, 60] or DFT+U [61, 62]. For a single-
+orbital AIM (as in the case of the simulations in this
+work) the DC correction can be reasonably approximated
+within the fully localized limit (FFL) [57, 63–65]
+�
+Hdc
+A
+�
+ii = (UA)ii,ii
+�
+nDFT
+i
+− 1
+2
+�
+,
+(21)
+where nDFT
+i
+is the DFT occupation of orbital i. Hence,
+we use this form of DC for the R-DMFT calculations.
+However, there’s no established method for the general
+case of multi-site and multi-orbital Coulomb interaction
+as is the case for ED. Here, we propose a self-consistent
+procedure in which a set of local parameters is optimized
+to fulfill the condition
+(ΣA)ii(|z| → ∞) = 0,
+(22)
+This approach ensures that the electronic properties at
+high-energies, which are well described by a one-particle
+approach, are restored to the DFT level.
+E.
+Correlated quantum transport
+To describe the electronic transport properties, we use
+the non-equilibrium Green’s function (NEGF) approach
+[66, 67].
+In NEGF, we identify a device region sur-
+rounding the nanojunction’s constriction and downfold
+the leads’ electrons by virtue of an efficient recursive al-
+gorithm [68]. The corresponding Green’s function reads
+GD(z) =
+�
+zSD−HD−ΣL(z)−ΣR(z)−ΣD(z)
+�−1, (23)
+where ΣL(R) is the self-energy describing the electrons in
+the left (right) electrodes, and
+ΣD(z) = SDAS−1
+A ΣA(z)S−1
+A SAD
+(24)
+projects the many-body self-energy of the active space
+ΣA (i.e., obtained within either ED or R-DMFT) onto
+the device region.
+Following the generalization of the
+Landauer formula proposed by Meir and Wingreen [69],
+the conductance is given by
+G = G0T(EF ),
+(25)
+where G0 = e2/h is the conductance quantum, and the
+transmission function is computed as
+T(E) = Tr[GD(z)ΓL(z)G†
+D(z)ΓR(z)],
+(26)
+with ΓL(R) the anti-hermitian part of ΣL(R)
+ΓL(R) = i
+�
+ΣL(R) − Σ†
+L(R)
+�
+.
+(27)
+While Eqs. (25)−(27) neglect the incoherent contribu-
+tions (i.e., due to inelastic scattering) to the transmis-
+sion that arises from the many-body self-energy [35, 70–
+74], they provide a good approximation of the low-bias
+transport properties, even in the presence of strong cor-
+relations within the A subspace [34, 69].
+active
+molecule
+screening
+scattering region
+(a)
+(b)
+tip layer
+slab
+pz LOs
+FIG. 1.
+(a) Schematics of the scattering region of the single-
+molecule junction, consisting of the molecular bridge and the
+Au electrodes. The screening region (R) and the active space
+within the molecule (A) are highlighted. (b) Detailed struc-
+ture of pentadienyl and benzyl radical, and Au electrodes. For
+pentadienyl, we also show schematically the mapping onto the
+C and N pz LOs.
+IV.
+COMPUTATIONAL DETAILS
+The structures were set up with the atomic simula-
+tion environment (ASE) software package [75] and the
+DFT calculations were performed with the GPAW pack-
+age [76–78]. We performed a geometry optimization, and
+the atomic positions were relaxed until the forces on each
+atom were below 0.001 Hartree/Bohr−1 (≈ 0.05 eV/˚A).
+For converging the electron density, we used an LCAO
+double-ζ basis set, with a grid spacing of 0.2 ˚A, and
+the Perdew–Burke–Ernzerhof exchange-correlation func-
+tional [79]. For the electron transport calculations, we
+followed the method described in [68]. The leads were
+modeled by a three-layer-thick Au(111) slab sampled
+with a 3×1×1 k-point grid along the transport direction.
+The scattering region also includes one Au slab and an
+additional Au layer terminated by a four-atom Au tip,
+to which the molecule anchoring groups are attached.
+For all structures, the A subspace describing the effec-
+tive model is composed of the pz LOs of the C and N
+atoms of the molecular bridge, while the R subspace for
+the cRPA calculation of the screened interaction includes
+the molecule and also extends to the Au atoms of the tip
+(see Fig. 1).
+
+6
+V.
+INIGHTS FROM AB-INITIO SIMULATIONS
+In order to understand the many-body effects arising
+in the open-shell configuration, it is useful to recall some
+chemical and electronic properties of the pentadienyl and
+benzyl radicals, and how those are reflected by ab-initio
+simulations.
+In particular, we look at the spatial dis-
+tribution of the SOMO and at the ab-initio Coulomb
+parameters projected onto the LOs of the active space.
+A.
+Structure of the SOMO
+The pentadienyl radical (C5H7) is a linear molecule,
+and the shortest polyene radical after allyl. It has three
+resonant structures. In each structure, the unpaired elec-
+tron is hosted on one of the odd C atoms.
+The delo-
+calization of the unpaired electron along the molecular
+backbone contributes to the thermodynamical stability
+of the molecule [80, 81].
+The structure we consider is
+obtained by substituting a hydrogen atom at each end
+of the chain by an amino group. By diagonalization of
+the AOs Hamiltonian in the subspace of the molecule,
+we find an eigenvalue just above the Fermi energy, corre-
+sponding to a partially occupied MO (i.e., the SOMO).
+The pentadienyl resonant structures and the projection
+of the SOMO onto the pz LOs of the active space are
+shown in Figs. 2(a,b), respectively. The SOMO reflects
+the resonant structures, with the largest projection on
+the odd- and nodes at even- C atoms. It also displays a
+significant projection onto the anchoring groups, suggest-
+ing a strong coupling to the electrodes in the junction.
+The benzene molecule (C6H6) is a cyclic aromatic hy-
+brocarbon and the archetypical building block for molec-
+ular electronics. For our analysis, we consider a related
+compound, the benzyl radical (C6H5CH2−), which is ob-
+tained by substituting a hydrogen atom with a methylene
+(CH2) group. The benzyl radical is also stabilized by res-
+onance but, unlike pentadyenil, in both resonant struc-
+tures the unpaired electron is hosted on the benzylic C,
+as illustrated in Fig. 2(c). We focus on the meta con-
+figuration, in which the amino groups are substituted at
+the 1,3-positions of the aromatic ring, while the methy-
+lene group is substituted in the 5-position, i.e., along the
+longer branch of the ring (see also Fig. 1). As expected,
+we find an eigenvalue lying at the Fermi energy, corre-
+sponding to the SOMO shown in Fig. 2(d). The SOMO
+displays the largest projection at the pz LO of the ben-
+zylic C atom and displays nodes at every other C (simi-
+larly to pentadienyl). However, it does not extend to the
+anchoring groups, thus suggesting a weak coupling to the
+electrodes.
+B.
+Coulomb parameters in the LO basis
+The partially screened Coulomb matrix projected onto
+the LO basis of the active space Uij = (UA)ij is shown in
+(c)
+(a)
+SOMO (pz LOs)
+SOMO (pz LOs)
+(b)
+(d)
+FIG. 2. Resonances and SOMO isosurface (from LOs pz) of
+pentadienyl (a,b) and benzyl (c,d) radicals. In pentadienyl,
+the unpaired electron is hosted by one of the odd C of the
+polyene chain, which also display the largest contributions in
+the isosurface, while the even C correspond to nodes. In both
+benzyl resonant structures, the unpaired electron is hosted
+by the benzylic C, and the isosurface displays nodes on every
+other C, similarly as in pentadienyl. Isovalues: ±0.03 au.
+1
+2
+3
+4
+5
+(b)
+N C C C C C C C N
+N
+C
+C
+C
+C
+C
+C
+C
+N
+(a)
+N
+N
+C
+C
+C
+C
+C
+C C C C C
+N
+N
+FIG. 3. Partially screened Coulomb parameters Uij = (UA)ij
+in the LO basis for the pentadienyl (a) and the benzyl (b)
+radicals.
+Figs. 3(a,b) for the pentadienyl and the benzyl radicals,
+respectively. In both cases, the intra-orbital couplings Uii
+are in the range of 4–5 eV and are slightly stronger for
+the atoms farther away from the metallic Au electrons,
+due to the weaker screening effects. Similar values of the
+Coulomb repulsion are found for the anchoring groups.
+However, as we shall see later, while the Cpz LOs are
+close to half-filling the Npz LOs are almost full, resulting
+in weak correlation effects.
+VI.
+ELECTRON TRANSPORT
+We start our analysis by looking at the electron trans-
+port properties of the pentadienyl and benzyl junctions.
+In particular, we compare the predictions of DFT and
+many-body simulations, where the Coulomb repulsion is
+treated at different levels of approximation.
+
+7CH2
+CH2
+CH
+H7CH2
+CH2
+CH
+H7CH2
+CH2
+CH
+H7CH2
+CH2
+CH
+H7CH2
+CH2
+CH
+H7
+A.
+Pentadienyl
+Within DFT, the transmission function displays a res-
+onance close to the Fermi energy (denoted by EF ) corre-
+sponding to ballistic transport through the SOMO. The
+resonance is found at ϵSOMO = 70 meV and has a width
+ΓSOMO ≈ 300 meV, reflecting a significant hybridization
+of the SOMO with the states of the electrodes.
+The
+slight misalignment between the SOMO resonance and
+EF , yield a conductance G = 5.7 × 10−1 G0 in each
+spin channel, see Fig. 4(a), This scenario changes as the
+SOMO resonance is split due to the Coulomb repulsion.
+However, depending on the splitting mechanism, we ob-
+serve fundamentally different transport properties.
+Within spin-unrestricted R-DMFT calculations, the
+spin rotational symmetry is broken.
+The doublet de-
+generacy is lifted as the SOMO is split into an occu-
+pied state in the majority-spin channel (e.g., ↓-SOMO)
+and an unoccupied state in the minority-spin channel (↑-
+SUMO). This approximation yields a magnetic insulator
+with a spin gap ∆s ≈ 1.3 eV and a magnetic moment
+⟨Sz⟩ ≃ 1/2 due to the single unpaired electron.
+The
+spin-dependent splitting of a transmission feature, e.g.,
+a resonance [16, 17, 82] or an antiresonance [30, 31], has
+been suggested as a suitable mechanism for the realiza-
+tion of organic spin filters. For pentadienyl, the splitting
+is approximately symmetric around the Fermi level, thus
+yielding a similar conductance in the two spin channels
+G↑ = 1.9 × 10−2 G0 and G↓ = 1.5 × 10−2 G0 and low
+spin-filtering efficiency. The spin-unrestricted R-DMFT
+transmission functions are shown in Fig. 4(a) .
+Another possible mechanism to split the SOMO is ob-
+tained without lifting the spin degeneracy (i.e., within
+10-4
+10-2
+1
+-2
+-1
+0
+1
+2
+R-DMFT
+R-DMFT
+10-6
+10-4
+10-2
+1
+-2
+-1
+0
+1
+2
+R-DMFT
+ED
+DFT
+(a)
+(b)
+node
+FIG. 4. Electron transmission function through the pentadi-
+enyl radical junction. DFT predicts a SOMO resonance close
+to EF . Taking into account the Coulomb repulsion beyond
+restricted DFT yields: (a) a splitting of the resonance into
+↓-SOMO and ↑-SUMO due to spin-symmetry breaking; (b) a
+splitting of the resonance without symmetry breaking and a
+transmission node due to many-body effects.
+either R-DMFT or ED). In this case, we find that the
+SOMO transmission resonance is split, revealing an un-
+derlying transmission node, see Fig. 4(b). Hence, many-
+body calculations predict a strong suppression of the con-
+ductance, by several order of magnitude, in stark contrast
+with the single-particle picture, in which electron trans-
+port is dominated by a nearly-resonant ballistic channel.
+Note that the splitting is substantially larger in ED than
+in R-DMFT, and considering that the antiresonance is
+not aligned with EF , it also results in a much stronger
+suppression of the conductance G = 8.1 × 10−4 G0 (ED)
+versus G = 4.9×10−1 G0 (R-DMFT). This suggests that
+non-local effects play an important role, as it can be ex-
+pected in low-dimensional systems [27, 32].
+Since a linear π-conjugated molecule does not display
+any topological node, the pentadienyl node has been sug-
+gested to arise from destructive interference between dif-
+ferent charged states of the molecule [14]. In Sec. VII, we
+discuss in detail the microscopic mechanism responsible
+for the splitting of the SOMO and for the transmission
+node, and show that they are intertwined.
+B.
+Benzyl
+In the case of benzene single-molecule junctions, there
+is more than one possible configuration for the ring to
+bridge the electrodes, depending on the position of the
+amino anchoring groups. We focus on the meta configu-
+ration (i.e., amino groups substituted at the 1,3-positions
+of the aromatic ring) which is particularly relevant in the
+context of molecular electronics.
+Within DFT, the transmission function displays two
+striking features which can be readily identified in
+Figs. 5(a,b): a narrow asymmetric Fano resonance at
+ϵFano < 10 meV, close to EF , and a wide antiresonance
+at ϵDQI ≈ −0.8 eV. Both features originate from quan-
+tum interference (QI) effects. Clarifying the nature of the
+resonances and highlighting their differences, will prove
+helpful in understanding how electronic correlations af-
+fect the transport properties and to shed light on the
+underlying microscopic mechanism.
+The Fano resonance has a characteristic asymmetric
+line shape and arises from the QI between the SOMO,
+which is mostly localized at the benzylic C atom, and
+the delocalized MOs on the molecular backbone, which
+have a strong overlap with the states of the metallic Au
+electrodes [83–85]. The antiresonance is the hallmark of
+destructive QI in the meta configuration and it is well-
+established in the literature, from both the experimen-
+tal [86–88] and theoretical [89–93] points of view. It arises
+from the interference between the HOMO and LUMO of
+the ring itself [93]. There is a subtle interplay between the
+antiresonance and the functional groups (not necessarily
+radical). It is well-established that substituents and ad-
+sorbates affect the relative position of destructive inter-
+ference features with respect to the Fermi energy. The
+chemical control of the antiresonance can be exploited
+
+8
+10-4
+10-2
+1
+-2
+-1
+0
+1
+2
+10-4
+10-2
+1
+-2
+-1
+0
+1
+2
+R-DMFT
+ED
+DFT
+R-DMFT
+R-DMFT
+(a)
+(b)
+Fano
+DQI
+FIG. 5. Electron transmission function through the benzyl radical junction, displaying the Fano and antiresonance originating
+by quantum interference effects. (a) Breaking the spin symmetry results in the spin-splitting of both the Fano and the DQI
+features. (b) Including many-body effects beyond DFT, the Fano resonance is split (without symmetry-breaking) while the
+DQI antiresonance is shifted to lower energies.
+for a wide range of applications ranging from nanoelec-
+tronics [94] to chemical sensing [95, 96] In principle, the
+position of the antiresonance is also influenced by the
+substitution position in the ring (see, e.g., [94] and refer-
+ences therein), but this effect is of marginal relevance to
+the scope of the present work.
+The Fano resonance is indeed the transport signa-
+ture of the SOMO. However, in contrast to pentedienyl,
+where the SOMO is delocalized along the molecular back-
+bone and dominates the electron transport, in benzyl,
+the SOMO is mostly localized on the methyl functional
+group. It is therefore interesting to investigate the effect
+of the Coulomb repulsion and highlight the differences
+between the two cases. Within restricted DFT simula-
+tions, the narrow Fano resonance is partially concealed by
+the wider QI antiresonance. Breaking the spin symmetry
+within spin-unrestricted R-DMFT yields a pair of spin-
+split Fano resonances, as shown in Fig. 5(a). In the ma-
+jority spin channel, ϵ↑
+Fano < 0 falls within the transmis-
+sion depletion caused by the antiresonance and the asym-
+metric Fano profile is clearly observable.
+Its counter-
+part in the minority spin channel is found above EF , i.e.,
+ϵ↓
+Fano > 0, and is still mostly concealed by the background
+transmission. Interestingly, the spin-symmetry breaking
+also induces spin-resolved QI antiresonances [30, 31, 97]
+but the splitting ϵ↓
+DQI − ϵ↑
+DQI is however weaker than in
+the Fano case, since the spin imbalance yields ⟨Sz⟩ ≃ 1/2
+on the pz LO of the benzylic C, and a weaker magneti-
+zation in the rest of the molecule.
+Not allowing breaking the spin symmetry in the many-
+body simulations reveal another scenario, as shown in
+Fig. 5(b). The difference is twofold. We observe a split-
+ting of the Fano resonance in both R-DMFT and ED
+(with the ED splitting being significantly larger) but no
+splitting is detected for the QI antiresonance, which is
+rather shifted further away from EF . This suggests that
+the microscopic mechanism behind the splitting with and
+without spin-symmetry breaking are fundamentally dif-
+ferent, as it distinguishes between the two QI features.
+Moreover, in contrast to the case of pentadienyl, the split-
+ting of the SOMO in benzyl does not result in a strong
+suppression of the transmission within the SOMO-SUMO
+gap. The two observations above are deeply connected,
+and eventually, they can both be rationalized in terms of
+the spatial distribution of the SOMO.
+VII.
+MICROSCOPIC MECHANISM
+A.
+Splitting of the SOMO
+So far, we have seen that the Coulomb repulsion in-
+duces a splitting of the SOMO of the organic radicals.
+In order to gain a deeper understanding of the electronic
+mechanism behind the splitting, and how it affects the
+transport properties of the junction, it is useful to look
+at the retarded self-energy in the LO basis Σij = (ΣA)ij,
+corresponding to ΣED
+A
+and Σσ,R−DMFT
+A
+in Eqs. (13, 20),
+respectively. The many-body effects encoded in the self-
+energy can be rationalized by interpreting the real part
+as an energy-dependent level shift, and the imaginary
+part as an effective electron-electron scattering rate. We
+argue that the mechanism discussed in the following is a
+common feature of organic radicals. Therefore, we dis-
+cuss the pentadienyl and benzyl radicals in parallel and
+highlight the differences whenever necessary.
+In order to compare the different approximations, it
+is convenient to look at the trace of the self-energy ma-
+trix. Within spin-unrestricted R-DMFT, which is shown
+in Figs. 6(a,d), the real part of the self-energy is weakly
+energy-dependent around EF , and determines a shift of
+the SOMO resonance in opposite directions for the two
+spin polarizations. The imaginary part is negligible (not
+shown) resulting in highly coherent SOMO and SUMO
+electronic excitations below and above EF .
+Note that
+the ground state of spin-unrestricted R-DMFT is two-
+fold degenerate, and it is invariant under a flip of all
+spins: {σi} → {¯σi}. This picture is qualitatively anal-
+ogous to what one can expect also at the single-particle
+level, i.e., within DFT+U. Many-body effects are weak,
+and the dominant effect arises from the spin-symmetry
+breaking, as both radicals are magnetic insulators with a
+
+9
+-6
+-4
+-2
+0
+2
+-60
+-40
+-20
+0
+20
+-4
+-2
+0
+2
+R-DMFT
+ED
+R-DMFT
+R-DMFT
+-0.5
+0
+0.5
+1
+-30
+-20
+-10
+0
+10
+-80
+-60
+-40
+-20
+0
+20
+40
+-1
+-0.5
+0
+0.5
+-12
+-10
+-8
+-6
+R-DMFT
+ED
+pentadienyl
+benzyl
+(a)
+(d)
+(b)
+(e)
+(c)
+(f)
+R-DMFT
+R-DMFT
+FIG. 6. Trace of the retarded self-energy Tr[Σ(E)] in the LO
+basis for the pentadienyl (a,b,c) and benzyl (d,e,f) radicals
+(the real and imaginary parts are denoted by solid and dashed
+lines, respectively). Within spin-unrestricted R-DMFT (a,d)
+the self-energy displays a weakly energy-dependent real part,
+which is different in each spin sector, while the imaginary part
+is negligible (not shown). Within both R-DMFT (b,e) and
+ED (c,f) the self-energy is dominated by a single resonance at
+energy ϵr (denoted by a solid grey line).
+spin SOMO-SUMO gap.
+The scenario is completely different within restricted
+R-DMFT and ED, as shown in Figs. 6(b,c,e,f). There,
+the self-energy is dominated by a single resonance and its
+energy dependence can be well described within a one-
+pole approximation (OPA)
+ΣOPA(E) =
+a
+E − EF − ϵr + ıγ .
+(28)
+The OPA self-energy has a Lorentzian shape, where ϵr
+and γ denote the resonant energy and the width of
+the resonance, whereas a controls the amplitude of the
+curve. The imaginary part of the self-energy plays the
+role of a giant electron-electron scattering rate and sup-
+presses electronic excitations around ϵr ≃ ϵSOMO, while
+the real part redistributes the spectral weight towards
+higher energies. This many-body mechanism, akin to the
+Mott metal-to-insulator transition as described within
+DMFT [42], is at the origin of the splitting of the SOMO
+resonance.
+In organic radicals, the following hierarchy of emergent
+energy scales is realized: ΓSOMO ≪ ∆ <∼ Uscreened, where
+the typical energy scale associated with the screened
+Coulomb repulsion Uscreened significantly exceeds the nar-
+row width of the SOMO resonance (∼ 10–100 meV),
+and the HOMO-LUMO single-particle gap ∆ controlled
+by the C-C π-bonds (∼ eV). This sets the electrons in
+the SOMO deep within the strongly correlated regime.
+Such a general condition suggests this mechanism to be
+common to organic radicals with a single unpaired elec-
+tron.
+Multi-radical molecules [98] and networks [99],
+may display different electronic and transport properties
+due to effective interactions between the unpaired elec-
+trons [7, 18–20].
+B.
+Spatial structure of the electronic correlations
+While R-DMFT and ED seem to qualitatively describe
+the same many-body mechanism for the splitting of the
+SOMO, it is also interesting to look at the whole self-
+energy matrix.
+As discussed in Sec. III C, within ED
+all elements Σij ̸= 0, whereas within R-DMFT Σij ∝ δij.
+Remarkably, all elements of the self-energy (irrespectively
+of the approximation) are well described by the OPA with
+the same resonant energy ϵr, as shown in Figs. 7(a,e).
+The off-diagonal elements (when non-zero) can have ei-
+ther sign since it is not determined by causality. It is then
+easy to have a comprehensive look at the self-energy by
+plotting the matrix Σij(ϵr), as shown in Figs. 7(c,d,g,h).
+Indeed, looking at the ED self-energy matrix, clear pat-
+terns emerge. Along the diagonal, some elements Σii are
+significantly larger than the others (note the logarithmic
+scale), and this asymmetry is mirrored by the off-diagonal
+elements. Upon close inspection, we can associate them
+with the pz LOs with the largest SOMO projection, thus
+confirming that the strongest many-body effects corre-
+late with the spatial distribution of the SOMO. Within
+R-DMFT, we find an analogous pattern along the diag-
+onal, as indicated in the insets.
+Despite its approximations (local Coulomb interaction,
+local correlations), it seems that R-DMFT tells qualita-
+tively the same story as the full ED simulations. This
+advocates for a substantially local character of the mi-
+croscopic mechanism, that can describe both the splitting
+of the SOMO and its consequences on electron transport,
+whereas non-local effects renormalize the splitting.
+C.
+Implications for electron transport
+The many-body mechanism behind the splitting of the
+SOMO is common to both the pentadienyl and benzyl
+radicals. However, its consequences on electron transport
+are dramatically different. In order to understand why,
+it is necessary to combine the insights from DFT with
+the knowledge about the spatial and energy structure of
+the self-energy.
+In pentadienyl, the SOMO is delocalized throughout
+the molecular backbone, and its large projection on the
+
+10
+0
+1
+2
+3
+4
+5
+6
+1
+2
+3
+4
+5
+0
+6
+-20
+-10
+0
+10
+20
+0
+1
+-10
+-5
+0
+5
+10
+0
+1
+(g)
+(h)
+(c)
+(d)
+N C C C C C C C N
+N
+C
+C
+C
+C
+C
+C
+C
+N
+N C C C C C C C N
+N
+C
+C
+C
+C
+C
+C
+C
+N
+N C C C C C N
+ED
+N C C C C C N
+R-DMFT
+N
+C
+C
+C
+C
+C
+N
+C
+C
+C
+C
+C
+N
+N
+ED
+R-DMFT
+0
+10-1
+101
+-10-1
+-101
+0
+10-1
+101
+-10-1
+-101
+-30
+-20
+-10
+0
+10
+20
+30
+0
+1
+-60
+-40
+-20
+0
+20
+40
+60
+0
+1
+(a)
+(b)
+(e)
+(f)
+ED
+ED
+ED
+ED
+0
+1
+2
+3
+4
+5
+6
+1
+2
+3
+4
+5
+0
+6
+0
+1
+2
+3
+4
+5
+6
+7
+8
+0
+1
+2
+3
+4
+5
+6
+7
+8
+0
+1
+2
+3
+4
+5
+6
+7
+8
+0
+1
+2
+3
+4
+5
+6
+7
+8
+FIG. 7. Component of the ED self-energy Σij(E) and its matrix representation at the resonant energy Im Σij(ϵr) in the LO
+basis for the pentadienyl (a,b,c,d) and benzyl (e,f,g,h) radicals. Each component of the self-energy (grey lines) is dominated by
+a single pole (a,b,e,f) at a resonant energy ϵr. Selected components (i, j) are highlighted (color lines) and are labeled according
+to their index in the matrix. The matrix structure of the self-energy reflects the spatial distribution of the SOMO, i.e., the
+largest local (Σii) and non-local (Σij̸=i) self-energy contributions are found for the LOs with the largest projections to the
+SOMO (denoted by arrows, see also Fig. 2). Within R-DMFT (d,h) the self-energy is diagonal in the LO indices Σij ∝ δij and
+displays the same pattern.
+pz LOs of the anchoring groups (see Fig.2(a)) ensures a
+substantial overlap with the states in the metallic elec-
+trodes. Hence, there is a transmission channel across the
+junction through the SOMO. The pole of the self-energy
+results in a zero of the corresponding Green’s function.
+The suppression of the Green’s function hinders electron
+transport at that energy and is at the origin of the trans-
+mission node [30, 31]. In contrast, in the benzyl radical,
+the SOMO has negligible projection on the amino groups
+(see Fig.2(d)) and transport is dominated by transmis-
+sion channels involving the frontier MOs. Therefore, the
+splitting of the Fano resonance weakly affects those chan-
+nels, and does not prevent the off-resonance transmission
+of electrons across the junction.
+The above picture can be essentially reproduced within
+the following tight-binding (TB) three-orbital model,
+which is schematically represented in Fig. 8(a). Let us
+consider three orbitals (ℓ, c, r) that can be interpreted as
+the amino groups, left (ℓ) and right (r), and the central
+molecule (c). The Hamiltonian in such a basis reads
+H =
+�
+�
+ϵℓ
+t
+t′
+t
+ϵc
+t
+t′
+t
+ϵr
+�
+� .
+(29)
+The hybridization to the electrodes is mediated by the
+external (ℓ, r) orbitals and, for the sake of this discussion,
+it is assumed to be energy-independent:
+ΓL =
+�
+�
+Γ 0 0
+0 0 0
+0 0 0
+�
+� , ΓR =
+�
+�
+0 0 0
+0 0 0
+0 0 Γ
+�
+� .
+(30)
+The Hamiltonian of the isolated system can be diagonal-
+ized to obtain the eigenvalues ϵHOMO, ϵSOMO, and ϵLUMO.
+In light of the results shown in Fig. 7, the Green’s func-
+tion of the device
+GD(z) =
+�
+z − H + ıΓL/2 + ıΓR/2 − ΣD(z)
+�−1
+(31)
+is dressed with an OPA self-energy
+ΣD(z) =
+�
+�
+0
+0
+0
+0 ΣOPA(z) 0
+0
+0
+0
+�
+�
+(32)
+which acts on the central part (see Fig. 7(a,e) for a con-
+nection with the ab-initio simulations) and has a pole at
+ϵSOMO. Within such a three-orbital model, the Landauer
+transmission in Eq. (26) simplifies to
+T(E) = Γ2|Gℓr(E)|2,
+(33)
+where Gℓr = (GD)ℓr is the upper-right element of the
+Green’s function, linking the orbitals connected to the
+
+11
+(c)
+10-8
+10-6
+10-4
+10-2
+1
+-2
+-1
+0
+1
+2
+10-8
+10-6
+10-4
+10-2
+1
+-2
+-1
+0
+1
+2
+-1
+0
+-2
+-1.5
+-1
+0
+1
+-0.5
+0
+0.5
+1
+-4
+0
+4
+-2
+-1.5
+-0.4
+0
+0.4
+-0.5
+0
+0.5
+1
+Fano
+splitting
+splitting
+node
+zero
+1
+(g)
+(d)
+(h)
+(e)
+(i)
+TB
+OPA
+TB
+OPA
+1
+(a)
+(b)
+(f)
+FIG. 8. Schematic representation of the three-orbital TB model with its parameter, and form of the OPA self-energy (a).
+Weight distribution and eigenvalues of the TB MOs for scenarios representative of the pentadienyl (b) and benzyl (f) radicals.
+The transmission function (c,g) obtained without (grey lines) and with (blue lines) the OPA self-energy captures all relevant
+features of the DFT and many-body simulations. The Green’s function Gℓr is shown for specific energy ranges, which are
+relevant to explaining the spectral features associated with the HOMOs (d,h) and the SOMOs (e,i), as discussed in the text.
+Model parameters [eV]: ϵ = 0.5, ϵc = 0.25, a = 0.25, Γ = 0.05, γ = 0.003, common to both scenarios, t = 0.5, t′ = 0 (b,c,d) and
+t = 0.1, t′ = 0.5 (e,f,g).
+electrodes, and describes the only transmission channel
+across the junction.
+For the sake of simplicity, one can take −ϵℓ = ϵr = ϵ,
+and ϵc ≪ ϵ, which together with a, Γ, and η are kept
+fixed, whereas we choose the parameters t and t′ to de-
+scribe two scenarios, which are representative of the pen-
+tadienyl and benzyl radicals. The results are shown in
+Fig. 8 and described in the following.
+The physics of the pentadienyl radical can be repro-
+duced by choosing t <∼ ϵ and t′ = 0. The correspond-
+ing TB MOs are fairly delocalized throughout the sys-
+tem, as shown in Fig. 8(b). Hence, electron transport
+happens through sequential hopping processes through
+the c orbital. The transmission function, Fig. 8(c), dis-
+plays a SOMO resonance which is split by including the
+OPA self-energy, revealing a transmission node within
+the SOMO-SUMO gap. The origin of the transmission
+node is ascribed to a zero of the Green’s function at the
+SOMO energy Gℓr(E ≃ ϵSOMO) [30, 31] as demonstrated
+in Fig. 8(e).
+Instead, with the choice of parameters t ≪ t′ <∼ ϵ, one
+can describe the physics of the benzyl radical, charac-
+terized by an orbital c, which is weakly coupled to the
+ℓ − r molecular backbone. The corresponding SOMO is
+fairly localized on the central orbital, see Fig. 8(f). The
+transmission function displays a Fano resonance which is
+split by the OPA self-energy see Fig. 8(g). In contrast to
+the previous case, Gℓr does not have a zero, and trans-
+port is dominated by a transmission channel that bridges
+the electrodes through the direct ℓ-r hopping t′. Finally,
+note that in both scenarios above, many-body effects are
+negligible for the HOMO and LUMO resonances (corre-
+sponding to states which are completely filled and empty,
+respectively) even when the “correlated” c orbital has a
+sizable hybridization with ℓ and r, cfr. Figs. 8(c,d,g,h).
+Hence, the three-orbital model can reproduce all fun-
+damental features of the radical junctions discussed in
+this work, and at the same time, provides a simple inter-
+pretation of the numerical simulations.
+D.
+Non-perturbative nature of the splitting
+Within ED and R-DMFT, the solution of the many-
+body problem (i.e., on the lattice or the auxiliary AIM)
+is numerically exact.
+This means that the Coulomb
+repulsion is taken into account in a non-perturbative
+way.
+It is interesting to compare these results to a
+perturbative approach, e.g., within the GW approxima-
+tion [100, 101], which has been extensively and success-
+fully applied to molecules [102–107]. However, the ques-
+tion arises to which extent many-body perturbation the-
+ory approaches are able to describe the physics of open-
+shell systems [108]. Within GW, the self-energy is com-
+puted to the lowest order in perturbation theory, as a
+convolution of the Green’s function and the screened in-
+
+12
+10-4
+10-2
+1
+-2
+-1
+0
+1
+2
+G0W0
+GW
+(a)
+FIG. 9. Electron transmission function through the pentadi-
+enyl radical junction. Both the G0W0 and the self-consistent
+GW approximations fail to predict the splitting of the SOMO,
+as described within ED and R-DMFT, cfr. Fig 4.
+teraction.
+We compute the GW self-energy correction
+projected onto the A region
+Σ(z) = GA(z)WA,
+(34)
+as described in [68], and we consider the case of the pen-
+tadienyl radical without loss of generality.
+In Fig. 9 we see that neither G0W0 nor the fully self-
+consistent GW approximation is able to induce a split-
+ting of the SOMO resonance, and the numerical simu-
+lations rather result in a shift of the corresponding res-
+onance above the Fermi energy. Hence, the many-body
+techniques we propose to investigate open-shell molecules
+are not only sufficient but also necessary for our goal,
+whereas less sophisticated approaches fall short in de-
+scribing the electronic and transport properties arising
+from the strong electronic correlations within the SOMO.
+VIII.
+CONCLUSIONS
+In this work, we have proposed a numerical method
+that that combines ab-initio with state-of-the-art many-
+body techniques and is able to address the complexity
+of a realistic chemical environment as well as electronic
+correlation effects beyond the single-particle picture.
+The deliverable of this project served to shed light on
+the mechanism governing the electronic and transport
+properties of quantum junctions with organic molecules
+in an open-shell configuration. By considering a linear
+and a cyclic radical molecule, we derive a general under-
+standing of the role of many-body effects in molecular
+radicals with a single unpaired electron, and we show
+that they have dramatic consequences on electron
+transport.
+We establish the microscopic mechanism
+behind the splitting of the SOMO resonance and unravel
+a clear link between the space-time structure of electron-
+electron correlations and the spatial distribution of the
+SOMO. We demonstrate this by proposing a minimal
+model, which is capable of grasping the microscopic
+mechanism and thus reproducing all relevant features of
+the transmission properties. Our work will pave the path
+toward a deeper and more comprehensive understanding
+of strongly correlated electron physics at the nanoscale.
+ACKNOWLEDGEMENTS
+We thank J. M. Tomczak for valuable discussions.
+This research is supported by the Austrian Science Fund
+(FWF) through project P 31631 (A.V., R.S.) and by the
+NCCR MARVEL funded by the Swiss National Science
+Foundation grant 51NF40-205602 (G.G., D.P., M.L).
+Computational support from the Swiss Supercomputing
+Center (CSCS) under project ID s1119 is gratefully ac-
+knowledged.
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+arXiv:2301.01030v1  [physics.plasm-ph]  3 Jan 2023
+Analytical study of ion-acoustic solitary waves in a
+magnetized plasma with degenerate electrons
+Moumita Indraa, K. K. Ghoshb, Saibal Rayc
+aSchool of Basic Science, Swami Vivekananda University, Kanthalia, Barrackpore, West
+Bengal, India
+bDepartment of Basic Science & Humanities, Abacus Institute of Engineering and
+Management, Magra, Chinchura, West Bengal, India
+cCentre for Cosmology, Astrophysics and Space Science (CCASS), GLA University,
+Mathura 281406, Uttar Pradesh, India
+Abstract
+The propagation of fully nonlinear ion acoustic solitary waves (IASW) in
+a magneto-plasma with degenerate electrons investigated by Abdelsalam et
+al. [[1] Physics Letters A 372 (2008) 4923]. Based on their work in the present
+work, a rigorous and general analytical study is presented. This confirms
+their implied assumption that (i) only hump and no cavity is possible and
+(ii) for humps, the algebraic equation for the maximum density N obtained
+by them determines it uniquely (naturally assumes N > 1). Here we confirm
+analytically their assertion that N decreases with lx (the direction cosine of
+the wave vector k along the x-axis) and N increases with the increase of the
+Mach number (M).
+Keywords: Waves; Plasmas; MHD; Hydrodynamics; Mach number
+1. Introduction
+Recently, the physics of an electron-positron-ion (EPI) plasma [2, 3, 4, 5, 6,
+7] has received considerable attention, mainly due to its importance in many
+systems in laboratory plasma as well as astrophysical arena. EPI plasma
+Email addresses: moumita.indra93@gmail.com (Moumita Indra),
+kkghosh1954@gmail.com (K. K. Ghosh), saibal.ray@gla.ac.in (Saibal Ray)
+Orcid ID: 0000-0002-7900-7947
+Orcid ID: 0000-0002-5909-0544
+Preprint submitted to Chinese Journal of PhysicsReceived 2022 September 22; accepted 2022 month day
+
+exists in places such as active galactic nuclei (AGNs) [8, 9], pulsar magneto-
+spheres [10, 11] and in many dense astronomical environments, namely, neu-
+tron stars and white dwarfs [12, 13] which is supposed to play a key role in
+understanding the origin and evolution of our entire universe [14]. This kind
+of plasma may also be practically produced in laboratories [15, 16, 17, 18].
+Electrons and positrons are assumed relativistic and degenerate, follow-
+ing the Fermi–Dirac statistics, whereas the warm ions are described by a
+set of classical fluid equations with an individual charge of Zie, (Zi denotes
+the ion-charge state, while e is the electron charge), subject to the influence
+of the electrostatic potential φ. Quantum hydrodynamics
+[23, 24, 25, 26],
+which describes quantum systems within a hydrodynamic framework, was
+first proposed by Madelung [19] and Bohm [20]. Although such a descrip-
+tion is formally accurate for a single particle, Manfredi and Haas [21] later
+expanded the idea to many-particle systems and it gained significant favour
+in the areas of the quantum plasma community.
+Considering the one-dimensional QHD model in the limit of the small
+mass ratio of the charge carriers, Hass et al. [22] were the first to study
+the ion-acoustic waves in unmagnetized quantum plasma. This model has
+been used in various investigations by several authors [23, 24, 25, 26] where
+generally a linear dispersion relation is derived in the linear approximation.
+Thus, ion-acoustic waves (IAW), a fundamental mode in plasma environ-
+ments, have been a subject of extensive research over several decades. Khan
+and Haque [27] showed that in the small (linear) limit of quantum diffraction
+parameter H (ratio of the plasmon energy to the Fermi energy), the system
+behaves as the classical IAW whereas in the non-linear regime the system
+behaves differently. One of the most interesting non-linear features of IAW
+is the existence of ion-acoustic solitary waves (IASW) [28, 29].
+In the weakly nonlinear limit, the quantum plasma is shown to support
+waves described by a deformed Korteweg–de Vries (KdV) equation which
+depends in a non-trivial way on the quantum parameter H. However, in the
+fully non-linear regime, the system exhibits travelling waves which show a
+periodic pattern. Hence there are two main approaches used to investigate
+IASWs, viz., the reductive perturbation technique (KdV method) [30] and
+the pseudo-potential technique for large-amplitude solitary waves (Sagdeev
+method) [31]. The theory of solitons in magneto-plasma was greatly improved
+by an intriguing work by Abdelsalam et al. [1] on completely non-linear IASW
+travelling obliquely to an external magnetic field in a collision-less dense
+Thomas-Fermi magneto-plasma with degenerate electrons.
+2
+
+The degenerate electrons in the above scenario may be described using
+the Thomas-Fermi approximation [32, 33] whereas the ion component can
+be thought of as a classical gas. They have obtained an energy balance-like
+equation involving the Sagdeev potential as follows:
+1
+2
+�dn
+dη
+�2
++ V (n) = 0,
+(1)
+where Sagdeev-like pseudo-potential V (n) is given by
+V (n) =
+9n6
+2(5an8/3 − 3)2[5an8/3 − 2an5/3−3a+1
+M2
++1 − 2n
+M2n2 + cM2(an10/3 − 2an5/3 − 2
+9 + a + 5)],
+(2)
+where
+a =
+3
+5M2,
+(3)
+c = 3lx
+2
+5M2,
+(4)
+η = lxx + lyy − Mt, lx
+2 + ly
+2 = 1 and n = n(η).
+Here M is the Mach number, lx and ly are the direction cosines of the
+wave vector k along the x and y axes respectively, n(= ne
+no) where ne is the
+electron density and no is the unperturbed electron density with ni as the
+ion density.
+As indicated by Abdelsalam et al. [1], the existence of IASW’s for which
+1 ≤ n ≤ N and dn
+dη = 0 at n = 1, N,
+(5)
+requires the following equations and inequality:
+V (n)|n=1 = 0,
+(6)
+dV
+dn |n=1 = 0,
+(7)
+3
+
+d2V
+dn2 |n=1 < 0,
+(8)
+and V (n)|n=N = 0.
+(9)
+Abdelsalam et al. [1] noted that Eqs. (6) and (7) are automatically satis-
+fied by Eq. (2) while the inequality (8) is satisfied if and only if
+lx < M < 1, i.e., c < 0.6 < a,
+in view of Eqs. (3) and (4).
+For the nonlinear dispersion relation, Eq.
+(9), they have numerically
+solved it for several specific values of lx (lx = 0.66, 0.68, 0.7) and on that
+basis argued that if Eq. (9) can be rewritten as
+N = N(lx, M),
+(10)
+where the maximum density N is a decreasing function of lx, i.e., ∂N
+∂lx < 0
+and N is an increasing function of M, i.e., ∂N
+∂M > 0.
+However, in the present work our motivation is to solve the problem of
+Abdelsalam et al. [1] with an analytical methodology under a more general
+treatment. For this we have considered a different format and have shown
+that some of their outcomes can be retrieved with a convincing way and can
+be demonstrated valid in the physical realm.
+2. An analytical methodology
+Putting n = x3 the equations (1) and (2) can be rewritten as
+�dx
+dη
+�2
++ x(1 − x)f(x, lx, M)
+(5ax8 − 3)2
+= 0,
+(11)
+where
+f(x, lx, M) = (1 + x + x2)2
++9lx
+2x6(1 + x + x2 + x3 + x4)2
+25M4
+−3x6(3 + 6x + 4x2 + 2x3)
+5M2
+−3lx
+2x3(2 + 4x + 6x2 + 3x3)
+5M2
+.
+(12)
+4
+
+Equations (5) and (9) are now rewritten as
+dx
+dη = 0 at x = 1, N1/3,
+(13)
+f(N1/3, lx, M) = 0.
+(14)
+The above Eqs. (3) and (4) and the inequality (8) remain unchanged
+except Eq. (9) which is to be replaced by Eq. (14). In other words, the
+question now is whether Eq.
+(14) can determine N (or x) uniquely.
+To
+answer this one needs the following observations on f(x, lx, M).
+3. Observations on f(x, lx, M)
+3.1. Observation 1:
+(i) f(0, lx, M) = 1,
+(ii) f(1) = (3 − 5a)(3 − 5c) < 0,
+(iii) f(∞) > 0.
+Proof: Trivial.
+3.2. Observation 2:
+For given a and c there exist a unique pair of (α, β) such that
+f(x, lx, M) > 0, for 0 < x < β,
+f(β, lx, M) = 0,
+f(x, lx, M) < 0, for β < x < α,
+f(α, lx, M) = 0,
+and f(x, lx, M) > 0 for x > α,
+where 0 < β < 1 < α.
+Corollary:
+∂f(x, lx, M)
+∂x
+> 0 at x = α.
+Proof: Trivial.
+5
+
+3.3. Observation 3:
+f(x, lx, M) < 0 at x = ( 3
+5a)1/8.
+Proof: See Appendix.
+Corollary:
+β < ( 3
+5a)1/8 < 1.
+Proof: Trivial from observation 2.
+3.4. Observation 4:
+∂f(x, lx, M)
+∂lx
+> 0 for x > 1.
+Proof: See Appendix.
+3.5. Observation 5:
+For any α > 1 there exists lx and M that satisfy f(α, lx, M) = 0 and also
+satisfy the inequality (9).
+Proof: See Appendix.
+With these observations one can uniquely determine x (or N) (> 1) satis-
+fying Eq. (12) and also deals with decreasing/increasing feature of x (or N)
+as well as for increase of lx or M. These are answered as follows.
+4. Proof of uniqueness of N
+From the Observation 1, we note that f(1) < 0 and f(∞) > 0. Owing to
+the continuity of f(x) there exists one x, such that f(x1) = 0 and x1 > 1. If
+possible, let there exist x1 and x2 such that
+f(x1) = f(x2) = 0 and x2 > x1 > 1.
+(15)
+6
+
+From Eq. (15), applying Rolle’s theorem, there exists x3 and x4, such
+that
+f ′(x3) = f ′(x4) = 0 and x2 > x4 > x1 > x3 > 1.
+(16)
+But f ′(x) is a polynomial of degree 13 such that f ′(−∞) < 0, f ′(0) > 0,
+f ′(1) < 0 and f ′(∞) > 0. So we can see that f ′(x) vanishes only for x > 1
+which contradicts Eq. (16).
+Hence there exists a unique x(> 1) such that f(x, lx, M) = 0, i.e. there
+exists unique N(> 1) satisfying Eq. (9).
+Now, we have to show analytically that the maximum density N is a
+decreasing function of lx and is an increasing function of M. Differentiating
+both sides of Eq. (12) with respect to M, one gets
+∂f
+∂M < 0, for x > 1.
+(17)
+For
+∂f
+∂M =
+6x3
+25M5[−6l2
+xx3(1 + x + x2 + x3 + x4)2 + 5x3M2(3 + 6x + 4x2 + 2x3)
++5l2
+xM2(2 + 4x + 6x2 + 3x3)]
+<
+6x3
+25M5[−6x3(1 + x + x2 + x3 + x4)2 + 5x3(3 + 6x + 4x2 + 2x3)
++5l2
+x(2 + 4x + 6x2 + 3x3)]
+(since lx < M < 1)
+=
+6x3
+25M5[−18(x9 − x4) − 30(x7 − x2) − 17(x8 − x3) − 7(x8 − x)
+−13(x6 − x) − 10(x10 − 1) − 2(x10 − x5) − x6 − 6x11]
+< 0,
+for x > 1.
+(18)
+From Eqs. (14) and (11), we obtain
+∂N1/3
+∂lx
+= −
+∂f
+∂lx
+∂f
+∂N1/3
+and ∂N1/3
+∂M
+= −
+∂f
+∂M
+∂f
+∂N1/3
+,
+which gives
+∂N1/3
+∂lx
+∂N1/3
+∂M
+=
+∂f
+∂lx
+∂f
+∂M
+, i.e.,
+∂N
+∂lx
+∂N
+∂M
+=
+∂f
+∂lx
+∂f
+∂M
+,
+i.e., ∂N
+∂lx
+< 0 for x > 1,
+(using observation 4 and Eq. (17)).
+7
+
+Again from Eq. (14), we have
+∂N1/3
+∂lx
+∂lx
+∂M
+∂M
+∂N1/3 = −1,
+and 1
+3N−2/3∂N
+∂lx
+5
+3MC = −1
+3N−2/3 ∂N
+∂M ,
+(by Observation (4))
+i.e., ∂N
+∂M > 0 for
+x > 1
+(since, ∂N
+∂lx < 0).
+4.1. Proof of Observation 5:
+From the observation 1 one can see that the equation f(x, lx, M) = 0 has
+at least one root between 0 and 1 and one root greater then 1. Also one can
+note that f(x, lx, M) regarded as a polynomial in x has two changes of sign
+and hence by Descarte’s rule of sign has at most two positive roots. Hence
+equation f(x, lx, M) = 0 has exactly one root between 0 and 1 and exactly
+one root greater than 1 which are called β and α respectively. The continuity
+of f(x, lx, M) ensures that the remaining part of the observation is true.
+5. Conclusion
+In the present work our main motivation was to provide an analytically
+performed rigorous base of the study of Abdelsalam et al. [1] on the propaga-
+tion of fully non-linear ion-acoustic waves in a collision-less magneto-plasma
+with degenerate electrons. The outcomes of the investigation are interesting
+and some explicit features can be exhibited as follows:
+(1) only hump and no cavity is possible;
+(2) for humps, (i) the algebraic equation for the maximum density N ob-
+tained by them determines it uniquely (under the assumption N > 1), (ii) N
+decreases with lx (the direction cosine of the wave vector k along the x-axis)
+and (iii) N increases with the increase of the Mach number (M). All these
+results yield simply from the maximum density N which can be uniquely
+determined by Eq. (9) under the constraint N > 1.
+Another motivation of the present work is related to the astrophysical
+relevance of an EPI plasma, especially in the cases of AGNs [8, 9], pulsar
+magneto-spheres [10, 11], neutron stars and white dwarfs [12, 13].
+A su-
+porting and confirmirmational results of Abdelsalam et al. [1] therefore will
+enhance to understand deeply the structural phenomena occuring in differ-
+ent astrophysical systems. In this connection we would like to mention the
+8
+
+very recent work of Piotrovich et al. [9] where they have hypothesized that
+the AGNs are wormhole mouths rather than supermassive black holes. Es-
+sentially due to bizzare gravitational formation wormholes may emit gamma
+radiation as a result of a collision of accreting flows inside it. Now the in-
+teresting fact is that the radiation has a distinctive spectrum much different
+from those of jets or accretion discs of AGNs. Hopefully an observation of
+such radiation via the EPI and hence IASW would serve as evidence of the
+existence of wormholes.
+Appendix
+Proof of Observation 3
+Let
+� 3
+5a
+�1/8 = γ so that a =
+3
+5γ8
+9
+
+Then at x = γ
+f(x, a, c) = (1 + γ + γ2)2 + 3c
+5γ2(1 + γ + γ2 + γ3 + γ4)2
+− 3
+5γ2(3 + 6γ + 4γ2 + 2γ3) − cγ3(2 + 4γ + 6γ2 + 3γ3
+=
+1
+5γ2[(5γ2(1 + γ + γ2)2 − 3(3 + 6γ + 4γ2 + 2γ3))
++c(3(1 + γ + γ2 + γ3 + γ4)2 − 5γ5(2 + 4γ + 6γ2 + 3γ3))]
+=
+1
+5γ2[(−9 − 18γ − 7γ2 + 4γ3 + 15γ4 + 10γ5 + 5γ6)
++c(3 + 6γ + 9γ2 + 12γ3 + 15γ4 + 2γ5 − 11γ6 − 24γ7 − 12γ8)]
+=
+1
+5γ2[(γ − 1)(9 + 27γ + 34γ2 + 30γ3 + 15γ4 + 5γ5)]
+−c(γ − 1)(3 + 9γ + 18γ2 + 30γ3 + 45γ4 + 47γ5 + 36γ6 + 12γ7)
+= γ − 1
+5γ2 [(9 + 27γ + 34γ2 + 30γ3 + 15γ4 + 5γ5)
+−c(3 + 9γ + 18γ2 + 30γ3 + 45γ4 + 47γ5 + 36γ6 + 12γ7)]
+< γ − 1
+5γ2 [(9 + 27γ + 34γ2 + 30γ3 + 15γ4 + 5γ5)
+−3(3 + 9γ + 18γ2 + 30γ3 + 45γ4 + 47γ5 + 36γ6 + 12γ7)]
+≤ γ − 1
+25γ2 [36 + 108γ + 116γ2 + 60γ3 − 60γ4 − 116γ5 − 108γ6 − 36γ7]
+< 0
+if γ < 1
+Proof of Observation 4
+∂f(x, a, c)
+∂c
+= ax6(1 + x + x2 + x3 + x4)2 − x3(2 + 4x + 6x2 + 3x3)
+≥ x3
+5 [3x3(1 + x + x2 + x3 + x4)2 − 5(2 + 4x + 6x2 + 3x3)] (since, a =
+3
+5M2)
+= x3
+5 [3x3(1 + 2x + 3x+4x3 + 5x4 + 4x5 + 3x6 + 2x7 + x8) − 5(2 + 4x + 6x2 + 3x3)]
+= x3
+5 [−10 − 20x − 30x2 − 12x3 + 6x4 + 9x5 + 12x6 + 15x7 + 12x8 + 9x9 + 6x10 + 3x11)
+> 0 for x > 1
+10
+
+Declaration of competing interest
+The authors declare that they have no known competing financial inter-
+ests or personal relationships that could have appeared to influence the work
+reported in this paper
+acknowledgement
+One of the authors, KKG would like to thank the authority of Abacus
+Institute of Engineering and Management for all the facilities and encourage-
+ment. We all are grateful to the anonymous referee for the useful comments
+which have enhanced the quality of the paper.
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+

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+page_content=' Uttar Pradesh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' India  Abstract  The propagation of fully nonlinear ion acoustic solitary waves (IASW) in  a magneto-plasma with degenerate electrons investigated by Abdelsalam et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [[1] Physics Letters A 372 (2008) 4923].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Based on their work in the present  work, a rigorous and general analytical study is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' This confirms  their implied assumption that (i) only hump and no cavity is possible and  (ii) for humps, the algebraic equation for the maximum density N obtained  by them determines it uniquely (naturally assumes N > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Here we confirm  analytically their assertion that N decreases with lx (the direction cosine of  the wave vector k along the x-axis) and N increases with the increase of the  Mach number (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Keywords: Waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Plasmas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' MHD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Hydrodynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Mach number  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Introduction  Recently, the physics of an electron-positron-ion (EPI) plasma [2, 3, 4, 5, 6,  7] has received considerable attention, mainly due to its importance in many  systems in laboratory plasma as well as astrophysical arena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' EPI plasma  Email addresses: moumita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='indra93@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='com (Moumita Indra),  kkghosh1954@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='com (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Ghosh), saibal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='ray@gla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='in (Saibal Ray)  Orcid ID: 0000-0002-7900-7947  Orcid ID: 0000-0002-5909-0544  Preprint submitted to Chinese Journal of PhysicsReceived 2022 September 22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' accepted 2022 month day  exists in places such as active galactic nuclei (AGNs) [8, 9], pulsar magneto-  spheres [10, 11] and in many dense astronomical environments, namely, neu-  tron stars and white dwarfs [12, 13] which is supposed to play a key role in  understanding the origin and evolution of our entire universe [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' This kind  of plasma may also be practically produced in laboratories [15, 16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Electrons and positrons are assumed relativistic and degenerate, follow-  ing the Fermi–Dirac statistics, whereas the warm ions are described by a  set of classical fluid equations with an individual charge of Zie, (Zi denotes  the ion-charge state, while e is the electron charge), subject to the influence  of the electrostatic potential φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Quantum hydrodynamics  [23, 24, 25, 26],  which describes quantum systems within a hydrodynamic framework, was  first proposed by Madelung [19] and Bohm [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Although such a descrip-  tion is formally accurate for a single particle, Manfredi and Haas [21] later  expanded the idea to many-particle systems and it gained significant favour  in the areas of the quantum plasma community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Considering the one-dimensional QHD model in the limit of the small  mass ratio of the charge carriers, Hass et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [22] were the first to study  the ion-acoustic waves in unmagnetized quantum plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' This model has  been used in various investigations by several authors [23, 24, 25, 26] where  generally a linear dispersion relation is derived in the linear approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Thus, ion-acoustic waves (IAW), a fundamental mode in plasma environ-  ments, have been a subject of extensive research over several decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Khan  and Haque [27] showed that in the small (linear) limit of quantum diffraction  parameter H (ratio of the plasmon energy to the Fermi energy), the system  behaves as the classical IAW whereas in the non-linear regime the system  behaves differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' One of the most interesting non-linear features of IAW  is the existence of ion-acoustic solitary waves (IASW) [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  In the weakly nonlinear limit, the quantum plasma is shown to support  waves described by a deformed Korteweg–de Vries (KdV) equation which  depends in a non-trivial way on the quantum parameter H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' However, in the  fully non-linear regime, the system exhibits travelling waves which show a  periodic pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Hence there are two main approaches used to investigate  IASWs, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=', the reductive perturbation technique (KdV method) [30] and  the pseudo-potential technique for large-amplitude solitary waves (Sagdeev  method) [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' The theory of solitons in magneto-plasma was greatly improved  by an intriguing work by Abdelsalam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [1] on completely non-linear IASW  travelling obliquely to an external magnetic field in a collision-less dense  Thomas-Fermi magneto-plasma with degenerate electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  2  The degenerate electrons in the above scenario may be described using  the Thomas-Fermi approximation [32, 33] whereas the ion component can  be thought of as a classical gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' They have obtained an energy balance-like  equation involving the Sagdeev potential as follows:  1  2  �dn  dη  �2  + V (n) = 0,  (1)  where Sagdeev-like pseudo-potential V (n) is given by  V (n) =  9n6  2(5an8/3 − 3)2[5an8/3 − 2an5/3−3a+1  M2  +1 − 2n  M2n2 + cM2(an10/3 − 2an5/3 − 2  9 + a + 5)],  (2)  where  a =  3  5M2,  (3)  c = 3lx  2  5M2,  (4)  η = lxx + lyy − Mt, lx  2 + ly  2 = 1 and n = n(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Here M is the Mach number, lx and ly are the direction cosines of the  wave vector k along the x and y axes respectively, n(= ne  no) where ne is the  electron density and no is the unperturbed electron density with ni as the  ion density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  As indicated by Abdelsalam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [1], the existence of IASW’s for which  1 ≤ n ≤ N and dn  dη = 0 at n = 1, N,  (5)  requires the following equations and inequality:  V (n)|n=1 = 0,  (6)  dV  dn |n=1 = 0,  (7)  3  d2V  dn2 |n=1 < 0,  (8)  and V (n)|n=N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (9)  Abdelsalam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [1] noted that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (6) and (7) are automatically satis-  fied by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (2) while the inequality (8) is satisfied if and only if  lx < M < 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=', c < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='6 < a,  in view of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  For the nonlinear dispersion relation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (9), they have numerically  solved it for several specific values of lx (lx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='66, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='68, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='7) and on that  basis argued that if Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (9) can be rewritten as  N = N(lx, M),  (10)  where the maximum density N is a decreasing function of lx, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=', ∂N  ∂lx < 0  and N is an increasing function of M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=', ∂N  ∂M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  However, in the present work our motivation is to solve the problem of  Abdelsalam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [1] with an analytical methodology under a more general  treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' For this we have considered a different format and have shown  that some of their outcomes can be retrieved with a convincing way and can  be demonstrated valid in the physical realm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' An analytical methodology  Putting n = x3 the equations (1) and (2) can be rewritten as  �dx  dη  �2  + x(1 − x)f(x, lx, M)  (5ax8 − 3)2  = 0,  (11)  where  f(x, lx, M) = (1 + x + x2)2  +9lx  2x6(1 + x + x2 + x3 + x4)2  25M4  −3x6(3 + 6x + 4x2 + 2x3)  5M2  −3lx  2x3(2 + 4x + 6x2 + 3x3)  5M2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (12)  4  Equations (5) and (9) are now rewritten as  dx  dη = 0 at x = 1, N1/3,  (13)  f(N1/3, lx, M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (14)  The above Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (3) and (4) and the inequality (8) remain unchanged  except Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (9) which is to be replaced by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' In other words, the  question now is whether Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (14) can determine N (or x) uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  To  answer this one needs the following observations on f(x, lx, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Observations on f(x, lx, M)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Observation 1:  (i) f(0, lx, M) = 1,  (ii) f(1) = (3 − 5a)(3 − 5c) < 0,  (iii) f(∞) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Proof: Trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Observation 2:  For given a and c there exist a unique pair of (α, β) such that  f(x, lx, M) > 0, for 0 < x < β,  f(β, lx, M) = 0,  f(x, lx, M) < 0, for β < x < α,  f(α, lx, M) = 0,  and f(x, lx, M) > 0 for x > α,  where 0 < β < 1 < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Corollary:  ∂f(x, lx, M)  ∂x  > 0 at x = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Proof: Trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Observation 3:  f(x, lx, M) < 0 at x = ( 3  5a)1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Proof: See Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Corollary:  β < ( 3  5a)1/8 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Proof: Trivial from observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Observation 4:  ∂f(x, lx, M)  ∂lx  > 0 for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Proof: See Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Observation 5:  For any α > 1 there exists lx and M that satisfy f(α, lx, M) = 0 and also  satisfy the inequality (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Proof: See Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  With these observations one can uniquely determine x (or N) (> 1) satis-  fying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (12) and also deals with decreasing/increasing feature of x (or N)  as well as for increase of lx or M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' These are answered as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Proof of uniqueness of N  From the Observation 1, we note that f(1) < 0 and f(∞) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Owing to  the continuity of f(x) there exists one x, such that f(x1) = 0 and x1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' If  possible, let there exist x1 and x2 such that  f(x1) = f(x2) = 0 and x2 > x1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (15)  6  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (15), applying Rolle’s theorem, there exists x3 and x4, such  that  f ′(x3) = f ′(x4) = 0 and x2 > x4 > x1 > x3 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (16)  But f ′(x) is a polynomial of degree 13 such that f ′(−∞) < 0, f ′(0) > 0,  f ′(1) < 0 and f ′(∞) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' So we can see that f ′(x) vanishes only for x > 1  which contradicts Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Hence there exists a unique x(> 1) such that f(x, lx, M) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' there  exists unique N(> 1) satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Now, we have to show analytically that the maximum density N is a  decreasing function of lx and is an increasing function of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Differentiating  both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (12) with respect to M, one gets  ∂f  ∂M < 0, for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (17)  For  ∂f  ∂M =  6x3  25M5[−6l2  xx3(1 + x + x2 + x3 + x4)2 + 5x3M2(3 + 6x + 4x2 + 2x3)  +5l2  xM2(2 + 4x + 6x2 + 3x3)]  <  6x3  25M5[−6x3(1 + x + x2 + x3 + x4)2 + 5x3(3 + 6x + 4x2 + 2x3)  +5l2  x(2 + 4x + 6x2 + 3x3)]  (since lx < M < 1)  =  6x3  25M5[−18(x9 − x4) − 30(x7 − x2) − 17(x8 − x3) − 7(x8 − x)  −13(x6 − x) − 10(x10 − 1) − 2(x10 − x5) − x6 − 6x11]  < 0,  for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (18)  From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (14) and (11), we obtain  ∂N1/3  ∂lx  = −  ∂f  ∂lx  ∂f  ∂N1/3  and ∂N1/3  ∂M  = −  ∂f  ∂M  ∂f  ∂N1/3  ,  which gives  ∂N1/3  ∂lx  ∂N1/3  ∂M  =  ∂f  ∂lx  ∂f  ∂M  , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=',  ∂N  ∂lx  ∂N  ∂M  =  ∂f  ∂lx  ∂f  ∂M  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=', ∂N  ∂lx  < 0 for x > 1,  (using observation 4 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  7  Again from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (14), we have  ∂N1/3  ∂lx  ∂lx  ∂M  ∂M  ∂N1/3 = −1,  and 1  3N−2/3∂N  ∂lx  5  3MC = −1  3N−2/3 ∂N  ∂M ,  (by Observation (4))  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=', ∂N  ∂M > 0 for  x > 1  (since, ∂N  ∂lx < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Proof of Observation 5:  From the observation 1 one can see that the equation f(x, lx, M) = 0 has  at least one root between 0 and 1 and one root greater then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Also one can  note that f(x, lx, M) regarded as a polynomial in x has two changes of sign  and hence by Descarte’s rule of sign has at most two positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Hence  equation f(x, lx, M) = 0 has exactly one root between 0 and 1 and exactly  one root greater than 1 which are called β and α respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' The continuity  of f(x, lx, M) ensures that the remaining part of the observation is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Conclusion  In the present work our main motivation was to provide an analytically  performed rigorous base of the study of Abdelsalam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [1] on the propaga-  tion of fully non-linear ion-acoustic waves in a collision-less magneto-plasma  with degenerate electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' The outcomes of the investigation are interesting  and some explicit features can be exhibited as follows:  (1) only hump and no cavity is possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (2) for humps, (i) the algebraic equation for the maximum density N ob-  tained by them determines it uniquely (under the assumption N > 1), (ii) N  decreases with lx (the direction cosine of the wave vector k along the x-axis)  and (iii) N increases with the increase of the Mach number (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' All these  results yield simply from the maximum density N which can be uniquely  determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (9) under the constraint N > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Another motivation of the present work is related to the astrophysical  relevance of an EPI plasma, especially in the cases of AGNs [8, 9], pulsar  magneto-spheres [10, 11], neutron stars and white dwarfs [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  A su-  porting and confirmirmational results of Abdelsalam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [1] therefore will  enhance to understand deeply the structural phenomena occuring in differ-  ent astrophysical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' In this connection we would like to mention the  8  very recent work of Piotrovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [9] where they have hypothesized that  the AGNs are wormhole mouths rather than supermassive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Es-  sentially due to bizzare gravitational formation wormholes may emit gamma  radiation as a result of a collision of accreting flows inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Now the in-  teresting fact is that the radiation has a distinctive spectrum much different  from those of jets or accretion discs of AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Hopefully an observation of  such radiation via the EPI and hence IASW would serve as evidence of the  existence of wormholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Appendix  Proof of Observation 3  Let  � 3  5a  �1/8 = γ so that a =  3  5γ8  9  Then at x = γ  f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' c) = (1 + γ + γ2)2 + 3c  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5γ2(1 + γ + γ2 + γ3 + γ4)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='− 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5γ2(3 + 6γ + 4γ2 + 2γ3) − cγ3(2 + 4γ + 6γ2 + 3γ3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5γ2[(5γ2(1 + γ + γ2)2 − 3(3 + 6γ + 4γ2 + 2γ3))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='+c(3(1 + γ + γ2 + γ3 + γ4)2 − 5γ5(2 + 4γ + 6γ2 + 3γ3))]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5γ2[(−9 − 18γ − 7γ2 + 4γ3 + 15γ4 + 10γ5 + 5γ6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='+c(3 + 6γ + 9γ2 + 12γ3 + 15γ4 + 2γ5 − 11γ6 − 24γ7 − 12γ8)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5γ2[(γ − 1)(9 + 27γ + 34γ2 + 30γ3 + 15γ4 + 5γ5)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='−c(γ − 1)(3 + 9γ + 18γ2 + 30γ3 + 45γ4 + 47γ5 + 36γ6 + 12γ7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='= γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5γ2 [(9 + 27γ + 34γ2 + 30γ3 + 15γ4 + 5γ5)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='−c(3 + 9γ + 18γ2 + 30γ3 + 45γ4 + 47γ5 + 36γ6 + 12γ7)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='< γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5γ2 [(9 + 27γ + 34γ2 + 30γ3 + 15γ4 + 5γ5)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='−3(3 + 9γ + 18γ2 + 30γ3 + 45γ4 + 47γ5 + 36γ6 + 12γ7)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='≤ γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='25γ2 [36 + 108γ + 116γ2 + 60γ3 − 60γ4 − 116γ5 − 108γ6 − 36γ7]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='< 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='if γ < 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='Proof of Observation 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='∂f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' c)  ∂c  = ax6(1 + x + x2 + x3 + x4)2 − x3(2 + 4x + 6x2 + 3x3)  ≥ x3  5 [3x3(1 + x + x2 + x3 + x4)2 − 5(2 + 4x + 6x2 + 3x3)] (since,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' a =  3  5M2)  = x3  5 [3x3(1 + 2x + 3x+4x3 + 5x4 + 4x5 + 3x6 + 2x7 + x8) − 5(2 + 4x + 6x2 + 3x3)]  = x3  5 [−10 − 20x − 30x2 − 12x3 + 6x4 + 9x5 + 12x6 + 15x7 + 12x8 + 9x9 + 6x10 + 3x11)  > 0 for x > 1  10  Declaration of competing interest  The authors declare that they have no known competing financial inter-  ests or personal relationships that could have appeared to influence the work  reported in this paper  acknowledgement  One of the authors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' KKG would like to thank the authority of Abacus  Institute of Engineering and Management for all the facilities and encourage-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' We all are grateful to the anonymous referee for the useful comments  which have enhanced the quality of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  References  [1] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
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+arXiv:2301.03120v1  [quant-ph]  8 Jan 2023
+On generating r-uniform subspaces with the isometric mapping method
+K. V. Antipin∗
+Faculty of Physics, M. V. Lomonosov Moscow State University,
+Leninskie gory, Moscow 119991, Russia
+(Dated: January 10, 2023)
+We propose a compositional approach to construct subspaces consisting entirely of r-uniform
+states, including the ones in heterogeneous systems. The approach allows one to construct new
+objects from old ones: it combines encoding isometries of pure quantum error correcting codes with
+entangled multipartite states and subspaces. The presented methods can be also used to construct
+new pure quantum error correcting codes from certain combinations of old ones. The approach is
+illustrated with various examples including constructions of 2-, 3-, 4-, 5-uniform subspaces. The
+results are then compared with analogous constructions obtained with the use of orthogonal arrays.
+I.
+INTRODUCTION
+Multipartite entanglement is crucial for realization of
+various protocols of quantum information processing [1–
+4].
+One important manifestation of this phenomenon
+is genuine multipartite entanglement (GME) [5–7].
+In
+GME states entanglement is present in every bipartite
+cut of a compound system, which makes them useful
+in communication protocols such as quantum telepor-
+tation and dense coding [8, 9].
+Another interesting
+form is r-uniform (also known as maximal) entangle-
+ment [3, 10, 11]. Each reduction of an r-uniform state
+to r subsystems is maximally mixed. This property is
+closely related to quantum secret sharing [12, 13] and
+quantum error correcting codes (QECCs) [14, 15].
+Recently the notion of entangled subspaces has been
+attracting much attention owing to its potential use in
+quantum information science. It was first described in
+Ref. [16], where the term “completely entangled sub-
+spaces” was coined.
+Later, depending on the form
+of multipartite entanglement present in each state of
+a subspace, several other types were introduced: gen-
+uinely entangled subspaces (GESs) [17], negative par-
+tial transpose (NPT) subspaces [18], r-uniform sub-
+spaces (rUSs) [15].
+In the present paper we concen-
+trate on construction of r-uniform subspaces, mostly
+for heterogeneous systems, i. e., those having differ-
+ent local dimensions.
+There are a number of tools
+for constructing r-uniform states in homogeneous sys-
+tems: graph states [19], elements of combinatorial de-
+sign such as Latin squares [20], symmetric matrices [21],
+orthogonal arrays (OAs) [22] and their variations [23–
+25].
+For construction of r-uniform states in heteroge-
+neous systems OAs were extended to mixed orthogonal
+arrays (MOAs) [26]. Recent developments of this method
+can be found in Refs. [27, 28].
+The main source for
+r-uniform subspaces in homogeneous systems are pure
+quantum error correcting codes [3, 15]. Little is known
+about construction of r-uniform subspaces in heteroge-
+neous systems (the only method we could find in liter-
+∗ kv.antipin@physics.msu.ru
+ature was based on Proposition 12 of Ref. [28]).
+De-
+velopment of new methods of construction of rUSs for
+this case is our main motivation for the present paper.
+R-uniform subspaces in heterogeneous systems have re-
+lation to QECCs over mixed alphabets [29] and quan-
+tum information masking [30]. To our knowledge, for a
+given system the largest possible dimension of rUSs is
+unknown, so building new instances of such subspaces
+could bring some insights in this question.
+We use compositional tools of diagrammatic reason-
+ing [31–33], which allow us to come up with new con-
+structions and provide further instances of states and
+subspaces with important properties. Tensor diagrams
+are widely used in quantum information theory, in par-
+ticular, in theory of QECCs. Recently a framework for
+the construction of new stabilizer QECCs from old ones
+with the use of tensor networks has been presented in
+Ref. [34].
+The paper is organized as follows. In Section II nec-
+essary definitions and some theoretical background are
+given. The main results of the current paper are provided
+in Section III. In Subsection III A we give diagrammatic
+representation of basic properties of rUSs upon which, in
+Subsection III B, we derive the methods of constructing
+rUSs in heterogeneous systems such as glueing several
+subspaces together, eliminating parties, combining pure
+error correcting codes and maximally entangled states
+and subspaces. In Subsection III C we compare our re-
+sults with the ones obtained with the use of the mixed
+orthogonal arrays method. Finally, in Section IV we con-
+clude with discussing possible directions of further re-
+search.
+II.
+PRELIMINARIES
+Let us first give the definition of r-uniform states of an
+n-partite finite-dimensional system with local dimensions
+d1, . . . , dn. Such a system is usually associated with the
+tensor product Hilbert space Cd1 ⊗. . .⊗Cdn. A state |ψ⟩
+in Cd1 ⊗ . . .⊗ Cdn is called r-uniform if all its reductions
+
+2
+FIG. 1. Doubling notation for the process of action of a linear
+operator V on a pure state ψ
+FIG. 2. Reduction of a bipartite pure state ψ to subsystem A
+at least to r parties are maximally mixed, i. e.,
+Tr{i1, ..., ir}c[|ψ⟩⟨ψ|] =
+1
+di1 · . . . · dir
+Ii1, ..., ir
+(1)
+for
+all
+r-element
+subsets
+{i1, . . . , ir}
+of
+the
+set
+{1, . . . , n}. Here {i1, . . . , ir}c denotes the complement
+of the given set in the set of all parties.
+It is clear
+that r-uniform state is also l-uniform for all l < r. By
+the properties of the Schmidt decomposition, the nec-
+essary condition for r-uniform states to exist is that
+di1 ·. . .·dir ⩽ dir+1 ·. . .·din is satisfied for each bipartition
+i1, . . . , ir|ir+1, . . . , in.
+An r-uniform subspace — a subspace of Cd1 ⊗. . .⊗Cdn
+consisting entirely of r-uniform vectors.
+For homogeneous systems, i. e., those having equal
+local dimensions, the existence of r-uniform subspaces
+can be deduced from the existence of certain quan-
+tum error correcting codes (QECCs).
+Recall that a
+QECC ((n, K, d))D is a special K-dimensional subspace
+of
+�
+CD�⊗n such that for each its state any error affect-
+ing not more than a certain number of subsystems can
+be corrected. For a code with distance d = 2t + 1 the
+number is equal to t. In addition, a code with distance d
+can detect d − 1 errors.
+In addition to the ((n, K, d))D notation for general
+QECCs, we will use the designation [[n, k, d]]D for stabi-
+lizer QECCs. While the symbols n and d from the latter
+notation have the same sense as those in the former one,
+the dimension of the codespace for the code [[n, k, d]]D
+is equal to Dk.
+A quantum error correcting code is called pure if
+⟨i| E |j⟩ = 0
+(2)
+for any states |i⟩ , |j⟩ from an orthonormal set spanning
+the code space and for any error operator E with weight
+strictly less than the distance of the code.
+It is known that each pure QECC ((n, K, d))D yields a
+K-dimensional (d − 1)-uniform subspace of (CD)⊗n, and
+vice versa [15].
+To address the case of heterogeneous systems, in the
+present paper we will use encoding isometries of the exist-
+ing pure QECCs in combination with various states and
+FIG. 3. Diagrammatic representation of the maximally mixed
+state (up to the normalization factor).
+subspaces of lower number of parties. A similar approach
+dealing with isometric mapping to entangled subspaces
+proved to be effective in constructing multipartite gen-
+uinely entangled subspaces [35].
+Throughout the paper we use tensor diagrams, in par-
+ticular, we use doubled-process theory notation adopted
+from Ref. [31]. The doubling notation indicates the pas-
+sage from pure state vectors to their associated density
+operators, as shown on Fig. 1.
+To deal also with mixed states, the discarding sym-
+bol (map) is used.
+Applying the discarding map to a
+subsystem of a multipartite state is equivalent to tracing
+out the subsystem, as shown on Fig. 2.
+The adjoint of the discarding map (see Fig. 3) denotes
+the identity operator, which is proportional to the max-
+imally mixed state.
+III.
+RESULTS
+A.
+Basic properties and their diagrammatic
+representation
+We start with pointing at an important basic prop-
+erty of subspaces under consideration. Let |φ⟩ and |χ⟩
+be two mutually orthogonal normalized vectors in an r-
+uniform subspace W. An arbitrary (normalized) linear
+combination |ψ⟩ = α |φ⟩ + β |χ⟩ is also in W, and hence
+its reduction to some r-element subset S of the set of all
+parties yields
+TrSc[|ψ⟩⟨ψ|] = N IS
+= N IS + αβ∗ TrSc[|φ⟩⟨χ|] + βα∗ TrSc[|χ⟩⟨φ|],
+(3)
+where N =
+��
+i∈S di
+�−1, the normalization factor. Con-
+sequently, the last two terms sum up to zero:
+αβ∗ TrSc[|φ⟩⟨χ|] + βα∗ TrSc[|χ⟩⟨φ|] = 0,
+∀ α, β ∈ C,
+|α|2 + |β|2 = 1.
+(4)
+Setting first α real and β imaginary and then both of
+them real, one can deduce that
+TrSc[|φ⟩⟨χ|] = 0.
+(5)
+Now we can formulate this property as
+Lemma 1. For any orthonormal set {|ψ⟩i} spanning r-
+uniform subspace it follows that
+TrSc[|ψi⟩⟨ψj|] ∼ δij IS
+(6)
+for any r-element subset S of the set of all parties.
+
+1TrB|)V
+A)A3
+FIG. 4. Action of V together with tracing out subsystems Sc
+results in a channel ΦS which discards its input and returns
+the maximally mixed state.
+Eq. (6) is known to hold for codewords of pure QECCs
+of distance r + 1, which correspond to r-uniform sub-
+spaces for homogeneous systems, but it is worth stress-
+ing that it is valid for more general case of heterogeneous
+systems.
+Let us think of r-uniform subspaces in terms of isome-
+tries and quantum channels. To each such subspace W
+with dimension K one can relate an isometry V : CK →
+Cd1 ⊗ . . . ⊗ Cdn, which maps an orthonormal basis {|i⟩}
+of CK to some orthonormal set {|ψ⟩i} spanning W:
+V |i⟩ = |ψi⟩ .
+(7)
+Hence, the range of the isometry V coincides with the
+subspace W. As before, let us choose an r-element subset
+S of the set of n parties. Applying the isometry to a state
+in CK with subsequent tracing out n − r subsystems in
+Sc results in action of a quantum channel on the state:
+TrSc[V |φ⟩⟨φ| V †] = ΦS(|φ⟩⟨φ|),
+|φ⟩ ∈ CK.
+(8)
+We
+thus
+obtain
+a
+family
+of
+quantum
+channels
+ΦS : L(CK) → L(Cdi1 ⊗. . .⊗Cdir ), where {i1, . . . , ir} =
+S; one channel for each choice of S. Since subspace W
+is r-uniform, a channel ΦS maps all states of CK to the
+identity on S:
+ΦS(|φ⟩⟨φ|) =
+1
+di1 · . . . · dir
+Ii1, ..., ir,
+|φ⟩ ∈ CK.
+(9)
+In other words, the channels discard the input and map
+everything to the maximally mixed state:
+ΦS(X) =
+Tr[X]
+di1 · . . . · dir
+IS,
+X ∈ L(CK).
+(10)
+By setting X = |i⟩⟨j| in this expression, with the use of
+Eqs. (7), (8), we recover Eq. (6).
+It should be stressed that the above property of map-
+ping to the maximally mixed state holds when the in-
+put dimension of the isometry (and of the corresponding
+channel) is not greater than the dimension of the range
+of the isometry, i. e., the dimension of the r-uniform sub-
+space. In fact, the input dimension can be strictly less
+than that dimension, in which case the isometry takes the
+input states to some subspace of the r-uniform space.
+FIG. 5.
+Construction of a 2-uniform state from an encod-
+ing isometry V of the ((5, 3, 3))3 pure code and a maximally
+entangled state |ψ⟩ in C2 ⊗ C3.
+Diagrammatic representation of Eqs. (8) and (10) is
+shown on Fig. 4, where the symbol ”∼” means that the
+two diagrams are equal up to a scalar factor (the normal-
+ization constant for the maximally mixed state). This
+construction will be crucial in further considerations.
+Now we can choose V to be an encoding isometry of
+some pure quantum error correcting code. Let us try to
+combine this isometry with some states.
+Example: 2-uniform state in heterogeneous systems
+Consider the ((5, 3, 3))3 pure code [36] and its encod-
+ing isometry V . Application of V to one of the parties
+of a bipartite pure state |ψ⟩ in C2 ⊗ C3 yields a 6-partite
+pure state in C2 ⊗
+�
+C3�⊗5, as shown on Fig. 5. The code
+has distance 3, so the code subspace is 2-uniform.
+In
+addition, the code subspace has dimension equal to 3,
+which matches the local dimension of the second party
+of the state |ψ⟩. Therefore, the property of Fig. 4 holds
+in this case with r = 2.
+If the bipartite state |ψ⟩ is maximally entangled, i. e.
+its reduction to the party with local dimension 2 is max-
+imally mixed, then the resulting state in C2 ⊗
+�
+C3�⊗5
+will be 2-uniform. This can be shown diagrammatically.
+One needs to consider the two cases of producing the
+two-party reduction of the state in question: a) all par-
+ties are traced out except some two output subsystems
+of V ; b) all parties are traced out except the first party
+of |ψ⟩ (with dimension 2) and some output subsystem of
+V .
+FIG. 6. The state |ψ⟩ is completely traced out - the part of
+the diagram on the bottom right is a scalar equal to 1.
+
+2
+3
+3
+J
+2
+3
+2
+32
+3
+3
+3
+3
+3
+V
+2
+3S
+S
+l1
+n
+V
+24
+FIG. 7. By the property on Fig. 4 and the maximal entangle-
+ment of |ψ⟩ the resulting state is proportional to I2 ⊗ I3.
+The case a) is presented on Fig. 6: the property on
+Fig. 4 being used, the state |ψ⟩ gets completely traced
+out and the resulting state is proportional to I3 ⊗ I3.
+The case b) is analyzed on Fig. 7: on the first step the
+property on Fig. 4 is used; the second step is due to the
+fact that |ψ⟩ is maximally entangled.
+It is interesting to note that C2⊗
+�
+C3�⊗5 was the small-
+est possible Hilbert space for which a 2-uniform state
+could be constructed with the methods of Ref. [26].
+B.
+Construction of r-uniform subspaces in
+heterogeneous systems
+The simplest method to produce an r-uniform sub-
+space in heterogeneous systems is to ”glue” together two
+r-uniform subspaces in homogeneous systems. By ”glue-
+ing” we mean taking tensor product of the two subspaces:
+this can be done by taking all possible tensor products
+of the vectors spanning the two subspaces, the resulting
+subspace will be spanned by such combinations.
+Lemma 2. Tensor product of an r-uniform subspace and
+a k-uniform subspace is an l-uniform subspace, where l =
+min(r, k).
+Proof. Let W1 be an r-uniform subspace with n parties
+and let W2 be a k-uniform subspace with m parties. Con-
+sider two isometries V1 : HA → HC1 ⊗ · · · ⊗ HCn and
+V2 : HB → HD1 ⊗· · ·⊗HDm, where dim(HA) = dim(W1)
+and dim(HB) = dim(W2). The first one, V1, maps the
+basis states {|i⟩}A of HA to an orthonormal system of
+vectors spanning W1. Similarly, V2 maps the basis states
+{|j⟩}B of HB to vectors spanning W2. Tensor product
+W1 ⊗ W2 is then spanned by the vectors
+(V1 ⊗ V2) (|i⟩A ⊗ |j⟩B) ,
+(11)
+as shown on Fig. 8. Now the property from Fig. 4 can
+be applied when one traces out any n + m − l of the
+parties C1, . . . , Cn, D1, . . . , Dm. As a result, a general
+state |φ⟩ ∈ HA ⊗HB gets completely traced out, and the
+l-party maximally mixed state is produced.
+FIG. 8.
+On the left: action of the isometry V1 ⊗ V2 on a
+particular basis state |i⟩A ⊗ |j⟩B of HA ⊗ HB. On the right:
+action of the isometry V1 ⊗V2 on a general state φ from HA ⊗
+HB, which is equal to a linear combination of basis states
+{|i⟩A ⊗ |j⟩B}. Action of V1 ⊗ V2 on each state in HA ⊗ HB
+generates W1 ⊗ W2, which is l-uniform.
+R-uniform subspaces can be used for r-uniform quan-
+tum information masking [30]. An operation V is said
+to r-uniformly mask quantum information contained in
+states {|i⟩} if it maps them to multipartite states {|ψi⟩}
+whose all reductions to r parties are identical.
+In the
+proof of Lemma 2 an instance of masking has been pro-
+vided: on the right part of Fig. 8 it is shown how each
+state φ from HA ⊗HB is “masked” by the two isometries
+V1 and V2 as an l-uniform state.
+As an example, combining encoding isometries of
+((5, 2, 3))2 and ((5, 3, 3))3 pure codes, by Lemma 2
+we obtain a 6-dimensional 2-uniform subspace of the
+�
+C2�⊗5 ⊗
+�
+C3�⊗5 Hilbert space.
+Can we reduce the number of parties?
+A structure
+similar to the one on Fig. 5 can be used. Let us take the
+encoding isometry V of the [[6, 2, 3]]3 (stabilizer) pure
+code (Ref. [37], Corollary 3.6). The range of the isome-
+try is a 2-uniform subspace of the
+�
+C3�⊗6 Hilbert space,
+which has dimension equal to 32 = 9. Consider a sub-
+space of C2 ⊗ C9, which consists entirely of states max-
+imally entangled with respect to the first party. Such a
+subspace can be easily constructed with the use of Propo-
+sition 3 of Ref. [38]. The dimension of the subspace is
+equal to ⌊ 9
+2⌋ = 4 (Corollary 4 of Ref. [38]). Now let us
+act with V on the second party (the one with dimension
+9) of each state in the subspace. This procedure will gen-
+erate a 4-dimensional subspace of the C2⊗
+�
+C3�⊗6 Hilbert
+space. The analysis, which is similar to that on Figs. 6
+and 7, shows that the resulting subspace is 2-uniform.
+The above construction can be viewed as an illustra-
+tion of subspace masking: only states that belong to a
+specific subspace of C2⊗C9 are masked by V as 2-uniform
+states in C2 ⊗
+�
+C3�⊗6.
+The next property provides an important way of gen-
+erating r-uniform subspaces from those of larger number
+of parties. It can be seen as the extension of Theorem 20
+of Ref. [39] to the case of heterogeneous systems.
+Theorem 1. Let W be an r-uniform subspace of Hilbert
+space with the set S = {d1, . . . , dn} of local dimensions.
+Let r ⩾ 1 and dim(W) = K. Then, for any di ∈ S,
+there exists an r − 1-uniform subspace of Hilbert space
+with local dimensions S \ {di}. The dimension of this
+
+m
+A
+B2
+35
+subspace is equal to diK.
+Proof. Consider an orthonormal set of vectors {|ψk⟩}
+which span W. Each vector |ψk⟩ is r-uniform, and so,
+in particular, its reduction to party i is proportional to
+the maximally mixed operator I{i}.
+Accordingly, the
+Schmidt decomposition of |ψk⟩ with respect to the bi-
+partition ”party i|the rest” reads
+|ψk⟩ =
+1
+√di
+di−1
+�
+j=0
+���φ(k)
+j
+�
+P ⊗
+���χ(k)
+j
+�
+P ,
+(12)
+where {
+���φ(k)
+j
+�
+P } and {
+���χ(k)
+j
+�
+P }, with j = 0, . . . , di − 1
+and k fixed, are two orthonormal sets of vectors in r − 1-
+partite and 1-partite Hilbert spaces with local parties
+P = {1, . . . , n} \ {i} and P = {i}, respectively. In the
+right part of Eq. (12) vectors with the same upper index
+satisfy the orthonormality condition, for example,
+�
+χ(k)
+m
+���χ(k)
+n
+�
+P = δmn.
+(13)
+Now consider an r − 1-element subset J ⊂ P and, for
+some numbers k, s ∈ {1, 2, . . . , K}, take the reduction
+of |ψk⟩⟨ψs| to the set J ∪ {i}:
+TrP \J [|ψk⟩⟨ψs|]
+= 1
+di
+di−1
+�
+j, l=0
+TrP \J
+����φ(k)
+j
+��
+φ(s)
+l
+���
+P
+�
+⊗
+���χ(k)
+j
+��
+χ(s)
+l
+���
+P .
+(14)
+By
+Lemma
+1,
+this
+reduction
+is
+proportional
+to
+δks IJ∪{i} = δks IJ ⊗ I{i}, and hence
+1
+di
+di−1
+�
+j, l=0
+TrP \J
+����φ(k)
+j
+��
+φ(s)
+l
+���
+P
+�
+⊗
+���χ(k)
+j
+��
+χ(s)
+l
+���
+P
+= δks
+1
+dJ
+IJ ⊗ 1
+di
+I{i},
+(15)
+where dJ is the product of local dimensions of the parties
+in J. Multiplying both parts of this equality by
+�
+χ(k)
+m
+���
+and
+���χ(s)
+n
+�
+, with the use of condition (13), we obtain
+TrP \J
+����φ(k)
+m
+��
+φ(s)
+n
+���
+P
+�
+= δksδmn
+1
+dJ
+IJ.
+(16)
+Taking trace over subsystem J in the last equation, one
+can see that the set of diK vectors {
+���φ(t)
+j
+�
+P }, j =
+0, . . . , di − 1, t = 1, . . . , K is an orthonormal system.
+Since Eq. (16) holds for any choice of an r − 1-element
+subset J ⊂ P, the states {
+���φ(t)
+j
+�
+P } are r − 1-uniform. In
+addition, from the same equation one can see that any lin-
+ear combination of these vectors is an r−1-uniform state.
+Therefore, the system {
+���φ(t)
+j
+�
+P } spans a diK-dimensional
+r − 1-uniform subspace. From Eq. (12) it follows that
+diTr{i}
+� K
+�
+s=1
+|ψs⟩⟨ψs|
+�
+(17)
+is the orthogonal projector on the subspace in question.
+A practical way to obtain an orthonormal system of
+vectors spanning the subspace defined by the projector
+in Eq. (17) is as follows. Let us suppose that party i
+with local dimension di is being eliminated, just as in the
+condition of Theorem 1. Consider an orthonormal system
+of one party vectors {|vj⟩}, j = 0, . . . , di−1, which spans
+Cdi and such that each partial scalar product
+���µ(s)
+j
+�
+≡ ⟨vj|ψs⟩ ,
+j = 0, . . . , di − 1,
+s = 1, . . . , K,
+(18)
+is a non-null vector, where the only input of the
+(co)vector ⟨vj| is joined with the i-th output of the vec-
+tor |ψs⟩ in each partial scalar product. Then diK vectors
+{
+���µ(s)
+j
+�
+} span the subspace in question. Indeed, the vec-
+tors are mutually orthogonal:
+�
+µ(t)
+l
+���µ(s)
+j
+�
+= ⟨ψt|vl⟩ ⟨vj|ψs⟩
+= ⟨vj| TrP [|ψs⟩⟨ψt|] |vl⟩ = 1
+di
+δts ⟨vj|vl⟩
+= 1
+di
+δtsδlj,
+(19)
+where the third equality follows from r-uniformity of the
+original vectors {|ψs⟩} and Lemma 1.
+Consequently,
+{√di
+���µ(s)
+j
+�
+} is an orthonormal system, and the corre-
+sponding projector
+�
+j, s
+�
+di
+���µ(s)
+j
+��
+µ(s)
+j
+���
+�
+di = di
+�
+j, s
+⟨vj|ψs⟩ ⟨ψs|vj⟩
+= diTr{i}
+��
+s
+|ψs⟩⟨ψs|
+�
+(20)
+coincides with the one in Eq. (17).
+As an example, from the obtained above 4-dimensional
+2-uniform subspace of C2 ⊗
+�
+C3�⊗6 Hilbert space one can
+produce a 12-dimensional 1-uniform subspace of C2 ⊗
+�
+C3�⊗5 Hilbert space by eliminating one of the parties
+with local dimension 3.
+When an initial r-uniform subspace is spanned just by
+1 vector, the described above practical method becomes
+similar to Proposition 12 of Ref. [28].
+As we have seen from Lemma 2, glueing two uniform
+subspaces together doesn’t increase the uniformity pa-
+rameter of the resulting subspace. This is in accordance
+
+6
+with the general principle that local operations cannot
+produce any entanglement over that present in origi-
+nal states. Let us show that making use of additional
+resources such as maximally entangled states can lead
+to larger uniformity parameters of the produced states
+and subspaces in comparison with original ones. At first
+we consider uniform subspaces in homogeneous systems,
+i. e., those corresponding to pure quantum error correct-
+ing codes.
+Recall that the parameters of a ((n, K, d))D code sat-
+isfy the inequality [36, 40]
+K ⩽ Dn−2(d−1),
+(21)
+which is called the quantum Singleton bound.
+If pa-
+rameters of a code saturate the bound in Eq. (21),
+the code is called quantum maximum distance separable
+code (QMDS) [36]. It is known that all QMDS codes are
+pure [36].
+In Ref. [15] an important observation about QMDS
+codes was made. We reformulate it here in a more general
+form and provide the proof.
+Lemma 3 (An observation in the proof of Proposition
+7 of Ref. [15]). Let ((n, K, d)) be a QMDS code. Con-
+sider the projector P = �K
+s=1 |ψs⟩⟨ψs| on the codespace,
+where {|ψs⟩} is an orthonormal set of vectors that span
+the codespace. Then each reduction of P to n − (d − 1)
+parties is proportional to the maximally mixed operator
+In−(d−1).
+Proof. According to Theorem 20 of Ref. [39] (or The-
+orem 1 here), tracing out one party yields a projector
+on a KD-dimensional subspace of
+�
+CD�⊗n−1, hence af-
+ter d − 1 such steps of tracing out we have a projec-
+tor on a subspace of
+�
+CD�⊗(n−(d−1)) with dimension
+KDd−1 = Dn−(d−1), i. e.
+the projector on the whole
+space
+�
+CD�⊗(n−(d−1)), the identity operator.
+We stress that this holds only for QMDS codes - those
+with K = Dn−2(d−1), and not for other pure codes.
+With the use of QMDS codes we can now formulate
+the following property.
+Theorem 2. Let ((n1, K1, d1))D1 and ((n2, K2, d2))D2
+be two QMDS codes with K1 = K2 ≡ K > 1. Denote
+r1 ≡ d1 − 1 and r2 ≡ d2 − 1.
+Then there exists an
+l-uniform state in
+�
+CD1�⊗n1 ⊗
+�
+CD2�⊗n2 Hilbert space
+with
+l = min (n1 − r1, n2 − r2, r1 + r2 + 1) .
+(22)
+Proof. Since the two codes are pure, there are two sub-
+spaces related to them: an r1-uniform subspace W1 of
+�
+CD1�⊗n1 and an r2-uniform subspace W2 of
+�
+CD2�⊗n2
+such that dim(W1) = dim(W2) = K.
+Consider two isometries V1 : CK →
+�
+CD1�⊗n1 and
+V2 : CK →
+�
+CD2�⊗n2 whose ranges coincide with W1 and
+FIG. 9. Steps to prove l-uniformity of the state |ψ⟩S1S2.
+W2, respectively. Now let us take any bipartite maxi-
+mally entangled state |φ⟩AB in CK ⊗ CK and construct
+the state
+|ψ⟩S1S2 = (V1 ⊗ V2) |φ⟩AB ,
+(23)
+which belongs to
+�
+CD1�⊗n1 ⊗
+�
+CD2�⊗n2. Here S1 and S2
+denote the sets of the output parties of the isometries V1
+and V2, respectively. We claim that the state |ψ⟩S1S2 is
+l-uniform, with l as in Eq. (22).
+The underlying principle is shown on Fig 9. Let us as-
+sume that all subsystems of |ψ⟩S1S2 are traced out except
+some m1 output subsystems of isometry V1 and some m2
+output subsystems of isometry V2 , as shown on Fig 9,
+a). If, for example, m2 ⩽ r2, the rule from Fig. 4 can be
+applied and we arrive at the situation shown on Fig 9, b),
+where isometry V2 is eliminated and the state |φ⟩AB gets
+partially traced out. Next, the state |φ⟩AB is maximally
+entangled, and so its reduction to A is the maximally
+mixed operator
+1
+K IA. As a result, isometry V1 acts on
+the identity operator IA, as shown on Fig 9, c). The steps
+b − c), without taking into account the trace over output
+parties of V1, can be written as
+V1TrB{|φ⟩⟨φ|AB}V †
+1 = 1
+K V1V †
+1 = 1
+K PW1,
+(24)
+where PW1 – the orthogonal projector on subspace W1,
+the first equality follows from maximal entanglement of
+|φ⟩AB, the second one – from the fact that the isometry
+V1 has subspace W1 as its range. Now if m1 ⩽ n1 − r1
+then, by Lemma 3, performing the trace over n1 − m1
+output subsystems of V1 (Fig 9, c)) produces the maxi-
+mally mixed state of m1 parties (Fig 9, d)) in addition to
+the maximally mixed state of m2 parties obtained earlier
+in the first step. We conclude that if m1 ⩽ n1 − r1 and
+m2 ⩽ r2, the reduced state of m1 + m2 parties is maxi-
+mally mixed. The roles of V1 and V2 can be interchanged,
+and we obtain that if m1 ⩽ r1 and m2 ⩽ n2 − r2, the
+reduced state is maximally mixed.
+Now we need to determine the maximal number l such
+that any partition of l = m1 +m2 into m1 output parties
+
+m2
+m1
+m2
+元.元
+元.元
+元.元
+Vi
+V2
+Vi
+A
+B
+A
+B
+a)
+6)
+m1
+m2
+m1
+m2
+V1
+A
+c)
+d)7
+of V1 and m2 output parties of V2 yields, after perform-
+ing the trace over the rest n1 + n2 − l subsystems, the
+maximally mixed state of l parties. Let us first consider
+partitions in which m2 = 0. In this case the maximal
+value of m1, for which the scheme on Fig. 9 can still be
+applied, is n1 − r1, as it was shown above. This number
+is then an upper bound on l. Interchanging the roles of
+V1 and V2 and setting m1 = 0, we obtain another bound,
+n2 − r2, and hence
+l ⩽ min(n1 − r1, n2 − r2).
+(25)
+Next, let us assume that r2 > r1. Let ⌈x⌉ denote the
+ceiling of x and ⌊x⌋ denote the floor of x. Consider a par-
+tition of l into m1 = ⌊ l
+2⌋ and m2 = ⌈ l
+2⌉. If ⌈ l
+2⌉ > r2 then
+⌊ l
+2⌋ > r1 also holds, and one cannot apply the scheme on
+Fig. 9 since neither V2 nor V1 can be eliminated in the
+first step a)-b) with the use of the rule from Fig. 4. Con-
+sequently, we can take into account only those values of
+l that satisfy ⌈ l
+2⌉ ⩽ r2. Accordingly, consider a partition
+of l into m2 = ⌈ l
+2⌉+α and m1 = ⌊ l
+2⌋−α for some integer
+α > 0 such that ⌈ l
+2⌉ + α = r2 + 1. In this case V2 cannot
+be eliminated in the first step of scheme on Fig. 9. On
+the other hand, the scheme can be initiated by apply-
+ing the rule from Fig. 4 with respect to V1 on condition
+that m1 = ⌊ l
+2⌋ − α ⩽ r1. If the condition is satisfied, in
+the step c)-d) of the scheme (with interchanged V1 and
+V2) the reduction of PW2 to m2 parties will be maximally
+mixed by Lemma 3, since m2 ⩽ l ⩽ n2 − r2 by Eq. (25).
+This principle continues to work for greater values of α
+(but bounded by the condition ⌈ l
+2⌉ + α = m2 ⩽ n2 − r2),
+as m1 gets smaller. It is clear that partitions with α < 0
+will also work, as the step a)-b) will be initiated with the
+use of V2. To sum up, the maximal possible value of l
+satisfies
+� l
+2
+�
++ α = r2 + 1,
+� l
+2
+�
+− α = r1.
+(26)
+Adding these two equalities, we obtain
+l =
+� l
+2
+�
++
+� l
+2
+�
+= r1 + r2 + 1.
+(27)
+When r1 = r2 ≡ r, we can choose (odd) l such that
+⌈ l
+2⌉ = r + 1 and ⌊ l
+2⌋ = r. For partitions with m1 = ⌊ l
+2⌋
+and m2 = ⌈ l
+2⌉ and, vice versa, m1 = ⌈ l
+2⌉ and m2 =
+⌊ l
+2⌋, the scheme on Fig. 9 is initiated with the use of
+V1 and V2, respectively. For all other partitions, which
+can be parameterized with integer α as m1 = ⌈ l
+2⌉ + α
+and m2 = ⌊ l
+2⌋ − α or vice versa, the scheme also works
+by the analysis similar to that in the above paragraph.
+Consequently,
+l =
+� l
+2
+�
++
+� l
+2
+�
+= 2r + 1,
+(28)
+which is just a special case of Eq. (27).
+Immediate application of Theorem 2, with the use
+of the correspondence between r-uniform states and 1-
+dimensional pure quantum codes, produces
+Corollary 2.1. Let ((n, K, d))D be a QMDS code with
+K > 1. Then there exists a pure ((2n, 1, d′))D code with
+distance
+d′ = min(n − d + 2, 2d).
+(29)
+As an example, consider a ((4, 4, 2))2 code, which can
+be the stabilizer [[4, 2, 2]]2 code obtained from the well-
+known [[5, 1, 3]]2 code with the use of Theorem 20 of
+Ref. [39]. Combining [[4, 2, 2]]2 with itself produces an
+8-qubit 3-uniform state. It is known that ⌊ n
+2 ⌋-uniform
+states of n qubits (absolutely maximally entangled (AME)
+states) don’t exist for n > 6 [3, 39, 41, 42], so the con-
+structed state has maximal possible uniformity parame-
+ter.
+To give an example with heterogeneous systems, let
+us combine pure codes ((4, 4, 2))2 and ((5, 4, 3))4. The
+latter code can be produced by tensoring ((5, 2, 3))2 with
+itself (Theorem 14 of Ref. [36]). By the construction in
+the proof of Theorem 2, the two codes yield a 3-uniform
+state in
+�
+C2�⊗4 ⊗
+�
+C4�⊗5. We can then obtain r-uniform
+subspaces by eliminating some parties of this state, but
+in this case some produced subspaces will demonstrate
+better values of r than those predicted by Theorem 1
+owing to the following observation.
+Corollary
+2.2.
+Let
+((n1, K1, d1))D1
+and
+((n2, K2, d2))D2
+be
+two
+QMDS
+codes
+with
+K1 = K2
+≡ K
+> 1.
+Denote r1
+≡ d1 − 1 and
+r2 ≡ d2 − 1.
+Then for any integers 0 ⩽ α ⩽ r1 and
+0 ⩽ β ⩽ r2 there exists an l-uniform subspace W of
+�
+CD1�⊗n1−α ⊗
+�
+CD2�⊗n2−β Hilbert space such that
+dim(W) = D α
+1 Dβ
+2 ,
+l = min(n1 − r1 − α, n2 − r2 − β,
+r1 + r2 + 1 − α − β).
+(30)
+Proof. By Theorem 2, we can construct a state in
+�
+CD1�⊗n1 ⊗
+�
+CD2�⊗n2 Hilbert space with the uniformity
+parameter given by Eq. (22). Next, we eliminate α + β
+parties of this state by the procedure described after the
+proof of Theorem 1. Let us take α orthonormal systems
+of vectors {
+���v(µ)
+i
+�
+}, µ = 1, . . . , α, i = 0, . . . , D1−1, each
+system being a basis for the corresponding CD1 Hilbert
+FIG. 10. Construction of the states which span the l-uniform
+subspace W of
+�
+CD1�⊗n1−α ⊗
+�
+CD2�⊗n2−β Hilbert space.
+
+Vi8
+space. Similarly, we take β orthonormal systems of vec-
+tors {
+���w(ν)
+j
+�
+}, ν = 1, . . . , β, j = 0, . . . , D2 − 1, each in
+its own CD2 Hilbert space. Next, we pick some specific
+vectors
+���v(1)
+i1
+�
+, . . . ,
+���v(α)
+iα
+�
+, one from each system, and
+eliminate α output parties of the isometry V1 by joining
+them with the inputs of the chosen vectors. Similarly,
+we pick β specific vectors
+���w(1)
+j1
+�
+, . . . ,
+���w(β)
+jβ
+�
+and elim-
+inate β output parties of the isometry V2 (see Fig. 10,
+the indices of vectors v, w are omitted). As a result, we
+obtain
+�
+v(1)
+i1
+��� . . .
+�
+v(α)
+iα
+���
+�
+w(1)
+j1
+��� . . .
+�
+w(β)
+jβ
+��� (V1 ⊗ V2) |φ⟩AB , (31)
+one of the D α
+1 Dβ
+2 states that span the subspace W of
+�
+CD1�⊗n1−α⊗
+�
+CD2�⊗n2−β Hilbert space. All such states
+are hence indexed by the numbers i1, . . . , iα, j1, . . . , jβ,
+which represent the correspondence between tuples of
+vectors v, w and the basis states of W.
+The uniformity of the state in Eq. (31) can be analyzed
+with the use of Fig. 10 and the same reasoning as in the
+proof of Theorem 2. The vectors v and w take up α and β
+positions out of n1 and n2 output parties of the isometries
+V1 and V2, respectively. The parties in these positions
+cannot be traced out, and this results in modifying the
+bounds on l in Eq. (25):
+l ⩽ min(n1 − r1 − α, n2 − r2 − β).
+(32)
+Eq. (26) is also modified, with r1 and r2 replaced by
+r1 − α and r2 − β, respectively. As a result, we obtain
+the expression for l in Eq. (30).
+Earlier a 3-uniform state in
+�
+C2�⊗4 ⊗
+�
+C4�⊗5 was
+obtained with the use of Theorem 2 from pure codes
+((4, 4, 2))2 and ((5, 4, 3))4. Eliminating one party with
+dimension 2 and one party with dimension 4, or, in terms
+of Corollary 2.2, setting α = β = 1, we produce an
+8-dimensional 2-uniform subspace of
+�
+C2�⊗3 ⊗
+�
+C4�⊗4
+Hilbert space. We stress that the original state has spe-
+cial structure and, as a result, after the elimination of 2
+parties the produced subspace has higher value l = 2 in
+comparison with l = 1 predicted by Theorem 1.
+C.
+Comparison with mixed orthogonal arrays
+method and further constructions
+Mixed orthogonal arrays (MOAs) [43, 44] in its specific
+form, irredundant MOAs (IrMOAs) [26], have become a
+powerful tool in construction of r-uniform states in het-
+erogeneous systems [26–28]. In this subsection we present
+several applications of the compositional approach that
+allow us to reproduce or extend some results obtained
+with the use of IrMOAs (in terms of the minimal num-
+ber of parties for a given uniformity parameter). Such a
+comparison also reveals some weaknesses of the presented
+in this paper approach.
+In general it becomes more difficult to find examples
+of r-uniform states in heterogeneous systems when the
+number of parties gets smaller.
+Let us consider some
+results from Ref. [27].
+Proposition 1 (Corollary 3.2 of Ref. [27].). 2-
+uniform states exist for the following configura-
+tions:
+1. C3 ⊗
+�
+C2�⊗N for N ⩾ 8.
+2.
+�
+C3�⊗2 ⊗
+�
+C2�⊗N for N ⩾ 12.
+3.
+�
+C3�⊗3⊗
+�
+C2�⊗N for N ⩾ 11 and
+�
+C3�⊗4⊗
+�
+C2�⊗N
+for N ⩾ 10.
+We can reproduce the first result for N = 8. The pro-
+cedure is as follows. The pure code ((5, 2, 3))2 is com-
+bined with itself by the construction of Theorem 2, which
+results in a 10-qubit 3-uniform state, i. e. a ((10, 1, 4))2
+pure code (Corollary 2.1). Next, by eliminating two par-
+ties, by Corollary 2.2 we obtain a 4-dimensional 8-qubit
+2-uniform subspace, i. e. a pure ((8, 4, 3))2 code. Now we
+have an encoding isometry which maps vectors from C4
+to the 2-uniform code space (briefly, the ”8-qubit isome-
+try”). Finally, we can take a maximally entangled state
+in C3⊗C3 and act on one of its parties with the obtained
+isometry, and the construction here will be similar to the
+one presented on Fig. 5. The resulting state, which be-
+longs to C3⊗
+�
+C2�⊗8, is 2-uniform. For larger values of N
+we can use the same auxiliary state and various combina-
+tions of isometries and, if necessary, glue them together
+with the use of Lemma 2. As an example, the isometry
+for N = 10, which maps C4 to 10-qubit 2-uniform sub-
+space, can be obtained from glueing the subspace of the
+code [[5, 1, 3]]2 with itself. In other words, 10-qubit isom-
+etry is obtained from glueing 5-qubit isometry with itself.
+Next, by eliminating one party of a ((8, 1, 4))2 state, we
+obtain an isometry which maps vectors from C2 to the
+2-uniform 7-qubit space (the ”7-qubit isometry”).
+By
+the same procedure, from the state ((10, 1, 4))2 we ob-
+tain the 9-qubit isometry. Now, the isometry for N = 12
+can be obtained from glueing 7-qubit and 5-qubit isome-
+tries, for N = 13 – from 5-qubit and 8-qubit ones, and
+so forth. We haven’t found any appropriate isometries to
+construct the states with N = 9 and N = 11 (those that
+we’ve found have input dimension 2, which is less than
+the local dimension of the second party of the auxiliary
+state). To conclude, we cannot reproduce the first result
+of Proposition 1 only for N = 9 and N = 11 with the
+current approach.
+The second result from Proposition 1 is harder to re-
+produce. The reason for that is as follows: we can take
+a 4-qutrit 2-uniform state, which can be, for example,
+the graph state of Ref. [19] or a QMDS code [[4, 0, 3]]3
+from Corollary 3.6 of Ref. [37], but now we need to act
+with an isometry on its two parties, i. e., on a compound
+subsystem with local dimension 3 × 3 = 9, as shown on
+Fig. 11. The 8-qubit isometry that was used before is
+
+9
+FIG. 11. An isometry V acting on a joint subsystem of two
+parties with local dimensions 3.
+not appropriate here since it has input dimension equal
+to 4, which is less than the output dimension of the two
+combined parties. A proper isometry can be constructed
+from other error correcting codes with the use of the split-
+ting property for r-uniform subspaces, which is a direct
+consequence of the splitting method for r-uniform states
+appeared earlier in Refs. [26, 28].
+Lemma 4. Let W be an r-uniform subspace of Hilbert
+space with the set S = {d1, . . . , dn} of local dimensions.
+Let di = d′
+id′′
+i for some i: 1 ⩽ i ⩽ n and some integer
+d′
+i, d′′
+i > 1. Then there exists an r-uniform subspace of
+Hilbert space with the set of local dimensions given by
+{d′
+i, d′′
+i } ∪ [S \ {di}] and having the same dimension as
+the original subspace.
+Proof. The subspace in question can be obtained from
+the original one by splitting the i-th subsystem of each
+state in W into two smaller ones, i′ and i′′, with local
+dimensions d′
+i and d′′
+i , respectively. Each newly obtained
+state is r-uniform, as follows from the splitting method
+described in Refs. [26, 28].
+Consequently, a subspace,
+which consists of such states, is r-uniform.
+Now we can return to the construction of a 2-uniform
+state in
+�
+C3�⊗2 ⊗
+�
+C2�⊗12. Consider a pure ((6, 16, 3))4
+code (Corollary 3.6 of Ref. [37]).
+By splitting each
+ququart into 2 qubits, by Lemma 4, the code is converted
+into a pure ((12, 16, 3))2 code. Since its encoding isome-
+try (the ”12-qubit isometry”) has input dimension equal
+to 16, we can act with it on a compound subsystem con-
+sisting of two combined parties of the state [[4, 0, 3]]3 (see
+Fig. 11). The resulting state is 2-uniform. The isome-
+tries for larger N can be obtained from glueing the 12-
+qubit isometry with the described above isometries. As
+an example, a 17-qubit isometry is obtained from glue-
+ing the 12-qubit and the 5-qubit ones (it doesn’t matter
+that the input dimension of the 5-qubit isometry is 2 –
+the input dimension of the 12-qubit isometry is 16, and
+the resulting one will have the input dimension equal to
+16 × 2 = 32).
+N = 16 is obtained from glueing the
+8-qubit isometry with itself. All other numbers N ⩾ 18
+can be otained similarly. In addition, N = 14 can be con-
+structed from splitting the code [[7, 3, 3]]4 (Corollary 3.6
+of Ref. [37]). We cannot reproduce the second result of
+Proposition 1 only for N = 13 and N = 15.
+As for the third result of Proposition 1, 2-uniform
+states in
+�
+C3�⊗3⊗
+�
+C2�⊗N can be obtained with action of
+the described above isometries on one party of the state
+[[4, 0, 3]]3. As earlier, the cases N = 9 and N = 11 are
+not covered by our approach, but we can construct a state
+with N = 8, which extends the proposition. The result
+for uniform states in
+�
+C3�⊗4 ⊗
+�
+C2�⊗N can be substan-
+tially extended. Consider a code [[4, 0, 3]]6, for example,
+from Corollary 3.6 of Ref. [37]. By Lemma 4, by split-
+ting each subsystem with local dimension 6 into qubit
+and qutrit subsystems, we obtain a 2-uniform state in
+�
+C3�⊗4 ⊗
+�
+C2�⊗4. The case N ⩾ 5 is trivial: we can glue
+the state [[4, 0, 3]]3 with a 2-uniform state of N qubits,
+which exists for N ⩾ 5 and can be obtained, for exam-
+ple, from graph states constructions. These observations
+extend the proposition from N = 10 to N = 4.
+Gathering the above results, we can formulate
+Proposition 2 (Combination of Corollary 3.2 of Ref. [27]
+with the current approach). 2-uniform states exist for the
+following configurations:
+1. C3 ⊗
+�
+C2�⊗N for N ⩾ 8.
+2.
+�
+C3�⊗2 ⊗
+�
+C2�⊗N for N ⩾ 12.
+3.
+�
+C3�⊗3 ⊗
+�
+C2�⊗N for N = 8 and N ⩾ 10 and
+�
+C3�⊗4 ⊗
+�
+C2�⊗N for N ⩾ 4.
+Let us also analyze some results of Ref. [28].
+Proposition 3 (Theorem 9 of Ref. [28]). For any d > 2,
+the following holds.
+1. There exists a 2-uniform state in
+�
+C2�⊗2 ⊗
+�
+Cd�⊗N
+for any N ⩾ 7 and N ̸= 4d + 2, 4d + 3.
+2. There exists a 2-uniform state in C2 ⊗
+�
+Cd�⊗N for
+any N ⩾ 5.
+We can start with the 2-uniform subspace of the code
+[[5, 1, 3]]2 and act with a proper isometry on three sub-
+systems, i. e. on a joint system of dimension 8, of each
+vector in the code. Therefore, in addition to having a
+2-uniform subspace as its range, an appropriate isometry
+must have input dimension greater or equal 8. The code
+family [[N, N − 4, 3]]d, 4 ⩽ N ⩽ d2 + 1, d > 2 (Corol-
+lary 3.6 of Ref. [37]) provides us with proper isometries
+for 6 ⩽ N ⩽ 10. In addition, the isometry with 5 out-
+put parties, which corresponds to [[5, 1, 3]]d, only works
+when d ⩾ 8, since in this case its input dimension is equal
+to d. Isometries for all other numbers, N > 10, can be
+obtained by glueing the codes with N < 10 (Lemma 2).
+As a result, we lift the constraint N ̸= 4d + 2, 4d + 3 and
+obtain 2-uniform subspaces instead of just states.
+We can only reproduce the second result of Proposi-
+tion 3. All the described above isometries, this time in-
+cluding the one with N = 5, can be used to act on one
+party of a maximally entangled state in C2 ⊗ C2.
+Instead of a maximally entangled state in C2 ⊗ C2 we
+could use maximally entangled subspaces of C2 ⊗ Cp,
+
+3
+3
+2
+2
+2
+A
+3
+3
+3
+310
+which have dimension equal to ⌊ p
+2⌋, by Corollary 4 of
+Ref. [7]. Now the isometry, which corresponds to code
+[[N, N − 4, 3]]d, acts on a party with local dimension p
+of each state in the maximally entangled subspace. The
+input dimension of the isometry, dN−4, hence must be
+greater or equal p, and we have the condition
+N ⩾ 4 + logd p.
+(33)
+Summing the results, we can formulate the extension
+of Proposition 3
+Proposition 4. The following holds.
+1. For 2 < d ⩽ 8 there exists a 2-uniform subspace of
+�
+C2�⊗2 ⊗
+�
+Cd�⊗N with dimension 2 for any N ⩾ 6.
+2. For d > 8 there exists a 2-uniform subspace of
+�
+C2�⊗2 ⊗
+�
+Cd�⊗N with dimension 2 for any N ⩾ 5.
+3. for d > 2 and p ⩾ 2 there exists a 2-uniform sub-
+space of C2 ⊗
+�
+Cd�⊗N with dimension ⌊ p
+2⌋ for any
+N ⩾ 4 + logd p.
+The above examples show that the presented approach
+is more effective in constructing r-uniform states and
+subspaces with larger local dimensions, i. e., qutrits or
+higher. Indeed, there are not many qubit isometries for
+a given value of the uniformity parameter, and, in repro-
+ducing some results of Proposition 1, we had to resort
+to splitting the codes of higher dimensionality. A simi-
+lar tendency was observed in Ref. [35] where genuinely
+entangled subspaces were constructed with the isometric
+mapping method: when local dimension goes to infinity,
+the dimension of the obtained subspaces asymptotically
+approaches the maximal possible value.
+Finally, let us provide some constructions with higher
+values of the uniformity parameter.
+Consider the pure QMDS code [[10, 4, 4]]3 from The-
+orem 13 of Ref. [45].
+From Corollary 2.2 with α =
+β = 1, we obtain a 5-uniform 9-dimensional subspace
+of
+�
+C3�⊗18. The corresponding isometry V has the input
+dimension equal to 9. Let us take a maximally entan-
+gled subspace of C2 ⊗ C9, which has dimension equal to
+⌊ 9
+2⌋ = 4 (Corollary 4 of Ref. [7]). Action of V on the sec-
+ond party of each state in this subspace yields a 5-uniform
+4-dimensional subspace of C2 ⊗
+�
+C3�⊗18. With the same
+isometry V we could act instead on the joint subsystem
+of three parties of each state in the code space [[5, 1, 3]]2,
+and this procedure yields a 5-uniform 2-dimensional sub-
+space of
+�
+C2�⊗2 ⊗
+�
+C3�⊗18.
+Consider the [[10, 0, 6]]4 code, which can be obtained
+from the classical [10, 5, 6] MDS code over GF(16) of
+Ref. [46] by the correspondence between stabilizer QMDS
+codes and self-dual classical MDS codes (Theorem 15 of
+Ref. [47], see also Proposition 15 of Ref. [15]). By elimina-
+tion of one party a code ((9, 4, 5))4 is constructed (The-
+orem 20 of Ref. [39]).
+By splitting the latter code we
+obtain a ((18, 4, 5))2 code whose encoding isometry has
+the input dimension equal to 4. Applying this isometry
+to one of the parties of a maximally entangled state in
+C3 ⊗ C3 produces a 4-uniform state in C3 ⊗
+�
+C2�⊗18.
+IV.
+DISCUSSION
+In this paper we’ve shown how new r-uniform states
+and subspaces can be constructed from combining al-
+ready known quantum error correcting codes, (maxi-
+mally) entangled states and subspaces.
+The isometric
+mapping method played the key role here: one takes an
+isometry, which, as its range, has a subspace with some
+useful property, and applies it to states or subspaces,
+perhaps with some other interesting property. This ap-
+proach allowed us to complement some results which were
+obtained with the mixed orthogonal arrays method. It
+would be interesting to continue this parallel with OAs.
+One example in this direction could be analyzing encod-
+ing isometries of the QECCs obtained with OAs, for in-
+stance, the ones from Ref. [25]. This could potentially
+lead to new OA and MOA constructions.
+The advantage of the presented approach is its exper-
+imental accessibility: whenever one can realize encoding
+isometries of QECCs as well as prepare auxiliary entan-
+gled states, one can construct uniform states in accor-
+dance with the described above procedures.
+The dis-
+advantage of the approach is that it doesn’t utilize the
+internal structure of the combined objects beyond their
+uniformity property. Taking more structural properties
+into account could result in constructing more classes of
+useful states and subspaces such as, for example, AME
+states, the ones we couldn’t produce with the current
+approach.
+This observation suggests another direction
+of further research: how to combine several QECCs in
+the most efficient way, with taking their specific proper-
+ties into account, to obtain a new QECC with “good”
+characteristics (in the sense similar to the recent “good
+quantum codes” constructions [48, 49]). We stress that
+the distance of the codes composed by the procedure of
+Theorem 2 doesn’t scale with the number of codes be-
+ing combined: the distance of the resulting code will al-
+ways be upper-bounded by the minimum of the number
+in Eq. (25) taken over all the codes being combined.
+It also would be interesting to apply the isometric map-
+ping method to construction of multipartite subspaces
+with another useful property – distillability and closely
+related to it non-positivity of partial transpose across
+each bipartition (distillable and NPT subspaces). This
+direction of research could complement the results ob-
+tained in Refs. [18, 50–52].
+ACKNOWLEDGMENTS
+The author thanks M. V. Lomonosov Moscow State
+University for supporting this work.
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+From discrete to continuous: Monochromatic
+3-term arithmetic progressions
+Torin Greenwood∗, Jonathan Kariv†, Noah Williams‡
+December 31, 2022
+Abstract
+We prove a known 2-coloring of the integers [N] := {1, 2, 3, ..., N}
+minimizes the number of monochromatic arithmetic 3-progressions under
+certain restrictions. A monochromatic arithmetic progression is a set of
+equally-spaced integers that are all the same color.
+Previous work by
+Parrilo, Robertson and Saracino conjectured an optimal coloring for large
+N that involves 12 colored blocks. Here, we prove that the conjecture is
+optimal among anti-symmetric colorings with 12 or fewer colored blocks.
+We leverage a connection to the coloring of the continuous interval [0, 1]
+used by Parrilo, Robertson, and Saracino as well as by Butler, Costello
+and Graham. Our proof identifies classes of colorings with permutations,
+then counts the permutations using mixed integer linear programming.
+1
+Introduction
+Consider coloring each of the integers in [N] with one of r colors. A κ-term
+arithmetic progression is any subset of κ equally-spaced integers, denoted a κ-
+AP. An arithmetic progression is monochromatic if every term is colored the
+same color. Can we color [N] in a way that avoids all monochromatic κ-APs?
+A classic result is van der Waerden’s Theorem:
+Theorem 1.1 (van der Waerden, [17]). For any integers r, κ ≥ 1, there exists
+a number N such that every r-coloring of [N] has a monochromatic κ-AP.
+Given that monochromatic κ-APs are guaranteed to exist when enough num-
+bers are colored, we ask a refined question: what is the minimum number of
+monochromatic κ-APs that could exist? To be more precise, define Cr(N) to
+be the set of r-colorings of [N]. For any c ∈ Cr(N), let mκ(c) be the number of
+∗Department
+of
+Mathematics,
+North
+Dakota
+State
+University,
+Fargo,
+ND
+USA,
+torin.greenwood@ndsu.edu
+†Isazi Consulting, Johannesburg, South Africa, jkariv@isaziconsulting.co.za
+‡Department of Mathematical Sciences, Appalachian State University, Boone, NC USA,
+williamsnn@appstate.edu
+1
+arXiv:2301.00336v1  [math.CO]  1 Jan 2023
+
+monochromatic κ-APs induced by c. Finally, let APκ(N) be the total number
+of κ-APs in [N], regardless of whether they are monochromatic or not. Then,
+we look at
+Pr,κ(N) :=
+min
+c∈Cr(N)
+mκ(c)
+APκ(N).
+The focus of this paper is to examine the minimum for monochromatic 3-APs
+within 2-colorings, P(N) := P2,3(N).
+In 1999, Ron Graham proposed that
+limn→∞ P(n) = β for some constant, β, and offered a $100 prize for finding
+β.
+Originally, it was not clear whether colorings could perform better than
+random in the long run: for large values of N, is it possible to color [N] so
+the probability that a randomly selected 3-AP is monochromatic is less than
+(1/2)3 + (1/2)3 = 1/4? It is notable that the analogous question for 2-colorings
+of Zp is answered negatively for p prime. Indeed, Lu and Peng [12] show that for
+a given 2-coloring of Zp, the fraction of 3-APs that are monochromatic depends
+only on the fraction of each color present in the coloring.
+For our question concerning 2-colorings of [N], Parrilo et al. [14] and Butler
+et al. [2] verified independently but nearly simultaneously that it is possible
+to do better than random, and they found upper and lower bounds for the
+minimum monochromatic APs. The upper bound was attained through simu-
+lating good colorings and finding one that performed well. They landed on the
+following 12-block coloring:
+Explicitly, when coloring [N], the blocks would be approximately of the following
+sizes:
+�28N
+548 , 6N
+548, 28N
+548 , 37N
+548 , 59N
+548 , 116N
+548 , 116N
+548 , 59N
+548 , 37N
+548 , 28N
+548 , 6N
+548, 28N
+548
+�
+(1)
+Due to this coloring, P(N) ≤
+117
+548 + o(1).
+Note that this coloring is anti-
+symmetric: the left half of the coloring is a mirror image of the right half
+but uses opposite colors. In [2], Butler et al. performed many computer sim-
+ulations using genetic algorithms to find the optimal coloring, and noted that
+this same 12-block coloring consistently appeared regardless of the seed coloring
+with which they started. They noted that a remaining challenge would be to
+analyze the case of rapidly alternating colorings.
+The goal of this paper is to show that as N → ∞, the 2-coloring of [N] that
+has alternating color blocks with sizes given in Equation (1) is globally optimal
+among anti-symmetric colorings with at most 12 blocks. As far as the authors
+are aware, this is the first result of optimality under any restrictions. Here, we
+let ˜C2(N) be the 2-colorings of [N] that are anti-symmetric and have at most
+12 contiguous segments of red or blue. Then, define
+˜P(N) =
+min
+c∈ ˜C2(N)
+m3(c)
+AP3(N).
+Our main result is as follows:
+2
+
+Theorem 1.2. Consider coloring each integer in [N] with either red or blue
+such that the coloring is anti-symmetric and has at most 12 contiguous blocks.
+Then, as N increases the minimum possible fraction of arithmetic progressions
+approaches 117
+548. That is, limN→∞ ˜P(N) = 117
+548.
+Below, we provide a proof sketch that outlines the sections in the paper.
+Sketch of proof. First, we will convert from discrete colorings of [N] to continu-
+ous colorings of [0, 1] with at most 12 contiguous segments, referred to as block
+colorings. After restricting the number of color changes that can occur within
+a coloring, it turns out that optimizing the discrete colorings is the same as
+optimizing the continuous colorings, as described rigorously in Lemma 3.6.
+When switching to the continuous realm, we let a continuous coloring be a
+function c : [0, 1] → {0, 1}, where 0 and 1 (in the range) represent red and blue,
+respectively. Then, we let f[0,1](c) be the fraction of arithmetic progressions in
+the coloring c that are monochromatic. We can represent this fraction geomet-
+rically by a BCG diagram, described by Butler, Costello, and Graham in [2] and
+illustrated in Figure 1 below. When c consists of 12 contiguous segments, we
+label the endpoints of the coloring as (x0 = 0, x1, . . . , x12 = 1). As we allow the
+coloring c to vary, f[0,1](c) is a piecewise quadratic function in the xi. Moreover,
+each piece of f[0,1](c) is determined completely by the relative ordering of the
+pairs of sums {xi + xj}, as described in Lemma 4.1.
+Next, we aim to identify every piece of the quadratic function over all color-
+ings c of [0, 1] with 12 intervals. Using the GNU Linear Programming Kit [9],
+we count 371, 219 possible arrangements of {xi + xj} that could give distinct
+quadratics in f[0,1], as proved in Lemma 4.2 with the help of our code available
+online at https://cocalc.com/TorinGreenwood/MonochromeSequences/Mo
+nochromaticProgressions.
+Finally, once we have identified the 371, 219 possible pieces in the quadratic
+function, we search for the global minimum of f[0,1] among all these pieces.
+Fortunately, from Lemma 4.3, it turns out that f[0,1] is a continuous function
+with continuous partial derivatives. Thus, we can minimize f[0,1] by search-
+ing for all critical points within each piece of the quadratic. Because f[0,1] is
+piecewise quadratic, its critical points are determined by systems of linear in-
+equalities (defining the domain of a piece of f[0,1]) and equalities (setting the
+partial derivatives of f[0,1] to zero), allowing us again to use linear programming
+to identify the critical points. We describe our search for these critical points
+in Lemma 4.4, completing the proof.
+A byproduct of our proof structure is that among colorings with a fixed
+number of contiguous blocks, there exist optimal colorings with rational end-
+points, as described in Corollary 4.5. In Section 5, we show that with respect
+to 2-colorings of the continuous unit circle S1, the fraction of monochromatic
+APs depends only on the measure of points colored red. This is analogous to
+the results in [5, 12] that concern colorings of Zp for p prime.
+3
+
+2
+Background
+When searching for bounds on the number of monochromatic arithmetic pro-
+gressions in [N], Frankl, Graham, and R¨odl developed the following theorem:
+Theorem 2.1 (Frankl, Graham, R¨odl, [7]). For fixed r and κ, there exists ℓ > 0
+so that the number of monochromatic κ-APs in any r-coloring of {1, 2, . . . , N}
+is at least ℓN 2 + o(N 2).
+This proved that a positive fraction of APs must be monochromatic in the
+long run, but gave no indication of how small ℓ could be.
+Datskovsky made progress on a related problem in [5], analyzing the minimal
+number of monochromatic Schur triples in [N]. A Schur triple (a, b, c) from
+[N] is any triple of integers where a + b = c.
+Datskovsky investigated the
+minimum possible number of monochromatic Schur triples when coloring each
+integer red or blue, and proved that asymptotically, the minimum is N 2/11. The
+proof relied on using a discrete Fourier transform, which yielded a combinatorial
+identity that broke down counts of Schur triples into a few easier to analyze
+sets. Although our proof does not use the discrete Fourier transform, it also
+will transform a discrete problem into a continuous space.
+In [14], Parrilo et al. applied some of the tools from Datskovsky’s work
+to arithmetic progressions. Again, the authors found a combinatorial identity
+breaking down sets of arithmetic progressions into simpler sets, but it was no
+longer possible to enumerate these sets exactly. Instead, the authors ended up
+with bounds on the minimum number of monochromatic progressions possible in
+[N]. They also identified the coloring shown in Equation (1) in the introduction
+above, and verified it was locally optimal among colorings with 12 intervals
+that are antisymmetric. Our paper aims to prove that this coloring is optimal
+globally among the same set of colorings.
+Constellations are a generalization of APs studied in [2], where instead of
+all points being equally spaced like in an AP, the consecutive differences of
+terms must satisfy some fixed proportions. Butler et al. analyzed constellations
+by representing sets of monochromatic constellations using integrals of indicator
+functions. This led them to represent monochromatic regions in two-dimensional
+diagrams which we refer to as BCG diagrams, as illustrated in Figure 1. Visu-
+alizing progressions via these diagrams is crucial to our proof, and provides the
+connection we need between discrete and continuous realms.
+One important aspect of our proof is enumerating the number of ways
+pairwise sums {xi + xj} can be ordered for a list of positive real numbers
+x0 ≤ x1 ≤ . . . ≤ xn with n even and xi + xn−i = 1.
+This problem could
+be framed as counting the number of chambers in a hyperplane arrangement,
+and there already exists a rich set of tools for counting chambers, as seen for ex-
+ample in [16]. However, in this paper, we use mixed integer linear programming,
+which is well-suited to determining whether a system of linear inequalities has
+a solution. This coding approach was also employed by Miller and Peterson in
+[13] when they counted more sums than differences sets, and also by Laaksonen
+4
+
+in [10] when he counted closely-related arrangements of sums of pairs. More
+details on this approach are given in Section 4.2 below.
+The current best known bounds on the minimum number of monochromatic
+κ-APs in the general (non-antisymmetric) case for κ > 3 are found using an
+“unrolling” strategy, described in [12] and [3].
+Here, an optimal coloring of
+some interval {1, . . . , ℓ} for ℓ ≪ N is found explicitly, and then repeated to fill
+the interval [N]. Although this strategy works well for κ > 3, when κ = 3, the
+colorings do no better than random in the long run.
+3
+Relationship between discrete and continuous
+case
+In this section, we define a precise connection between discrete 2-colorings of
+[N], and a natural continuous analogue of 2-coloring [0, 1]. First, we pause to
+define a 3-AP in [N] formally: a 3-AP is any set of 3 terms (a, a+d, a+2d) each
+in [N] where d is any integer including negative values or zero. It is convenient
+for us to include the case where d ≤ 0 in our arguments, although this choice
+ultimately does not change which colorings minimize monochromatic APs nor
+the minimum they attain.
+For the interval [0, 1], we identify any 3-AP (a, a + d, a + 2d) by its first and
+last term (a, a + 2d) in [0, 1] × [0, 1], now allowing d to be any real number.
+We obtain a measure on the set of 3-APs in [0, 1] by choosing the starting and
+ending point of the progressions uniformly. A coloring of the interval is defined
+to be a function c : [0, 1] → {0, 1}.
+In this section, we begin by discussing measurable colorings of [0, 1], which
+can be approximated in a standard way by bead colorings, defined below. Then,
+we show that minimizing monochromatic APs over all measurable colorings of
+[0, 1] is the same as minimizing all APs over just bead colorings, as formalized
+in Lemmas 3.1, 3.2, and 3.3 below.
+Next, we justify that every discrete coloring of [N] has a corresponding con-
+tinuous coloring of [0, 1], and that the fraction of monochromatic APs in the
+continuous coloring is a function of both the monochromatic APs and monochro-
+matic off-by-1 APs in the discrete coloring, as explained above and in Lemma
+3.4. Using this connection, we find that when the number of blocks of contiguous
+runs of colors in a coloring is bounded by n, the fraction of APs in a continuous
+coloring versus its discrete analogue is small as N grows large, formalized in
+Lemma 3.5. Finally, this allows us to prove our main result of the section: that
+minimizing over discrete colorings with a fixed number of blocks is the same as
+minimizing over continuous colorings with the same number of blocks, stated
+rigorously in Lemma 3.6.
+Now, we begin stating our results formally, starting with the definition of a
+Lebesgue-measurable coloring.
+Definition 1. A coloring of [0, 1] is Lebesgue-measurable if c−1(0) is Lebesgue-
+measurable (or equivalently c−1(1) is Lebesgue-measurable).
+5
+
+Definition 2. A bead coloring of [0, 1] is a coloring where for some ℓ, each
+of the intervals ( i
+ℓ, i+1
+ℓ ) is monochrome for i = 0, 1, . . . , ℓ − 1. Each interval
+( i
+ℓ, i+1
+ℓ ) is called a bead, and we sometimes refer to such a coloring as an ℓ-bead
+coloring.
+We introduce bead colorings because they are the continuous analogue of
+coloring the integers [N] obtained by fattening each integer into an interval.
+Our goal is to show that when optimizing colorings over the interval [0, 1], we
+may restrict our attention to bead colorings. We call the set of bead colorings B
+and the set of Lebesgue-measurable colorings M. Observe that B ⊂ M. Finally
+we define a difference between two colorings as follows.
+Definition 3. For two colorings ca ∈ M and cb ∈ M we define d(ca, cb) :=
+µ({x | ca(x) ̸= cb(x)}), where µ is the usual Lebesgue measure on R.
+Recall that we identify an arithmetic progression in [0, 1] by the pair of
+starting and ending points in [0, 1]. For a coloring c on [N], we define f[N](c) to
+be the fraction of arithmetic progressions that are monochromatic. Analogously,
+when c is a coloring of [0, 1], we have the following definition:
+Definition 4. For a coloring c : [0, 1] → {0, 1}, let f[0,1](c) be the Lebesgue
+measure of the set of monochromatic arithmetic 3-term progressions (viewed as
+a subset of [0, 1]2) induced by the coloring c.
+We justify our restriction to bead colorings with the following standard
+measure-theoretic lemmas (proved for completeness momentarily):
+Lemma 3.1. For any two measurable colorings c1 and c2 of [0, 1], if d(c1, c2) <
+ϵ, then |f[0,1](c1) − f[0,1](c2)| < 4ϵ.
+Lemma 3.2. For any measurable coloring cm of [0, 1] and any ϵ > 0 there exists
+a bead coloring cb such that cm and cb disagree on a set of measure at most ϵ.
+As B ⊂ M, Lemma 3.2 immediately implies the following:
+Lemma 3.3. Optimizing monochromatic 3-APs over bead colorings is the same
+as optimizing over all measurable colorings in the following sense:
+inf
+cb∈B f[0,1](cb) =
+inf
+cm∈M f[0,1](cm).
+We begin with the proof of Lemma 3.1.
+Proof of Lemma 3.1. Let A ⊂ [0, 1] be a set of measure ϵ, and consider flipping
+the colors of all elements in A. There are three classes of monochromatic 3-APs
+that could be created or destroyed: the APs where the first, middle or last
+element is flipped (where some APs may belong to more than one class). We
+consider the measure of each of these three classes. As the first and last elements
+of a progression are chosen uniformly, the corresponding classes have measure
+ϵ. The middle element is the average of two uniform random variables, and so
+6
+
+has a triangular distribution on [0, 1] with maximum density 2. Therefore the
+set of monochrome progressions whose middle term is in A would have measure
+at most 2ϵ. Summing the measures of these three classes yields an upper bound
+for their union of 4ϵ.
+We now justify Lemma 3.2, whose proof is a standard measure-theoretic
+argument.
+Proof of Lemma 3.2. By hypothesis, the set Xbl := c−1
+m (0) of blue-colored el-
+ements of [0, 1] is measurable with finite measure. So, a standard result from
+measure theory (e.g.
+[15, Theorem 12]) establishes the existence of a finite
+disjoint collection of open intervals I1, . . . , Iℓ ⊂ [0, 1] satisfying
+µ
+�� ℓ�
+i=1
+Ii
+�
+\ Xbl
+�
++ µ
+�
+Xbl \
+ℓ�
+i=1
+Ii
+�
+< ϵ
+2.
+Since the rationals are dense in [0, 1], we can perturb the 2ℓ endpoints of the
+intervals {Ii}, each by some amount less than
+ϵ
+4ℓ, to find a disjoint collection
+I′
+1, I′
+2, . . . , I′
+ℓ of open intervals with rational endpoints. Let Ubl be the union of
+these intervals. Then, Ubl and Xbl have a symmetric difference of measure at
+most ϵ. It follows that the coloring cb defined by coloring each interval of Ubl
+blue is a bead coloring for which d(cb, cm) < ϵ.
+Call a progression an off-by-1 AP if it is of the form (a, a + d, a + 2d ± 1).
+We will show that we can easily compute f[0,1](cb) for a bead coloring cb with N
+beads by considering the colored beads as an integer coloring of [N], computing
+the number of 3-term APs in this sequence, and adding half of the off-by-1 APs.
+Recall that for a discrete coloring c, m3(c) is the number of monochromatic
+3-APs induced by c. Let m′
+3(c) be the number of monochromatic off-by-1 APs.
+Then, we have the following comparison between colorings of [0, 1] with exactly
+N beads (of not necessarily alternating colors) and corresponding colorings of
+[N].
+Lemma 3.4. Let cb be an N-bead coloring of [0, 1], and let c∗
+b be the discrete
+coloring of [N] corresponding to cb, where the number i is colored blue if and
+only if the ith bead in cb is colored blue. Then,
+f[0,1](cb) = m3(c∗
+b) + m′
+3(c∗
+b)/2
+N 2
+.
+Proof. Consider a randomly chosen progression in [0, 1] identified by its end-
+points (a, b), and a fixed N-bead coloring cb. We use a probabilistic proof, so
+we rewrite
+(µ × µ)((a, b) ∈ [0, 1]2 : (a, b) is monochromatic) =: P((a, b) monochromatic),
+where µ × µ is the usual Lebesgue measure on R2. We will condition on which
+beads S and E contain a and b. Let M be the bead containing the middle
+7
+
+element of the progression. Given a bead coloring of [0, 1], it is useful to define
+the distance between two beads A and B, db(A, B) as 0 when A = B and as one
+more than the number of other beads strictly between A and B otherwise. Note
+that when db(S, E) is even, then S, M, and E must form a 3-AP of beads. On the
+other hand, when db(S, E) is odd, S, M, and E form an off-by-one progression
+and M could be two possible beads depending on the internal positioning of a
+and b within S and E. Formally, letting {Bi}N
+i=1 be the set of beads,
+f[0,1](cb) =
+�
+i,j
+P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)
+=
+�
+d(Bi,Bj) even
+P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)
++
+�
+d(Bi,Bj) odd
+P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)
+(2)
+Now, P(a ∈ Bi, b ∈ Bj) = 1/N 2 for each i and j since a and b are indepen-
+dently and uniformly distributed among the beads. Also, because our coloring
+is fixed, when db(Bi, Bj) is even P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) is 0 or
+1 depending on whether or not the beads S, M, and E form a monochromatic
+3-term AP. Thus,
+�
+d(Bi,Bj) even
+P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)
+= m3(c∗
+b) · 1
+N 2 .
+(3)
+When db(Bi, Bj) is odd, there are two choices for M: B(i+j−1)/2 or B(i+j+1)/2.
+Thus, we can condition on these two choices:
+P((a, b) monochromatic|a ∈ Bi, b ∈ Bj)
+= P((a, b) mono.|a ∈ Bi, b ∈ Bj, M = B(i+j−1)/2) · P(M = B(i+j−1)/2|a ∈ Bi, b ∈ Bj)
++ P((a, b) mono.|a ∈ Bi, b ∈ Bj, M = B(i+j+1)/2) · P(M = B(i+j+1)/2|a ∈ Bi, b ∈ Bj)
+Here, P(M = B(i+j−1)/2|a ∈ Bi, b ∈ Bj) = P(M = B(i+j+1)/2|a ∈ Bi, b ∈ Bj) =
+1/2 because a and b are positioned uniformly within S and E. Additionally,
+P((a, b) monochromatic|a ∈ Bi, b ∈ Bj, M = B(i+j−1)/2)
+is 0 or 1 depending on whether the off-by-1 progression in c∗
+b is monochromatic
+or not. Hence, these two terms combined simplify to
+�
+d(Bi,Bj) odd
+P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)
+=
+1
+2N 2 m′
+3(c∗
+b).
+(4)
+8
+
+Plugging in Equations (3) and (4) into Equation (2) completes the proof.
+Much of the rest of this paper will deal with a particular class of colorings
+called “block colorings” which we now define. Informally, they are partitions of
+I into disjoint intervals which are alternately colored red and blue.
+Definition 5. For a finite collection of endpoints {xi} such that 0 = x0 <
+x1 < x2 < · · · < xn−1 < xn = 1, we define the associated “block” coloring as
+the coloring where the n intervals Ji = (xi−1, xi) (for i ∈ {1, 2, . . . , n}) are all
+monochrome and alternate in color.
+Note that the colors assigned to the endpoints {xi} (or indeed to any points
+within a measure zero set) do not matter. With these definitions, we can now
+compare the performance of discrete colorings with their continuous analogues.
+Lemma 3.5. Let C(N, n) be the set of 2-colorings of [N] with at most n contigu-
+ous blocks of colors. For any coloring c ∈ C(N, n), let c∗ be the corresponding
+block coloring of [0, 1] where the interval [(i − 1)/N, i/N) is colored blue by c∗ if
+and only if i ∈ [N] is colored blue by c. Then,
+max
+c∈C(N,n)
+��f[0,1](c∗) − f[N](c)
+�� = O
+� n
+N
+�
+.
+Here, there exists a C > 0 independent of N and n such that |O(n/N)| < Cn/N
+for all positive integers n and N.
+Proof. Our proof will use Lemma 3.4 to rewrite f[0,1](c∗) in terms of f[N](c).
+Before proceeding with this, we will interpret the number of off-by-1 monochro-
+matic APs induced by c, m′
+3(c), in terms of the regular monochromatic APs,
+m3(c). We claim the following:
+m′
+3(c) = 2m3(c) + O(nN).
+(5)
+To verify this, note that each AP (a, a + d, a + 2d) in [N] corresponds almost
+bijectively to a pair of off-by-1 APs by moving the first or last endpoint inwards
+by one: (a + 1, a + d, a + 2d) or (a, a + d, a + 2d − 1). (When N is odd, this
+misses exactly two off-by-1 APs: (1, (N −1)/2, N) and (1, (N +1)/2, N). When
+N is even, this is truly a bijection.)
+Using this near bijection, we now compare when APs and off-by-1 APs are
+monochromatic. Under this contraction action, the only time an AP (a, a +
+d, a + 2d) is monochromatic while one of its corresponding off-by-1 APs is not
+monochromatic is when a or a + 2d is adjacent to a number of the opposite
+color, and the same could be said if the original AP is not monochromatic but
+the off-by-1 AP is. If our coloring only has n intervals total, there are only n−1
+ways to position a immediately before a color change, and similarly only n − 1
+ways to position a + 2d immediately after a color change. Since d can still be
+chosen freely, there are O(nN) possible off-by-1 APs that disagree with their
+corresponding APs on being monochromatic, verifying our claim.
+9
+
+Now, in the notation of Lemma 3.4, we see that (c∗)∗ = c. Thus,
+f[0,1](c∗) = m3(c)
+N 2
++ m′
+3(c)/2
+N 2
+= 2m3(c) + O(nN)
+N 2
+= m3(c)
+N 2/2 + O
+� n
+N
+�
+.
+The proof of the lemma will be complete if we can verify the following:
+m3(c)
+N 2/2 = f[N](c) + O
+� 1
+N
+�
+.
+To see this, recall that by definition f[N](c) = m3(c)/ AP3(N), so that
+m3(c)
+N 2/2 − f[N](c) =
+m3(c)
+AP3(N) · AP3(N) − N 2/2
+N 2/2
+.
+(6)
+We have m3(c) ≤ AP3(N), so that the first fraction on the right in Equation (6)
+is at most 1. Next, note that AP3(N) = N 2/2+O(N): it is easy to compute this
+explicitly for when N is even or odd. But, intuitively, if we pick two numbers x
+and y from [N] at random, there are N 2 ways to do this, and about half the time
+x − y is even and these correspond to the start and end of a 3-AP. Therefore,
+AP3(N) − N 2/2 = O(N), and plugging this into Equation (6) completes the
+proof with
+m3(c)
+N 2/2 − f[N](c) = O
+� 1
+N
+�
+.
+Finally, we end this section with the result rigorously justifying our conver-
+sion between discrete and continuous colorings.
+Lemma 3.6. Let Sn be the block 2-colorings of [0, 1] with at most n blocks, and
+let C(N, n) be the 2-colorings of [N] with at most n contiguous blocks, where
+n = o(N) as N approaches infinity.
+Then, minimizing monochromatic APs
+over Sn is the same as minimizing monochromatic APs over C(N, n) in the
+following sense:
+lim
+N→∞
+���� inf
+c∈Sn f[0,1](c) −
+min
+c∈C(N,n) f[N](c)
+���� = 0.
+Here, we consider block colorings of [0, 1] where the edge of a block is at a
+possibly irrational number. However, as we will see later, all optimal colorings
+of [0, 1] with a fixed number of blocks must have rational endpoints.
+10
+
+Proof. This is mostly a standard ϵ argument, so let ϵ > 0 be given. We aim to
+show for all N sufficiently large,
+���� inf
+c∈Sn f[0,1](c) −
+min
+c∈C(N,n) f[N](c)
+���� ≤ ϵ.
+We prove this in two halves, first proving the infimum is nearly bounded above
+by the minimum, and then arguing the reverse. Consider any coloring ˜c ∈ Sn.
+Then, by Lemma 3.1, for every N sufficiently large, we can find a n-block
+coloring ˜cN of [0, 1] with endpoints of the form r/N for r an integer such that
+��f[0,1](˜c) − f[0,1](˜cN)
+�� < ϵ/4.
+(7)
+This is true because we can round each endpoint to the nearest 1/N. Then,
+we define ˜c∗
+N to be the coloring of [N] where i is colored blue if and only if the
+ith block of ˜cN is blue. Note that ˜c∗
+N still only has at most n blocks, and that
+using the notation from Lemma 3.5, (˜c∗
+N)∗ = ˜cN. So, from Lemma 3.5, for N
+sufficiently large (independent of the colorings ˜c, ˜cN, ˜c∗
+N),
+|f[0,1](˜cN) − f[N](˜c∗
+N)| = O(n/N).
+(8)
+By choosing N sufficiently large (independent of the colorings ˜c, ˜cN, ˜c∗
+N), Equa-
+tions (7) and (8) imply
+min
+c∈C(N,n) f[N](c) ≤ f[N](˜c∗
+N) < f[0,1](˜c) + ϵ
+where this bound holds for all N sufficiently large and for all ˜c ∈ Sn. Therefore,
+for all N sufficiently large,
+min
+c∈C(N,n) f[N](c) ≤ inf
+c∈Sn f[0,1](c) + ϵ.
+Now, we prove the reverse inequality: consider any coloring ˆc of [N], and
+let ˆc∗ be the coloring of [0, 1] induced by ˆc. Again, from Lemma 3.5, for N
+sufficiently large,
+��f[N](ˆc) − f[0,1](ˆc∗)
+�� < ϵ,
+and since this is true for any coloring ˆc ∈ C(N, n), this proves that for N
+sufficiently large,
+inf
+c∈Sn f[0,1](c) ≤
+min
+c∈C(N,n) f[N](c) + ϵ.
+Combining this with the complementary inequality above completes the proof.
+At this point, we have justified that once bounding the number of blocks
+in our coloring, optimizing colorings of [N] is the same as optimizing colorings
+of [0, 1]. We only make use of this result when the number of blocks n = 12
+because that is the conjectured global optimal number of blocks. But, the same
+proof shows that switching to the continuous realm works whenever n = o(N)
+as N → ∞.
+11
+
+first term in progression
+third term in progression
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+coloring
+of [0, 1]
+Figure 1: Below the horizontal axis a coloring, c, is depicted. The horizontal
+axis represents the first term a in an arithmetic progression, and the vertical
+axis represents the third term a+2d in the progression. Whenever a point in the
+diagram is colored red (blue), this corresponds to the progression (a, a+d, a+2d)
+being colored red (blue) by c.
+4
+Proofs for the continuous case
+4.1
+Colorings can be represented by BCG diagrams
+Consider any block coloring c : [0, 1] → {0, 1} of the interval with endpoints
+of the blocks given by {x0, x1, . . . , xn} with x0 = 0 and xn = 1. Without loss
+of generality, assume that the first block (x0, x1) is colored blue, and alternate
+colors for each remaining interval. Recall that the colors of the endpoints of the
+blocks can be assigned in any way, since this does not change the probability of
+selecting a monochromatic progression.
+In [2], Butler, Costello, and Graham proposed a method of visualizing the
+monochromatic arithmetic progressions associated to a coloring in terms of di-
+agrams like in Figure 1. Any arithmetic progression (a, a + d, a + 2d) can be
+identified uniquely by its first and last coordinates, which are represented by
+the horizontal and vertical axes of such a diagram. Note that the diagram is
+divided into vertical strips, horizontal strips, and northwest/southeast diagonal
+strips. Consider any region identified as the intersection of one horizontal, one
+vertical, and one diagonal strip. For a block coloring, this region corresponds
+12
+
+0.8
+0.6
+0.4
+0.2 
+0
+0
+0.2
+0.4
+0.6
+0.8
+xto a collection of monochromatic arithmetic progressions if and only if the in-
+dices of the vertical, horizontal, and diagonal strips defining the region all have
+matching parities.
+Because the total area of the square in any diagram like Figure 1 is one, the
+measure of the set of monochromatic sequences is equal to the sum of the areas
+of the red and blue regions. In Theorem 2.1 of [2], Butler et al. express the
+total colored area as the sum of two integrals involving an indicator function.
+Their work applied to constellations, a generalization of arithmetic progressions.
+Here, we instead derive explicit polynomial equations for the areas. Consider
+any one colored region in such a diagram. As the endpoints xi are perturbed
+slightly, the region remains the same type of polygon although its dimensions
+may change.
+This implies that the area of each region can be represented
+locally as a quadratic in the variables {xi}. Denote a block coloring c by its list
+of endpoints x := (x0, . . . , xn). Then, summing over all monochromatic regions
+shows that the measure of the monochromatic progressions, f[0,1](x), is locally
+quadratic in the {xi}, too. We now denote f(x) := f[0,1](x). When we restrict
+f to act on colorings with exactly n blocks, we will write f(xn).
+As x varies, the regions in the diagram change polygon type. Thus, for each
+n, f(xn) is a piecewise function that is locally quadratic. In order to minimize
+f globally, we wish to identify the boundaries of these pieces in terms of x. The
+following lemma describes how to identify the polygons in such a diagram.
+Lemma 4.1. The region that is the intersection of the ith vertical strip, jth
+horizontal strip, and kth diagonal strip of a diagram is empty or forms a closed
+polygon. The type of polygon is determined by testing whether each of the four
+values {xi + xj, xi + xj+1, xi+1 + xj, xi+1 + xj+1} is greater than or less than
+the two values {2xk, 2xk+1}.
+If this ordering is known, the area of the cor-
+responding region can be expressed as a quadratic polynomial in the variables
+{xi, xi+1, xj, xj+1, xk, xk+1}.
+Proof. In the diagrams like in Figure 1, the horizontal lines all are given by
+{y = xi}n
+i=0 and the vertical lines by {x = xi}n
+i=0.
+At any point (x, y) in
+the diagram, the middle value in the corresponding arithmetic progression is
+(x+y)/2, and setting this equal to any endpoint in our coloring implies that the
+diagonal lines are given by {y = 2xi − x}n
+i=0. As described above, for any triple
+(i, j, k) where i, j, k ∈ {0, . . . , 12} all have matching parities, the intersection of
+the ith vertical strip, jth horizontal strip, and kth diagonal strip corresponds
+to a region of monochromatic arithmetic progressions.
+To determine the shape of the region of the monochromatic progressions,
+first consider the rectangle formed by the intersection of the ith vertical strip
+and jth horizontal strip. The corners of this rectangle have coordinates (xi, xj),
+(xi+1, xj), (xi, xj+1), and (xi+1, xj+1), as labelled in Figure 2. In order for the
+intersection of this rectangle with the kth diagonal strip to be non-empty, we
+need the upper diagonal line y = 2xk+1 − x to be above the lower left corner of
+the rectangle, (xi, xj), and the lower diagonal line y = 2xk − x to be below the
+upper right corner of the rectangle, (xi+1, xj+1). This is the same as requiring
+the inequalities 2xk+1 ≥ xi + xj and 2xk ≤ xi+1 + xj+1.
+13
+
+ith vertical strip
+jth
+horizontal
+strip
+kth diagonal strip
+(x    , x    )
+i+1
+j+1
+(x    , x )
+i+1
+j
+(x , x )
+i
+j
+(x , x    )
+i
+j+1
+y = 2x  - x
+k
+y = 2x     - x
+k+1
+Figure 2: Above is the intersection of the ith vertical strip, jth horizontal
+strip, and kth diagonal strip determined by a block coloring with endpoints
+x = (x0, x1, . . . , xn). Whether the intersection is empty can be determined by
+comparing the diagonal lines {y = 2xk − x, y = 2xk+1 − x} to the corners of
+the box {(xi, xj), (xi+1, xj+1)}.
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi+1 + xj ≤ 2xk ≤ xi + xj+1 ≤ xi+1 + xj+1 ≤ 2xk+1
+Region Area:
+(xi+1 − xi)(xj+1 + xi/2 + xi+1/2 − 2xk)
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+2xk ≤ xi + xj,
+max(xi + xj+1, xi+1 + xj) ≤ 2xk+1 ≤ xi+1 + xj+1
+Region Area:
+(xi+1 − xi)(xj+1 − xj) − (2xk+1 − xj+1 − xi+1)2/2
+Figure 3: Illustrated here are two different ways that the kth diagonal strip
+can intersect with the ith horizontal and jth vertical strip in a coloring. The
+resulting monochromatic region is shaded in gray, and the area of the region is
+given as a quadratic in x below the diagram. The type of polygon is determined
+by the partial permutation given below each diagram. The other 18 possibilities
+are enumerated in Appendix A.
+14
+
+Additionally, the type of polygon formed by the intersection of the strips is
+determined by whether the two diagonal lines y = 2xk − x and y = 2xk+1 − x
+are above or below each of the four corners of the box. For any specific relation-
+ship between the lines and the four corners, some basic geometric arguments
+allow us to find the area of the polygon enclosed by the strips in terms of
+{xi, xi+1, xj, xj+1, xk, xk+1}. It turns out that there are 20 possible arrange-
+ments of the lines that yield distinct polygons. In Figure 3, two possibilities are
+given, along with the corresponding quadratic equations for their areas. The
+full list of 20 polygons is given in Appendix A.
+4.2
+Enumerating BCG diagrams
+Now that we have identified criteria that allow us to determine the shape of each
+monochromatic region in a diagram, we wish to compute how many collections
+of shapes are possible between all diagrams.
+In other words, we now know
+that for each fixed n the function f(xn) is a piecewise quadratic function in
+the endpoints xn, but we would like to identify how many pieces it has. From
+Lemma 4.1, we have that the ordering of the pairwise sums {xi + xj}0≤i<j≤n
+completely determines the shapes in the diagram. This is sequence A237749
+in the On-Line Encyclopedia of Integer Sequences. Currently only 9 elements
+in the sequence are known, ending with 771, 505, 180 possible orderings for the
+pairwise sums with n = 8. Thus, this sequence grows much too quickly to be
+useful in checking every piece of f(xn) for n = 12.
+Note that if f(xn) were everywhere concave up, it could only have a single
+local minimum, which would necessarily be the global minimum as well. Since
+the conjectured optimum solution is a local minimum, the proof would be com-
+plete for any coloring with a finite number of intervals regardless of whether
+the coloring is antisymmetric. Additionally, a gradient descent algorithm would
+quickly lead to the global minimum even if it were unknown in advance. Unfor-
+tunately, through computational search, it is easy to find pieces of f(xn) that
+are not concave up. For this reason, we must search for local minima on each
+piece of f(xn) individually in order to guarantee the conjectured coloring is
+globally minimal.
+Counting pairwise orderings of {xi + xj}0≤i<j≤n is closely related to other
+combinatorial problems. In Figure 4, we find that any ordering of {xi + xj}
+could be encoded within a standard Young tableau of inverse staircase shape.
+Here, the filling of the (r, s) entry of the tableau (where the top of the tableau
+is the (0, 0) entry) is equal to the position of xr + xs when all pairs {xi + xj}
+are placed in increasing order.
+To simplify our computations, we make two restrictions. First, because of
+numerical simulations in [2] and our own, we consider only colorings that are
+antisymmetric: when reflected about the middle of the unit interval, almost
+every point in the coloring is swapped to the opposite color. Equivalently, we
+require that xk +xn−k = 1 for 0 ≤ k ≤ n. Additionally, we do not need to know
+all of the relations in the total ordering of {xi+xj}0≤i<j≤n in order to determine
+a configuration. Instead, for each pair (i, j) with i ̸= j, we search for the value of
+15
+
+x0
+x1
+x2
+x3
+x4
+x5
+x0
+x1
+x2
+x3
+x4
+x5
+1
+2
+4
+5
+3
+6
+11
+9
+7
+8 10
+14
+13
+15
+12 17
+16
+19 20
+18
+21
+Figure 4: Above is pictured a standard Young diagram corresponding to the
+choice of endpoints x0 = 0, x1 = 0.19, x2 = 0.9, x3 = 0.6, x4 = 0.65, and x5 = 1.
+The entry labelled 3 tells us that x0 + x2 is the third smallest in the ordering
+of pairs {xi + xj}5
+i,j=0. Additionally, the non-crossing lattice paths from the
+line y = −x to the lower left of the Young diagram partition the diagram into
+regions where all the corresponding pairwise sums are between two consecutive
+values 2xk−1 and 2xk for some k.
+16
+
+k where 2xk−1 ≤ xi + xj ≤ 2xk. Thus, we look for the number of ways to insert
+the pairwise sums {xi + xj}i̸=j into the line, 0 = 2x0 ≤ 2x1 ≤ · · · ≤ 2xn = 2.
+In Figure 4, this reframing corresponds to not needing to know the entire filling
+of the diagram, but instead having a family of non-crossing lattice paths each
+starting at different points down the diagonal. This set-up is very similar to the
+Lindstr¨om-Gessel-Viennot Lemma counting non-intersecting lattice paths, [11,
+8], which was instrumental in proving the conjecture counting the number of
+n × n alternating sign matrices, the story of which is told in [1].
+To determine the number of such partial permutations, we develop an al-
+gorithm that works recursively to verify whether growing partial permutations
+are possible. Our implementation is similar to Miller and Peterson’s geomet-
+ric approach to solving questions about More Sums Than Differences sets, [13,
+Lemma 2.1], and also similar to Laaksonen’s approach to enumerating OEIS
+sequence A237749, [10]. Both of these problems plus the problem we study here
+could be phrased in terms of enumerating chambers in a hyperplane arrange-
+ment, potentially including a restriction to a specific cone within the hyperplane
+arrangement. Enumerating chambers is a stream of research on its own (e.g.
+[16, 6]), and there are existing theorems counting chambers by using M¨obius
+inversion, [20, 19]. However, here we do not need to enumerate every chamber
+without restrictions because this is again equivalent to counting the possible
+total orderings of {xi + xj}0≤i<j≤n.
+Lemma 4.2. Consider antisymmetric block colorings with endpoints (x0, x1, . . . , xn)
+for n even, so that xk + xn−k = 1 for 0 ≤ k ≤ n. Then, the number of ways to
+insert the pairs {xi + xj}i̸=j into the ordering 0 = 2x0 < 2x1 < · · · < 2xn = 2
+grows as follows, starting with n = 0 and with n increasing by twos:
+1, 1, 3, 23, 357, 9391, 371219, . . .
+Proof. The key computational tool in our proof is linear programming: with
+existing linear programming packages like the GNU Linear Programming Kit
+(GLPK, [9]) we can easily check whether a single system of inequalities has a
+valid solution. Thus, we create a running list of partial systems of inequalities
+that have valid solutions, and count in how many ways it is possible to extend
+each system with a single additional inequality. Below, we give pseudocode for
+the algorithm we use, followed by a brief explanation of some of the technicalities
+required to make this code run correctly and efficiently. The full code is posted
+online at https://cocalc.com/TorinGreenwood/MonochromeSequences/Mo
+nochromaticProgressions.
+Pseudocode to Enumerate BCG Diagrams
+1
+\\ Initialize a running list of partial systems of inequalities
+2
+PartialInequalitiesOld =
+�
+{0 = x0, xk + xn−k = 1 for 0 ≤ k ≤ n,
+3
+xk ≤ xk+1 for 0 ≤ k ≤ n − 1}
+�
+4
+5
+\\ For each partial system of inequalities (i.e. for each set of
+6
+partial constraints), find all ways to add a new inequality
+17
+
+7
+2xk ≤ xi + xj ≤ 2xk+1 by deciding where xi + xj fits between
+8
+successive 2xk
+9
+for (i, j) with i ̸= j:
+10
+PartialInequalitiesNew = {}
+11
+for k from i to j − 1:
+12
+for constraints in PartialInequalitiesOld:
+13
+if constraints ∪ {2xk < xi + xj < 2xk+1} is valid:
+14
+PartialInequalitiesNew + =
+�
+constraints
+15
+∪ {2xk < xi + xj < 2xk+1}
+�
+16
+PartialInequalitiesOld = PartialInequalitiesNew
+17
+return PartialInequalitiesOld
+We now discuss some important aspects of our implementation with GLPK
+that ensured the code ran efficiently and correctly. The uninterested reader may
+skip the rest of this proof without a loss of continuity. First, mixed integer linear
+programs typically search to optimize a linear objective function in the variables
+x over a region of linear inequalities written in terms of x. Here, our goal was
+simply to check whether a system of linear inequalities was feasible, meaning
+that a solution exists. This can be achieved with linear programming by setting
+the objective function to be any constant, C, because the linear program will
+return a certificate x∗ where the maximum is achieved. When the objective
+function is constant, this is simply any feasible solution.
+As an added layer of complexity, linear programming typically only al-
+lows for inequalities that are not strict.
+However, exponentially many ar-
+rangements of the pairs {xi + xj}i̸=j can be achieved trivially by the solution
+x∗ = (0, 1/2, 1/2, . . . , 1/2, 1), since any sum of distinct endpoints xi + xj would
+equal 1/2, 1, or 3/2. In fact, many such arrangements can only be achieved
+by these trivial solutions. If we allow such solutions, it is not possible for the
+program to finish due to an explosion in the number of possible systems of
+inequalities. To avoid this scenario, we force all inequalities in every system
+to be strict.
+Thus, we introduce a single auxiliary variable ϵ that converts
+strict inequalities into weak inequalities. For example, the strict inequalities
+2xk < xi + xj < 2xk+1 become a pair of weak inequalities 2xk + ϵ ≤ xi + xj and
+xi + xj + ϵ ≤ 2xk+1. After adding ϵ to every inequality, we change the objective
+function from a constant C to the variable ϵ, and search for the maximum value
+of ϵ within the region where the inequality system is true. As long as a value of
+ϵ > 0 is found, the set of inequalities is feasible.
+Generally, linear programming implementations work with floating point
+arithmetic, leading to rounding errors. Because there is no way to bound how
+small a feasible region could be, we used the version of GLPK that works using
+rational arithmetic.
+Even still, GLPK returns its solutions as floating point
+numbers, occasionally with roundoff errors. Thus, we set the threshold for ϵ to
+be near the limits of floating point arithmetic at 5 × 10−15. We found that the
+smallest ϵ value above this threshold was on the order of 10−3, illustrating that
+any value below 5 × 10−15 was due to precision error.
+Unfortunately, rational solvers tend to be much slower than their floating
+18
+
+point counterparts. To address this, we needed to optimize our code. One factor
+that impacted runtime significantly was the order in which pairs (i, j) were
+checked in the for loop in Line 9 of the pseudocode above. After experimenting
+with different orderings, we found that checking the pairs in decreasing order of
+j − i was several times faster than checking the pairs in lexicographic order.
+Additionally, the rational solver became stuck in an infinite loop for 26 of
+the millions of feasibility checks it ran on systems of inequalities for the n = 12
+case. This issue was resolved by changing the order of the inequalities within
+these problematic systems of inequalities before they were input into GLPK. We
+did not find a single ordering that avoided infinite loops for all of the feasibility
+checks. Instead, we found that for any specific set of inequalities, there always
+existed some ordering where GLPK would halt rapidly.
+4.3
+Optimizing over all BCG diagrams
+Now that we have found the number of possible BCG diagrams for f(xn) for
+each n ≤ 12 and xn that are antisymmetric, we can finally leverage the power
+of calculus.
+Despite being a piecewise function, we soon find that f(xn) is
+continuous with continuous partial derivatives.
+This implies that its global
+maximum happens either at a critical point, or at a boundary point of the
+domain of the function. In Lemma 4.3, we prove that f(xn) has continuous
+partial derivatives for any fixed n, after which we can finish the proof of Theorem
+1.2.
+Lemma 4.3. Consider all block colorings with n blocks and endpoints x =
+(x0, x1, . . . , xn).
+In the region 0 = x0 < x1 < . . . < xn = 1, f(xn) is a
+continuous function with continuous partial derivatives in each variable xj.
+Proof. From the diagram representation in Figure 1, it is clear that f is a
+continuous function of the endpoints, xn. To verify that the partial derivatives
+are continuous, we give a geometric argument: consider a single region R in the
+diagram, like those drawn in Figure 3. Let fR(xn) be the area of this single
+region as a function of the endpoints. The region has up to 6 sides, and each
+side is a line whose position is determined by some single endpoint xi. Thus,
+∂
+∂xi fR(xn) is equal to the total length of the boundaries of R determined by the
+variable xi. (Indeed, moving a single xi by a small ∆xi changes the area of the
+polygon R by ∆xi · ℓi + O(∆xi)2 as ∆xi → 0, where ℓi is the total length of the
+boundaries of R determined by xi.)
+Now, we consider several cases. As xn varies, R may do any of the following:
+stay the same type of polygon, change polygon types, or enter or leave the
+diagram altogether. It is clear that when R stays the same type of polygon, its
+side lengths change continuously in xn, so fR(xn) has continuous partials in this
+case. When R changes polygon type, the change must occur when a diagonal
+line crosses over a corner of the box formed by the horizontal and vertical strips
+shown in Figure 3. This means that any time a region changes polygon type, the
+side that enters or leaves the region does so with initial length 0, again implying
+that the partials are continuous. Finally, we consider when R enters or leaves
+19
+
+the diagram. There are two ways this can happen: either a horizontal, vertical,
+or diagonal strip collapses to width 0, or a diagonal line crosses over the corner
+of the box described above. When a strip collapses to width 0, this means that
+there are two consecutive endpoints xi and xi+1 where (xi+1 −xi) tends to zero.
+Thus, although the partial derivative is not continuous in this case, it is on the
+boundary of the region of xn values we consider. On the other hand, when a
+diagonal line crosses over the corner of a box, all the side lengths of the polygon
+approach zero, so the partials are again continuous.
+The diagram representation of f(xn) makes it clear that f(xn) has a bounded
+number of regions: at most one for each intersection of a horizontal, vertical, and
+diagonal strip. Since fR(xn) is continuous with continuous partial derivatives
+for every region R, f(xn) is too.
+Now that we have shown that f(xn) is continuous with continuous partial
+derivatives for a fixed n, we are ready to complete the proof of Theorem 1.2
+with the following lemma.
+Lemma 4.4. Let x12 = (x0, . . . , x12) with 0 ≤ x0 ≤ · · · ≤ x12 = 1 and x12
+antisymmetric. The global minimum of f(x12) over all such x12 is 117/548,
+occurring uniquely at the coloring from Equation 1.
+Proof. Because f(x12) is a C1 function on the polytope 0 = x0 < x1 < . . . <
+x12 = 1, its global minimum occurs on the boundary of the polytope or at a
+critical point within the interior of the polytope. The boundary of this polytope
+is the union of polytopes of the same form with fewer variables. For this reason,
+we find the critical points for f(xn) for each even value of n between 0 and 12.
+Lemma 4.1 implies that f(xn) is a piecewise-quadratic function for each
+n. Fix n, and consider any piece of this function, which can be extended to a
+function f ∗(xn) on all of Rn/2−1 (since x1 through xn/2−1 determine the coloring
+because it is anti-symmetric). The partial derivatives of f ∗(xn) are piecewise
+linear functions. The critical points of this everywhere-defined quadratic are the
+solution to a linear system of equations. Therefore, there are either no critical
+points, or a vector space of critical points. In the case that the vector space
+has positive dimension, the value of f ∗(xn) must be constant among all of its
+critical points. Thus, when f ∗(xn) has critical points, it suffices to check the
+value of f ∗(xn) at a single critical point when checking for the values of local
+optima.
+This leads us to the following pseudocode to search for the global minimum
+of f(x12) on the polytope 0 = x0 ≤ x1 ≤ · · · ≤ x12 = 1. (The full version of the
+code is posted at https://cocalc.com/TorinGreenwood/MonochromeSequen
+ces/MonochromaticProgressions.)
+Pseudocode to find the minimum value of f(x12)
+18
+\\ We search the interior of f(xn) for n = 2, 4, 6, 8, 10, and 12.
+19
+>> for n from 0 to 12 by twos:
+20
+20
+
+21
+>> for each piece f ∗(xn) of the piecewise function f(xn)
+22
+(identified by Lemma 4.2):
+23
+24
+>> calculate the quadratic polynomial corresponding to
+25
+f ∗(xn) (by using Lemma 4.1)
+26
+27
+\\In the next line, we can feed into GLPK all of the
+28
+inequalities defining the configuration for f ∗(xn) plus
+29
+the equalities that set each of the partial derivatives
+30
+of f ∗(xn) to zero.
+31
+>> use GLPK to check the existence of a critical point
+32
+of f ∗(xn) within the region of xn-values where
+33
+f(xn) ≡ f ∗(xn)
+34
+35
+>> if critical points exist:
+36
+>> evaluate f ∗(xn) at any critical point cn
+37
+>> store cn and f ∗(cn) if this is a new record minimum
+38
+39
+>> return the minimum cn and f ∗(cn) values
+This code verifies that the global minimum of f(xn) when n is at most 12 is
+117/548, which is attained only at the coloring with endpoints given in Equation
+(1) (without the N in each coordinate).
+The number of pieces of the function f(xn) for n ≥ 14 grows very rapidly,
+making an analysis of its critical points increasingly challenging. However, we
+can guarantee that the optimal is always rational:
+Corollary 4.5. For each n ∈ Z+, the minimum value of f(xn) is rational,
+regardless of whether xn is restricted to be anti-symmetric or not.
+Proof. This is nearly immediate from our proof structure: the minimum of
+f(xn) occurs at some critical point of f(xℓ) with xℓ in the interior of where
+f(xℓ) is defined, for an ℓ ≤ n. These critical points are defined by a system
+of linear equations with rational coefficients. Whenever there are only finitely
+many critical points, they all must have rational coordinates. On the other hand,
+if there is a piece f ∗(xℓ) of the piecewise function f(xℓ) that has infinitely many
+critical points, all of the critical points of f ∗(xℓ) attain the same constant value.
+This implies that there still exists a critical point with rational coordinates where
+the minimum is attained. Finally, because each piece of f(xℓ) is a quadratic
+with rational coefficients, the minimum is thus also rational.
+5
+Circle colorings
+As a variation on the theme of enumerating monochromatic progressions within
+colorings of [N], some authors have also investigated properties of arithmetic
+21
+
+progressions within colorings of the cyclic group ZN. For example, given a fixed
+red and blue 2-coloring of Zp for p prime, the fraction of monochromatic 3-term
+progressions that are red or blue depends only on the proportion of elements
+colored red, and not on the exact positioning of the red and blue elements, [5, 12].
+Even when N is not prime, the fraction of monochromatic 3-term progressions
+in ZN is bounded below by the quantity given if N were prime, [12].
+Inspired by these results, we now explore a continuous analogue to the enu-
+meration of monochromatic progressions within 2-colorings of ZN. Color each of
+the numbers in the unit circle S1 = {e2πiθ : θ ∈ [0, 1)} with red or blue, and con-
+sider 3-term arithmetic progressions of the form (e2πix1, e2πi(x1+d), e2πi(x1+2d))
+for x1, d ∈ [0, 1). To properly discuss the “fraction” of these that are monochro-
+matic for a given coloring, we introduce the uniform probability measure µ on
+[0, 1) and randomly sample arithmetic progressions by independently choosing
+x1, d ∈ [0, 1) according to µ. Using this framework, the probability of selecting a
+monochromatic progression depends only on the Lebesgue measure of the set of
+points colored red (i.e. the likelihood that, say, e2πix1 is red) and not on which
+points were colored red, which is an analogous result to the one for 2-colorings
+of the discrete group Zp.
+Lemma 5.1. Let C : S1 → {0, 1} be any measurable coloring of S1 with
+p := µ
+��
+θ ∈ [0, 1) : C
+�
+e2πiθ�
+= 0
+��
+defined as the proportion of points colored red, and let m(C) be the set containing
+all pairs (x1, d) ∈ [0, 1) × [0, 1) such that (e2πix1, e2πi(x1+d), e2πi(x1+2d)) are
+monochromatic. Then,
+(µ × µ)(m(C)) = 1 − 3p + 3p2.
+In particular, if we randomly select a starting point x1 and an increment d
+independently from each other according to the uniform distribution on S1, then
+the probability that the associated 3-AP is monochrome depends only on the
+proportion p of red points and not on how these points are distributed around
+S1.
+Proof. We take a probabilistic approach that follows the proof structure of The-
+orem 6 from [12]. To that end, let x1 and d be independent draws from µ and
+for i = 1, 2, 3 let Ai (respectively, Bi) be the event that the ith term in the
+progression (e2πix1, e2πi(x1+d), e2πi(x1+2d)) is red (respectively, blue). Then, via
+inclusion/exclusion, we have
+P(A1 ∪ A2 ∪ A3) =
+� 3
+�
+i=1
+P(Ai)
+�
+−
+�
+�
+�
+1≤i<j≤3
+P(Ai ∩ Aj)
+�
+� + P(A1 ∩ A2 ∩ A3).
+We note that P(A1 ∪ A2 ∪ A3) = 1 − P(B1 ∩ B2 ∩ B3).
+Since P(m(C)) =
+P(A1 ∩ A2 ∩ A3) + P(B1 ∩ B2 ∩ B3), we can rearrange the above to obtain
+P(m(C)) = 1 −
+3
+�
+i=1
+P(Ai) +
+�
+1≤i<j≤3
+P(Ai ∩ Aj).
+(9)
+22
+
+The random variables e2πix1, e2πi(x1+d), and e2πi(x1+2d) are pairwise indepen-
+dent and uniformly distributed on S1.
+This is true based on the rotational
+invariance of the uniform distribution on S1 and the fact that e2πix1 and e2πid
+are independent and uniformly distributed on S1. It follows that P(Ai) = p and
+P(Ai ∩ Aj) = p2 for i, j = 1, 2, 3. Substituting these into (9) yields
+(µ × µ)(m(C)) = P(m(C)) = 1 − 3p + 3p2.
+6
+Future Work
+In Sections 3 and 4 above, we outlined an approach to identifying the optimal
+coloring of [N] and the interval [0, 1] that minimizes the fraction of monochro-
+matic 3-APs for any fixed upper bound on the number of blocks n. A natural
+question is whether we can show that the optimal coloring for any n > 12 is the
+same as the optimal coloring for n = 12. One possibility is to prove that the
+colorings are no better for n = 14, and then argue that adding arbitrarily more
+intervals is no better than adding just two more intervals.
+Besides investigating how colorings of [N], ZN, [0, 1], and S1 affect the preva-
+lence of monochromatic arithmetic progressions of length 3, there are other re-
+lated problems that have yet to be explored. Perhaps the most natural question
+to ask is how the analysis changes if we consider longer arithmetic progressions,
+and the articles [18, 2, 12, 3] make partial progress in this direction for several
+different lengths of progressions. A slightly less obvious question is to ask what
+happens when we consider arithmetic progressions of color-dependent lengths.
+For example, we could attempt to color [0, 1] or [N] in a way that simultane-
+ously minimizes the fractions of monochromatic blue progressions of length 3
+and monochromatic red progressions of length 4.
+Another natural generalization is to add more colors. What do the 3- and 4-
+colorings of [0, 1] that minimize monochrome arithmetic progressions of length
+3 look like?
+Can anything be said about the rate at which the fraction of
+monochrome progressions decays as the number of colors increases? All of these
+questions have natural analogues in the setting of Ramsey theory as applied
+to graphs, and of course these generalizations might interact in any number of
+ways.
+When the problems studied in this paper were first posed, it was unclear
+whether or not colorings could perform better than random. Although they
+can perform better than random in the cases we present in detail above, is this
+also true for related problems? Recent work in [4] gives interesting insights into
+some classes of problems where solutions must be better than random.
+In addition to changing the number of colors or length of the progressions
+we study, we could also consider colorings in other geometries. For example,
+we wonder how to color an interval that has a gap in the middle in order
+to minimize monochromatic APs therein. By varying the length of the gap,
+we might gain insight into why antisymmetry is seemingly important in the
+23
+
+optimal block colorings of [0, 1] that we discuss above. Furthermore, we have
+already seen that in the contexts of Zp for p prime and the continuous circle, the
+performance of colorings with respect to 3-term progressions depends only on the
+ratios of the colors present. What other algebraic and geometric settings exhibit
+similar behavior? Alternatively, what would happen if we were to consider S1
+as in Section 5 but sample 3-APs by choosing the start point and increment
+according to a different distribution than uniform?
+7
+Acknowledgments
+Computations were performed using High Performance Computing infrastruc-
+ture provided by the Mathematical Sciences Support unit at the University of
+the Witwatersrand, and for this the authors are thankful.
+Additionally, the
+authors are grateful for invaluable tips from Professor Antti Laaksonen on how
+to optimize the code in Lemma 4.2.
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+A
+Appendix: 20 Polygonal Regions
+The 20 possible regions from Lemma 4.1 are given below.
+1
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+2xk ≤ xi + xj ≤ xi+1 + xj+1 ≤ 2xk+1
+Region Area:
+(xi+1 − xi)(xj+1 − xj)
+2
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj ≤ 2xk ≤ min(xi + xj+1, xi+1 + xj),
+xi+1 + xj+1 ≤ 2xk+1
+Region Area:
+(xi+1 − xi)(xj+1 − xj) − (2xk − xi − xj)2/2
+3
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi+1 + xj ≤ 2xk ≤ xi + xj+1 ≤ xi+1 + xj+1 ≤ 2xk+1
+Region Area:
+(xi+1 − xi)(xj+1 + xi/2 + xi+1/2 − 2xk)
+4
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj+1 ≤ 2xk ≤ xi+1 + xj ≤ xi+1 + xj+1 ≤ 2xk+1
+Region Area:
+(xj+1 − xj)(xi+1 + xj/2 + xj+1/2 − 2xk)
+26
+
+5
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+max(xi + xj+1, xi+1 + xj) ≤ 2xk ≤ xi+1 + xj+1 ≤ 2xk+1
+Region Area:
+(2xk − xi+1 − xj+1)2/2
+6
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+2xk ≤ xi + xj,
+max(xi + xj+1, xi+1 + xj) ≤ 2xk+1 ≤ xi+1 + xj+1
+Region Area:
+(xi+1 − xi)(xj+1 − xj) − (2xk+1 − xj+1 − xi+1)2/2
+7
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj ≤ 2xk ≤ min(xi + xj+1, xi+1 + xj),
+max(xi + xj+1, xi+1 + xj) ≤ 2xk+1 ≤ xi+1 + xj+1
+Region Area:
+(xi+1 − xi)(xj+1 − xj) − (2xk − xi − xj)2/2
+− (2xk+1 − xi+1 − xj+1)2/2
+8
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi+1 + xj ≤ 2xk ≤ xi + xj+1 ≤ 2xk+1 ≤ xi+1 + xj+1
+Region Area:
+(xi2 − xi1)(xj2 + xi1/2 + xi2/2 − 2xk1)
+− (2xk2 − xi2 − xj2)2/2
+9
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj+1 ≤ 2xk ≤ xi+1 + xj ≤ 2xk+1 ≤ xi+1 + xj+1
+Region Area:
+(xj+1 − xj) · (xi+1 + xj/2 + xj+1/2 − 2xk)
+− (2xk+1 − xi+1 − xj+1)2/2
+10
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+max(xi + xj+1, xi+1 + xj) ≤ 2xk ≤ 2xk+1 ≤ xi+1 + xj+1
+Region Area:
+(2xk − xi+1 − xj+1)2/2 − (2xk+1 − xi+1 − xj+1)2/2
+27
+
+11
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+2xk ≤ xi + xj ≤ xi+1 + xj ≤ 2xk+1 ≤ xi + xj+1
+Region Area:
+(xi+1 − xi)(2xk+1 − xj − xi/2 − xi+1/2)
+12
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj ≤ 2xk ≤ xi+1 + xj ≤ 2xk+1 ≤ xi + xj+1
+Region Area:
+(xi+1 − xi)(2xk+1 − xj − xi/2 − xi+1/2)
+− (2xk − xi − xj)2/2
+13
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi+1 + xj ≤ 2xk ≤ 2xk+1 ≤ xi + xj+1
+Region Area:
+(xi+1 − xi)(2xk+1 − 2xk)
+14
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+2xk ≤ xi + xj ≤ xi + xj+1 ≤ 2xk+1 ≤ xi+1 + xj
+Region Area:
+(xj+1 − xj)(2xk+1 − xj/2 − xj+1/2 − xi)
+15
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj ≤ 2xk ≤ xi + xj+1 ≤ 2xk+1 ≤ xi+1 + xj
+Region Area:
+(xj+1 − xj)(2xk+1 − xj/2 − xj+1/2 − xi)
+− (2xk − xi − xj)2/2
+16
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj+1 ≤ 2xk ≤ 2xk+1 ≤ xi+1 + xj
+Region Area:
+(xj+1 − xj)(2xk+1 − 2xk)
+28
+
+17
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+2xk ≤ xi + xj ≤ 2xk+1 ≤ min(xi + xj+1, xi+1 + xj)
+Region Area:
+(2xk+1 − xi − xj)2/2
+18
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj ≤ 2xk ≤ 2xk+1 ≤ min(xi + xj+1, xi+1 + xj)
+Region Area:
+(2xk+1 − xi − xj)2/2 − (2xk − xi − xj)2/2
+19
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+2xk+1 ≤ xi + xj
+Region Area:
+0
+20
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi+1 + xj+1 ≤ 2xk
+Region Area:
+0
+29
+

GtE3T4oBgHgl3EQfWQqL/content/tmp_files/2301.04467v1.pdf.txt

@@ -0,0 +1,1351 @@
+FrustumFormer: Adaptive Instance-aware Resampling for Multi-view 3D
+Detection
+Yuqi Wang1,2
+Yuntao Chen3
+Zhaoxiang Zhang1,2,3
+1Institute of Automation, Chinese Academy of Sciences (CASIA)
+2 School of Artificial Intelligence, University of Chinese Academy of Sciences
+3 Centre for Artificial Intelligence and Robotics, HKISI CAS
+{wangyuqi2020,zhaoxiang.zhang}@ia.ac.cn
+chenyuntao08@gmail.com
+Abstract
+The transformation of features from 2D perspective
+space to 3D space is essential to multi-view 3D object de-
+tection. Recent approaches mainly focus on the design of
+view transformation, either pixel-wisely lifting perspective
+view features into 3D space with estimated depth or grid-
+wisely constructing BEV features via 3D projection, treat-
+ing all pixels or grids equally. However, choosing what to
+transform is also important but has rarely been discussed
+before. The pixels of a moving car are more informative
+than the pixels of the sky.
+To fully utilize the informa-
+tion contained in images, the view transformation should
+be able to adapt to different image regions according to
+their contents. In this paper, we propose a novel frame-
+work named FrustumFormer, which pays more attention to
+the features in instance regions via adaptive instance-aware
+resampling. Specifically, the model obtains instance frus-
+tums on the bird’s eye view by leveraging image view object
+proposals. An adaptive occupancy mask within the instance
+frustum is learned to refine the instance location. Moreover,
+the temporal frustum intersection could further reduce the
+localization uncertainty of objects. Comprehensive exper-
+iments on the nuScenes dataset demonstrate the effective-
+ness of FrustumFormer, and we achieve a new state-of-the-
+art performance on the benchmark. Codes will be released
+soon.
+1. Introduction
+Perception in 3D space has gained increasing attention in
+both academia and industry. Despite the success of LiDAR-
+based methods [14,33,41,44], camera-based 3D object de-
+tection [19, 35, 36, 43] has earned a wide audience, due to
+the low cost for deployment and advantages for long-range
+detection. Recently, multi-view 3D detection in Bird’s-Eye-
+View (BEV) has made fast progresses. Due to the unified
+representation in 3D space, multi-view features and tem-
+poral information can be fused conveniently, which leads
+to significant performance improvement over monocular
+methods [5,28,35,39].
+Transforming perspective view features into the bird’s-
+eye view is the key to the success of modern BEV 3D de-
+tectors [12,18,19,22]. As shown in Fig. 1, we categorize the
+existing methods into lifting-based ones like LSS [30] and
+BEVDet [12] and query-based ones like BEVFormer [19]
+and Ego3RT [25]. However, these methods mainly focus
+on the design of view transformation strategies while over-
+looking the significance of choosing the right features to
+transform during view transformation. Regions containing
+objects like vehicles and pedestrians are apparently more in-
+formative than the empty background like sky and ground.
+But all previous methods treat them with equal importance.
+We suggest that the view transformation should be adaptive
+with respect to the image content. Therefore, we propose
+Adaptive Instance-aware Resampling (AIR), an instance-
+aware view transformation, as shown in Fig. 1c. The core
+idea of AIR is to reduce instance localization uncertainty by
+focusing on a selective part of BEV queries. Localizing in-
+stance regions is difficult directly on the BEV plane but rel-
+atively easy in the image view. Therefore, the instance frus-
+tum, lifting from instance proposals in image views, gives
+geometrical hints of the possible locations of objects in the
+3D space. Though the instance frustum has provided initial
+prior locations, it is still a large uncertain area. We propose
+an occupancy mask predictor and a temporal frustum fusion
+module to further reduce the localization uncertainty. Our
+model learns an occupancy mask for frustum queries on the
+1
+arXiv:2301.04467v1  [cs.CV]  10 Jan 2023
+
+(a) Grid Sampling in Image.
+(b) Grid Sampling in BEV.
+(c) Instance-aware Sampling in Frustum.
+Figure 1. Comparison of different sampling strategies for the feature transformation from image view to bird’s eye view. (a)
+represents the sampling in image view and lift features [12] to BEV with pixel-wise depth estimation. (b) shows the grid sampling in BEV
+and queries back [19] to obtain image features via camera projection. (c) illustrates our proposed strategy: instance-aware sampling in the
+frustum, which is adaptive sampling according to the view content. More attention will focus on instance regions.
+BEV plane, predicting the possibility that a region might
+contain objects. We also fuse instance frustums across dif-
+ferent time steps, where the intersection area poses geomet-
+ric constraints for actual locations of objects.
+We propose a novel framework called FrustumFormer
+based on the insights mentioned previously, which effec-
+tively enhances the learning of instance-aware BEV fea-
+tures via Adaptive Instance-aware Resampling. Frustum-
+Former utilizes the instance frustum to establish the con-
+nection between perspective and bird’s eye view regions,
+which contains two key designs: (1) frustum encoder to en-
+hance instance-aware features via adaptive instance-aware
+resampling within the instance frustum. (2) frustum fusion
+module to aggregate history instance frustum features for
+accurate localization and velocity prediction. In conclusion,
+the contributions of this work are as follows:
+• We propose FrustumFormer, a novel framework that
+exploits the geometric constraints behind perspective
+view and birds’ eye view by instance frustum.
+• We propose that choosing what to transform is also im-
+portant during view transformation. The view transfor-
+mation should adapt to the view content. Instance re-
+gions should gain more attention rather than be treated
+equally.
+Therefore, we design Adaptive Instance-
+aware Resampling (AIR) to focus more on the in-
+stance regions, enhancing the learning of instance-
+aware BEV features.
+• We evaluate the proposed FrustumFormer on the
+nuScenes dataset. We achieve improved performance
+compared to prior arts. FrustumFormer achieves 58.9
+NDS and 51.6 mAP on nuScenes test set without bells
+and whistles.
+2. Related Work
+2.1. Frustum-based 3D Object Detection
+Frustum indicates the possible locations of 3D objects in
+a 3D space by projecting 2D interested regions. The frustum
+is commonly used to aid fusion [8,27,31,37,42] in 3D ob-
+ject detection when RGB images and LiDAR data are avail-
+able. Frustum PointNets [31] takes advantage of mature 2D
+object detectors and performs 3D object instance segmenta-
+tion within the trimmed 3D frustums. Frustum Fusion [26]
+leverages the intersection volume of the two frustums in-
+duced by the 2D detection on stereo images. To deal with
+LiDAR sparsity, Faraway-Frustum [42] proposes a novel fu-
+sion strategy for detecting faraway objects. In this paper, we
+introduce the idea of frustum into camera-only 3D detection
+for enhancing instance-aware BEV features.
+2.2. Multi-view 3D Object Detection
+Multi-view 3D object detection aims to predict the 3D
+bounding boxes and categories of the objects with multi-
+view images as input. Current methods can be divided into
+two schemes: lifting 2D to 3D and Querying 2D from 3D.
+Lifting 2D to 3D. Following the spirit of LSS [30],
+BEVDet [12] lifts multi-view 2D image features into a
+depth-aware frustum and splats into a unified bird’s-eye-
+view (BEV) representation and applies to the detection task.
+BEVDepth [18] utilizes LiDAR points as supervision to
+learn reliable depth estimation. BEVDet4D [11] incorpo-
+rates the temporal information and extends the BEVDet to
+the spatial-temporal 4D working space. Recently, STS [38],
+BEVStereo [16] and SOLOFusion [29] further attempt to
+improve the depth learning by combining temporal geomet-
+ric constraints.
+2
+
+T
+T-1
+Backbone
+Frustum Fusion
+Frustum Encoder
+Backbone
+BEV Feature
+Detection Head
+AIR
+Cross 
+Attention
+Cross 
+Attention
++
+Instance Feature
+Scene Feature
+Frustum Encoder
+AIR
+Cross 
+Attention
+Cross 
+Attention
++
+Instance Feature
+Scene Feature
+AIR
+Instance Frustum
+Occupancy Mask
+BEV Feature
+(a) Overall architecture of FrustumFormer.
+T"
+T#
+(b) Temporal Frustum Fusion.
+Figure 2. Illustration of our proposed FrustumFormer. (a) shows the overall pipeline. The image backbone first extracts the multi-view
+image features. The frustum encoder transforms the multi-view image features into a unified BEV feature by integrating temporal infor-
+mation from frustum fusion. Then the detection head decodes the BEV feature to the final outputs. Adaptive Instance-aware Resampling
+(AIR) is utilized to adaptively adjust the sampling area and points according to the view content. It consists of two parts: instance frustum
+query generation and frustum occupancy mask prediction. (b) illustrates the hints for object locations during temporal frustum fusion.
+Querying
+2D
+from
+3D.
+Following
+DETR
+[2],
+DETR3D [36] predicts learnable queries in 3D space
+and projects back to query the corresponding 2D im-
+age features.
+PETR [22, 23] proposes to query directly
+with 3D position-aware features, which are generated
+by encoding the 3D position embedding into 2D image
+features.
+Ego3RT [25] introduces the polarized grid of
+dense imaginary eyes and sends rays backward to 2D visual
+representation.
+BEVFormer [19] learns spatiotemporal
+BEV features via deformable attention, which explicitly
+constructs the BEV grid samples in 3D space and queries
+back to aggregate multi-view image features.
+Polar-
+Former [13] further generates polar queries in the Polar
+coordinate and encodes BEV features in Polar space.
+3. Method
+In multi-view 3D object detection task, N monocular
+views of images I = {Ii ∈ R3×H×W }N
+i=1, together with
+camera intrinsics K = {Ki ∈ R3×3}N
+i=1 and camera extrin-
+sics T = {Ti ∈ R4×4}N
+i=1 are given. The objective of the
+model is to output the 3D attributes (locations, size and ve-
+locity) and the corresponding category of objects contained
+in multi-view images.
+As shown in Fig. 2a, FrustumFormer mainly focuses
+on the feature transformation process and is composed of
+four components: an image backbone, a frustum encoder,
+a frustum fusion module, and a detection head. The im-
+age backbone first extracts the image features of multi-view
+images. Aiding by the frustum fusion module, the image
+features transform into a unified BEV feature via the frus-
+tum encoder. Finally, a query-based detection head decodes
+the BEV feature to the 3D outputs of the detection task.
+3.1. Frustum Encoder
+Frustum encoder transforms the multi-scale multi-view
+image features F to a unified BEV feature B. Instead of
+treating all regions equally during the feature view trans-
+formation, our frustum encoder adaptively transforms the
+image features according to the view content. As shown
+in Fig. 2a, we use two types of BEV queries to construct
+the final BEV features, the scene queries Qs and the in-
+stance queries Qi. The learning process of the scene query
+is similar to BEVFormer [19]. However, the instance query
+is only learned inside instance regions, and the learned in-
+stance features are further combined with the scene feature
+to form the final instance-aware BEV features.
+Specifi-
+cally, the selection of instance regions is made via adaptive
+instance-aware resampling, which consists of (1) instance
+frustum query generation and (2) frustum occupancy mask
+prediction. Finally, the instance feature is learned by (3)
+instance frustum cross attention computed in selected in-
+stance regions. We will introduce these three parts in the
+following.
+Instance Frustum Query Generation. This section in-
+troduces the query generation for a single instance frus-
+tum Qf, which is a subset of instance query Qi.
+The
+core insight is to leverage the instance mask from perspec-
+tive views and select the corresponding region on the BEV
+plane. Following the query-based [13,19] view transforma-
+tion, we define a group of grid-shape learnable parameters
+Qi ∈ RH×W ×C as the instance queries. H, W are the spa-
+tial shape of BEV queries, and C is the channel dimension.
+We first generate sampling points {pk
+i = (xi, yi, zk), i ∈
+H × W, k ∈ K} corresponding to a single BEV query Qpi
+at grid region center pi = (xi, yi), and then project these
+3
+
+points to different image views. K is the number of sam-
+pling points in the vertical direction of pillars. The projec-
+tion between sampling points pk
+i and its corresponding 2D
+reference point (uk
+ij, vk
+ij) on the j-th image view is formu-
+lated as:
+πj(pk
+i ) = (uk
+ij, vk
+ij)
+(1)
+dk
+ij · [uk
+ij, vk
+ij, 1]T = Tj · [xk
+i , yk
+i , zk
+i , 1]T
+(2)
+where πj(pk
+i ) denotes the projection of the k-th sampling
+point at location pi on the j-th camera view. Tj ∈ R3×4
+is the projection matrix of the j-th camera. xk
+i , yk
+i , zk
+i rep-
+resents the 3D location of the sampling point in the vehicle
+frame. uk
+ij, vk
+ij denotes corresponding 2D reference point
+on j-th view projected from 3D sampling point pk
+i . dk
+ij is
+the depth in the camera frame.
+Predicting the instance frustum region directly in bird’s
+eye view is challenging, but detecting the objects in per-
+spective view [9, 32] is relatively mature. Inspired by this,
+we take advantage of object masks on the image plane and
+leverage its geometric clues for the BEV plane. The in-
+stance frustum queries Qf of a specific 2D instance could
+be defined as all instance BEV queries Qi with image plane
+projection points inside the object mask S Eq. (3):
+Qf = {Qpi ∈ Qi|
+∃
+π(pk
+i ) ∈ S}
+(3)
+Qpi ∈ R1×C is the query located at pi = (xi, yi). pk
+i rep-
+resents the k-th sampling points in the pillars at pi. π(pk
+i )
+denotes the projection points of pk
+i to the image plane.
+Frustum Occupancy Mask Prediction. Although instance
+frustum provides potential locations for objects, it still cor-
+responds to a large area on the BEV plane due to depth un-
+certainty. Therefore, we propose to predict an occupancy
+mask for all frustum to further reduce the localization un-
+certainty of instances within the instance frustum. Besides
+perspective constraints from instance frustum, another con-
+straint we could utilize is the supervision directly on the
+BEV plane. Specifically, given the union of all instance
+frustum queries ∪Qf, we design a AdaMask module to pre-
+dict a binary occupancy mask Obev ∈ RH×W ×1 on the
+BEV plane for all instance frustum queries in a single shot.
+The occupancy mask reflects the probability of a grid-wise
+region containing the objects, and is computed by Eq. (4):
+Obev = AdaMask(∪Qf)
+(4)
+AdaMask is a learned module composed of 2D convolu-
+tions. The supervision comes from the projection of the
+ground truth 3D bounding boxes on the BEV plane. We
+choose the focal loss [21] for learning occupancy mask in
+Eq. (5):
+Lm = Focal Loss(Obev, Ω)
+(5)
+where Ω denotes the projection mask of 3D bounding boxes
+on the BEV plane.
+We choose the minimum projecting
+bounding box on the BEV plane as the supervise signal.
+The minimum projecting bounding box is composed of the
+outermost corners of the objects, considering the rotation.
+Furthermore, the supervision for Obev adds to each layer of
+the frustum encoder for iterative refinement, together with
+the BEV instance feature learning. We utilize the last out-
+put instance frustum query for the current layer to predict
+the current occupancy mask. Therefore, the sampling areas
+adapt to the previous layer output.
+Instance Frustum Cross Attention.
+Instance Frustum
+Cross Attention (IFCA) is designed for the feature interac-
+tion between instance queries Qi and image view features
+F. The instance queries Qi is selected by Eq. (6):
+Qi = {Qpi ∈ ∪Qf|Obev(i) = 1}
+(6)
+Instance queries are selected from the instance frustum
+queries Qf. Obev(i) denotes the occupancy value at po-
+sition pi on the BEV plane. For each query Qpi in Qf, if
+Obev(i) predicts the occupancy value is 1, then the query
+Qpi is marked as instance query. The process of instance
+frustum cross-attention (IFCA) can be formulated as:
+IFCA(Qpi
+i , Fj) =
+M
+�
+m=1
+DeformAttn(Qpi
+i , πj(pm
+i ), Fj)
+(7)
+where Qpi
+i
+is an instance query at location pi, πj(pm
+i ) is
+the projection to get the m-th 2D reference point on the j-
+th camera view. M is the total number of sampling points
+in an instance query. Fj is the image features of the j-th
+camera view.
+3.2. Frustum Fusion Module
+Temporal information plays an important role in camera-
+based 3D object detection, especially in inferring the mo-
+tion state of objects and recognizing objects under heavy
+occlusions. Beyond learning occupancy mask on the BEV
+plane, another solution for eliminating the location uncer-
+tainty in the instance frustum is to fuse the temporal infor-
+mation.
+Temporal Frustum Intersection. As shown in Fig. 2b, the
+intersection area of the instance frustum at different times-
+tamps leaves hints for the accurate location of 3D objects.
+Inspired by this, we constrain the query interaction within
+instance frustum regions, implicitly learning features from
+interaction areas. Given instance frustum queries Qf at cur-
+rent timestamp t and history instance frustum queries Hf
+preserved at timestamp t′. For a query ∪Qpi
+f at position
+pi, we use the information from ego-motion (∆x, ∆y, ∆θ)
+to compute the corresponding position p′
+i at timestamp t′.
+The cross attention for query Qpi
+f only compute the his-
+tory queries around position p′
+i of Hf. Following [19], we
+adopt an RNN-like [6] way to fuse the historical instance
+4
+
+frustum queries sequentially. In this way, the long-range
+hints for the intersection area can be aggregated.
+Temporal Frustum Cross Attention. Temporal Frustum
+Cross Attention (TFCA) aggregates the information of his-
+tory instance frustum queries Hf into the current instance
+frustum queries Qf. Since the objects might be movable
+in the scene, causing the misalignment if only computing
+the query at p′
+i. Deformable attention [46] is utilized to
+reduce the influence of object movement. The process of
+temporal frustum cross attention (TFCA) can be formulated
+as follows:
+TFCA(Qpi
+f , Hf) =
+M
+�
+m=1
+DeformAttn(Qpi
+f , p′m
+i , Hf) (8)
+where Qpi
+f denotes the instance frustum query located at
+pi = (xi, yi). Hf represents the history instance frustum
+query. p′
+i is the aligned position by ego-motion. For each
+query at location p′
+i, we sample M points p′m
+i to query the
+history instance frustum feature.
+4. Experiment
+4.1. Datasets
+We conduct experiments on the challenging public au-
+tonomous driving datasets, namely nuScenes dataset [1].
+nuScenes dataset [1]. The nuScenes dataset provides 1000
+sequences of different scenes collected in Boston and Singa-
+pore. These sequences are officially split into 700/150/150
+ones for training, validation, and testing. Each sequence is
+roughly about 20s duration, and the key samples are anno-
+tated at 2Hz, contributing to a total of 1.4M objects bound-
+ing boxes.
+Each sample consists of RGB images from
+6 cameras covering the 360-degree horizontal FOV: front,
+front left, front right, back, back left, and back right. The
+image resolution is 1600×900 pixels in all views. 10 classes
+are annotated for the object-detecting task: car, truck, bus,
+trailer, construction vehicle, pedestrian, motorcycle, bicy-
+cle, barrier, and traffic cone.
+Evaluation metrics. For the official evaluation protocol
+in the nuScenes dataset, the metrics include mean Average
+Precision (mAP) and a set of True Positive (TP) metrics,
+which contains the average translation error (ATE), average
+scale error (ASE), average orientation error (AOE), average
+velocity error (AVE), and average attribute error (AAE). Fi-
+nally, the nuScenes detection score (NDS) is defined to con-
+sider the above metrics as in Eq. (9):
+NDS = 1
+10
+�
+5mAP +
+�
+mTP∈T P
+(1 − min (1, mTP))
+�
+(9)
+4.2. Experimental Settings
+Implementation Details. Following previous methods [19,
+35, 36], we utilize two types of backbone: ResNet101-
+DCN [7, 10] that initialized from FCOS3D [35], and
+VoVnet-99 [15] that initialized from DD3D [28]. We uti-
+lize the output multi-scale features from FPN [20] with
+sizes 1/8, 1/16, 1/32, and 1/64, and the feature dimension
+is 256. The frustum encoder has 6 layers, and we imple-
+ment it based on BEVFormer [19]. The default size of BEV
+queries is 200 × 200, and the perception ranges are [-51.2m,
+51.2m] for the X and Y axis and [-3m, 5m] for the Z axis.
+We sample K = 8 points for each pillar-like region of the
+BEV query. We adopt learnable position embedding for
+BEV queries. For the 2D instance proposals, we utilized the
+Mask R-CNN [9] pre-trained on the nuImages [1]. We use
+the output bounding boxes to generate object mask regions,
+and the score threshold is set to 0.5. The loss weight for
+Lm is set to 5. For the frustum fusion module, the tempo-
+ral window size W is set to 8, and we randomly sampled 4
+keyframes in the training phase. We utilized a query-based
+detection head [36] to decode the BEV features. The num of
+the object query is set to 600 and has 3 groups of queries [4]
+during training.
+Training. We train the model on 8 NVIDIA A100 GPUs
+with batch size 1 per GPU. We train our model with
+AdamW [24] optimizer for 24 epochs, an initial learning
+rate of 2 × 10−4 with a cosine annealing schedule. The
+input of the images is cropped to 1600 × 640. We adopt
+data augmentations like image scaling, flipping, color dis-
+tortion, and GridMask [3]. For the ablation study, we train
+the model with a total batch size of 8 for 24 epochs with-
+out data augmentation. We use the ResNet-50 [10] as the
+backbone. The image resolution is resized at a scale of 0.8,
+which is 1280 × 512.
+Inference. During inference, the previous BEV features
+are saved and used for the next, corresponding to the infi-
+nite temporal window of a sequence. This online inference
+strategy is time-efficient. Since we adopted three groups of
+queries during training, only one group is utilized at infer-
+ence time. We do not adopt model-agnostic tricks such as
+model ensemble and test time augmentation when evaluat-
+ing our model on both val and test sets.
+4.3. 3D Object Detection Results
+We compare our method with the state of the art on both
+val and test sets of nuScenes.
+nuScenes test set.
+Table 1 compares the results on the
+nuScenes test set. We achieved 51.6 mAP and 58.9 NDS
+without utilizing extra depth supervision from LiDAR. Un-
+der the setting without utilizing LiDAR as supervision, our
+method outperforms the previous state of the art. We eval-
+uate our model in two types of backbone mentioned in
+the implementation details.
+With R101-DCN [7] as the
+backbone, we could achieve 47.8 mAP and 56.1 NDS, a
+significant improvement (+2.1 mAP and 1.8 NDS) over
+previous methods.
+For the final performance, we train
+5
+
+Methods
+Backbone
+CBGS
+LiDAR
+mAP↑
+NDS↑
+mATE↓
+mASE↓
+mAOE↓
+mAVE↓
+mAAE↓
+FCOS3D‡ [35]
+R101†
+0.358
+0.428
+0.690
+0.249
+0.452
+1.434
+0.124
+PGD [34]
+R101†
+0.386
+0.448
+0.626
+0.245
+0.451
+1.509
+0.127
+BEVFormer [19]
+R101†
+0.445
+0.535
+0.631
+0.257
+0.405
+0.435
+0.143
+PolarFormer [13]
+R101†
+0.457
+0.543
+0.612
+0.257
+0.392
+0.467
+0.129
+FrustumFormer
+R101†
+0.478
+0.561
+0.575
+0.257
+0.402
+0.411
+0.132
+DD3D [28]‡
+V2-99*
+0.418
+0.477
+0.572
+0.249
+0.368
+1.014
+0.124
+DETR3D‡ [36]
+V2-99*
+✓
+0.412
+0.479
+0.641
+0.255
+0.394
+0.845
+0.133
+Ego3RT [25]
+V2-99*
+0.425
+0.473
+0.549
+0.264
+0.433
+1.014
+0.145
+M2BEV [40]
+X-101
+0.429
+0.474
+0.583
+0.254
+0.376
+1.053
+0.190
+BEVDet4D‡ [11]
+Swin-B
+✓
+0.451
+0.569
+0.511
+0.241
+0.386
+0.301
+0.121
+UVTR [17]
+V2-99*
+0.472
+0.551
+0.577
+0.253
+0.391
+0.508
+0.123
+BEVFormer [19]
+V2-99*
+0.481
+0.569
+0.582
+0.256
+0.375
+0.378
+0.126
+PolarFormer [13]
+V2-99*
+0.493
+0.572
+0.556
+0.256
+0.364
+0.440
+0.127
+PETRv2 [23]
+V2-99*
+0.490
+0.582
+0.561
+0.243
+0.361
+0.343
+0.120
+BEVDepth‡ [18]
+V2-99*
+✓
+✓
+0.503
+0.600
+0.445
+0.245
+0.378
+0.320
+0.126
+BEVStereo [16]
+V2-99*
+✓
+✓
+0.525
+0.610
+0.431
+0.246
+0.358
+0.357
+0.138
+FrustumFormer
+V2-99*
+0.516
+0.589
+0.555
+0.249
+0.372
+0.389
+0.126
+Table 1. Comparison to state-of-art on the nuScenes test set. * notes that VoVNet-99(V2-99) [15] was pre-trained on the depth
+estimation task with extra data [28]. †Initialized from FCOS3D [35] backbone. ‡ means utilizing test-time augmentation during inference.
+The commonly used scheme for training is 24 epochs, and CBGS [45] would increase the training epochs by nearly 4.5×. LiDAR means
+training depth branch utilizing extra modality supervision from LiDAR.
+Methods
+Backbone
+CBGS
+LiDAR
+mAP↑
+NDS↑
+mATE↓
+mASE↓
+mAOE↓
+mAVE↓
+mAAE↓
+FCOS3D [35]
+R101†
+0.295
+0.372
+0.806
+0.268
+0.511
+1.315
+0.170
+DETR3D [36]
+R101†
+✓
+0.349
+0.434
+0.716
+0.268
+0.379
+0.842
+0.200
+PGD [34]
+R101†
+0.358
+0.425
+0.667
+0.264
+0.435
+1.276
+0.177
+PETR [22]
+R101†
+✓
+0.370
+0.442
+0.711
+0.267
+0.383
+0.865
+0.201
+UVTR [17]
+R101†
+0.379
+0.483
+0.731
+0.267
+0.350
+0.510
+0.200
+BEVFormer [19]
+R101†
+0.416
+0.517
+0.673
+0.274
+0.372
+0.394
+0.198
+PolarFormer [13]
+R101†
+0.432
+0.528
+0.648
+0.270
+0.348
+0.409
+0.201
+BEVDepth [18]
+R101
+✓
+✓
+0.412
+0.535
+0.565
+0.266
+0.358
+0.331
+0.190
+STS [38]
+R101
+✓
+✓
+0.431
+0.542
+0.525
+0.262
+0.380
+0.369
+0.204
+FrustumFormer
+R101†
+0.457
+0.546
+0.624
+0.265
+0.362
+0.380
+0.191
+Table 2. Comparison to state-of-art on the nuScenes val set. †Initialized from FCOS3D [35] backbone. Our model is trained for 24
+epochs without CBGS [45]. LiDAR means training depth branch utilizing extra modality supervision from LiDAR.
+FrustumFormer on the trainval split for 24 epochs without
+CBGS [45], with VoVNet(V2-99) as backbone architecture
+with a pre-trained checkpoint from DD3D [28].
+nuScenes validation set. Table 2 shows that our method
+achieves leading performance on the nuScenes val set. We
+achieved 45.7 mAP and 54.6 NDS without bells and whis-
+tles. Unlike the evaluation on test set, all the methods are
+compared with a fair backbone here. Since BEVDepth [18]
+and STS [38] utilized extra modality supervision in training,
+our NDS metric only improved slightly compared to them,
+but our mAP improved significantly. The translation error
+would be reduced with LiDAR supervision for the depth es-
+timation, but this required extra modality data from LiDAR.
+Besides, our model is trained for 24 epochs, while they actu-
+ally trained 90 epochs if using CBGS [45]. More qualitative
+results are shown in supplementary materials.
+4.4. Ablation Study
+We
+conduct
+several
+ablation
+experiments
+on
+the
+nuScenes val set to validate the design of FrustumFormer.
+As mentioned in the implementation details, we use the
+ResNet-50 [10] as the backbone, and the image resolution
+is resized to 0.8 scales for all ablation experiments.
+Ablation of Components in FrustumFormer. Table 3 ab-
+lates the components designed in FrustumFormer. (a) is the
+baseline setting of our method. (b) is the baseline with the
+6
+
+instance frustum query, which resamples the points in the
+whole instance frustum region. (c) is the baseline with the
+occupancy mask, which gets supervision on the BEV plane.
+(d) is the baseline with adaptive instance-aware resampling,
+which consists of instance frustum query and occupancy
+mask. By utilizing the adaptive instance-aware resampling
+to enhance the instance-aware BEV feature, both mAP and
+NDS can be significantly improved. (e) is based on (d) and
+further adds the history frustum information to incorporate
+temporal clues. Above all, our FrustumFormer could im-
+prove 4.2 mAP and 9.7 NDS compared to the baseline.
+IF
+OM
+FF
+mAP↑
+NDS↑
+mATE↓
+(a)
+0.318
+0.366
+0.771
+(b)
+✓
+0.326
+0.373
+0.765
+(c)
+✓
+0.328
+0.381
+0.759
+(d)
+✓
+✓
+0.337
+0.383
+0.749
+(e)
+✓
+✓
+✓
+0.360
+0.463
+0.719
+Table 3. Ablation of components in FrustumFormer. IF denotes
+instance frustum, OM denotes occupancy mask, and FF means
+temporal frustum fusion. Adaptive instance-aware resampling is
+the combination of IF and OM, shown in (d).
+Ablation of Instance-aware Sampling. Table 4 proves the
+effectiveness of instance-aware sampling. (a) represents the
+baseline setting, which treats all regions equally and sam-
+ples 1x points for a cell region. (b) increases the sampling
+points to 2x for all regions. (c) selectively resamples the
+points inside instance regions. Compared with (b) and (c),
+we found that instance-aware sampling is more effective
+since simply increasing the sampling points for all regions
+has no gain.
+Total
+Scene
+Instance
+mAP↑
+NDS↑
+(a)
+1×
+1×
+-
+0.318
+0.366
+(b)
+2×
+2×
+-
+0.318
+0.362
+(c)
+2×
+1×
+1×
+0.326
+0.373
+Table 4. Ablation of instance-aware sampling. In our method,
+1× means sampling 8 points for a cell region on the BEV plane.
+Ablation of Occupancy Mask Learning. Table 5 com-
+pares different supervision for learning occupancy masks
+on the BEV plane. (a) is the baseline without explicit su-
+pervision. (b) adds the supervision on the BEV plane, in
+which the instance area can be obtained by projecting an-
+notated bounding boxes. To ease the learning, we slightly
+enlarge the bounding boxes by 1.0 meters in this setting. (c)
+uses the strict projection area (without enlargement) from
+the bounding boxes. (d) increases the loss weight for Lm
+from 5.0 to 10.0. We choose the (c) as our default setting.
+Ablation for Temporal Frustum Fusion. In Table 6, we
+first demonstrate the effectiveness of utilizing frustum in
+Supervision
+α
+mAP↑
+NDS↑
+mATE↓
+(a)
+w/o
+-
+0.318
+0.366
+0.771
+(b)
+w/ BEV box*
+5.0
+0.324
+0.374
+0.756
+(c)
+w/ BEV box
+5.0
+0.328
+0.381
+0.759
+(d)
+w/ BEV box
+10.0
+0.322
+0.381
+0.749
+Table 5. Ablation of occupancy mask learning. BEV box means
+utilizing the ground truth bounding boxes’ projection on the BEV
+plane as supervision. * denotes enlarged bounding box. α is the
+loss weight for learning occupancy mask.
+temporal fusion, and then we ablate the parameters for win-
+dow size W and keyframes K. (a) is the baseline for com-
+parison, with window size 4 and 2 keyframes. (b) adds the
+temporal frustum information.
+(c) enlarges the temporal
+window size to 8 and uses 4 keyframes in the temporal win-
+dow, which achieves the best performance. Both mAP and
+NDS would improve with a longer temporal window. We
+use the parameters in (d) as the default setting.
+W
+K
+Frustum
+mAP↑
+NDS↑
+mAVE↓
+(a)
+4
+2
+0.353
+0.454
+0.497
+(b)
+4
+2
+✓
+0.355
+0.457
+0.479
+(c)
+8
+4
+✓
+0.360
+0.463
+0.463
+Table 6. Ablation for temporal frustum fusion. W means the
+history window size. K determines the key frames sampled in tem-
+poral window during model training.
+0.35
+0.45
+0.55
+0.65
+0.75
+0.85
+car
+traffic cone
+barrier
+pedestrian motorcycle
+truck
+bicycle
+trailer
+bus
+construction 
+vehicle
+Baseline
+w/ AIR
+Figure 3. Improvement of Recall Under Low Visibility. We
+compute the recall under the visibility at 0-40% for all categories
+on the nuScenes val set. The recall for bus, bicycle, trailer, and
+construction vehicle categories improved significantly.
+Improvement of Recall Under Low Visibility.
+The
+nuScenes dataset provides the visibility labels of objects
+in four subsets {0-40%, 40-60%, 60-80%, 80-100%}. As
+shown in Fig. 3, we compare the recall between baseline
+and baseline with adaptive instance-aware resampling un-
+der low visibility (0-40%). We found that the recall for cat-
+egories of bicycle, trailer, bus, and construction vehicle im-
+proved a lot under the low visibility. Since nearly 29% of
+7
+
+objects belong to the visibility of 0-40%, such improvement
+is crucial for a better 3D object detector. More quantitative
+results and visualizations under heavy occlusions are in sup-
+plementary materials.
+4.5. Qualitative Analysis
+Visualization of Recall Improvement. As shown in Fig. 4,
+we illustrate the recall improvement with AIR from the pre-
+diction on the nuScenes val set. The prediction boxes are
+marked in blue, while the ground truth boxes are marked in
+green. In the red circle region, the objects would be discov-
+ered by utilizing AIR to enhance the learning of instance
+features.
+v
+Figure 4. Visualization of Recall Improvement. The left side is
+the baseline, and the right is the baseline with AIR. Examples are
+selected from the prediction on the nuScenes val set. The predic-
+tion boxes are marked in blue, while the ground truth boxes are
+marked in green. In the red circle region, more objects can be
+discovered by our methods.
+Visualization of Instance-aware Sampling. As shown in
+Fig. 5, we illustrate the instance-aware sampling points of
+our method on both perspective view and bird’s eye view.
+The sampling points are highly related to the instance re-
+gions, enhancing the learning of the instance-aware feature.
+Examples are selected from the nuScenes dataset.
+Visualization of Instance-aware BEV Feature. As shown
+in Fig. 6, we visualize the BEV feature output by our frus-
+tum encoder. Examples are selected from the prediction
+on the nuScenes val set. The BEV feature learned by our
+frustum encoder is instance-aware and has strong relations
+to the real positions. Here we visualize the corresponding
+ground truth boxes on the right side. Furthermore, we com-
+pare the BEV feature between the baseline and the base-
+line with adaptive instance-aware resampling. When utiliz-
+ing AIR, more instance regions would be discovered (cor-
+Figure 5. Visualization of Instance-aware Sampling. We visual-
+ize the instance-aware sampling on the perspective view and bird’s
+eye view. Ground truth bounding boxes are marked in green color.
+Examples are selected from the nuScenes dataset.
+responding to the recall improvement), and the features are
+more instance discriminative in the dense areas.
+Figure 6.
+Visualization of the instance-aware BEV feature.
+From left to right, we compare the feature heatmaps output by
+frustum encoder (w/o AIR), with AIR, and ground truth boxes in
+green shown on the right. The colors for the feature heatmap corre-
+spond to the norm value. The BEV feature learned by our frustum
+encoder is instance-aware and has strong relations to the actual po-
+sitions in 3D space. By using AIR, more instance regions would
+be discovered, and the features are more instance discriminative in
+the dense areas. Examples are selected from the prediction on the
+nuScenes val set.
+5. Conclusion
+In this paper, we propose FrustumFormer, a novel
+framework for multi-view 3D object detection.
+The
+core insight of FrustumFormer is to transform adaptively
+according to the view contents. To this end, we designed
+adaptive instance-aware resampling, paying more attention
+to the instance regions during feature view transformation.
+By utilizing adaptive instance-aware resampling in the
+frustum encoder and temporal frustum fusion module, the
+model can better locate the instance regions while learning
+the instance-aware BEV features.
+Experimental results
+on the nuScenes dataset demonstrate the effectiveness
+of our method for multi-view 3D object detection.
+Our
+method significantly improved on mAP over the previous
+methods due to the focus on the instance regions.
+We
+hope our framework can serve as a new baseline for
+8
+
+XI厂l:
+.:80
+口:following 3D perception work, and let more people pay
+attention to the view content during feature transformation.
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+ITA-ELECTION-2022: A multi-platform dataset of social media conversations
+around the 2022 Italian general election
+Francesco Pierri, Geng Liu, Stefano Ceri
+Dipartimento di Elettronica, Informazione e Bioingegneria,
+Politecnico di Milano, Milano, Italy
+E-mail: {name.surname}@polimi.it
+Abstract
+Online social media play a major role in shaping public dis-
+course and opinion, especially during political events. We
+present the first public multi-platform dataset of Italian-
+language political conversations, focused on the 2022 Italian
+general election taking place on September 25th. Leveraging
+public APIs and a keyword-based search, we collected mil-
+lions of posts published by users, pages and groups on Face-
+book, Instagram and Twitter, along with metadata of TikTok
+and YouTube videos shared on these platforms, over a period
+of four months. We augmented the dataset with a collection
+of political ads sponsored on Meta platforms, and a list of so-
+cial media handles associated with political representatives.
+Our data resource will allow researchers and academics to
+further our understanding of the role of social media in the
+democratic process.
+Introduction
+Online social media provide researchers and academics with
+unprecedented opportunities to observe a wide range of po-
+litical and societal phenomena (Rossi, Righetti, and Marino
+2021). They also play a critical role in shaping public opin-
+ion during political events (Vitak et al. 2011), and represent
+a rich source of data to study the interplay between politi-
+cal actors’ campaigns (Sahly, Shao, and Kwon 2019), media
+outlets’ agenda settings (Kim et al. 2016), and users’ news
+consumption (Allcott and Gentzkow 2017).
+In Italy, as of 20221, YouTube is the platform used by the
+largest amount of internet users (88%), followed by Meta
+platforms (64%) and TikTok (54%), whereas Twitter only
+accounts for approximately 7%2. However, previous studies
+of online social media during Italian elections and referen-
+dum mostly focused on Twitter (Rossi, Righetti, and Marino
+2021), due to the large availability of data via its APIs. In
+this work, instead, we present a public data resource of polit-
+ical conversations and user-generated content shared around
+the 2022 Italian general election, which allows researchers
+and academics to study multiple social platforms simultane-
+ously.
+Copyright © 2022, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+1www.statista.com/statistics/1311549/top-social-platforms-
+italy/
+2datareportal.com/reports/digital-2022-italy
+The 2022 Italian general election was the first ever to take
+place in autumn, as a consequence of the fall of the govern-
+ment of national unity led by Mario Draghi in July3. The
+election had a record-low voter turnout and it was won by
+the right-wing coalition of Giorgia Meloni with over 43% of
+the vote share. Among the opponents, the Centre-left coali-
+tion led by Enrico Letta obtained approximately 25% of
+the voters, the populist Movimento 5 Stelle led by former
+PM Giuseppe Conte reached less than 16%, and the liberal
+and centrist Third Pole, which included former PM Matteo
+Renzi, obtained almost 8% of the vote share.
+We present ITA-ELECTION-2022, the first public
+multi-platform dataset of Italian-language political conver-
+sations taking place on online social media, with a focus
+on the 2022 Italian general election. We collected millions
+of social media posts from Facebook, Instagram and Twit-
+ter, as well as advertisements sponsored on Meta platforms
+and metadata for TikTok and YouTube videos shared on the
+aforementioned platforms. We finally augment the dataset
+with a collection of social media handles associated with
+Italian political representatives. To collect the data, we em-
+ployed a snowball sampling procedure and curated a list
+of relevant terms to accordingly perform a keyword-based
+search during a period of four months (July 2022 - October
+2022). We provide public access to the data via GitHub and
+DataVerse repositories, as detailed next.
+The outline of this paper is the following: in the next sec-
+tion we review existing public data resources related to the
+present work; then, we describe the data collection proce-
+dure(s) carried out to build the dataset; next, we describe a
+few potential applications of the collected data; finally, we
+discuss limitations, draw conclusions and provide some eth-
+ical remarks.
+Related Work
+There are several public datasets that allow to study social
+media conversations around political issues. We focus our
+literature review on the Italian context, and then describe a
+few datasets related to other countries. We also refer the in-
+terested reader to (Rossi, Righetti, and Marino 2021) for an
+overview of studies that describe the interplay between so-
+cial media and Italian politics.
+3en.wikipedia.org/wiki/2022 Italian general election
+arXiv:2301.05119v1  [cs.SI]  12 Jan 2023
+
+(Basile, Lai, and Sanguinetti 2018) collect tweets in the
+Italian language continuously from 2012 to 2018, extracting
+a number of smaller datasets enriched with different kinds
+of annotations for linguistic purposes. They provide access
+to tweet IDs and annotations in a public repository.
+(Pierri, Artoni, and Ceri 2020) analyze the prevalence
+of Italian disinformation spreading on Twitter in the five
+months preceding the 2019 European Parliament election.
+They collect over 300 k tweets sharing thousands of news
+articles originating from websites flagged as unreliable by
+journalists and fact-checkers, providing public access to
+tweet IDs and lists of websites. The same authors provide
+a similar dataset collected in a different period of 2019, and
+that contains tweets sharing links to mainstream and tradi-
+tional news websites, both in the Italian and French language
+(Pierri 2020).
+(Di Giovanni and Brambilla 2021) study the polarization
+around the 2020 Italian constitutional referendum. They col-
+lect a dataset of 1.2 M tweets discussing the event – and
+provide access to their IDs –, with the goal of designing
+a hashtag-based semi-automatic approach to label Twitter
+users’ stance towards the referendum.
+Following the COVID-19 pandemic, several researchers
+collected social media data to study conversations around
+the crisis, with a particular focus on the impact of vaccine
+misinformation. (Crupi et al. 2022) study the evolution of
+Italian Twitter conversations around vaccines during the pe-
+riod 2019-2021, whereas (Di Giovanni et al. 2022) collect
+tweets in multiple languages (French, German and Italian)
+during the first year of world vaccination programs. Both
+contributions give public access to tweet IDs, with the lat-
+ter providing also a set of labeled pro/anti-vaccines tweets
+and hashtags that can be used for training machine learning
+classifiers.
+(Calisir and Brambilla 2020) provide a dataset of tweets
+discussing Brexit for a period of 45 months, from January
+2016 until September 2019. The data, which comprises 50.8
+million tweets and 3.97 million users, is enriched with meta-
+data such as the bot score of users, sentiment score of tweets,
+and political stance labels predicted by a classifier developed
+by the authors.
+There is a large number of datasets that focus on the U.S.
+elections (both presidential and midterms), and we provide
+here a non-exhaustive list of available resources. (Hanna
+et al. 2011) mapped candidates from the 2010 U.S. Midterm
+election with their Twitter accounts and a random sample
+of their followers. (Bovet and Makse 2019) collected over
+171 M tweets in the English language, mentioning Donald
+Trump and Hillary Clinton during the 2016 U.S. Presiden-
+tial election. (Deb et al. 2019) and (Yang, Hui, and Menczer
+2022) collected tweets discussing the 2018 U.S. Midterm
+election, both using a hashtag-based search (e.g. tweets shar-
+ing the hashtag ”#ivoted” on election day) and querying
+Twitter APIs with general keywords related to the midterm
+election. (Chen, Deb, and Ferrara 2022) provide a longitudi-
+nal dataset of over 1.2 billion U.S. politics- and election-
+related tweets shared around the period of the 2020 U.S.
+Presidential election. Related to the same election, (Abilov
+et al. 2021) released a multi-modal dataset of 7.6 M tweets
+Figure 1: Example of an ad run on Meta platforms along
+with the information provided by Meta Ad Library API.
+and 25.6 M retweets from 2.6 M users related to voter fraud
+claims. They augmented the data with cluster labels, users’
+suspension status, and perceptual hashes of tweeted images
+as well as aggregate data from external links and YouTube
+videos shared on Twitter.
+Data Collection
+This section describes the data collection procedure(s) car-
+ried out to gather data from different social media platforms.
+We remark that we employed the same list of keywords
+related to the Italian election, which we obtained through
+a snowball sampling approach using Twitter data only, to
+query different APIs. Our dataset conforms with FAIR prin-
+ciples: it is Findable, Accessible and Reusable as it is pub-
+licly accessible in an online Github4 and DataVerse reposi-
+tory5, where we provide the means to recreate it almost com-
+pletely (see limitations discussed next). It is also Interopera-
+ble as the data files are released in “.csv” and “.txt” formats.
+We summarize some statistics of the dataset in Table 1.
+Twitter
+We collected all tweets in the Italian language related to the
+election by using tweepy Python library to query Twitter
+v1.1 Filter streaming API endpoint6 in the period September
+4github.com/frapierri/ita-election-2022
+5doi.org/10.7910/DVN/EALXH2
+6developer.twitter.com/en/docs/twitter-api/v1/tweets/filter-
+realtime/overview
+
+Inactive
+8Sep2022-23Sep2022
+Platforms 
+Categories
+:Estimatedaudiencesize:100K-500kpeople
+自Amount spent (EUR):<E100
+@Impressions:4K-5k
+ID:376439398025966
+See ad details
+Sebastiano Valenti
+Sponsored·PaidforbySebastianoMarioValenti
+SonoufficialmentecandidatoalleelezioniregionalicolMoVimento5
+Stelle ☆★
+LaureatoinInformaticaehosemprelavoratoinquestoambito
+Nonfacciopartedinessunacordatapolitica,
+hosempremessoimpegnonelportareavantiproposteeleggiche
+migliorinolavitadellapersone,checomeme,tuttiigiornisialzano..
+COMESIVOTA
+0050
+NELLASCHEDA VERDE
+SICILIA
+EGNAAL SIMBOLO
+SCRIVI WALENTI
+SEGNADIPAOL
+2022
+DIPAOLA
+VALENTI
+NUCCIO
+SEBASTIANG
+VALENTI
+DIPAOLA
+Sebastiano Valenti
+Send Messa...
+Personal blogFigure 2: Daily number of social media posts and ads col-
+lected in our dataset, for different platforms. Solid lines
+show 3-day moving averages.
+2nd, 2022 - October 20th, 2022. We also leveraged Twit-
+ter’s historical Search API v2 endpoint7 to collect tweets
+retrospectively in the period July 1st, 2022 - September 2nd,
+2022. To query Twitter’s APIs we employed a snowball sam-
+pling approach, following existing work (Di Giovanni et al.
+2022; DeVerna et al. 2021), and generated a list of relevant
+keywords starting with seed terms such as “elezioni2022”
+and “elezioni”8; the final list contains 62 keywords and it
+is available in the repository associated with this paper. A
+sample is provided in Table 2. The total collection of tweets
+contains 19,087,594 tweets shared by 618,089 unique users.
+We remark that to abide by Twitter’s terms of service we
+only share tweet IDs publicly. These can be “re-hydrated”
+to retrieve tweet objects, with the exception of removed or
+protected tweets, by querying Twitter API directly or using
+tools like Hydrator9 or twarc.10
+Facebook and Instagram posts
+We collected Facebook and Instagram data by employ-
+ing CrowdTangle, a public tool owned and operated by
+Meta (CrowdTangle Team 2022) that allows retrieving posts
+7developer.twitter.com/en/docs/twitter-
+api/tweets/search/introduction
+8In the Italian language “elezioni” means elections.
+9github.com/DocNow/hydrator
+10github.com/DocNow/twarc
+Twitter
+19,087,594 tweets
+618,089 unique accounts
+Facebook
+1,142,812 posts
+445,461 unique accounts
+Instagram
+68,078 posts
+5,274 unique accounts
+Meta
+29,211 ads
+3,750 unique sponsors
+YouTube
+22,754 unique videos (Twitter)
+17,401 unique videos (FB)
+TikTok
+1,903 unique videos (Twitter)
+1,744 unique videos (FB)
+Table 1: Statistics of the dataset.
+elezioni
+partito democratico
+berlusconi
+renzi
+movimento 5 stelle
+salvini
+calenda
+di maio
+politiche2022
+meloni
+elezioni2022
+conte
+Table 2: A sample of Italian language keywords related to
+the 2022 election that were used to retrieve social media
+posts in our dataset.
+shared by public pages and groups with a certain amount of
+followers or that were manually added by other researchers
+on the platform.11 We queried the /posts/search end-
+point12 using the same list of keywords employed for col-
+lecting Twitter data. For each post, the API returns several
+attributes related to the post and the account (page or group)
+that shared it; the full list of attributes is available in the
+official documentation13. We retained only posts in the Ital-
+ian language by filtering on the languageCode param-
+eter: the final dataset contains 1,142,812 Facebook posts,
+shared by 445,461 unique pages and groups and generating
+over 233 M interactions (shares, comments, reactions), and
+68,078 Instagram posts, shared by 5,274 unique pages and
+generating over 97 M interactions (likes and comments). We
+provide access to the URLs and IDs14 of these posts, which
+can be used to access and retrieve those that are not removed
+or deleted, in the repository associated with this paper.
+TikTok and YouTube videos
+We augmented our dataset of social media posts by extract-
+ing metadata for TikTok and YouTube videos shared in Face-
+book15 and Twitter messages present in our dataset. For what
+concerns YouTube, we identified all external links to the
+11More
+details
+are
+available
+in
+the
+official
+documenta-
+tion:
+help.crowdtangle.com/en/articles/1140930-what-data-is-
+crowdtangle-tracking
+12github.com/CrowdTangle/API/wiki/Search
+13github.com/CrowdTangle/API/wiki
+14For each post we provide both platform and Crowdtangle ID
+that can be given as input to the GET Post ID endpoint accessi-
+ble here: github.com/CrowdTangle/API/wiki/Posts#get-postid
+15There were no links shared on Instagram.
+
+Election day
+600000
+eets
+400000
+TWe
+200000
+80000
+osts
+od
+60000
+B
+40000
+1500
+posts
+1000
+500
+1000
+ads
+750
+Meta
+500
+250
+130
+Jul 25
+Aug 19
+Sep 13
+un
+0ct 08platform and employed the official YouTube API16 to ex-
+tract video information such as the author, channel id, video
+title, description, Top 10 popular comments, etc. The result-
+ing collection yields metadata for 22,754 unique YouTube
+videos shared on Twitter and 17,401 unique YouTube videos
+shared on Facebook. For what concerns TikTok, given the
+lack of an official API, we employed pyktok Python li-
+brary17 to collect metadata about TikTok videos such as the
+title, description, length as well as information about the au-
+thor of the video. The resulting collection yields metadata
+for 1,903 unique TikTok videos shared on Twitter and 1,744
+unique TikTok videos shared on Facebook.
+Facebook and Instagram ads
+We leveraged Meta Ad Library API18 to collect all ads about
+“social issues, elections or politics” that were active on Meta
+platforms19 in the period July 1st, 2022 - October 20th,
+2022. We provide an example of a sponsored ad in Fig-
+ure 1. We queried the API with the same set of keywords
+mentioned beforehand; the API allows to search ads using
+one keyword at a time, and we queried the endpoint multi-
+ple times eventually discarding duplicated ads. The resulting
+collection contains 29,211 unique ads paid by 3,750 unique
+sponsors. For each ad, the API provides several different at-
+tributes: date of creation, period when the ad is active, name
+of the sponsor, message, platform on which the ad is active,
+lower and upper bound for the amount spent and the number
+of impressions generated, etc. In the repository associated
+with this dataset we provide access to the ID of ads, which
+can be then used to retrieve ads through Meta Ad Library
+interactive search console or API. In particular, to abide by
+Meta’s terms of use, an identification procedure is required
+to access the API endpoint, whereas the interactive search
+console only requires a Meta account to access it.
+Social media handles of political representatives
+We compiled a list of Facebook, Instagram and Twitter
+handles of elected members in the Senate and Chamber
+of deputies based on the official list released by the Ital-
+ian Ministry of Interior20. Specifically, for each represen-
+tative, we manually checked whether their official account
+was present on the three platforms. Insofar, we were able to
+match around 500 Twitter accounts, and approximately 100-
+150 Facebook and Instagram accounts. The full list is avail-
+able in the repository associated with this paper. We refer the
+interested reader to a similar useful resource presented by
+(Haman and ˇSkoln´ık 2021), who provide an online running
+database of politicians’ activity on social media (currently
+only Twitter is supported) spanning multiple countries.
+16developers.google.com/youtube/v3
+17github.com/dfreelon/pyktok
+18www.facebook.com/ads/library/api
+19These are: Facebook, Instagram, Messenger, and the Audience
+Network. Notice that only a dozen ads were placed on platforms
+other than Facebook and Instagram.
+20github.com/ondata/elezioni-politiche-2022
+Figure 3: Distributions of the number of tweets/posts and
+retweets/interactions for each Twitter, Facebook and Insta-
+gram account. Dashed lines indicate median values: 2 tweets
+and 3 retweets for Twitter accounts; 2 posts and 8 interac-
+tions for Facebook accounts; 1 post and 174 interactions for
+Instagram accounts.
+Data Characterization
+In this section, we provide a few basic descriptive statistics
+of the data presented in this work and leave more detailed
+analyses for future research.
+In Figure 2, we show the daily number of social media
+posts and ads collected in our dataset, for each platform. We
+can observe a significant increasing trend (Mann-Kendall
+P < 0.001) toward election day in all cases, with a sharp
+drop in the weeks afterward. We also notice that Twitter ac-
+tivity in our dataset is much more represented than other
+platforms, followed by Facebook and Instagram.
+In Figure 3, we show the distribution of account-wise met-
+rics for Twitter, Facebook and Instagram. Specifically, we
+show the Cumulative Distribution Function (CDF) for the
+number of tweets/posts created and retweets/interactions re-
+ceived by accounts on each platform. All distributions show
+an exponential-like behavior, with most of the accounts be-
+ing very rarely active and receiving little engagement, and
+only a minority of them exhibiting a large number of posts
+created and engagement received. Median values are shown
+by dashed lines and are available in the caption of the figure.
+In Figure 4, we show distributions of metrics for Meta ads.
+Specifically, we show the CDF of the mean amount spent
+and the mean number of impressions generated at both the
+
+1.0
+1.0
+0.8
+0.8
+0.6
+0.6
+0.2
+0.2
+-
+0.0
+0.0
+101
+103
+101
+103
+105
+Number of tweets
+Number of retweets
+per Twitter account
+per Twitter account
+1.0
+1.0
+-
+0.8
+0.8
+uoI
+0.6
+0.6
+orti
+0%0.4
+0.2
+0.2
+-
+0.0
+0.0
+101
+102
+103
+101
+103
+105
+107
+100
+Number of posts
+Number of interactions
+per Facebook account
+per Facebook account
+1.0
+1.0
+-
+0.8
+0.8
+tion
+-
+20.6
+0.6
+or
+0
+0.2
+0.2
+-
+-
+0.0
+0.0
+101
+100
+102
+103
+101
+103
+105
+Number of posts
+Number of interactions
+per Instagram account
+per Instagram accountFigure 4: Distributions of the mean amount spent and the
+mean number of impressions generated at the ad and spon-
+sor level for Meta ads. Dashed lines indicate median values:
+148.5 EUR spent for ads and by sponsors; 7498.5 impres-
+sions generated by ads and 12498.5 impressions by spon-
+sors.
+ad and sponsor level; the mean value is computed by taking
+the average between the lower and upper bound estimates of
+the amount/impressions provided by Meta Ad Library API
+for each ad. Median values are shown by dashed lines and
+are available in the caption of the figure. We refer the reader
+to (Pierri 2022) for a more detailed analysis of political ad-
+vertising on Meta platforms during the 2022 Italian general
+election.
+In Figure 5, we show the top 10 accounts on Twitter, Face-
+book and Instagram ranked by the number of tweets/posts
+created and the number of retweets/interactions received.
+We can observe notable politicians from the entire politi-
+cal spectrum as well as journalists and news outlets, but also
+supporting pages and groups. We also notice that some of the
+most active accounts do not appear among the most engaged
+ones. Finally, we can see that Italian PM Giorgia Meloni is
+the most engaged account on the three platforms.
+Potential Applications
+There are several potential applications for our dataset,
+which can consider a single platform or multiple ones at the
+same time.
+Interested researchers could further the current under-
+standing of polarization processes taking place during elec-
+tion seasons by analyzing content shared on multiple so-
+cial platforms at once. They could study whether “echo-
+chamber” effects take place on different platforms, high-
+lighting similarities and differences in their formation pro-
+cess.
+Other researchers might leverage the data in order to study
+how political candidates interacted with potential voters on
+social media platforms, thus analyzing in detail the politi-
+cal communication strategies put in place by different can-
+Figure 5: Top 10 accounts ranked by the number of
+tweets/posts created and retweets/interactions received on
+Twitter, Facebook and Instagram. Due to space limitations,
+some account names are truncated.
+didates. They could also investigate the presence of correla-
+tional effects between online signals and electoral outcomes,
+or detect the presence of toxic and hateful speech originating
+in communities of political supporters.
+Some researchers could investigate the presence of
+mis/disinformation and astroturfing campaigns taking place
+in the run-up to the election, studying patterns of similari-
+ties and differences among different platforms. They could
+also analyze how fringe and harmful content spreads across
+communities present on different platforms, and whether in-
+fluential accounts play a role in amplifying certain malicious
+narratives.
+Discussion
+We released ITA-ELECTION-2022, a large-scale dataset
+of social media posts in the Italian language discussing the
+2022 Italian General election, which took place on 25th
+September 2022, spanning multiple online platforms and
+covering a period of four months. In addition to gathering
+posts shared on Twitter, Facebook and Instagram, we col-
+lected ads sponsored on Meta platforms, we extracted meta-
+data for YouTube and TikTok videos shared on different
+platforms during the collection period, and we compiled a
+
+1.0
+1.0
+0.8
+0.8
+0.6
+0.6
+0.2
+0.2
+-
+0.0
+0.0
+103
+104
+104
+105
+106
+Mean amount (EUR) per ad
+Mean impressions per ad
+1.0
+1.0
+0.8
+0.8
+uoI
+uoI
+0.6
+0.6
+porti
+0.4
+0.4
+0.2
+0.2
+0.0
+0.0
+103
+104
+105
+106
+Mean amount (EUR) per sponsor
+Mean impressions per sponsorTwitter
+Twitter
+infoitinterno
+GiorgiaMeloni
+danieledv79
+CarloCalenda
+Infinitolsacco
+GiuseppeContelT
+Divorex8
+ultimora_pol
+CenturrinoLuigi
+Mov5Stelle
+OM_VA_SH
+Fratellidltalia
+Giancar70336148
+matteosalvinimi
+Hattoriando
+ilruttosovrano
+naladrof53
+jacopo_iacoboni
+GianmarioAngius
+matteorenzi
+0.0
+0.5
+1.5
+1.0
+0.00
+0.25
+0.50
+0.75
+1.00
+Number of tweets
+1e4
+Number of retweets
+ 1e5
+Facebook
+Facebook
+CONTE E CUORE I..
+Giuseppe Conte
+Amore per Cont...
+Giorgia Meloni
+Noi con Salvini.
+II Fatto Quotid.
+NOI E MATTEO RE..
+Matteo Salvini
+Lega - Salvini
+..·
+Andrea Scanzi
+Marco Travaglio...
+la Repubblica
+Raccolta firme
+Fanpage.it
+Gianluigi Parag...
+Ultime Notizie
+.
+II Fatto Quotid...
+W IL M5S
+Conte President...
+Tu e I informaz...
+0
+2
+4
+6
+0
+4
+6
+8
+Number of posts
+1e3
+Number of interactions le6
+Instagram
+Instagram
+Affaritaliani.i...
+Giorgia Meloni
+Antonella Faggi...
+Matteo Salvini
+Roberta Ferrero...
+Sveglia Italia...
+AQTR
+Erica Rivolta
+MoVimento 5 Ste...
+Fanpage.it
+CRONACHE DI SPO...
+Carlo Calenda
+Italia Viva
+CALCIATORIBRUTT...
+AQTR
+Cronache di bas...
+Il Giornale
+Fratelli d'ltal...
+Tg2 Rai
+la Repubblica
+0
+2
+4
+6
+8
+0
+3
+1
+Number of posts
+1e2
+Number of interactions le6list of social media handles associated to political represen-
+tatives that can be used to gather further data. We described
+in detail the collection procedures carried out to build the
+dataset, and provided a few basic statistics about the col-
+lected data. We also suggested promising directions for fu-
+ture research.
+Our work is not without limitations. First, our keyword-
+based search might entail results that are not completely
+accurate, e.g., one of the terms employed for the query is
+“conte”, which might refer both to former PM Giuseppe
+Conte and football manager Antonio Conte. From another
+perspective, election-related terms might have been em-
+ployed for marketing campaigns and promoting content that
+is not pertinent to the election. However, while we are un-
+able to address these issues, which would require non-trivial
+efforts, researchers can further refine our data collection
+to meet their needs. Moreover, we performed a backward
+search to retrieve Twitter, Facebook and Instagram posts
+shared from July to September 2022, and we missed those
+that were deleted or removed during the same period. Simi-
+larly, by providing access only to IDs and URLs of collected
+posts, posts that have been removed or made private by users
+cannot be retrieved, thus limiting reproducibility analyses.
+Furthermore, we did not filter out the activity of automated
+and inauthentic accounts that might have polluted organic
+conversations around the election. Another limitation con-
+cerns Meta which, as highlighted in (Le Pochat et al. 2022),
+might not accurately label all political ads as such and our
+collection might be missing some data. Finally, the user base
+of different platforms analyzed in this work might not be
+fully representative of the actual Italian population, and this
+should be taken into consideration by future research.
+Despite these limitations, we believe that our dataset pro-
+vides fertile ground for a number of intriguing and interest-
+ing research applications, and we hope that this resource can
+advance our understanding of the interplay between online
+social media and democratic processes.
+Ethical considerations
+We performed our data collection and public release in com-
+plete agreement with the platforms’ terms of service. We ac-
+knowledge that TikTok metadata was scraped from the plat-
+form, thus potentially violating the platform’s terms of ser-
+vice, but this was due to the lack of an official API(Freelon
+2018). We do not directly share the content of social media
+posts, but rather provide access to IDs and URLs that can
+be used to retrieve the original data, with the exception of
+posts that have been deleted by platforms, and removed or
+made private by their author. We did not cause any harm nor
+expose information about individual users in the process of
+collecting and releasing the data, with the only exception of
+political representatives and a handful of popular accounts
+shown in the descriptive statistics. We understand that dis-
+closing their social media accounts might open up to poten-
+tial abuse by malicious actors, but at the same time, it en-
+ables researchers, journalists and other stakeholders to put
+important public actors, such as the members of the Italian
+Parliament and Senate, to scrutiny in order to better under-
+stand the influence of social media platforms on the demo-
+cratic process.
+Acknowledgments
+We are thankful to M.Sc. students Valeria Pant´e and Ilaria
+Saini for helping match social media accounts to politi-
+cal representatives. Work supported in part by PRIN grant
+HOPE (FP6, Italian Ministry of Education).
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+

IdA0T4oBgHgl3EQfCP85/content/tmp_files/2301.01986v1.pdf.txt

@@ -0,0 +1,1016 @@
+ 
+A mempolar transistor made from tellurium 
+Yifei Yang†, Lujie Xu†, Mingkun Xu†, Huan Liu, Dameng Liu, Wenrui Duan, Jing Pei, 
+Huanglong Li* 
+ 
+Y. Yang, M. Xu, J. Pei, H. Li 
+Department of Precision Instrument, Center for Brain Inspired Computing Research, 
+Tsinghua University, Beijing, 100084, China. 
+E-mail: li_huanglong@mail.tsinghua.edu.cn   
+ 
+L. Xu, W. Duan 
+School of Instrument Science and Opto Electronics Engineering, Beijing Information 
+Science and Technology University, Beijing, 100192, China. 
+ 
+L. Xu 
+Application Technology Department, Dongfang Jingyuan Electron Limited, Beijing, 
+100176, China. 
+ 
+H. Liu, D. Liu 
+State Key Laboratory of Tribology, Tsinghua University, Beijing, 100084, China. 
+            
+H. Li 
+Chinese Institute for Brain Research; Beijing, 102206, China. 
+ 
+Keywords: transistors, memristors, tellurium, reconfigurable polarity, mempolar, 
+ternary content-addressable memories, regularization methods 
+ 
+Abstract  
+The classic three-terminal electronic transistors and the emerging two-terminal ion-
+based memristors are complementary to each other in various nonconventional 
+information processing systems in a heterogeneous integration approach, such as 
+DRAM/storage-class memory hierarchy, hybrid CMOS/memristive neuromorphic 
+crossbar arrays, and so on. Recent attempts to introduce transitive functions into 
+memristors have given rise to gate-tunable memristive functions, hetero-plasticity and 
+mixed-plasticity functions. However, it remains elusive under what application 
+scenarios and in what ways transistors can benefit from the incorporation of ion-based 
+memristive effects. Here, we introduce a new type of transistor named ‘mempolar 
+transistor’ to the transistor family that has included the well-known unipolar and 
+ambipolar transistors. As its name suggests, mempolar transistor has polarity with 
+memory, reminiscent of memristor having resistance with memory. Specifically, its 
+polarity can be converted reversibly, in a nonvolatile fashion, between n-type and p-
+type depending on the history of the applied electrical stimulus. This is achieved by 
+the use of the emerging semiconducting tellurium as the electrochemically active 
+source/drain contact material, in combination with monolayer two-dimensional MoS2 
+channel, which results in a gated lateral Te/MoS2/Te memristor, or from a different 
+perspective, a transistor whose channel can be converted reversibly between n-type 
+MoS2 and p-type Te. With this unique mempolar function, our transistor holds the 
+promise for reconfigurable logic circuits and secure circuits, addressing a fundamental 
+limitation in previous implementations, that is, polarity reconfiguration was volatile 
+
+ 
+and required additional gate terminals. In addition to this manifest advantage, we 
+propose and demonstrate experimentally, a ternary content-addressable memory made 
+of only two mempolar transistors, which used to require a dozen normal transistors, 
+and by simulations, a device-inspired and hardware matched regularization method 
+‘FlipWeight’ for training artificial neural networks, which can achieve comparable 
+performance to that achieved by the prevalent ‘Dropout’ and ‘DropConnect’ methods. 
+This work represents a major advance in diversifying the functionality of transistors.  
+ 
+1. Introduction 
+Polarity is one of the most fundamental aspects of a transistor’s identity, according to 
+which transistors are most famously categorized into either n-type or p-type. The 
+pairing of n-type and p-type transistors has resulted in complementary metal-oxide-
+semiconductor (CMOS) technology, the leading technology for digital integrated 
+circuits over the past four decades. The polarity of a transistor is a reflection of the 
+comprehensive effect of its entire materials system, including the channel 
+semiconductor, impurities, source/drain contact, and so on. Typically, polarity identity 
+is determined at the fabrication stage and cannot be altered afterwards. 
+ 
+Traditionally, the performance improvements of the integrated circuits have depended 
+almost solely on the miniaturization of transistors (Moore’s Law). As miniaturization 
+is approaching its physical limits, transistor scaling is delivering performance 
+improvements at a slower pace. However, software is evolving extremely rapidly with 
+emerging applications1,2, such as artificial intelligence. Given this, the hardware that 
+cannot adapt to software will suffer from a short lifecycle and high nonrecurring 
+engineering cost3. In addition, with the ever increasing density of transistors in 
+microprocessors, power dissipation has become a huge factor challenging the 
+continuous success of the CMOS technology. Overall, both flexibility and energy 
+efficiency have become main criteria for computing fabrics. In this context, 
+incorporation into devices of functionalities that do not necessarily scale according to 
+Moore’s Law but provide additional value in different ways has become increasingly 
+pursued by semiconductor industry and academia4. 
+ 
+Different from transistor in many key aspects, memristor, experimentally discovered 
+less than two decades ago5, is one such emerging device that holds great promise for 
+low-power and adaptive electronics. Memristor is best-known as a two-terminal 
+resistor with long-term memory. In contrast to the transient electronic switching 
+(volatile) during the operation of transistor, enduring atomic structure change in the 
+switching medium can be elicited when suitable electrical stimulus is applied to 
+memristor, giving rise to nonvolatile reconfigurability of the resistance state. This 
+unique property of memristor has made it a key complementary device to transistor in 
+forming DRAM/storage-class memory hierarchy in conventional von Neumann 
+architecture and in enabling the emerging computing paradigms, such as in-memory 
+computing and neuromorphic computing6-8, where transistors and memristors are 
+integrated heterogeneously. 
+ 
+As one step further, attempts have been made to introduce transitive functions into 
+memristors. In such devices, additional control gates are positioned aside and 
+electrically insulated from the memristive channels9. These have given rise to gate-
+tunable memristive functions10-12, enabling the emulation of hetero-plasticity13-15. 
+
+ 
+Apparently, memristors and memristor-centered circuits benefit from the additional 
+transistive functions to further enhance their flexibility and augment their 
+functionalities16,17. 
+ 
+At this point, an interesting question arises as to under what application scenarios and 
+in what ways transistors can benefit from the incorporation of ion-based memristive 
+effects. Here, we propose and demonstrate that the fusion of transistive and 
+memristive functions can result in a novel type of transistor which we name 
+‘mempolar transistor’. As its name suggests, mempolar transistor has polarity with 
+memory, reminiscent of memristor. Specifically, its polarity can be converted 
+reversibly, in a nonvolatile fashion, between n-type and p-type depending on the 
+history of the applied electrical stimulus. Mempolar transistors bring major 
+improvements over the existing reconfigurable transistors18-21 in the following key 
+aspects: first, mempolar transistors are more energy-efficient because polarity 
+conversion takes place in a nonvolatile fashion, whereas in previous devices control-
+gate voltages have to be applied persistently; second, mempolar transistors have 
+smaller footprints and less fabrication complexity because they retain the classic 
+three-terminal structures without the addition of more control gates.  
+ 
+To enable mempolarity, a key design idea is to first make a two-terminal memristor 
+whose two resistance states are of n-type and p-type semiconductivity, respectively, 
+then introduce a third terminal to provide gate control over the memristive channel in 
+either polarity, as in a normal transistor. To this end, we choose monolayer two-
+dimensional (2D) MoS2 as the pristine n-type memristive medium with lateral p-type 
+tellurium (Te) electrodes, making a two-terminal lateral Te/MoS2/Te device. The 
+memristive function of this device can be foreseen (demonstrations provided) based 
+on the previous demonstrations of the electrochemically activity of Te and other 
+memristors made from it22-25. With a back gate, a mempolar transistor that can be 
+reconfigured between an n-type MoS2 transistor and a p-type Te transistor is created. 
+In addition to the aforementioned apparent advantages over the existing 
+reconfigurable transistors in logic and secure circuits, our mempolar transistor also 
+demonstrate additional value in other applications, such as ternary content-
+addressable memory (TCAM) cell made of two mempolar transistors, which used to 
+require a dozen normal transistors. Inspired by the device properties, a hardware-
+matched regularization method for mitigating the over-fitting problems in artificial 
+neural networks is also developed, which can achieve comparable performance to that 
+achieved by the prevalent ‘Dropout’ and ‘DropConnect’ methods. Our proposed 
+mempolar transistor is a valuable addition to the transistor family, enabling 
+nonvolatile fine-grain reconfigurability and supporting general-purpose hardware 
+design. 
+2. Results and discussion 
+2.1. Mempolar function and its mechanism 
+The schematic structure and the optical image of the mempolar transistor are shown in 
+figure 1a and 1b, respectively. To fabricate the device, a monolayer MoS2 flake (the 
+monolayer characteristics verified by Raman spectroscopy are shown in 
+supplementary figure S1) is mechanically exfoliated onto a heavily p-type doped Si 
+substrate with 300-nm-thick thermally oxidized SiO2. The two Te electrodes are then 
+deposited by magnetron sputtering as the source and drain terminals, followed by the 
+deposition of platinum protective layers (see Methods).  
+
+ 
+The n-type transfer characteristic of the as-fabricated device is depicted in figure 1c. 
+Under a constant drain-source voltage (Vds=Vd-Vs) of +1 V, the drain current (Id) is 
+increased by about 200 times as Vg is swept from -10 V to +10 V. Certain degree of 
+clockwise hysteresis can also be observed as Vg is swept back and forth, indicating 
+that trapping/detrapping of electrons in/from the gate oxide (SiO2) takes place during 
+the course of Vg sweeping. To elicit polarity conversion, a sufficiently large Vds pulse 
+of +30 V in amplitude and 30 s in duration is applied to the device (under Vg=0 V). 
+As aforementioned, the Te electrode under negative bias (source electrode) with 
+respect to the other can be electrochemically reduced. The induced Te2- anions may 
+migrate towards the counter electrode where they will be re-oxidized to elemental Te. 
+As this electrochemical process proceeds, this accumulated elemental Te may 
+eventually bridge the source and drain electrodes by forming local Te filament or 
+wider Te sheet. Because Te is a narrow-bandgap semiconductor with native p-type 
+conductivity, we have previously exploited this electrochemical mechanism to enable 
+filamentary resistance switching in vertical memristors where wider-bandgap (more 
+insulating) dielectrics are sandwiched between the Te electrodes23. Here, the 
+formation of Te sheet of the width as close to that of the channel as possible is 
+preferred in order to best maintain the channel geometry. Figure 1d shows the transfer 
+curve of the device after large Vds is applied. It is seen that Id is now decreased with 
+increasing Vg, verifying that transistor polarity is conversed to p-type. With the 
+conversion of polarity, the hysteresis in the transfer curve also changes direction from 
+clockwise to anti-clockwise. The p-type transfer characteristics can still be reproduced 
+after one month without degradation, confirming the nonvolatile nature of the polarity 
+conversion. The p polarity can be reversibly switched back to n polarity by applying a 
+Vds pulse of -30 V under Vg=+5 V. Gate biasing here reduces the number of free hole 
+carriers in the p-type Te channel and thus mitigates channel field screening for 
+enabling electrochemical reaction and ion drift. The reversible polarity change is 
+robustly reproduced under 100 times of polarity switching operations, as shown in 
+supplementary figure S2. Supplementary figure S3 also shows the evolutions of Id 
+during the periods of polarity-switching Vds pulses. Gradual increase (decrease) in Id 
+with time is observed in n-to-p (p-to-n) switching, which is consistent with the higher 
+conductivity of p-type Te than that of n-type MoS2.  
+In contrast to other commonly used source/drain electrode materials, Te is known to 
+be electrochemically active. In addition, Te anions are also shown to be mobile in 
+various solids. These two materials properties are key in enabling Te filament-based 
+resistance switching, as previously reported22,23. To explore the mechanisms behind 
+the unique mempolar phenomenon, we fabricate two control devices with Ti and Pt 
+source/drain electrodes that are comparatively inert. Similar measurements on their 
+transfer characteristics before and after the applications of large (+40 V, 30 s) Vds 
+pulses are performed. As shown in supplementary figure 4, polarity conversion occurs 
+in neither device.  
+ 
+
+ 
+ 
+Fig. 1 Schematic and electrical performance of the mempolar transistor. a. 
+Schematic and b. optical image of the mempolar transistor (scale bar: 2 μm). c. n-type 
+transfer curve of an as-fabricated device. d. p-type transfer curves of the device after 
+polarity switching. 
+To further identify if the atomic constitution of the channel of the mempolar transistor 
+is changed after polarity conversion, Auger electron spectroscopy (AES) analyses of 
+elements in the channel area for the as-fabricated n-type device, the p-type device 
+after polarity conversion and the restored n-type device are conducted, as shown in 
+figure 2a-c. For the as-fabricated device, it is seen that only Mo and S elements exist 
+in the channel and no observable Te element is found. However, a remarkable amount 
+of Te can be observed in the channel area after polarity conversion. Its distribution 
+looks uniform along the channel width and length except, unsurprisingly, an obvious 
+enrichment near the Te electrodes. The formation of uniform Te sheet in the channel 
+area verifies the hypothesis that polarity conversion is due to electrochemically 
+induced Te inclusion in MoS2. When the device is switched back to n-type, the 
+concentration of Te in the channel area dramatically decreases, indicating that Te 
+atoms are electrochemically extracted from the channel.  
+Figure 2d, e show the atomic force microscopy (AFM) images of the source and drain 
+electrodes before and after the device has undergone polarity conversion. Notches at 
+the electrodes can be clearly seen, which can be understood as related to 
+electrochemical reduction during n-to-p conversion and incomplete Te replenishment 
+during p-to-n conversion.  In line with the deformation of electrodes, AFM images of 
+a local area in the channel for device before and after polarity conversion reveal that 
+the as-exfoliated flat MoS2 surface (figure 2f) turns into a pretty rough  surface as the 
+device has been converted to a p-type transistor (figure 2g). This results from Te 
+
+a
+b
+Monolaver MoS,
+.V.
+MoS2
+Pt
+Pt
+Te
+Sio,
+Pt/Te
+P++ Si
+c
+p
+10-5
+10~5
+n type (pristine)
+Clockwise
+10~6
+10~6
+3
+10-7
+10-7
+Anti-clockwise
+-ptype
+p type (30 days after switch)
+108
+10~8
+-10
+-5
+0
+5
+10
+-10
+-5
+0
+5
+10
+Vg (V)
+Vg (V) 
+inclusion. Sizes and heights of the protrusions in sight can be as great as 60 μm and 6 
+nm, respectively. These protrusions disappear after the device has been converted 
+back to an n-type transistor (figure 2h).  
+To identify the chemical structures of the channel area for the as-fabricated device 
+and the converted p-type device, Raman spectroscopic studies are carried out. As seen 
+in figure 2i, the as-fabricated MoS2 channel shows two characteristic peaks at ~ 380 
+cm-1 and ~ 405 cm-1, in consistence with those of the in-plane Mo-S vibrational mode 
+(E12g) and the out-of-plane S-S vibrational mode (A1g) in monolayer MoS2, 
+respectively26. No Te-related peak is observed. However, characteristic peaks 
+corresponding to basal plane vibration (A1) and bond-stretching vibration (E2) of Te 
+chains at 123 cm−1 and 140 cm−1, respectively27, emerge after the device has been 
+converted to a p-type transistor (figure 2j). These results further support the 
+conjectured polarity conversion mechanism (schematic diagram in supplementary 
+figure S5). 
+  
+Fig. 2 Materials characterizations of the mempolar transistors. AES elemental 
+mapping images of the channel area of a mempolar transistor a. before polarity switch, 
+b. after n-to-p polarity switching and c. after being switched back from p-type to n-
+type. Scale bars for a-c: 500 nm. AFM images of the mempolar transistor d. before 
+and e. after polarity switching (scale bar: 1.5 μm). Closed-up AFM image of the 
+channel area of the mempolar transistor f. before polarity switching, g. after n-to-p 
+polarity switching and h. after being switched back from p-type to n-type (scale bar: 
+120 nm). Raman spectra collected from three different areas labeled in the SEM 
+image (scale bar: 500 nm) of the channel area of the mempolar transistor i. before 
+polarity switching and j. after n-to-p polarity switching. 
+ 
+ 
+
+a
+4.7nm
+Position1
+Mos
+Position2
+Intensity(a.u.)
+Position3
+Te
+Mo
+S
+b
+100
+200
+300
+400
+0.8nm
+--
+Ramanshift(cm-1)
+8.8nm
+C
+Positionl
+Position
+Position3
+1.5nm
+p
+e
+90nm
+95nm
+6.5mm
+Te
+Position1
+Position2
+(a.u.)
+Position3
+Intensity
+MoS,
+100
+200
+300
+400
+3nm
+5 nm
+1.2nm
+Ramanshift (cm-1) 
+2.2. A TCAM cell made of two mempolar transistors 
+Transistors with reconfigurable polarities find applications in reconfigurable logic 
+circuits20, neuromorphic circuits20,28, secure circuits21, and so on. Here, we present a 
+new application of mempolar transistors, that is, making TCAM cells. TCAM is a 
+specialized type of computer memory used in certain very-high-speed searching 
+applications. It is considered as an opposite of the more widely known random access 
+memory (RAM). In a RAM, the user supplies a memory address and the RAM returns 
+the data word stored at that address. By contrast, a TCAM is designed such that the 
+user supplies a data word and the TCAM searches its entire memory based on pattern 
+matching to see if that data word is stored anywhere in it, just like associative memory 
+in the brain. The term “ternary” refers to the ability of the memory to store and query 
+data using three different inputs: 0, 1 and X. The “X” input, which is often referred to 
+as a “don’t care” state, enables TCAM to perform broader searches, as opposed to 
+binary CAM, which performs exact-match searches using only 0s and 1s. 
+TCAM is much faster than RAM in search-intensive applications. However, there are 
+cost disadvantages to TCAM. Unlike a RAM that has simple storage cells, 
+conventional TCAMs normally have more complex circuits with large physical sizes 
+and increased power dissipation. Specifically, a single TCAM cell based on standard 
+CMOS transistor technology requires 16 transistors. Though TCAMs based on the 
+emerging nonvolatile memory device (or simply, memristor) technology have simpler 
+2T-2R cells, each employing two transistors and two memristors29,30, the memristors 
+must be integrated via the back-end-of-line process and thus the electrical parasitics is 
+worsened. In this regard, our mempolar transistor as the product of the fusion of 
+transistive and memristive functions may further simplify the structure and fabrication 
+of the TCAM cell. 
+The schematic diagram of our proposed two mempolar transistor-based TCAM cell 
+and its workings are shown in figure 3a. In this schematic, ‘ML’ refers to the match 
+line which will get charged up to the supply voltage Vdd (1 V) before the search 
+operation. WL1 and WL2 are the two write lines through which polarity switch 
+voltages (±30 V) are applied to the respective mempolar transistors during the 
+memory encoding stage. They also serve as the paths for ML discharging when a 
+match is detected during the search stage, as will soon be introduced. For the storage 
+of the 1 and 0 states, complementary polarity configurations are written (encoded) 
+into the two mempolar transistors. For the storage of the X state, both mempolar 
+transistors are written to the same polarity configuration, either p-type or n-type. SL 
+and SL
+��� are two search lines through which the searching signal and its inverse are 
+applied to the gate terminals of the respective mempolar transistors. The searching 
+signal is presented as either +10 V or -10 V voltage bias, representing data 1 or 0, 
+respectively. With the above encoding and search schemes, if a match between the 
+searching signal and the stored memory state is detected, both mempolar transistors 
+are in their ON states, which discharges the ML to the ground. However, if a 
+mismatch is detected, both mempolar transistors are in their OFF states and thus the 
+ML stays high. In the case that the TCAM cell is in the X state, no matter what the 
+searching signal is, one mempolar transistor must be in the ON state while the other is 
+turned off, thereby discharging the ML.  
+A proof-of-concept TCAM cell made with two mempolar transistors are shown in 
+figure 3b. These two devices are fabricated from MoS2 layers exfoliated on two 
+different silicon dies which are then connected with copper wires by elargol. We 
+
+ 
+experimentally demonstrate the storage of data 0/1 in this TCAM cell by converting 
+the polarity of one mempolar transitor (mempolarT1)/(mempolarT2) to p-type while 
+keeping the other (mempolarT2)/(mempolarT1) unchanged. Alternating 0 and 1 
+search data are then fed into the cell through a pair of SL and SL
+���. For each memory 
+state, the evolution of the resistance between the ML and the ground during the search 
+process is shown. Large resistance indicates that both mempolar transistors are in the 
+OFF states and therefore a mismatch is detected; on the other hand, small resistance 
+indicates that one of the two devices is turned on, corresponding to a match. The 
+resistance contrast retains as high as 102 in one thousand searches (figure 3c, d). 
+Unlike the recently reported ferroelectric TCAM cell31 in which both writing and 
+searching signal transductions share the same pathway (i.e., SL/SL
+���), our cell uses two 
+signal transduction pathways for these two operations, i.e., the SL/SL
+��� for searching 
+and the WL1/WL2 for writing. The decoupling of the search and write operations 
+prevents the stored state from being disturbed by the searching signals and therefore 
+may enable long retention time and enhance reliability.  
+ 
+Fig. 3 Two-mempolar transistors-based TCAM cell. a. Circuit diagram of the two-
+mempolar transistors-based TCAM cell during the search stage and its operating 
+mode. b. A proof-of-concept two-mempolar transistors-based TCAM cell (scale bar: 2 
+μm). Evolutions of the measured resistance between the ML and the ground with the 
+
+a
+PP△!
+b
+ML
+WL
+7S2
+(grounded)
+Mempolar
+ Mempolar
+Mempolanni
+T1
+T2
+ML(Vaa)
+MempolarT2
+WL1
+WL2
+ML
+SL
+TS
+Polarity
+Matched searching
+signal
+MempolarT1
+MempolarT2
+Stored
+Mempolar
+Mempolar
+7S
+SL
+value
+T1
+T2d
+-IUV
+1
+n
++10V
+-10V
+X
+n
+n
++10V
+-10V
+-10V
++10V
+C
+d
+Resistance between
+10M
+Resistance between
+10M
+Mismatch
+Mismatch
+Stored value 'o'
+1M
+1M
+Stored value'1'
+VML= 1V
+VML= 1V
+Match
+Match
+100k
+100k
+0
+200
+400
+600
+800
+1000
+0
+200
+400
+600
+800
+1000
+Read cycle
+Read cycle 
+searching signals alternating between matched and mismatched signals when c. ‘0’ 
+and d. ‘1’ are stored in the TCAM cell.  
+    
+2.3. Mempolar transistor-inspired method for regularizing neural networks 
+As mentioned before, the mempolar transistor with n (p) polarity can show certain 
+degree of clockwise (counter-clockwise) hysteresis in its transfer curve (figure 1c and 
+1d), which can be attributed to trapping/detrapping of carriers in/from the gate oxide 
+during the course of Vg sweeping. As seen from the transfer curves in figure 4a 
+(figure 4b), the channel conductance (under Vg=0 V) of the mempolar transistor with 
+n (p) polarity keeps decreasing (increasing) with successive forward and backward 
+voltage sweepings between 0 V to +10 V. Pulse measurements are also carried out, 
+revealing similar trends of conductance changes in devices with n and p polarities, 
+respectively (supplementary figure S6). These phenomena have been widely exploited 
+by neuromorphic engineers for emulating the long-term plasticity of synapses in the 
+training phase of neural networks16. 
+As shown in figure 4c, 10 cycles of forward and backward Vg sweeps between 0 V to 
++10 V applied to an as-fabricated n-type device (1→2) lead to dramatic decrease in its 
+baseline Id (Id under Vg=0) from about 0.7 μA to 100 nA. After the induction of this 
+long-term synaptic depression (LTD), a positive polarity switching Vds is applied 
+(2→3). The resulting p-type device has large baseline Id about 10 μA. Recall that 
+polarity conversion is elicited by applying a voltage across the source and drain 
+terminals. Therefore, this operation, in principle, should not influence the charge 
+trapping state of the gate oxide which only depends on the history of gate inputs. To 
+verify that the charge trapping state of the gate oxide is not influenced by polarity 
+conversion, we apply a negative polarity switching Vds to convert the device back to 
+n-type (3→4). It is seen that the transfer curve of the present n-type device overlap 
+that of the long-term depressed device before polarity conversion. Likewise, long-
+term synaptic potentiation (LTP) is induced in the p-type device converted from the 
+as-fabricated n-type device, followed by p-to-n conversion and then n-to-p conversion, 
+as shown in figure 4d. Results from these measurements consistently indicate that 
+polarity conversion does not influence the charge trapping state of the gate oxide but 
+can give rise to reverse effect to what long-term plasticity induces before polarity 
+conversion. Specifically, the strength of the reverse effect is monotonically dependent 
+on the pre-accumulated long-term plasticity. 
+Drawing inspiration from this device behavior, we propose a new algorithm 
+‘FlipWeight’ and demonstrate its application by simulations in the context of 
+regularizing neural networks. Traditionally, training a deep neural network (DNN) 
+that can generalize well to new data is a challenging task. This is because a typical 
+DNN has so many parameters (over-parameterized) and limited available data, 
+exhibiting a significant tendency toward overfitting on the training dataset. 
+Approaches to reduce error in generalizing to out-of-sample data points are referred to 
+as regularization methods32. Generally speaking, the rationale behind regularization is 
+constraining the complexity of the DNN model by either reducing the number of 
+synaptic connections or reducing values of synaptic weights. Dropout33,34 and its 
+variant DropConnect35 are two of the most widely used regularization methods. As 
+their names suggest, these two methods randomly drop a number of neurons or 
+connections for each batch during training. With this, not all but only a fraction of 
+weights are updated. After the training on a batch of samples, the omitted neurons 
+(and connections attached to it) or omitted connections in the last batch of training are 
+
+ 
+recovered. A new set of neurons or connections are randomly selected and omitted in 
+the next training batch. Unlike these two methods, our method adapts the idea of 
+reducing values of synaptic weights to improve generalization performance.    
+ 
+ 
+Fig. 4 Gradual channel conductance changes in mempolar transistors and 
+flipping between the high and low conductance states via polarity switching. a. 
+Gradual decrease in the baseline Id (Id under Vg=0) of an n-type mempolar transistor 
+under successive positive Vg sweep. b. Gradual increase in the baseline Id  of a p-type 
+mempolar transistor under successive positive Vg sweep. Flipping back and forth 
+between the high and low conductance states c. in an n-type mempolar transistor 
+before which the channel conductance has been gradually tuned to be relatively low 
+and d. in a p-type mempolar transistor before which the channel conductance has been 
+gradually tuned to be relatively high.  
+Large weights in a DNN are a sign of a complex network that has a tendency to 
+overfit the training data36. Therefore, we consider randomly selecting large weights 
+that are over a threshold and scaling them down according to a certain rule before 
+each training batch starts. A weight scaling rule that naturally matches the properties 
+of our mempolar transistor is inverse scaling with respect to the initial weight value. 
+After this pre-treatment, the standard backpropagation (BP) method is used to 
+calculate the gradients for all the weights in the DNN. Except those pre-treated 
+weights, all the other weights are directly updated using the calculated gradients. The 
+pre-treated weights are first recovered to their original values (also supported by 
+device functions) and then updated using the corresponding gradients calculated from 
+the pre-treated DNN. In the next training batch, a new set of large weights are 
+randomly selected and scaled down, followed by the same procedure of BP 
+calculations and weight update.  
+
+a
+b
+3x106
+4x10°
+Sweep
+Sweep
+1
+1
+3x10-6
+2
+2x10~6
+-3
+2x10-6
+4
+5
+5
+1x106
+1x10~6
+0
+0
+0
+2
+4
+6
+8
+10
+0
+2
+4
+6
+8
+10
+V (V)
+Vg (V)
+c
+n-type
+1-2LTDtraining
+d
+2
+p-type1-2LTPtraining
+3
+1x10~5
+2-3 n-p switching
+1x10
+2--3 p-n switching
+3-4 p-n switching
+3—-4 n-p switching
+10 times
+sweep
+1x106
+1x106
+La
+10timessweep
+3
+1x10-7
+1x10
+2
+1x10*8
+-2
+0
+2
+4
+6
+8
+10
+12
+-2
+0
+2
+4
+6
+8
+10
+12
+Vg(V)
+Vg(V) 
+We benchmark this FlipWeight method against the Dropout and DropConnect 
+methods on a five-layer convolutional neural network (CNN), as shown in figure 5a. 
+We point out that the FlipWeight method is only used in the fully-connected (FC) 
+layers, like Dropout and DropConnect. As presented in figure 5b and 5c, CNN trained 
+with either Dropout or DropConnect or FlipWeight technique can achieve higher 
+validation accuracies and lower loss values compared to the baseline that is 
+implemented without any regularization technique. The gaps between the training 
+curves and the validation curves are also reduced dramatically with the use of these 
+regularization 
+techniques, 
+verifying 
+their 
+effectiveness 
+in 
+improving 
+the 
+generalization performance. Notably, although the final validation accuracy and loss 
+value of CNN implemented with FlipWeight method are similar to those of the CNNs 
+implemented with Dropout and DropConnect, it is evident that our FlipWeight 
+method results in faster convergence than do the other two methods. This can be 
+understood from the fact that all the weights are updated in each training iteration in 
+CNN regularized by our FlipWeight method, while connections omitted in the 
+Dropout or DropConnect approach are simply not involved in weight updating. 
+We also compare the mean, standard deviation and sum of absolute value of weights 
+between CNN models regularized by different methods, as shown in figure 5d-f. It is 
+seen that our FlipWeight method leads to overall excitatory connections (positive 
+mean) while the connections in baseline CNN without regularization and CNNs 
+regularized by Dropout and DropConnect are overall inhibitory (negative mean). Our 
+FlipWeight method also leads to the largest standard deviation of weights and sum of 
+absolute values of weights. The weight distributions are also visualized. The truncated 
+Gaussian distribution for weight initialization is shown in figure 5g. After training, the 
+baseline model without regularization shows a wide distribution of weights among 
+which many have large values (figure 5g). For Dropout and DropConnect, the weight 
+distributions are tighter and peak near zero (figure 5h, i). In stark contrast, the weight 
+distribution in CNN regularized by FlipWeight method is multimodal and even tighter, 
+whose several peaks are close to zero (figure 5j). These results demonstrate the 
+uniqueness of our FlipWeight method that can achieve state-of-the-art performance.   
+
+ 
+ 
+ 
+
+a
+Layerl: Convolution
+Layer2: Convolution
+Layer3: FC
+Layer4: FC Layer5: FC
+Kernel: 4x4
+Kernel:4x4
+12 channels
+Output
+64 units
+12 channels
+(8x8x12)
+Input
+256 units
+(16x16x12)
+(32x32x3)
+b
+c
+100
+Train:Baseline
+Train:Dropout
+3
+Train:Dropconnect
+Train:Flipw
+Accuracy
+Train:Baseline..
+OSS
+Val:Baseline
+60
+Val:Dropout
+TrainDropout
+Val:Dropconnect
+Train:Dropconnect
+Val:Flipw
+40
+Val:Baseline
+1
+Val:Dropout
+20
+Val:Dropconnect
+Val:Flipw
+0
+0
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+100
+Enoch
+Epoch1e-2
+le-1
+1e4
+4
+6
+ of absolute values
+6
+2
+Standard deviation
+5
+5
+0
+4
+4
+3
+Sum
+2
+OV
+g
+lel
+h
+1e2
+Weight Histogram: Initial , Baseline
+Weight Histogram: Dropout
+0.8
+6
+5
+0.6
+4
+0.4
+3
+2
+0.2
+0
+-0.4
+-0.2
+0.0
+0.2
+0.4
+-0.4
+-0.2
+0.0
+0.2
+0.41e2
+le3
+Weight Histogram: Dropconnect
+Weight Histogram: Flip W
+0.8
+0.8
+0.6
+0.6
+0.4
+0.4
+0.2
+0.2
+0.0
+0.0
+-0.4
+-0.2
+0.0
+0.2
+0.4
+-0.4
+-0.2
+0.0
+0.2
+0.4 
+Fig. 5 Performance and characteristics of the CNNs regularized by different 
+methods. a. Schematic illustration of the adopted network structure for image 
+recognition. Comparison of the b. convergence curves and c. loss curves obtained 
+from unregularized model and models regularized by different methods. Comparison 
+of the d. mean, e. stand deviation and f. sums of absolute values of synaptic weights 
+among different models after training, where “Initial” denotes the untrained model, 
+“Base” denotes unregularized baseline model, “Dout”, “DCon” and “FlipW” denote 
+models regularized by Dropout, DropConnect and FlipW methods, respectively. 
+Comparison of weight distribution among g. the untrained model and the 
+unregularized baseline model, and models regularized by h. Dropout, i. DropConnect, 
+and j. FlipWeight methods, respectively. 图 j FlipWeight 
+ 
+3. Conclusion 
+In summary, we introduce the emerging ion-based memristive functions into the 
+purely electronic transistors and demonstrate a new type of transistor named 
+‘mempolar transistor’ whose polarity can be run-time switched between n-type and p-
+type in a non-volatile manner. This novel transistor function is achieved by the use of 
+the emerging semiconducting Te as the electrochemically active source/drain contact 
+material, in combination with monolayer MoS2 channel, which results in a gated 
+lateral Te/MoS2/Te memristor, or from a different perspective, a transistor whose 
+channel can be converted reversibly between n-type MoS2 and p-type Te. Mempolar 
+transistors address a key drawback of the previously showcased transistors with 
+reconfigurable polarities, that is, polarity reconfiguration was volatile and realized via 
+electrostatic control from additional gate terminals. When used in reconfigurable logic 
+circuits or secure circuits, mempolar transistors will potentially mitigate the problems 
+of excessive energy consumption, extensive hardware overhead and massive 
+interconnections. In addition to these manifest advantage, we design and demonstrate 
+experimentally a TCAM made of only two mempolar transistors, which used to 
+require a dozen normal transistors. We also develop and demonstrate by simulations a 
+device-inspired regularization method for training ANNs, which achieves state-of-the-
+art performance. This work broadens the functionality of transistors and provides the 
+implication that rich technological opportunities are available for the fusion between 
+electronics and ionics. 
+ 
+4. Methods 
+Device fabrication: The MoS2 flakes were mechanically exfoliated from bulk crystals 
+(purchased from Six Carbon Technology, Inc.) onto the SiO2 (300 nm)/Si substrate. 
+The source and drain electrodes made from 50 nm Te and 20 nm Pt protective layers 
+were deposited by magnetron sputtering after the standard electron-beam lithography 
+patterning.  
+Electrical measurements: Cyclic quasi-DC voltage sweep measurements were 
+performed by the Keysight B1500A semiconductor analysis system. The Keysight 
+B1530A waveform generator/fast measurement unit is used to perform the pulse 
+measurements.  
+
+ 
+Materials characterizations: The AES analyses were performed by a scanning auger 
+microprobe (PHI710, ULVAC). The morphology characterizations of the devices 
+were performed by an AFM (DIMENSION ICON, BRUKER) in ScanAsyst mode. 
+The Raman spectra were obtained on a single-gating micro-Raman spectrometer 
+(Horiba-JY T64000) excited with 532 nm laser. 
+Neural network model parameterization, training and tests: We evaluated the 
+performance of various regularization methods in CIFAR-10 pattern classification 
+tasks37. The adopted network structure was [Input-12C4-12C4-768FC-256FC-64FC-
+10] (C: convolution, FC: fully-connected layer). We modelled the flipping of a 
+connection weight from its current large value to a small value according to the 
+physical process of polarity switching-induced channel conductance change in the 
+mempolar transistor and by simplifying this process as obtaining the reciprocal of the 
+initial large value (inverse scaling of weight). All CNN models were trained using the 
+adaptive moment estimation (Adam) optimizer38 for 160 epochs with batch size of 
+100 and an initial learning rate of 0.0005. The retaining proportions for models 
+regularized by Dropout or DropConnect were set to 0.5. For the FlipWeight method, 
+connections in the FC layers with weights larger than a threshold value are inversely 
+scaled to their reciprocals with the effect of a scaling factor before each training epoch 
+starts. 0.06 was found to be a suitable threshold value. The hyper-parameters were 
+kept the same for all tested models in this work. The simulations were performed by 
+Tensorflow1.15.0 on 4 RTX 2080Ti GPUs.   
+Supporting Information  
+Supporting Information is available from the author. 
+ 
+Acknowledgments  
+Y. Y., L. X. and M. X. contributed equally to this work. H. L. conceived the idea. Y.Y. 
+and L.X. performed the device fabrication and measurements under the supervision of 
+H.L. and W.D.. M.X. conducted the neural network simulations under the supervision 
+of J.P.. H. Liu and D.L. assisted the device fabrication. Y.Y., M.X. and H.L. wrote 
+this manuscript. This research was supported by National Natural Science Foundation 
+(grant nos. 61974082, 61704096, 61836004), National Key R&D Program of China 
+(2021ZD0200300, 2018YFE0200200), Youth Elite Scientist Sponsorship (YESS) 
+Program of China Association for Science and Technology (CAST) (no. 
+2019QNRC001), Tsinghua-IDG/McGovern Brain-X program, Beijing science and 
+technology program (grant nos. Z181100001518006 and Z191100007519009), 
+Suzhou-Tsinghua innovation leading program 2016SZ0102, CETC Haikang Group-
+Brain Inspired Computing Joint Research Center.  
+ 
+Competing interests 
+The authors declare no competing interests. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+References 
+[1] J. Dean, The deep learning revolution and its implications for computer 
+architecture and chip design. In 2020 IEEE International Solid-State Circuits 
+Conference-(ISSCC). 2020, pp. 8-12. 
+[2] C. E. Leiserson, N. C. Thompson, J. S. Emer, B. C. Kuszmaul, B. W. Lampson, D. 
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+

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+1
+Cost-Effective Two-Stage Network Slicing for
+Edge-Cloud Orchestrated Vehicular Networks
+Wen Wu‡, Kaige Qu⋆, Peng Yang∗, Ning Zhang†, Xuemin (Sherman) Shen⋆, and Weihua Zhuang⋆
+Frontier Research Center, Peng Cheng Laboratory, Shenzhen, China‡
+Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Canada⋆
+School of Electronic Information and Communications, Huazhong University of Science and Technology, China∗
+Department of Electrical and Computer Engineering, University of Windsor, Windsor, Canada†
+Email: wuw02@pcl.ac.cn‡, {k2qu, sshen, wzhuang}@uwaterloo.ca⋆,
+yangpeng@hust.edu.cn∗, and ning.zhang@uwindsor.ca†
+Abstract—In this paper, we study a network slicing problem
+for edge-cloud orchestrated vehicular networks, in which the edge
+and cloud servers are orchestrated to process computation tasks
+for reducing network slicing cost while satisfying the quality
+of service requirements. We propose a two-stage network slicing
+framework, which consists of 1) network planning stage in a large
+timescale to perform slice deployment, edge resource provision-
+ing, and cloud resource provisioning, and 2) network operation
+stage in a small timescale to perform resource allocation and
+task dispatching. Particularly, we formulate the network slicing
+problem as a two-timescale stochastic optimization problem to
+minimize the network slicing cost. Since the problem is NP-hard
+due to coupled network planning and network operation stages,
+we develop a Two timescAle netWork Slicing (TAWS) algorithm
+by collaboratively integrating reinforcement learning (RL) and
+optimization methods, which can jointly make network planning
+and operation decisions. Specifically, by leveraging the timescale
+separation property of decisions, we decouple the problem into
+a large-timescale network planning subproblem and a small-
+timescale network operation subproblem. The former is solved
+by an RL method, and the latter is solved by an optimization
+method. Simulation results based on real-world vehicle traffic
+traces show that the TAWS can effectively reduce the network
+slicing cost as compared to the benchmark scheme.
+I. INTRODUCTION
+To make autonomous driving from a mere vision to reality,
+future vehicular networks are required to support various
+Internet of vehicles (IoV) services, such as object detection,
+in-vehicle infotainment, and safety message dissemination [1].
+Those IoV services have diversified quality of service (QoS)
+requirements in terms of delay, throughput, reliability, etc.
+Emerging network slicing is deemed as a de-facto solution to
+support diversified IoV services in vehicular networks. Its ba-
+sic idea is to construct multiple isolated logical sub-networks
+(i.e., slices) for different services on top of the physical
+network, thereby facilitating flexible, agile, and cost-effective
+service provisioning. Starting from the fifth-generation (5G)
+era, standardization efforts from the 3rd generation partnership
+project (3GPP) body, e.g., Releases 15-17 [2]–[4], and proof-
+of-concept systems, e.g., Orion [5], have fuelled the maturity
+of network slicing. In the coming 6G era, advanced network
+slicing techniques are expected to play an increasingly impor-
+tant role [6]–[8].
+In the literature, significant research efforts have been
+devoted to network slicing. Ye et al. investigated a radio
+spectrum resource slicing problem, in which radio spectrum
+is sliced between macro base stations (MBSs) and small
+BSs (SBSs) [9]. To achieve efficient resource allocation, a
+deep learning-based algorithm was proposed to jointly allo-
+cate radio spectrum and transmit power in a slicing-based
+network [10]. The previous work in
+[11] considered the
+resource provisioning problem and proposed a constrained
+learning algorithm to solve it. However, this work differs from
+the existing works in several important aspects. Firstly, the
+existing works focus on utilizing resources on the network
+edge, low-cost cloud resources are yet to be considered. As
+a remedy, a certain amount of computation tasks processed
+at the congested BSs can be dispatched to the remote cloud,
+i.e., task dispatching, such that system cost can be reduced.
+Secondly, network slicing includes two stages: 1) network
+planning stage to provision network resources for slices in
+the large timescale, and 2) network operation stage to allocate
+the reserved resources to end users in the small timescale [3],
+[12]. The existing works mainly decouple network slicing
+into two independent stages, while the interaction between
+them is seldom considered. Hence, designing a cost-effective
+network slicing scheme should take cloud resources and such
+interaction relationship into consideration.
+Optimizing network slicing performance in dynamic vehic-
+ular networks faces the following challenges. Firstly, network
+planning and operation decisions are nested. Large-timescale
+network planning decisions (e.g., resource reservation), will
+condition small-timescale network operation decisions (e.g.,
+resource allocation). Meanwhile, the performance achieved
+in the network operation stage will also affect the decision-
+making in the network planning stage, which is difficult to
+be solved by conventional optimization methods. Secondly,
+since vehicle traffic density varies temporal-spatially, net-
+work planning decisions need to be made to optimize long-
+term performance in the slice lifecycle while accommodating
+such network dynamics. Deep reinforcement learning (RL)
+is considered as a plausible solution for long-term stochastic
+optimization.
+In this paper, we first propose a cost-effective two-stage
+arXiv:2301.03358v1  [cs.NI]  31 Dec 2022
+
+2
+network slicing framework for edge-cloud orchestrated vehic-
+ular networks, by considering nested network planning and
+operation stages and effectively leveraging cloud resources.
+We then apply a network slicing cost model that accounts
+for slice deployment, resource provision, slice configuration
+adjustment, and QoS satisfaction. Based on the model, we
+formulate the network slicing problem as a two-timescale
+stochastic optimization problem to minimize the network
+slicing cost. Second, to solve the problem, we develop a
+learning-based algorithm, named Two timescAle netWork
+Slicing (TAWS). The TAWS exploits the timescale separation
+structure of decision variables and decouples the problem into
+two subproblems in different timescales. Regarding the large-
+timescale network planning subproblem, an RL algorithm is
+designed to minimize network slicing cost via optimizing slice
+deployment, edge resource provisioning, and cloud resource
+provisioning. Regarding the small-timescale network operation
+subproblem, an optimization algorithm is designed to mini-
+mize average service delay via optimizing resource allocation
+and task dispatching. In addition, the achieved service delay in
+the network operation stage is incorporated into the reward of
+the RL-based network planning algorithm, thereby capturing
+the interaction between two stages and enabling closed-loop
+network control. Simulation results on real-world vehicle
+traces demonstrate that the proposed algorithm outperforms
+the benchmark scheme in terms of reducing network slicing
+cost.
+The remainder of this paper is organized as follows. The
+system model and problem formulation are presented in Sec-
+tions II and III, respectively. Section IV describes the proposed
+TAWS algorithm. Simulation results are given in Section V,
+along with the conclusion in Section VI.
+II. SYSTEM MODEL
+A. Network Model
+As shown in Fig. 1, the network slicing framework consists
+of several components.
+Physical network: A two-tier cellular network is deployed
+for serving on-road vehicles. The set of BSs is denoted by
+M, including the set of MBSs denoted by Mm and the set
+of SBSs denoted by Ms, i.e., M = Mm ∪Ms. Each BS has
+a circular coverage and is equipped with an edge server. In
+the considered scenario, vehicles driving on the road generate
+computation tasks over time, which are offloaded to roadside
+BSs. Those tasks can be either processed at edge servers or
+dispatched to the remote cloud server via backbone networks.
+Once completed, computation results are sent back to vehicles.
+Network slice: Multiple network slices are constructed
+on top of the physical vehicular network. We consider K
+delay-sensitive services with differentiated delay requirements,
+denoted by set K. Let θk, ∀k ∈ K denote the tolerable delay
+of service k. For example, the tolerable delay of objective
+detection service is 100 ms [13], whereas the tolerable delay
+of in-vehicle infotainment can be up to several hundreds of
+milliseconds.
+Network controller: A hierarchical network control archi-
+tecture is adopted, including an upper-layer software defined
+Slice 1
+Slice N
+Network 
+Slices
+...
+Physical
+Network 
+SBS
+MBS
+Backbone
+Transmission
+Cloud 
+Server
+Edge Server
+Switch
+Vehicle
+Computation 
+Task
+Control
+ Link
+SDN 
+Controller
+Fig. 1.
+Network slicing for edge-cloud orchestrated vehicular networks.
+networking (SDN) controller that connects to all BSs, and
+lower-layer local network controllers located at BSs. Those
+controllers are in charge of network information collection and
+making network slicing decisions.
+B. Two-Stage Network Slicing Framework
+We present a two-stage network slicing framework for the
+considered network. Firstly, a network planning stage operates
+in the large timescale (referred to as planning windows) to
+reserve resources at specific network nodes for the constructed
+slices. The duration of each planning window is denoted by
+Tp. At each planning window, the SDN controller collects the
+average vehicle traffic density information in the considered
+area, based on which planning decisions are made. Secondly,
+the network operation stage operates in the small timescale
+(referred to as operation slots) to dynamically allocate the
+reserved resources to vehicles according to real-time vehicles’
+service requests and network conditions. The duration of each
+operation slot is denoted by To. A planning window includes
+multiple operation slots, i.e., Tp/To ∈ Z+. At each operation
+slot, the local network controller at each BS collects real-time
+service requests and channel conditions of its associated vehi-
+cles, based on which operation decisions are made. Decision
+structures in two stages are detailed respectively as follows.
+1) Network Planning Decision Structure: The planning
+window is indexed by w ∈ W = {1, 2, ..., W}, and plan-
+ning decisions in planning window w include the following
+components.
+Slice deployment decision, denoted by ow ∈ RMs×1. Each
+element is a binary variable, i.e.,
+ow
+m ∈ {0, 1}, m ∈ Ms.
+(1)
+If SBS m is activated for slice deployment, we have ow
+m = 1;
+otherwise, ow
+m = 0. When service demands are low, deploying
+slices at a selective subset of BSs can reduce network slicing
+cost as compared to deploying slices at all BSs while guaran-
+teeing slices’ service level agreements (SLAs). This is because
+running network slicing requires resource virtualization, which
+incurs network operating costs. For service continuity consid-
+eration, we assume that MBSs that cover the entire area are
+always activated. Note that only when a BS is activated for
+slice deployment, edge resources at the BS can be provisioned.
+Edge resource provisioning decision, including radio spec-
+trum and computing resource provisioning at all BSs for
+all slices, denoted by Bw ∈ RK×M and Cw ∈ RK×M,
+
+3
+respectively. The corresponding elements
+{bw
+k,m, cw
+k,m} ∈ Z+, ∀k ∈ K, m ∈ M,
+(2)
+represent the number of subcarriers and edge virtual machine
+(VM) instances provisioned for slice k at BS m, where Z+
+denotes the set of positive integers.1 The bandwidth of a
+subcarrier is denoted by β, and the computing capability of
+an edge VM is denoted by Fe. Due to the limitation of edge
+resources, the following capacity constraints are imposed:
+ow
+m
+�
+k∈K
+bw
+k,m ≤ Bm, ow
+m
+�
+k∈K
+cw
+k,m ≤ Cm, ∀m ∈ M,
+(3)
+where Bm and Cm represent the total numbers of subcarriers
+and VM instances at BS m, respectively.
+Cloud resource provisioning decision, denoted by hw ∈
+RK×1. Each element
+hw
+k ∈ Z+, ∀k ∈ K
+(4)
+denotes the number of cloud VM instances reserved for slice k.
+The computing capability of a cloud VM is denoted by Fc.
+2) Network Operation Decision Structure: Let t ∈ T =
+{1, 2, ..., T} denote the index of operation slots within a
+planning window. At operation slot t, the following decisions
+are determined for each slice k.
+Radio spectrum allocation decision, denoted by yt
+k
+∈
+RN t×1. The reserved radio spectrum at each BS is allocated to
+active vehicles within BS’s coverage for task offloading. Due
+to vehicle mobility, the number of vehicles varies across time.
+Let N t denote the set of active vehicles in operation slot t,
+and N t = |N t|. For simplicity, each vehicle associates to the
+nearest BS. Let N t
+m denote the set of active vehicles associated
+to BS m at operation slot t, and yt
+k,n ∈ R+ represents the
+fraction of radio spectrum allocated to vehicle n. The total
+amount of the allocated bandwidth should not exceed the
+reserved number of subcarriers at the corresponding BS, i.e.,
+�
+n∈N tm
+yt
+k,n ≤ bw
+k,m, ∀m ∈ Mw.
+(5)
+Here, Mw denotes the set of the activated BSs in window w.
+Task dispatching decision, denoted by xt
+k
+∈
+ZM w×1.
+The BS receives computation tasks uploaded from its as-
+sociated vehicles. The task arrivals of vehicles follow an
+arbitrary stochastic process. Let at
+k,n denote the number of
+the generated tasks of vehicle n in operation slot t, and
+the aggregated computation workload at BS m is given by
+At
+k,m = �
+n∈N tm at
+k,n. Processing all tasks at BSs with limited
+computing resources may incur prohibitive high queuing delay,
+and hence a portion of computation tasks can be dispatched to
+the remote cloud via backbone networks. Let xt
+k,m represent
+the number of dispatched tasks from BS m in slice k, i.e.,
+xt
+k,m ∈ {0, 1, 2, ..., At
+k,m}, ∀m ∈ Mw.
+(6)
+The operation decisions impact service delay at each oper-
+ation slot, which is analyzed in the following subsection.
+1Memory resource is also allocated to the VM instance to enable task
+processing, which is matched to its allocated computing resource.
+C. Service Delay Model
+The service delay includes task offloading delay and task
+processing delay at either the edge or the cloud. For service
+k, the following delay analysis is adopted.
+Task offloading delay: The transmission rate of one sub-
+carrier from vehicle n to its associated BS is given by
+Rt
+n = β log2
+�
+1 +
+Pvgt
+n
+βNo+βI
+�
+, where Pv, gt
+n, No, and I repre-
+sent vehicle’s transmission power, instantaneous channel gain,
+noise spectrum density, and interference spectrum density,
+respectively. With the allocated radio spectrum yt
+k,nbw
+k,m, the
+task offloading delay of vehicle n is given by dt
+k,n,o =
+ξk
+yt
+k,nbw
+k,mRtn , ∀n ∈ N t
+m, where ξk (in bits) denotes the task
+data size of service k.
+Edge processing delay: Given the task dispatching de-
+cision, At
+k,m − xt
+k,m tasks are processed at BS m. Let
+Qt
+k,m (in bits) denote the amount of the backlogged tasks
+at BS m. Taking task computation delay and queuing delay
+into account, edge processing delay at BS m is given by
+dt
+k,m,e = (Qt
+k,m+(At
+k,m−xt
+k,m+1)ξk/2)ηk
+cw
+k,mFe
+, ∀m ∈ Mw, where ηk
+(in cycles/bit) denotes task computation intensity of service k,
+and cw
+k,mFe is the computing capability of BS m with cw
+k,m
+provisioned edge VMs. The task backlog at BS m is updated
+by Qt+1
+k,m =
+�
+Qt
+k,m + (At
+k,m − xt
+k,m)ξk − cw
+k,mFeTo/ηk
+�+
+,
+where [x]+ = max {x, 0}.
+Cloud processing delay: For BS m, xt
+k,m tasks are dis-
+patched via backbone networks and then processed at the
+cloud, whose delay is given by dt
+k,m,c = dt
+r + ξkηk
+hw
+k Fc , where
+dt
+r denotes the round trip time in the backbone network. The
+second term represents the task processing delay in the cloud.
+Note that the queuing delay at the cloud is negligible as multi-
+core cloud servers can parallelly process different tasks.
+As such, the average delay for each computation task is
+given by
+Dt
+k(xt
+k, yt
+k) =
+�
+m∈Mw
+�
+n∈N tm
+dt
+k,n,o
+�
+m∈Mw N tm
++
+�
+m∈Mw
+dt
+k,m,e
+�
+At
+k,m − xt
+k,m
+�
++ dt
+k,m,cxt
+k,m
+�
+m∈Mw At
+k,m
+.
+(7)
+In the above equation, the first term represents the average task
+offloading delay for each task, and the second term represents
+the average task processing delay taking workload distribution
+between the edge and cloud servers into account. By averaging
+all operation slots, the average service delay is given by ¯Dw
+k =
+1
+T
+�T
+t=1 Dt
+k(xt
+k, yt
+k).
+D. Network Slicing Cost Model
+The following network slicing cost model is adopted for
+slicing performance evaluation, including several components.
+Slice deployment cost: The cost is because running network
+slices at BSs incurs the overhead of resource virtualization,
+which is given by Φw
+d = qd
+�
+m∈Ms ow
+m. Here, qd denotes the
+unit cost of deploying network slices at a BS.
+Resource provisioning cost: The cost component character-
+izes resource provisioning cost of radio spectrum resources,
+
+4
+edge computing resources, and cloud computing resources.
+For simplicity, we assume the unit costs of a subcarrier, an
+edge VM instance, and a cloud VM instance are the same,
+denoted by qr > 0. The resource provisioning cost is given
+by Φw
+p = qr
+�
+k∈K
+�
+hw
+k + �
+m∈M
+�
+ow
+mbw
+k,m + ow
+mcw
+k,m
+��
+.
+Slice adjustment cost: The cost component characterizes the
+difference between two subsequent planning decisions, i.e., the
+cost for adjusting the amount of the reserved spectrum and
+computing resources. For computing resources, VM instances
+can be resized via advanced virtualization techniques in prac-
+tical systems, e.g., Kubernetes [14]. Here, qs represents the
+unit price of adjusting a unit of reserved network resources.
+Hence, the slice adjustment cost is given by
+Φw
+s =qs1
+�
+ow−1
+k,m = 1 ∧ ow
+k,m = 1
+�
+·
+�
+k∈K
+��
+hw
+k − hw−1
+k
+�+
++
+�
+m∈M
+��
+bw
+k,m − bw−1
+k,m
+�+
++
+�
+cw
+k,m − cw−1
+k,m
+�+��
+,
+(8)
+where
+1 {·}
+is
+an
+indicator
+function
+and
+1
+�
+ow−1
+k,m = 1 ∧ ow
+k,m = 1
+�
+indicates that slice k is deployed
+in the previous and current planning windows.
+SLA revenue: The cost component characterizes the benefit
+caused by QoS satisfaction, i.e., the achieved service delay of
+each slice. The piece-wise SLA revenue function is denoted
+by
+Ωk (D) =
+�
+�
+�
+�
+�
+�
+�
+qb,
+if D < θ
+′
+k,
+qb
+�
+D−θ
+′
+k
+θk−θ′
+k
+�
+,
+if θ
+′
+k ≤ D ≤ θk,
+−qp,
+if D > θk.
+(9)
+Here, qb > 0 is the highest unit revenue once a slice’s SLA
+is satisfied, and qp > 0 is the unit penalty once the slice’s
+SLA is violated. Obviously, qp > qb for discouraging slice’s
+SLA violation. In addition, θ
+′
+k < θk represents the threshold
+achieving the highest revenue. For simplicity, we set θ
+′
+k =
+θk/2 in the simulation. The overall SLA revenue of all slices
+is given by Φw
+q = �
+k∈K Ωk
+� ¯Dw
+k
+�
+.
+Taking all cost components into account, the overall network
+slicing cost in the entire slice lifecycle (i.e., all planning win-
+dows) is given by Φ (ow, Bw, Cw, hw, {xt
+k, yt
+k}t∈T ,k∈K) =
+�
+w∈W
+�
+Φw
+d + Φw
+p + Φw
+s − Φw
+q
+�
+, which is adopted to evalu-
+ate network slicing performance.
+III. PROBLEM FORMULATION
+The network slicing problem aims to minimize the network
+slicing cost via determining network planning decisions at
+each planning window and network operation decisions at each
+operation slot for each slice, which is formulated as:
+P0 :
+min
+{ow,Bw,Cw,hw}w∈W
+{xt
+k,yt
+k}t∈T ,k∈K,w∈W
+�
+w∈W
+Φ (ow, Bw, Cw, hw)
+s.t. (1), (2), (3), (4), (5), and (6). (10a)
+In Problem P0, the network planning and operation decision
+making are coupled in two timescales, which should be jointly
+optimized. To address the challenge, we first decouple the
+problem into a large-timescale network planning subproblem
+and multiple small-timescale network operation subproblems.
+Subproblem 1: Network planning subproblem is to mini-
+mize the network slicing cost across all the planning windows,
+which is formulated as:
+P1 :
+min
+{ow,Bw,
+Cw,hw}w∈W
+�
+w∈W
+Φ (ow, Bw, Cw, hw)
+s.t. (1), (2), (3), and (4).
+(11a)
+Addressing the above subproblem requires network traffic
+information of all planning windows, which is difficult to
+be known a priori. To solve it, we leverage an RL method
+to design a network planning algorithm, which makes online
+decisions under spatial-temporally varying vehicle traffic.
+Subproblem 2: Network operation subproblem is to sched-
+ule network resources of each slice to active vehicles with
+random task arrivals with the objective of minimizing average
+service delay, which is formulated as:
+P2 : min
+xt
+k,yt
+k
+Dt
+k(xt
+k, yt
+k)
+s.t. (5) and (6).
+(12a)
+In the above subproblem, radio spectrum resource allocation
+and task dispatching decisions jointly impact the service
+delay performance. To solve the problem, we analyze the
+subproblem property and design an optimization algorithm to
+make real-time network operation decisions.
+IV. LEARNING-BASED NETWORK SLICING ALGORITHM
+In this section, we solve two subproblems in Sections IV-A
+and IV-B, respectively. Finally, we present the TWAS algo-
+rithm for jointly optimizing planning and operation decisions
+in Section IV-C.
+A. Network Operation Optimization
+We can observe that the radio spectrum allocation de-
+cision only impacts offloading delay component, and the
+task dispatching decision only impacts the computation delay
+component. Moreover, both decisions are independent in each
+BS. Hence, the radio spectrum allocation and task dispatching
+decisions can be optimized individually at each BS.
+1) Radio Spectrum Allocation Optimization: From (7), the
+radio spectrum allocation optimization problem is equivalent
+to minimizing the task offloading delay at each BS, i.e.,
+Pr
+m : min
+yt
+k
+�
+n∈N tm
+ξk
+yt
+k,nbw
+k,mRtn
+s.t. (5).
+(13a)
+The objective function can be proved to be convex since its
+second-order derivative is positive. In addition, the constraint
+is convex. Hence, problem Pr
+m is a convex optimization
+problem. Using the Karush-Kuhn-Tucker conditions [15], the
+optimal radio spectrum resource allocation decision is
+(yt
+k,n)⋆ =
+�
+1/Rtn
+�
+i∈N tm
+�
+1/Rt
+i
+, ∀n ∈ N t
+m.
+(14)
+
+5
+2) Task Dispatching Optimization: Similarly, from (7), task
+dispatching optimization is to minimize the task processing
+delay, which is formulated as:
+Pw
+m : min
+xt
+k,m
+dt
+k,m,e
+�
+At
+k,m − xt
+k,m
+�
++ dt
+k,m,cxt
+k,m
+s.t. (6).
+(15a)
+The above objective function can be rewritten as
+Ψ(xt
+k,m) = dt
+k,m,e
+�
+At
+k,m − xt
+k,m
+�
++ dt
+k,m,cxt
+k,m
+= ν1ξk
+2
+(xt
+k,m)2 +
+�
+νt
+2 − ν1ν3 − ξkAk,mν1
+2
+�
+xt
+k,m
++ ν1νt
+3At
+k,m.
+(16)
+Here, ν1 =
+ηk
+cw
+k,mFe
+> 0, νt
+2 = dt
+r +
+ηkξk
+hw
+k Fc , and ν3 =
+Qk,m + Ak,m+1
+2
+ξk. Since the second-order derivative of the
+objective function ∂2Ψ(xt
+k,m)/∂2xt
+k,m = νt
+1ξk > 0, the
+problem is a convex optimization problem [15]. The optimal
+task dispatching decision is given by
+(xt
+k,m)⋆ = 2νt
+2 + ξkν1Ak,m − 2ν1νt
+3
+2ν1ξk
+, ∀m ∈ Mw.
+(17)
+B. Network Planing Optimization
+The network planning problem is a stochastic optimization
+problem to minimize the network slicing cost, which can be
+transformed into a Markov decision process (MDP) [11]. The
+components of the MDP are defined as follows.
+1) Action, which is consistent with planning decisions,
+including slice deployment, radio spectrum and computing
+resource provisioning at BSs, and cloud computing resource
+provisioning, i.e., Aw
+=
+{ow, Bw, Cw, hw}. The action
+dimension is Ms + 2KM + K.
+2) State, which includes average vehicle traffic density in
+the current planning window and the planning decisions in the
+previous window due to the switching cost. The entire area is
+divided into J disjoint regions, and the average vehicle traffic
+density of all regions is denoted by Λw ∈ RJ×1. As such, the
+state is given by Sw = {Λw, ow−1, Bw−1, Cw−1, hw−1}. The
+state dimension is 2KM + M + K + J.
+3) Reward, which is defined as the inverse of the net-
+work slicing cost in the current planning window, i.e.,
+Rw (Sw, Aw) = −Φ (ow, Bw, Cw, hw) . Note that minimiz-
+ing the network slicing cost is equivalent to maximizing the
+cumulative reward.
+Upon state Sw, the learning agent takes action Aw, and the
+corresponding reward Rw (Sw, Aw) is obtained, along with
+the state evolves into new state Sw+1. With the above setting,
+our goal is to obtain an optimal planning policy π⋆ ∈ Π
+which makes decisions based on the observed state, thereby
+maximizing the expected long-term cumulative reward. As
+such, problem P2 can be formulated as the following MDP:
+P′
+2 : max
+π∈Π E
+�
+lim
+W →∞
+W
+�
+w=1
+(ϕ)wRw (Sw, Aw) |π
+�
+,
+(18a)
+where ϕ > 0 is the discount factor. Since vehicle traffic density
+is continuous, the action-state space can be prohibitively large.
+To address this issue, an RL algorithm can be adopted.
+Algorithm 1: TAWS algorithm.
+1 for training episode =1, 2, ... do
+2
+for planning window w = 1, 2, ..., W do
+3
+Generate planning decisions via the actor network;
+4
+for each slice in parallel do
+5
+for operation slot t = 1, 2, ..., T do
+6
+for each BS in parallel do
+7
+Make radio spectrum allocation and task
+dispatching decisions by (14) and (17);
+8
+Calculate the instantaneous service delay;
+9
+Measure the average service delay within the
+planning window;
+10
+Collect vehicle traffic density of all regions, and
+observe reward Rw and new state Sw+1;
+11
+Store transition {Sw, Aw, Rw, Sw+1} in the
+experience replay buffer;
+12
+Sample a random minibatch of transitions from the
+experience replay buffer;
+13
+Update the weights of neural networks;
+C. Proposed TAWS Algorithm
+We present the TAWS algorithm to jointly solve the entire
+network slicing problem P0, collaboratively integrating RL
+and optimization methods. The core idea of TAWS is to adopt
+an RL method for network planning decision making and an
+optimization method for network operation decision making.
+The service delay performance is measured at the end of each
+planning window and then incorporated into the reward in
+the RL framework, such that the interaction between network
+planning and operation stages can be captured. The TAWS
+algorithm is shown in Algorithm 1.
+The RL method is based on the deep deterministic policy
+gradient (DDPG) algorithm [16], [17], which consists of
+four neural networks, i.e., actor evaluation network, critic
+evaluation network, actor target network, and critic target
+network. In the initialization phase, all neural networks and
+the environment are initialized. The procedure of the TAWS
+is two-step: 1) Network slicing decisions are generated and
+executed. The actor network outputs the planning decisions
+Aw, which is clipped to feasible decision space. The network
+operation decisions are generated via the optimization method,
+and the service delay performance is measured at the end
+of each planning window. The reward Rw can be obtained
+and the new state can be observed Sw+1. The transition tuple
+{Sw, Aw, Rw, Sw+1} is stored in the experience replay buffer
+for updating neural networks; and 2) Neural networks are
+updated. A mini-batch of transitions are randomly sampled
+from the experience replay buffer to update the weights of
+neural networks. Specifically, the critic network is updated
+by minimizing the loss function, and the actor network is
+updated via the policy gradient method. Then, actor and critic
+target networks are updated by slowly copying the weights of
+evaluation networks.
+V. SIMULATION RESULTS
+We evaluate the performance of the proposed algorithm on
+real-world vehicle traffic traces in urban vehicular networks.
+We consider a 1,000×1,000 m2 simulation area, which is
+
+6
+Table I
+SIMULATION PARAMETERS.
+Parameter
+Value
+Parameter
+Value
+No
+−174 dBm
+I
+−164 dBm
+Pv
+27 dBm
+β
+20 MHz
+dr
+0.15 sec
+J
+16
+To
+1 sec
+Tp
+10 min
+Fc
+100 GHz
+Fe
+10 GHz
+Bm
+10
+Cm
+10
+ξ1, ξ2
+{0.6, 2} Mbit
+η1, η2
+{1000, 200} cycles/bit
+θ1, θ2
+{100, 200} ms
+θ
+′
+1, θ
+′
+2
+{50, 100} ms
+0
+100
+200
+300
+400
+500
+Training Episodes
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+Overall System Cost
+Five-Point Moving Average
+(a) Convergence
+1.4
+1.6
+1.8
+2
+Task Arrival Rate (Packet/sec)
+0
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+Overall System Cost
+Proposed
+Short Term Optimization
+(b) Network slicing cost
+Fig. 2.
+Performance of the proposed TWAS algorithm.
+covered by two SBSs and an MBS. Each SBS has a coverage
+radius of 300 m, and the MBS located in the centre covers
+the entire simulation area. The vehicle traffic density of the
+simulation area is measured by a unit of a small region
+of 250×250 m2, i.e., J = 16. This dataset is collected by
+Didi Chuxing GAIA Initiative2 and contains vehicle traces in
+the second ring road in Xi’an collected from taxis that are
+equipped with GPS devices. The periods of a planning window
+and an operation slot are set to 10 minutes and 1 second,
+respectively. The period of the slice lifecycle is set to 4
+hours, including 24 planning windows. The task arrivals of
+two services both follow Poisson processes with different task
+arrival rates. We construct two slices for supporting two types
+of delay-sensitive services. One is an object detect service
+whose service delay requirement is 100 ms, while the other
+is an in-vehicle infotainment service whose service delay
+requirement is 200 ms. Regarding the TWAS algorithm, the
+neuron units in hidden layers of both actor and critic networks
+are set to 128 and 64. Important simulation parameters are
+summarized in Table I.
+As shown in Fig. 2(a), we present the overall network slicing
+cost with respect to training episodes. All simulation points are
+processed by a five-point moving average in order to highlight
+the convergence trend of the proposed algorithm. It can be
+seen that the proposed algorithm converges after 500 training
+episodes.
+As shown in Fig. 2(b), we compare the performance of
+the proposed algorithm and a short term optimization bench-
+mark. The basic idea of the benchmark is to minimize the
+network slicing cost at each individual planning window. Since
+planning decisions are discrete, a simple exhaustive searching
+method is adopted to obtain the optimal one-shot planning
+decisions. Firstly, it can be seen that the proposed algorithm
+can greatly reduce the network slicing cost as compared to the
+benchmark. Specifically, when the task arrival rate is 2 packets
+per second, the proposed algorithm can reduce the network
+slicing cost by 23%. The reason is that the proposed algorithm
+takes the switching cost between two consequent planning
+windows into account, while the benchmark scheme does
+2Didi Chuxing Dataset: https://gaia.didichuxing.com.
+not. Secondly, the overall network slicing cost increases with
+the increase of the task arrival rate, because more radio and
+computing resources are consumed in heavy traffic scenarios.
+VI. CONCLUSION
+In this paper, we have investigated a network slicing prob-
+lem in edge-cloud orchestrated vehicular networks. A two-
+stage network slicing algorithm, named TWAS, has been
+proposed to jointly make network planning and operation
+decisions in an online fashion. The TAWS can adapt to
+network dynamics in different timescales, including spatial-
+temporally varying vehicle traffic density and random task
+arrivals. Simulation results demonstrat that the TAWS can re-
+duce the network slicing cost as compared to the conventional
+scheme. For the future work, we aim to determine the optimal
+planning window size for minimizing the network slicing cost
+under vehicular network dynamics.
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