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+Semantic Segmentation via Pixel-to-Center Similarity Calculation
+Dongyue Wu1, Zilin Guo1, Aoyan Li1, Changqian Yu2, Changxin Gao1, Nong Sang1
+1National Key Laboratory of Science and Technology on Multispectral Information Processing, School of
+Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, China
+{dongyue wu,zilin guo,aoyanli,cgao,nsang}@hust.edu.cn
+2Meituan Inc., Beijing, China
+changqianyu@meituan.com
+Abstract
+Since the fully convolutional network has achieved
+great success in semantic segmentation, lots of works
+have been proposed focusing on extracting discrimina-
+tive pixel feature representations. However, we observe
+that existing methods still suffer from two typical chal-
+lenges, i.e. (i) large intra-class feature variation in dif-
+ferent scenes, (ii) small inter-class feature distinction in
+the same scene. In this paper, we first rethink semantic
+segmentation from a perspective of similarity between
+pixels and class centers. Each weight vector of the seg-
+mentation head represents its corresponding semantic
+class in the whole dataset, which can be regarded as the
+embedding of the class center. Thus, the pixel-wise clas-
+sification amounts to computing similarity in the final
+feature space between pixels and the class centers. Un-
+der this novel view, we propose a Class Center Similarity
+layer (CCS layer) to address the above-mentioned chal-
+lenges by generating adaptive class centers conditioned
+on different scenes and supervising the similarities be-
+tween class centers. It utilizes a Adaptive Class Center
+Module (ACCM) to generate class centers conditioned
+on each scene, which adapt the large intra-class vari-
+ation between different scenes. Specially designed loss
+functions are introduced to control both inter-class and
+intra-class distances based on predicted center-to-center
+and pixel-to-center similarity, respectively. Finally, the
+CCS layer outputs the processed pixel-to-center simi-
+larity as the segmentation prediction. Extensive experi-
+ments demonstrate that our model performs favourably
+against the state-of-the-art CNN-based methods.
+Keywords: semantic segmentation, similarity, adaptive
+class center, intra-class variation, intra-class distinction.
+1. Introduction
+Semantic segmentation aims to assign each pixel with a
+semantic category, which is a fundamental and challenging
+task in the computer vision field. Benefited from the devel-
+opment of deep convolutional networks [23, 14, 22, 12], the
+fully convolutional network (FCN) [18] has been the domi-
+nant solution in the semantic segmentation task.
+Due to the simplicity of the architecture of FCN, exist-
+ing methods mainly focus on enhancing the feature repre-
+sentations to improve visual recognition capability. They
+aggregate rich contextual information via large receptive
+field [31, 6], multi-scale methods [37, 4], or attention mech-
+anisms [25, 9, 38, 8, 26]. However, we observe that these
+methods still suffer from two challenges. (i) Large intra-
+class feature variation between different scenes. Accord-
+ing to Fig. 1, the features of the “trees” (colored in red) near
+the wall in Sample A greatly differ from those on the bank
+in Sample B, despite that they belong to the same semantic
+category. Similarly, plant pixels of Sample A locate signif-
+icantly far from those of Sample B, which also suggesting
+large intra-class variation between pixels in different scenes.
+(ii) Small inter-class distinction inside each scene. Take
+Sample A in Fig. 1 for example, it is difficult for the seg-
+mentation network to tell pixels belonging to the “tree” and
+pixels belonging to the “plant” apart, since pixels belong-
+ing to the “tree” are near to pixels of the “plant”, indicating
+that the two groups of pixels of different semantic classes
+bear similar features. These two challenges make it hard
+to category pixels from the same class but different scenes
+into the correct class while telling groups of pixels from the
+different classes but the same scene apart from each other.
+To solve the aforementioned challenges, we rethink pre-
+vious methods from a perspective of similarity between pix-
+els and class centers. Most previous methods utilize a 1 × 1
+convolutional layer as the segmentation head to get the fi-
+nal prediction. The weights of the convolutional layer are
+trained through all training samples and conduct a convolu-
+tion operation on the feature maps in the inference phase. In
+other words, the learned weights perform correlation calcu-
+lation with each pixel and output the map of inner produc-
+tion which is widely used to measure the similarity between
+features. As each vector of the weights can be regarded as
+1
+arXiv:2301.04870v1  [cs.CV]  12 Jan 2023
+
+Figure 1. Illustration of the overall distribution of “tree” and “plant” pixels on the ADE20K validation set. We show the distribution of the
+two classes in feature space. For clarity, we only label the pixels belonging to “tree” and “plant”, which are in red and blue, respectively,
+while the pixels that belong to other classes keep their original color. Each dot in our plot represents a randomly sampled pixel in the
+feature space. The light-colored dots denote pixels sampled from other samples(scenes) in the whole dataset, while the dark-colored ones
+are sampled from Sample A and Sample B. According to the plots, the features of both “tree” and “plant” in Sample A are quite different
+from those in Sample B, suggesting large intra-class variation between different scenes. The features of “tree” and “plant” are hard to
+distinguish in Sample A, because of the small inter-class distinction. The dimension of features is reduced for illustration using t-SNE [24].
+(a) Images
+(b) Baseline
+(c) Ours
+Figure 2. Comparison of pixel distribution in feature space on the
+ADE20K validation set. Baseline: ResNet-101 + DeeplabV3+.
+Ours: Baseline + CCS layer. Pixels that randomly sampled form
+the “tree” and “plant” are colored in red and blue, respectively.
+Our distribution of each class is more compact than the baseline.
+There are also clear boundaries between different classes in the
+feature space of our method, when the pixels of the two classes
+are mixed up in the feature space of the baseline.
+a learned representation of the corresponding class, the fi-
+nal segmentation prediction map can also be regarded as a
+pixel-to-center similarity map. These learned representa-
+tions contain the common information of their class on the
+whole dataset. Therefore, we call them global class centers.
+In this view, the classification can be remodeled as follows:
+A pixel will be assigned to the class whose global class cen-
+ter is the most similar to the pixel among all classes. In con-
+clusion, semantic segmentation can be viewed as a task to
+predict the similarity between pixels and class centers.
+Motivated by this perspective, we argue that there are
+two limitations of previous methods responsible for the
+above-mentioned challenges: (i) The global class centers
+are incapable of adapting the large intra-class variations be-
+tween different scenes, since the global class centers are
+learned based on the whole dataset and are kept unchanged
+and identical for different scenes during the inference stage.
+(ii) Since there is no constraint on the similarities between
+global class centers, these centers which represent different
+classes may share excessively similar representations, lead-
+ing to difficulties in correctly distinguishing pixels. These
+limitations correspond to the two challenges of the large
+intra-class variation between scenes and the small inter-
+class distinction inside each scene, respectively.
+To solve the challenges, we propose a novel and flexi-
+ble Class Center Similarity (CCS) layer, which replaces
+the segmentation heads of networks and transfers the pixel-
+wise classification task to a pixel-to-center similarity pre-
+diction task. Our CCS layer consists of three parts: Adap-
+tive Class Center Module (ACCM), Similarity Calculation
+Module (SCM), and Class Distance (CD) Loss.
+First,
+Adaptive Class Center Module (ACCM) generates adaptive
+
+Tree
+Sample B
+Plant囍class centers conditioned on each scene (image), which ac-
+commodates the large intra-class variance between scenes.
+Thus, the prediction only depends on the similarity between
+each pixel and the scene-specific adaptive class centers, in-
+stead of the immutable global class centers. Then, the adap-
+tive class centers are forward Similarity Calculation Mod-
+ule (SCM) to compute the pixels-to-centers similarity be-
+tween pixels and class centers and the mutual similarity of
+class centers as the inter-class similarity. Finally, our CD
+Loss is applied to the inter-class similarity and pixels-to-
+centers similarity in each scene to supervise the segmenta-
+tion prediction while increasing the lack of inter-class dis-
+tinction inside each scene simultaneously. The CCS layer
+can be integrated into almost arbitrary semantic segmen-
+tation architectures substituting for the final segmentation
+head, namely the 1 × 1 convolution layer. To demonstrate
+the effectiveness of the proposed Class Center Similarity
+layer, we add the CCS layer to existing segmentation net-
+works and carry out extensive experiments on ADE20K and
+Pascal Context. Fig. 2 and Fig. 3 also show that our method
+leads to clear and compact clusters for each semantic class
+in each scene. In summary, the following contributions are
+made in this paper:
+• We rethink the semantic segmentation task from a
+pixel-to-center (P-C) similarity perspective. Semantic
+segmentation can be viewed as a task with two stages:
+compute P-C similarity for each pixel and categorize
+pixels to the most similar semantic class.
+• We propose a class Center Similarity layer that can
+generate conditional class centers for each scene under
+the constraint of our Class Distance Loss. Our CCS
+layer is easy to plug into almost any FCN-based se-
+mantic segmentation network.
+• We conduct extensive experiments to analyze the ef-
+fectiveness of our approach. The proposed method,
+called CCSNet, achieves state-of-the-art performance
+on two challenging datasets.
+Based on ResNet-
+101 [11], our model achieves 47.76% mIoU on
+ADE20K, and 54.9% mIoU on PASCAL Context.
+2. Related Work
+Semantic segmentation.
+The development of deep neu-
+ral networks dramatically boosts semantic segmentation.
+Since FCN [18] replaces the fully connected layer in the
+traditional classification network with a convolutional to
+get pixel-wise predictions, FCNs achieves great success in
+semantic segmentation.
+Segnet [1], UNet [21] and Re-
+fineNet [16] adopt encoder-decoder structure to recover the
+spatial information that lost by downsample operation via
+cascading upsampling. In order to capture long-range de-
+pendencies, lots of works have been done by introducing
+CRF [3, 2] and MRF [17] into segmentation tasks. Dilated
+convolution [31] and deformable convolution[6], which in-
+crease the resolution of feature maps, is used to enlarge
+the receptive field. We revisit these typical networks and
+find that semantic segmentation can be regarded as catego-
+rizing each pixel to the semantic category with the great-
+est pixel-to-center similarity (or the smallest pixel-to-center
+distance).
+Contextual information.
+Context is believed of great
+significance in semantic segmentation.
+Recently, plenty
+of works focusing on mining richer context information.
+Multi-scale representations are used to capture more context
+in PSPNet [37]. the family of DeepLab [3, 4, 5] also cap-
+tures the context information from multi-scales. Based on
+these approaches, many extensions have been proposed,e.g.
+DenseASPP [27] and APCNet [10]. The Attention mecha-
+nism is also adopted to capture global contextual informa-
+tion [13, 15]. Other studies pay their attention to the sim-
+ilarity of pixels to help aggregate contextual information.
+OCNet [33], DANet [9] and CFNet [36] aggregate context
+that computed on all pixels and augment context represen-
+tation to the representation of each pixel. Based on the self-
+attention mechanism, these approaches calculate similarity
+(or relation) between pixels and aggregate representations
+according to similarity. Although a great number of studies
+explore discriminate representations to help segmentation,
+the large variance between pixels of the same category from
+different scenes and the lack of distinction in each scene still
+remains. Our work addresses these two challenges based on
+similarity as well, but we aim to get better similarity maps
+while those utilize context to get better features. In addition,
+thanks to our ACCM which generates the varying adaptive
+class centers for classification, our method is more effective
+with minor computational cost.
+Center-based methods in semantic segmentation.
+Re-
+cently, several works propose approaches with centers in
+semantic segmentation networks. In contrast to previous
+works, ACFNet [34] presents the concept of class centers
+as the global context from a categorical perspective, which
+describes the overall representation of each class in a scene.
+Then, different class centers are adaptively concatenated
+with features according to each pixel for aggregating class-
+wise context. OCRNet [32] presents a simple method char-
+acterizing a pixel by exploiting the representation of the
+corresponding object class.
+Under supervision, OCRNet
+learns object regions and generate representations of ob-
+ject regions, which can be viewed as the object centers of
+each region. Pixel-region relation is computed to aggre-
+gate information from object centers. Similar to ACFNet,
+the object-contextual representations generated from object
+centers and pixel-region relation augment the original pixel
+representations via concatenation.
+Compared with these
+methods that also introduce the concept of center in their
+
+(a) Images
+(b) Baseline
+(c) Ours
+(d) Baseline distribution
+(e) Our distribution
+Figure 3. Visualization results on ADE20K validation set. We visualize the distribution results to demonstrate our model learns a better
+scene-level feature distribution than previous works. Comparing our method with the Baseline DeeplabV3+, we find that our boundaries
+between different clusters are far more clear, with most pixels of the same class located in the same cluster.
+works, our concern is how to get conditional centers to ad-
+dress the drawback of global centers, while they follow the
+thought of exploring context from a different aspect. Be-
+sides, we apply our inter CD Loss and intra CD Loss on
+class centers to enlarge distinction and optimize the distri-
+bution of different classes while they do not have any super-
+vision on the generated centers.
+3. Method
+In this section, we first have a review of typical segmen-
+tation models in semantic segmentation. Then, we expli-
+cate our rethinking-modeling semantic segmentation as a
+task to assign each pixel to a category directly based on the
+pixel-to-center (P-C) similarity. After a brief overview of
+the proposed prediction pipeline, we introduce the details
+
+wall
+floor
+ceiling
+windowpane
+table
+plant
+curtain
+chair
+painting
+sofa
+rug
+lamp
+cushion
+blind
+coffee table
+pot
+blanket
+sconcesky
+tree
+water
+rockCOGNAC
+JACOUETwall
+floor
+ceiling
+windowpane
+cabinet
+door
+table
+plant
+chair
+painting
+lamp
+blind
+potcase
+basket
+food
+traywall
+floor
+ceiling
+chair
+shelf
+rug
+box
+book
+light
+playthingbuilding
+sky
+treeK
+H
+W
+𝑀𝑎𝑠𝑘
+1 × 1
+𝐶𝑜𝑛𝑣
+⊗
+𝑟𝑒𝑠ℎ𝑎𝑝𝑒
+K×HW
+𝑟𝑒𝑠ℎ𝑎𝑝𝑒
+&
+𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑒
+HW×D
+𝑨𝒅𝒂𝒑𝒕𝒊𝒗𝒆 𝑪𝒍𝒂𝒔𝒔 𝑪𝒆𝒏𝒕𝒆𝒓 𝑴𝒐𝒅𝒖𝒍𝒆
+𝑪
+K
+𝑃 − 𝐶 𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝑀𝑎𝑝𝑠
+H
+W
+D
+H
+W
+𝐹𝑒𝑎𝑡𝑢𝑟𝑒𝑠
+K
+H
+W
+𝑖𝑛𝑡𝑒𝑟 𝐶𝐷 𝐿𝑜𝑠𝑠
+𝑟𝑒𝑠ℎ𝑎𝑝𝑒
+D×HW
+𝐺𝑇
+…
+D
+K
+𝐴𝑑𝑎𝑝𝑡𝑖𝑣𝑒 𝐶𝑙𝑎𝑠𝑠 𝐶𝑒𝑛𝑡𝑒𝑟𝑠
+K-1
+K-1
+𝐶 − 𝐶 𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝑀𝑎𝑡𝑟𝑖𝑥
+𝑺𝒊𝒎𝒊𝒍𝒂𝒓𝒊𝒕𝒚 𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝑴𝒐𝒅𝒖𝒍𝒆
+𝐶𝑒𝑛𝑡𝑒𝑟 − 𝐶𝑒𝑛𝑡𝑒𝑟
+𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦
+𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛
+𝑃𝑖𝑥𝑒𝑙 − 𝐶𝑒𝑛𝑡𝑒𝑟
+𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦
+𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛
+𝑖𝑛𝑡ra 𝐶𝐷 𝐿𝑜𝑠𝑠
+𝑪𝑫 𝑳𝒐𝒔𝒔
+Figure 4. Structure of the proposed CCS layer. The Mask is a spatial weight map used for aggregating features for all the centers of each
+semantic class. C ∈ RK×D denote the generated K adaptive class centers, where D is the number of channels and K is the number
+of classes. Then, the CCS layer calculates the pixel-to-center (P-C) similarity maps of each semantic class and center-to-center (C-C)
+similarity between different classes. Finally, the proposed inter-class distance loss and intra-class distance loss are applied. The final
+prediction is P-C similarity Maps normalized by the softmax function.
+of the Adaptive Class Center Module, Similarity Calcula-
+tion Module and Class Distance Loss.
+3.1. Revisiting prediction of semantic segmentation
+Before we go to our proposed approach, let’s revisit
+the architecture of typical segmentation networks. A typ-
+ical segmentation network consists of two parts: feature
+extractor and classifier.
+The feature extractor is usually
+a deep convolutional network, which takes images as in-
+puts, then extracts high-dimensional representations of each
+pixel. The classifier takes the output features of the ex-
+tractor as inputs and computes the score maps indicating
+the probabilities that every element belongs to each class.
+The output of the feature extractor is noted as F ∈ RD×N,
+where N is the total number of elements, D is the number
+of output channels in the extractor. The representation of
+the i-th pixel in the input image is noted as fi ∈ RD×1.
+The weights of the classifier are noted as Wc ∈ RK×D,
+and wj ∈ R1×D (j = 1, 2, ..., K) denotes the kernel
+weight which performs correlation on feature maps to get
+the prediction of j-th class. Following the work of FCN,
+most segmentation networks replace the FC classifier with
+a 1 × 1 convolutional layer, which computes the score map
+S ∈ RK×N of input feature F. The computation can be
+formulated as:
+S = W ⊛ F,
+(1)
+where ⊛ is the convolution operation. S is then normalized
+by soft-max to get a soft-margin prediction of the whole
+image.
+As classifier employs a 1×1 conv layer, the computation
+of the j-th class’ score sij of i-th pixel can be formulated as:
+sij = w⊤
+j · fi,
+(2)
+where · denotes the inner product, sij is the score of i-
+th pixel at j-th class, and w⊤
+j is the transposed wj, i =
+1, 2, ..., N. The prediction pi assigned to i-th pixel is made
+by performing a argmax operation to each location in the
+image:
+pi = arg max
+j
+(sij) = arg max
+j
+(w⊤
+j · fi).
+(3)
+Hence, for a single pixel, the score sij is actually the in-
+ner production similarity, or so-called correlation, between
+its feature fi and the transposed weight vector w⊤
+j of class
+j. According to the pixel-to-weight similarity map S, the
+final prediction is made by categorizing pixels to the se-
+mantic class which has the greatest similarity value in fea-
+ture space.
+Thus, the weight vectors can be deemed as
+learned class centers, and the category assignment is equiv-
+alent to finding the nearest class center for each pixel. Con-
+sequently, the overall inference procedure can be summa-
+rized as Alg. 1, where I ∈ R3×N is the input RGB features
+of an image, θ is the parameter of network, Wc ∈ RK×D is
+the weights of the classifier, namely the learned global class
+centers, i = 1, 2, ..., N, j = 1, 2, ..., K.
+Algorithm 1 Inference with Global Class Centers
+Input: I, network Net(, ) and parameters θ, weights of
+classifier Wc.
+Extract feature map: F ← Net(I, θ).
+Calculate P-C similarity: S ← Simi(F, Wc)
+Get prediction: pi ← arg maxj(sij)
+Output: Final prediction P.
+If pixel i belongs to the j-th class according to ground
+truth, we hope that we can get greater relative sij at the j-th
+channel compared with other channels and smaller scores at
+
+Algorithm 2 Inference with Adaptive Class Centers
+Input: I, network Net(, ), the parameters θ, and the
+ACCM G(·).
+Extract feature map: F ← Net(I, θ).
+Gnerate Adaptive Class Centers: C ← G(F).
+Calculate P-C similarity: S ← Simi(F, C)
+Get prediction: pi ← arg maxj(sij)
+Output: Final prediction P.
+irrelevant channels, which correspond to irrelevant classes.
+So, the network is trained under the supervision of Cross-
+entropy (CE) Loss that applied on the normalized scores
+maps:
+LCE
+=
+�
+i
+−log
+� exp(si · yi)
+�
+j exp(sij)
+�
+=
+�
+i
+−(si · yi) + log
+� �
+j
+exp(sij)
+�
+,
+(4)
+where yi is the one-hot label of element i.
+In our understanding, CE Loss mainly focuses on enlarg-
+ing the relative inner production similarity between the cor-
+responding weight vector and pixels of each class by push-
+ing pixels and corresponding weight vectors closer in fea-
+ture space. Accordingly, we can regard the weight vectors
+as a set of global class centers learned by the network on
+the whole dataset. The training stage is a process that the
+network learns to find the global class centers of each class,
+best fitting the distribution of all the pixels which belong to
+the same class on the whole training set. Moreover, these
+global class centers are identical for all input scenes.
+However, semantic segmentation networks are still fac-
+ing two challenges: (i) pixels of the same category yet dif-
+ferent scenes are significantly different in feature space,
+leading to difficulty for networks in categorizing all these
+pixels to the same semantic class. (ii) pixels from different
+categories but the same scene lack enough distinction, caus-
+ing trouble finding distinctive separating plane to tell pixels
+belonging to different classes apart.
+To demonstrate the challenges, we conduct experiments
+and visualize the feature representations, which are the in-
+puts of the final convolution, in Fig. 1 of randomly sampled
+pixels from “tree” and “plant” classes on ADE20K valida-
+tion set, with sampling ratio set as 1%. We adopt t-SNE [24]
+for a clear visualization. As shown in Fig. 1, the pixels of
+“trees” and “plants” in Sample A are far from those in Sam-
+ple B in feature space despite the fact that they belong to
+the same class. This phenomenon indicates the large vari-
+ance of features between different scenes, making it hard
+to assign pixels of the same class yet different scenes with
+the same class based on the immutable global class centers.
+Moreover, for those pixels in the same scene, they are much
+closer to each other despite that they belong to different cat-
+egories. These short distances in feature space between pix-
+els from the same scene but different classes corroborate the
+scarcity of enough distinction to categorize them correctly.
+3.2. Overview of our proposed method
+As illustrated above, considering the learned class cen-
+ters Wc are unchangeable for different input images I, the
+first challenges are unavoidable. To solve it, we propose a
+new pipeline for semantic segmentation prediction pipeline
+as shown in Alg. 2. The pixel-to-center (P-C) similarity
+is calculated between pixels in each scene and the adap-
+tive class centers which are generated based on the feature
+map of the scene. Compared with Wc which is immutable
+during inference for every input image, the adaptive class
+center should be capable of varying between scenes to ac-
+commodate the large intra-class feature variation in differ-
+ent scenes.
+On the other hand, there is no constraint on the similar-
+ity between different class centers inside each scene. So,
+the second challenge is aggravated, even though the adap-
+tive class centers can mitigate the problem. Therefore, we
+propose the Class Distance (CD) Loss to enlarge the inter-
+class feature variation in the same scene. CD Loss directly
+requires large similarities between pixels and the adaptive
+class center of the ground truth class and small mutual sim-
+ilarities among adaptive class centers.
+Extensive experi-
+ments and ablation studies in Sec. 4 corroborate the effec-
+tiveness of our proposed prediction pipeline.
+3.3. Adaptive Class Center Module
+Previous approaches take the transposed learned weight
+Wc of the final classifier as global class centers, which are
+the representations of each class at the dataset-level. On
+account of the feature variances on the dataset from scene
+to scene, these approaches are impeded by their global
+class centers which are not able to vary between scenes.
+Therefore, we propose the Adaptive Class Center Module
+(ACCM) to generate unique class centers for each scene
+as class representations at the scene-level instead of the
+dataset-level.
+ACCM performs matrix multiplication of pixel features
+and a learned mask to generate coarse class centers for each
+input image. This module refines the coarse class centers
+and outputs a set of adaptive centers as scene-level repre-
+sentations for all classes. The whole computation process
+of ACCM can be formulated as:
+C = A(M ⊗ F⊤),
+(5)
+where C ∈ RK×D is the matrix of generated conditional
+adaptive class centers, ⊗ is matrix multiplication, A(·) is
+the adaptive module consisting of several convolutional lay-
+ers, and M ∈ RK×N is a learned weight map, based
+
+on which ACCM aggregates information and generates the
+coarse class centers. In practice, we find that for the adap-
+tive module, a 1 × 1 convolution works well to generate
+the adaptive class centers. We also conduct experiments to
+study the impact of different M and eventually choose the
+one supervised by Dice loss [19] for better performance.
+The Dice loss is defined as:
+LDice = 1 −
+2 �N
+i mi · yi
+�N
+i ||mi||2 + �N
+i ||yi||2 + ϵ
+,
+(6)
+where mi ∈ RK×1 is the i-th column vector in M corre-
+sponding to i-th pixel, ||a|| is the second norm of vector a,
+and ϵ is set as 1e−3 to prevent division by zero.
+(a) traditional centers
+(b) our centers
+Figure 5. Illustration of our adaptive class centers. During infer-
+ence, traditional approaches calculate pixel-to-centers similarity
+maps with fixed centers, while our approach generates adaptive
+centers for each input image, respectively.
+Based on the adaptive class centers, we perform pixel-to-
+center similarity calculation on feature maps to get absolute
+P-C similarity maps, followed by soft-max at the dimension
+of different classes to generate the relative P-C similarity
+maps as the soft-margin prediction. The structure of the
+model is shown in Fig. 4.
+Moreover, to address the problem that features of differ-
+ent classes but the same scene lack enough distinction, pre-
+vious methods try to aggregate more context information
+into pixel representations, but there is no supervision im-
+posed on the representations of class centers. We propose
+the Class Distance Loss composed of inter-class distance
+loss and intra-class distance loss imposed on class centers
+to make features in the same scene more discriminative. We
+will introduce it in the next section.
+3.4. Similarity Calculation Module
+We introduce of similarity function to measure the prox-
+imity of two embedding a ∈ RD×1 and b ∈ RD×1 in the
+feature space. we define the inner production similarity as:
+Simi(a, b)inn = a · b.
+(7)
+To encourage class centers to be linearly independent
+and obviate the influence of the embedding’s norm, we also
+introduce the cosine similarity as :
+Simi(a, b)cos = abs(a · b)
+||a|| ||b|| ,
+(8)
+where the abs(·) denotes the absolute value.
+To calculate the C-C similarity, we employ consine sim-
+ilarity (Eq. 8) to get rid of the influence of the norms of
+class centers. We simply replace a and b in Eq. 8 with
+adaptive class centers c⊤
+p and c⊤
+p as Simi(c⊤
+p , c⊤
+q ), where
+cq ∈ R1×D is the q-th row in C, namely the adaptive class
+center of the q-th class.
+For P-C similarity calculation, we employ inner pro-
+duction simialrity (Eq. 7). As the prediction is made by
+the argmax operation on P-C similarity maps, the absolute
+value of similarity between a pixel with a class center alone
+is meaningless unless compared with similarities between
+the pixel and other class centers. Hence, we introduce rel-
+ative similarity as the probability that i-th pixel belongs to
+q-th class:
+RSimi(fi, c⊤
+q ) =
+exp
+�
+Simi(fi, c⊤
+q )
+�
+�
+j exp
+�
+Simi(fi, c⊤
+j )
+�,
+(9)
+where j = 1, 2, ..., K.
+3.5. Class Distance Loss
+Since there is no constraint on the similarity between
+different class centers inside each scene, the challenge that
+different-classes pixels in the same scene lack enough dis-
+tinction is aggravated consequently. We believe that it is
+easy for pixel-wise classification if pixel representations in
+the same scene have large inter-class distinction and small
+intra-class distinction. Motivated by this notion, we propose
+the Class Distance (CD) Loss. We introduce the inter-class
+distance and intra-class distance at first, then explicate the
+details of the Class Distance Loss.
+Definition of inter-class and intra-class distance.
+The
+inter-class distance between the p-th and the q-th class is
+defined as:
+d(p,q)
+inter = D(c⊤
+p , c⊤
+q ),
+(10)
+where p and q are the numbers of two different classes, c⊤
+p
+and c⊤
+q are the class center vectors of p-th and q-th class
+respectively, D(·, ·) is the distance function. The intra-class
+distance of the q-th class is defined as:
+dq
+intra =
+N
+�
+i=1
+1[yiq = 1]D(fi, c⊤
+q ),
+(11)
+1[condition] =
+�
+1
+condition is True
+0
+condition is False ,
+(12)
+
+Conditional
+CentersWe introduce our distance function based on the relative
+similarity. The distance should be negative correlated with
+relative similarity. Therefore, we define our distance as:
+D(fi, c⊤
+q ) = −log
+�
+RSimi(fi, c⊤
+q )
+�
+.
+(13)
+We apply inner production similarity and cosine similarity
+for dintra and dinter, respectively.
+Inter-class and intra-class Distance Loss.
+As illustrated
+above, we propose inter-class distance loss and intra-class
+distance loss to enlarge inter-class distances and diminish
+intra-class distance. Our loss functions are as:
+Lintra =
+K
+�
+q=1
+dq
+intra,
+(14)
+Linter =
+K
+�
+p=1
+K
+�
+q=1
+1[q ̸= p]exp(−dp,q
+inter),
+(15)
+where Lintra aims at diminishing intra-class distinction un-
+der the supervision of GT, and Linter focuses on enlarging
+the differences between class centers that can operate with-
+out GT.
+The weighted summation of the Lintra and Linter is
+called Class Distance (CD) Loss for convenience, which
+can be formulated as:
+LCD = Lintra + αLinter,
+(16)
+where α is the a hyper-parameter that set empirically.
+For dataset-level CD Loss, we regard each w⊤
+j
+as
+dataset-level class center just as our rethinking of traditional
+segmentation networks. So, the dataset-level intra-class dis-
+tance loss and inter-class distance loss are calculated re-
+spectively as:
+Ldataset
+intra
+=
+K
+�
+q=1
+N
+�
+i=1
+1[yiq = 1]D(fi, w⊤
+q ),
+(17)
+Ldataset
+inter
+=
+K
+�
+p=1
+K
+�
+q=1
+1[q ̸= p]exp
+�
+− D(w⊤
+p , w⊤
+q )
+�
+.(18)
+Please notice that when combining Eq. 13 with Eq. 11, Eq. 9
+and Eq. 7, the Lintra is in the same format with CE Loss
+shown in Eq. 4:
+Ldataset
+intra
+=
+K
+�
+q=1
+N
+�
+i=1
+1[yiq = 1]
+�
+log
+� �
+j
+exp(fi·c⊤
+j )
+�
+−fi·c⊤
+q
+�
+.
+(19)
+This shows that the CE Loss is a special case of our pro-
+posed intra-class distance loss.
+For scene-level CD Loss, we use the proposed ACCM to
+calculate the class centers conditioned on each scene. So,
+the scene-level CD Loss can be formulated as follows:
+Lscene
+intra =
+K
+�
+q=1
+N
+�
+i=1
+1[yiq = 1]D(fi, c⊤
+q ),
+(20)
+Lscene
+inter =
+K
+�
+p=1
+K
+�
+q=1
+1[q ̸= p]exp
+�
+− D(c⊤
+p , c⊤
+q )
+�
+.(21)
+Overall loss function.
+The over-all loss function of our
+model are as follows:
+L
+=
+LCD + βLDice
+=
+Lintra + αLinter + βLDice,
+(22)
+where LDice is the Dice loss imposed on the mask M and
+β is the weight of LDice. Since the scene-level CD Loss is
+much more powerful, we employ the Lscene
+intra and Lscene
+inter in-
+stead of Ldataset
+intra
+and Ldataset
+inter
+. We conduct experiments to
+show the effectiveness of Lscene
+intra and Lscene
+inter over Ldataset
+intra
+and Ldataset
+inter
+in Tab. 5.
+4. Experiment
+We evaluate our approach on two challenging semantic
+segmentation datasets: ADE20K and Pascal Context. We
+perform a comprehensive ablation study on the ADE20K
+dataset and report the comparison with other methods on the
+ADE20K validation set and the Pascal Context validation
+set.
+4.1. Settings
+Dataset.
+ADE20K [40] dataset is a large-scale scene pars-
+ing benchmark with 150 fine-grained objects and stuff cate-
+gories, containing 20,210 images for training, 2,000 images
+for validation, and 3352 images for testing.
+Pascal Context dataset [20] is a scene parsing dataset that
+provides semantic labels for whole scene(both “things” and
+“stuff” classes), which augments 10,103 images from PAS-
+CAL VOC 2010 [7]. It has 4,998 training and 5,105 vali-
+dation images. We use the 59 most common categories for
+evaluation.
+Training.
+We
+conduct
+our
+experiments
+using
+four
+NVIDIA GTX 2080 ti GPUs with four images per GPU. All
+of our models are optimized by SGD optimizer with 0.9 mo-
+mentum. The initial learning rate is set 1e−2 for ADE20K
+and 4e−3 for PASCAL Context. We adopt the polynomial
+learning rate decay strategy in training following previous
+works [30, 29, 5]. The initial learning rate is multiplied by
+(1−
+iter
+maxiter)0.9. We apply random resizing with a ratio be-
+tween 0.5 and 2 in training, random cropping input images
+
+into (512,512), and random horizontal flipping during train-
+ing for all the experiments. We set the total iterations on
+ADE20K to 80K and 160K for our ResNet-50 and ResNet-
+101 models, respectively. For Pascal Context, models run
+80k iterations during training.
+Auxiliary loss.
+Follow previous work [39], we adopt aux-
+iliary segmentation loss to help train our model. We add
+an auxiliary FCN head, which outputs prediction under the
+supervision of CE Loss multiplied by 0.4.
+Evaluation.
+During the evaluation, we average the pre-
+dictions of multiple scaled following the previous work [37,
+29, 5]. Each image is then flipped horizontally, then scaled
+to a uniform size with scaling factor (0.5, 0.75, 1.0, 1.25,
+1.5, 1.75) for better performance. Besides, we use Synchro-
+nized BN in our models. Additionally, we report mean In-
+tersection over Union (mIoU) and pixel accuracy (Acc) for
+ADE20K and mIoU for PASCAL context.
+4.2. Ablation Study On ADE20K
+We conduct ablation studies on ADE20K to demonstrate
+the effectiveness of our approach. Models are trained on
+ADE20K train set and evaluated on val set. All the models
+are pretrained on Image-Net without extra data.
+Upper-bound verification.
+To verify the feasibility of
+our proposed methods, we first carry out a simple upper-
+bound verification experiment. We directly replace the pre-
+dicted Mask, which is supervised by CE loss, of our trained
+model with processed ground truth during inference. Ac-
+cording to the results shown in Tab. 1, our method pro-
+vides a huge improvement compared with the baseline,
+while the upper bound method Ours-GT greatly outper-
+forms our method when the predicted Mask is replaced by
+GT. Since the GT provides better weight maps than the pre-
+dicted Mask to generate the adaptive class centers, the per-
+formance is dramatically improved, indicating that there is
+still a lot of room for improvement.
+Method
+CCS
+mIoU(%)
+Acc(%)
+Baseline
+37.94
+77.98
+Ours
+✓
+43.57
+81.06
+Ours-GT
+✓
+47.21
+84.12
+Table 1. Upper-bound verification.
+Baseline: ResNet-50 FCN.
+Ours: Baseline + CCS layer. Ours-GT: Baseline + CCS layer
+whose Mask is supplanted by GT to verify the upper bound of
+CCS layer.
+Ablation study of class centers.
+To demonstrate the
+effectiveness of the adaptive class centers, we compare
+Figure 6. P-C distance comparison between MC & AC. Left:
+inter-class P-C distance. Right: intra-class P-C distance. We de-
+note pixel-to-mean centers and pixel-to-adaptive centers as MC
+and AC, respectively.
+the P-C similarity, depending on which we generated the
+probability maps.
+Taking “tree” and “plant” as exam-
+ples, we show the frequency histograms of intra-class
+pixel-to-adaptive centers distance and pixel-to-mean cen-
+ters (namely the global centers that learned from the whole
+dataset) distance in Fig. 6. As we introduced before, the
+results demonstrate that the adaptive class centers make
+prominent success in increasing the inter-class distance be-
+tween “tree” pixels and “plant” class center, while the intra-
+class distances of “plant” pixels and adaptive class center of
+“plant” are dramatically smaller than the distance between
+pixels and global class center.
+Ablation study of ACCM.
+We conduct ablation studies
+of ACCM to explore the best performance. We compared
+different loss functions on Mask in order to generate better
+adaptive class centers. We conduct experiments on FCN
+based on ResNet-50 and report the results in Tab. 2.
+Res50 FCN CCSNet
+w/o loss
+w/ CE loss
+w/ Dice loss
+mIoU(%)
+43.01
+41.56
+43.57
+Acc(%)
+80.69
+80.45
+81.06
+Table 2. Ablation study of defferent loss fucntion on Mask. Our
+baseline in this experiment is ResNet-50 FCN, with our CS replace
+the final convolution of FCN under supervision of scene-level CD
+Loss.
+Ablation study of hyper-parameter and architecture.
+We first conduct experiments with different segmentation
+heads. The multi-scale mIoU results of different segmen-
+tation heads based on ResNet-50 are reported in Tab. 3.
+Among three different segmentation heads, deeplabv3+ has
+the best performance with 44.25% ms mIoU. We also ex-
+plore proper hyper-parameters for CCSNet. We take FCN
+Methods
+FCN
+PSP
+Deeplabv3+
+baseline
+37.94
+41.94
+43.57
++CCS
+43.57
+44.07
+44.25
+Table 3. Ablation study of segmentation heads.
+CCS layer is
+equipped on different segmentation heads based on ResNet-50 are
+trained on ADE20K training set for 80k iterations. The mIoU re-
+sults on ADE20K validati set of each model are reported.
+
+0.20
+inter P-MC
+Frequency
+inter P-AC
+0.15
+0.10
+0.05
+0.00
+2
+4
+6
+8
+10
+12
+14
+Inter Distance (tree-to-plant)0.15
+intra P-MC
+intra P-AC
+Frequ
+0.10
+aA
+lati
+0.05
+Rel
+0.00
+2
+4
+6
+8
+10
+12
+14
+Intra Distance (plant-to-plant)mIoU
+β = 0.1
+β = 0.5
+β = 1.0
+α = 0.1
+42.79
+42.86
+42.85
+α = 0.5
+43.01
+43.05
+43.57
+α = 1.0
+43.14
+43.50
+43.47
+Table 4. Alblation Study of weight of Lscene
+inter and weight of dice
+loss. We vary the value of α and β and find when α = 0.5, β =
+1.0, model has the best performance.
+based on ResNet-50 as a baseline to exploit the best weight
+of dinter and the weight of Dice loss.
+As shown in
+Tab. 4, our approach achieves the best performance when
+the weight of dinter is empirically set as 0.5. The best FCN
+based model equipped with our approach improves the pix-
+Acc and mIoU by 2.59% and 5.63%.
+Ablation study of CCS layer.
+We break down the
+improvements of our work over ResNet-101 based on
+DeeplabV3+, which has the best performance.
+We add
+the proposed components in our approaches step by step
+to the DeeplabV3+ baseline. Experiments are conducted
+on ADE20K and run 160k iterations. By simply replacing
+the global class centers Wc with our adaptive class cen-
+ters conditionally generated by ACCM, our network im-
+proves the performance by 0.86% in mIoU. This provides
+strong support for our assertion that global class centers are
+inferior compared to conditional class centers. Moreover,
+together with the image-level inter-class distance loss, our
+network achieves 47.76% mIoU on the ADE20K validation
+set, which demonstrates the effectiveness of our inter-class
+distance loss.
+4.3. Results on ADE20K
+We train the ResNet-101 on the ADE20K training set
+and report the mIoU results on the validation set (results are
+shown in Tab. 6). Our CCSNet uses a pre-trained backbone
+network on ImageNet and achieves 47.76% mIoU, which
+outperforms DeeplabV3+ by 1.41% mIoU using the same
+backbone network. Visual comparison examples are shown
+in Fig. 7.
+4.4. Results on PASCAL Context
+Tab. 7 reports the comparison results of our net-
+work and other state-of-the-the-art approaches. Based on
+ResNet-101, our method makes favourable performance
+and achieves 54.9% mIoU. CCSNet outperforms previous
+methods using the same ResNet-101 backbone.
+5. Conclusion
+In this paper, we provide a novel perspective to view typ-
+ical semantic segmentation models, and re-model the prob-
+lem as a task that models compute the similarity maps be-
+tween pixels and class centers on each scene, then assign
+Method
+centers
+Linter
+Lintra
+mIoU
+Deeplabv3+
+Wc
+Wc
+46.35
++ Ldataset
+inter
+Wc
+Wc
+Wc
+46.45
++ACCM&Lscene
+inter
+Wc
+C
+Wc
+46.94
++CCS w/o Linter
+C
+C
+47.21
+CCSNet (Ours)
+C
+C
+C
+47.76
+Table 5. Ablation study of CCS layer and CD Loss. The “centers”
+are those used to calculate the P-C similarity for prediction. For
+example, Wc denotes the learned weights of classifier, namely
+global class centers, used for final prediction, while the C indicate
+that the prediction is based on the P-C similarity using adaptive
+class centers. “Linter” and “Linter” shows whether the loss is
+applied on Wc (Ldataset
+inter
+and Ldataset
+intra
+) or C (Lscene
+inter and Lscene
+intra).
+Method
+Baseline
+mIoU(%)
+CascadeNet [40]
+VGG-16
+34.90
+RefineNet [16]
+ResNet-152
+40.7
+UperNet [16]
+ResNet-101
+42.66
+PSPNet [37]
+ResNet-101
+43.51
+PSPNet [37]
+ResNet-269
+44.94
+PSANet [38]
+ResNet-269
+43.77
+EncNet [35]
+ResNet-101
+43.77
+CFNet [36]
+ResNet-101
+44.65
+ANL [41]
+ResNet-101
+45.24
+OCRNet [32]
+ResNet-101
+45.28
+APCNet [10]
+ResNet-101
+45.38
+RGNet [28]
+ResNet-101
+45.80
+CPNet [29]
+ResNet-101
+46.27
+DeeplabV3+ [5]
+ResNet-101
+46.35
+CCSNet
+ResNet-101
+47.76
+Table 6. Results on the ADE20K validation set. Our model based
+on ResNet-101 achieves 47.76% in mIoU and outperforms all pre-
+vious methods using the same backbone network.
+Method
+Baseline
+mIoU(%)
+RefineNet [16]
+ResNet-152
+47.3
+PSPNet [37]
+ResNet-101
+47.8
+DeeplabV3+ [5]
+ResNet-101
+48.3
+EncNet [35]
+ResNet-101
+51.7
+DANet[9]
+ResNet-101
+52.6
+ANL [41]
+ResNet-101
+52.8
+CFNet [36]
+ResNet-101
+54.0
+APCNet [10]
+ResNet-101
+54.7
+RGNet [28]
+ResNet-101
+53.9
+CPNet [29]
+ResNet-101
+53.9
+OCRNet [32]
+ResNet-101
+54.8
+CCSNet
+ResNet-101
+54.9
+Table 7. Results on the PASCAL Context validation set. We report
+our result evaluated on 59 class without background. Our model
+based on ResNet-101 achieves 54.9% in mIoU.
+pixels to the semantic category with the highest P-C sim-
+ilarity. Based on this perspective, we provide solutions to
+address the two typical issues, i.e. same category yet dif-
+ferent scenes features could be of large variance, while fea-
+
+(a) Images
+(b) GT
+(c) FCN
+(d) DeeplabV3+
+(e) CCSNet (Ours)
+Figure 7. Qualitative results on ADE20K validation set.
+tures of different categories but the same scene may be quite
+similar to each other. Based on P-C similarity maps, we
+propose ACCM to generate adaptive class centers condi-
+tioned on each scene to deal with the feature variances of
+different scenes and design inter-class and intra-class dis-
+tance loss at scene-level for more inter-class distinction in-
+side each scene. Our approach is easy yet effective and can
+be plugged into most FCN-based architectures. Finally, CC-
+SNet achieves state-of-the-the-art performance on two chal-
+lenging semantic segmentation datasets.
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+Learning Data Representations with Joint Diffusion Models
+Kamil Deja 1 Tomasz Trzcinski 1 2 3 Jakub M. Tomczak 4 †
+Abstract
+We introduce a joint diffusion model that simul-
+taneously learns meaningful internal representa-
+tions fit for both generative and predictive tasks.
+Joint machine learning models that allow synthe-
+sizing and classifying data often offer uneven per-
+formance between those tasks or are unstable to
+train. In this work, we depart from a set of em-
+pirical observations that indicate the usefulness
+of internal representations built by contemporary
+deep diffusion-based generative models in both
+generative and predictive settings. We then intro-
+duce an extension of the vanilla diffusion model
+with a classifier that allows for stable joint train-
+ing with shared parametrization between those
+objectives. The resulting joint diffusion model
+offers superior performance across various tasks,
+including generative modeling, semi-supervised
+classification, and domain adaptation.
+1. Introduction
+Training a single machine learning model that can jointly
+synthesize new data as well as to make predictions about
+input samples remains a long-standing goal of machine
+learning (Jebara, 2012; Lasserre et al., 2006). Shared rep-
+resentations created with a combination of those two ob-
+jectives promise benefits on many downstream problems
+such as calibration of model uncertainty (Chapelle et al.,
+2009), semi-supervised learning (Kingma et al., 2014), un-
+supervised domain adaptation (Ilse et al., 2020) or continual
+learning (Masarczyk et al., 2021).
+Since the introduction of deep generative models such
+as Variational Autoencoders (VAEs) (Kingma & Welling,
+2014), a growing body of work takes advantage of shared
+deep neural network-based parameterization and latent vari-
+ables to build joint models. For instance, Ilse et al. (2020);
+† Work done at Vrije Universiteit Amsterdam 1Warsaw Univer-
+sity of Technology, Poland 2IDEAS NCBR, Poland 3Tooploox,
+Poland 4Eindhoven University of Technology, The Netherlands.
+Correspondence to: Kamil Deja <kamil.deja.dokt@pw.edu.pl>,
+Jakub M. Tomczak <j.m.tomczak@tue.nl>.
+Preprint.
+Tulyakov et al. (2017); Knop et al. (2020); Yang et al. (2022)
+stack a classifier on top of latent variables sampled from a
+shared encoder. Similarly, (Nalisnick et al., 2019; Perugachi-
+Diaz et al., 2021) use normalizing flows to obtain an invert-
+ible representation that is further fed to a classifier. However,
+these approaches require modifying the log-likelihood func-
+tion by scaling either the conditional log-likelihood or the
+marginal log-likelihood. This idea, known as hybrid mod-
+eling (Lasserre et al., 2006), leads to the situation where
+models concentrate either on synthesizing data or predicting
+but not on both of those tasks simultaneously.
+We address existing joint models’ limitations and lever-
+age the recently introduced diffusion-based deep generative
+models (DDGM) (Sohl-Dickstein et al., 2015; Dhariwal &
+Nichol, 2021; Kingma et al., 2021). This new family of
+methods has become popular because of the unprecedented
+quality of the samples they generate. However, relatively
+little attention was paid to their inner workings, especially
+to the internal representations built by the DDGMs. In this
+work, we fill this gap and empirically analyze those repre-
+sentations, validating their usefulness for predictive tasks
+and beyond. Then, we introduce a joint diffusion model,
+where a classifier shares the parametrization with the UNet
+encoder by operating on the extracted latent features. This
+results in meaningful data representations shared across
+discriminative and generative objectives.
+We validate our approach in several use cases where we
+show how one part of our model can benefit from the other.
+First, we investigate how DDGMs benefit from the addi-
+tional classifier to conditionally generate new samples or
+alter original images. Next, we show the performance im-
+provement our method brings in the classification task. Fi-
+nally, we extend the evaluation of our joint diffusion model
+to semi-supervised learning, domain adaptation, and coun-
+terfactual explanations. For all of those tasks, our method
+does not require any problem-specific adjustments, which
+confirms the flexibility of our approach.
+We can summarize the contributions of our work as follows:
+• We provide empirical observations with insight into
+representations built internally by diffusion models,
+on top of which we introduce a joint classifier and
+diffusion model with shared parametrization.
+arXiv:2301.13622v1  [cs.LG]  31 Jan 2023
+
+Diffusion Models Learn Data Representations
+• We introduce a conditional sampling algorithm where
+we optimize internal diffusion representations with a
+classifier.
+• We prove that our solution work with several use cases
+including the semi-supervised learning, domain adap-
+tion and counterfactual explanations.
+2. Background
+Joint models
+Let us consider two random variables: x ∈
+X and y ∈ Y. For instance, in the classification problem we
+can have X = RD and Y = {0, 1, . . . , K − 1}. The joint
+distribution over these random variables could be factorized
+in one of the following two manners:
+p(x, y) = p(x|y)p(y)
+(1)
+= p(y|x) p(x).
+(2)
+In Eq. (2), we get the conditional distribution p(y|x) (e.g., a
+classifier) and the marginal distribution p(x). For prediction,
+it is enough to learn the conditional distribution, which is
+typically parameterized with neural networks. However,
+training the joint model with shared parametrization has
+many advantages since one part of the model can positively
+influence the other.
+Diffusion-based Deep Generative Models
+In this work,
+we follow the formulation of Diffusion-based deep gen-
+erative models as presented in (Ho et al., 2020; Sohl-
+Dickstein et al., 2015). Given a data distribution x0 ∼
+q(x0), we define a forward noising process q that pro-
+duces a sequence of latent variables x1 through xT by
+adding Gaussian noise at each time step t, with a vari-
+ance of βt ∈ (0, 1), defined by a schedule β1, ..., βT ,
+namely, q(x1, . . . , xT |x0) = �T
+t=1 q(xt|xt−1), where
+q(xt|xt−1) = N(xt; √1 − βtxt−1, βtI).
+Following (Huang et al., 2021; Kingma et al., 2021;
+Tomczak, 2022; Tzen & Raginsky, 2019), we consider
+DDGMs as infinitely deep hierarchical VAEs with
+a specific family of variational posteriors;
+namely,
+Gaussian diffusion processes (Sohl-Dickstein et al.,
+2015).
+Therefore, for data point x0, and latent vari-
+ables x1, . . . , xT , we want to optimize the marginal
+likelihood
+pθ(x0)
+=
+�
+pθ(x0, . . . , xT )dx1, . . . , xT .
+We
+define
+the
+backward
+diffusion
+process
+as
+pθ(x0, . . . , xT )
+=
+p(xT )
+�T
+t=0 pθ(xt−1|xt), where
+pθ(xt−1|xt) = N(xt−1; µθ(xt, t), Σθ(xt, t)).
+We can define the variational lower bound as follows:
+ln pθ(x0) ≥ Lvlb(θ) :=
+Eq(x1|x0)[ln pθ(x0|x1)]
+�
+��
+�
+−L0
+− DKL [q(xT |x0)∥p(xT )]
+�
+��
+�
+LT
+−
+−
+T
+�
+t=2
+Eq(xt|x0)DKL [q(xt−1|xt, x0)∥pθ(xt−1|xt)]
+�
+��
+�
+Lt−1
+. (3)
+that we further optimize with respect to the parameters of
+the backward diffusion.
+Training objective
+In (Ho et al., 2020), authors notice
+that instead of estimating the probability of previous la-
+tent variable p(xt−1|xt), we can predict the added noise ϵ.
+Therefore, a single part of the variational lower bound is
+equal to:
+Lt(θ) =Ex0,ϵ
+�
+β2
+t
+2σ2
+t αt (1 − αt)×
+��ϵ − ϵθ
+�√αtx0 +
+√
+1 − αtϵ, t
+���2�
+,
+(4)
+where ϵ ∼ N(0, I) and ϵθ(·, ·) is a neural network predict-
+ing the noise ϵ from xt.
+In (Ho et al., 2020), it is also suggested to train the model
+with a simplified objective that is a modified version of
+equation (4) without scaling, namely:
+Lt,simple(θ) = Ex0,ϵ
+���ϵ − ϵθ
+�√αtx0 + √1 − αtϵ, t
+���2�
+.
+(5)
+In practice, a single shared neural network is used for mod-
+eling ϵθ. For that end, most of the works (Ho et al., 2020;
+Kingma et al., 2021; Nichol & Dhariwal, 2021) use UNet
+architecture (Ronneberger et al., 2015) that can be seen as a
+specific type of an autoencoder. This is particularly relevant
+for this work since we benefit from the Encoder – Decoder
+structure of the denoising DDGM model.
+3. Related Work
+Diffusion models There are several extensions to the base-
+line DDGM setup that aim to improve the quality of sampled
+generations (Ho et al., 2020; Huang et al., 2021; Kingma
+et al., 2021; Song & Ermon, 2019; Song et al., 2020). This
+includes noising and sampling schedulers and modified
+training objectives. Several works propose to improve the
+quality of samples from DDGMs by conditioning the gen-
+erations with class identities (Tashiro et al., 2021; Ho &
+Salimans, 2022; Huang et al.). Among those works, (Dhari-
+wal & Nichol, 2021) introduces a classifier-guided genera-
+tion, where a gradient from an externally and independently
+trained classifier is added in the process of backward diffu-
+sion to guide the generation towards a target class. On top
+
+Diffusion Models Learn Data Representations
+of this approach, (Augustin et al., 2022) present a tool for
+investigating the decision of a classifier by generating visual
+counterfactual explanations with a diffusion model. In this
+work, we simplify both of those methods benefiting from
+training a joint model with representations shared between
+a diffusion model and a classifier.
+Diffusion models and UNet representations Some works
+tackle the problem of data representation with diffusion
+models. (Abstreiter et al., 2021) introduce additional en-
+coded information to the score estimator, which allows them
+to use the score matching loss function for learning data rep-
+resentations. (Baranchuk et al., 2021) use activations from
+the pre-trained diffusion UNet model for the image segmen-
+tation task. Other works consider data representations from
+the UNet model within other generative models. (Esser
+et al., 2018) introduce a conditional UNet-based variational
+autoencoder, while (Falck et al., 2022) show the connection
+between the UNet architecture and wavelet transformation,
+applying it to the hierarchical VAEs. We show that indeed
+diffusion models learn useful representations. We further
+take advantage of that in utilizing a shared parameterization
+between a diffusion model and a classifier in a joint model.
+Joint training Apart from latent variable joint models,
+(Grathwohl et al., 2019b) show that it is possible to use a
+shared parameterization (a neural network-based classifier)
+to formulate an energy-based model. This Joint Energy-
+based Model (JEM) could be seen as a classifier if a softmax
+function is applied to logits or a generator if a Markov-chain
+Monte Carlo method is used to sample from the model.
+Although it obtains strong empirical results, gradient estima-
+tors used to train JEM are unstable and prone to diverging
+when optimization parameters are not perfectly tuned, which
+limits the robustness and applicability of this method. Alter-
+natively, Introspective Neural Networks could be used for
+generative modeling and classification by applying a single
+parameterization (Jin et al., 2017; Lazarow et al., 2017; Lee
+et al., 2018). The idea behind this class of models relies
+on utilizing a training procedure that combines adversarial
+learning and contrastive learning. Similarly to JEMs, sam-
+pling is carried out by running an MCMC method. Here,
+we propose to combine diffusion models with a classifier
+by sharing parameterization. Thus, our training is entirely
+based on the log-likelihood function, and sampling is carried
+out by backward diffusion instead of any MCMC algorithm.
+4. Diffusion models learn data representations
+As mentioned earlier, learning useful data representations is
+important for having a good generator or classifier. Ideally,
+we would like to train a joint model that allows us to obtain
+proper representations for both p(y|x) and p(x) simulta-
+neously. In this work, we investigate parameterizations
+of DDGMs and, in particular, the use of an autoencoder
+𝐱𝐭−𝟏
+𝐱𝐭
+Encoder
+Decoder
+𝐳𝐭
+𝟏
+𝐳𝐭
+𝟐
+𝐳𝐭
+𝟑
+𝐳𝐭
+Pooling
+Figure 1. Data representation zt in a UNet-based diffusion model.
+as a denoising decoder pθ(xt−1|xt). Within this architec-
+ture, the denoising function can be decomposed into two
+parts: encoding of the image at the current timestep into a
+set of features Zt = e(xt) and then decoding it to obtain
+xt−1 = d(Zt). In particular, for the UNet architecture, a set
+of features obtained from an input is a structure composed
+of several tensors with image features encoded to different
+levels, Zt = {z1
+t, z2
+t . . . zn
+t }. For simplification, for all fur-
+ther experiments, we propose to pool features encoded by
+the same filter and concatenate the averaged representations
+into a single vector zt, as presented in Fig. 1 for n = 3.
+In particular, we can use average pooling to select average
+convolutional filter activations to the whole input. Details
+of this procedure are described in Appendix A.1.
+4.1. UNet representations are useful for prediction
+First, we would like to verify whether averaged representa-
+tions z0 extracted from an original image x0 by the UNet
+contain information that is in some sense predictive. For
+that, we measure the performance of an MLP-based classi-
+fier fed with z0.
+As presented in Fig 2, representations encoded in z0 are
+indeed very informative and, in some cases (e.g., CIFAR-
+10), could lead to performance comparable to a stand-alone
+classifier with the same architecture as the combination of
+the UNet encoder and MLP but trained with the standard
+cross-entropy loss function. This observation is in line
+with (Baranchuk et al., 2021), where the same activations
+from the pre-trained diffusion model were used for semantic
+image segmentation.
+4.2. Diffusion models learn features of increasing
+granularity
+The next question is how the data representations zt differ
+with diffusion timesteps t. To investigate this issue, we train
+an unsupervised DDGM on the CelebA dataset, which we
+then use to extract the features zt at different timesteps. On
+top of those representations, we fit a binary logistic regres-
+sion classifier for each of the 40 attributes in the dataset. In
+Fig. 3, we show the performance of those regression models
+
+Diffusion Models Learn Data Representations
+Figure 2. The test-set accuracy of a stand-alone classifier compared
+to a classifier trained on top of data representations from a pre-
+trained diffusion model extracted from original images x0.
+for 6 different attributes when calculated on top of represen-
+tations from ten different diffusion timesteps. We observe
+that the model learns different data features depending on
+the amount of noise added to the original data sample. As
+presented in Fig. 3, high-grained data features such as hair
+color start to emerge at late diffusion steps (closer to the
+noise), while low-grained features (e.g., necklace or glasses)
+are not present until the early steps. This observation is in
+line with the works on denoising autoencoders where au-
+thors observe similar behavior for denoising with different
+amounts of added noise (Chandra & Sharma, 2014; Geras
+& Sutton, 2014; Zhang & Zhang, 2018).
+200
+400
+600
+800
+1000
+Timestep
+0.5
+0.6
+0.7
+0.8
+0.9
+AUC
+Blond hair
+Black hair
+Necklace
+Mouth slightly open
+Pointy nose
+Eyeglasses
+Figure 3. The area under the ROC curve (AUC) for logistic regres-
+sion models fit on data representations extracted with a pre-trained
+diffusion model at ten different diffusion timesteps. High-grained
+features are already distinguishable at late diffusion steps (closer
+to random noise), while low-grained features are only represented
+at the earlier stage of the forward diffusion.
+5. Method
+Taking into account the observations described in Section
+4, we propose to train a joint model that is composed of a
+classifier and a DDGM. Specifically, we propose to use a
+shared parameterization, namely, a shared encoder of the
+UNet architecture that serves as the generative part and for
+calculating pooled features for the classifier.
+5.1. Joint Diffusion Models: DDGMs with classifiers
+Following the procedure introduced in Sec. 4, we pool the
+latent representations of the data from different levels of the
+UNet architecture into one vector z. On top of this vector,
+we build a classifier model trained to assign a label to the
+data example represented by the vector z.
+In particular, we consider the following parameterization
+of a denoising diffusion model within a single diffusion
+timestep t, pθ(zt−1|zt). We distinguish the encoder eν
+with parameters ν that maps input xt into a set of vectors
+Zt = eν(xt), where Zt = {z1
+t, z2
+t . . . zn
+t }, i.e., a set of
+representation vectors derived from each depth level of the
+UNet architecture. The second component of the denoising
+diffusion model is the decoder dψ with parameters ψ that
+reconstructs feature vectors into a denoised sample, xt−1 =
+dψ(Zt). Together the encoder and the decoder form the
+denoising model pθ with parameters θ = {ν, ψ}. Next, we
+introduce a third part of our model, which is the classifier
+gω with parameters ω that predicts target class ˆy = gω(Zt).
+The first layer of the classifier is the average pooling that
+results in a single representation zt.
+ො𝐲
+𝐱𝟎
+𝐱𝐓
+𝐱𝐭−𝟏
+𝐱𝐭
+(a) The parameterization of our joint diffusion
+𝐱𝟎
+𝐱𝐓
+𝐱𝐭−𝟏
+𝐱𝐭
+ො𝐲𝐓
+ො𝐲𝐭
+ො𝐲𝐭−𝟏
+(b) Additional noisy classifiers
+Figure 4. The parameterization of our joint diffusion model.
+(a) Each step in the backward diffusion is parameterized by a
+shared UNet. The classifier uses the encoder of the UNet together
+with the average pooling (green) and additional layers (yellow).
+(b) An alternative training that additionally uses the classifier for
+noisy images xt (t > 0).
+In our approach, we consider a classifier that takes the origi-
+nal image x0 for which a vector of probabilities is returned ϕ
+and eventually the final prediction is calculated, ˆy = gω(x0).
+The visualization of our shared parameterization is pre-
+sented in Figure 4(a). As a result, our model could be writ-
+ten as follows pν,ψ,ω(x0:T , y) = pν,ω(y|x0) pν,ψ(x0:T ),
+and applying the logarithm yields:
+ln pν,ψ,ω(x0:T , y) = ln pν,ω(y|x0) + ln pν,ψ(x0:T ).
+(6)
+The logarithm of the joint distribution (6) could serve as
+the training objective in which ln pθ(x0:T ) could be either
+
+100%
+75%
+Accuracy
+50%
+25%
+0%
+FashionMNIsT
+SVHN
+CIFAR-10
+CIFAR-100
+I Standard classfier
+: Classifier on pre-trained DDGMDiffusion Models Learn Data Representations
+approximated by the ELBO for the diffusion-based model
+in (3) or the simplified objective with (5)). In this paper, we
+follow the simplified objective:
+Lt,diff(ν, ψ) = Ex0,ϵ
+�
+∥ϵ − ˆϵ∥2�
+,
+(7)
+where ˆϵ is a prediction from the decoder:
+{z1
+t, z2
+t . . . zn
+t } =eν
+�√αtx0 +
+√
+1 − αtϵ, t
+�
+(8)
+ˆϵ =dψ({z1
+t, z2
+t . . . zn
+t }).
+(9)
+For the classifier, we use the logarithm of the categorical
+distribution, i.e., the cross-entropy loss:
+Lclass(ν, ω) = −Ex0,y
+��K−1
+k=0 1[y = k] log
+exp(ϕk)
+�K−1
+c=0 exp(ϕc)
+�
+,
+(10)
+where y is the target class, ϕ is a vector of probabilities
+returned by the classifier gω(eν(x0)), and 1[y = k] is the
+indicator function that is 1 if y equals k, and 0 otherwise.
+The final loss function in our approach is then the following:
+L(ν, ψ, ω) = Lclass(ν, ω)+
+(11)
+− L0(ν, ψ) −
+T
+�
+t=2
+Lt,diff(ν, ψ) − LT (ν, ψ).
+We optimize the objective in (11) jointly with a single opti-
+mizer over parameters {ν, ψ, ω}.
+5.2. An alternative training of joint diffusion models
+The training of the proposed approach over a batch of data is
+straightforward. For a sampled pair (x0, y), the example x0
+is first noised with a forward diffusion to a random timestep,
+xt so that the training loss for the denoising model is a
+Monte-Carlo approximation of the sum over all timesteps.
+Then x0 is fed to a classifier that returns probabilities ϕ, and
+the cross-entropy loss is calculated for given y.
+However, as discussed in Section 4.2, the diffusion model
+trained even in a fully unsupervised manner provides data
+representations related to the different granularity of input
+features at various diffusion timesteps. Considering this, we
+can improve the robustness of our method by applying the
+same classifier to intermediate noisy images xt (0 < t <
+T), which by reason adds the cross-entropy losses for xt,
+namely:
+Lt
+class(ν, ω) = −Ex0,y
+��K−1
+k=0 1[y = k] log
+exp(ϕt
+k)
+�K−1
+c=0 exp(ϕtc)
+�
+,
+(12)
+where ϕt
+k is a vector of probabilities given by gω(eν(xt)).
+Then the extended objective (11) is the following:
+LT (ν, ψ, ω) = L(ν, ψ, ω) +
+�
+t∈T
+Lt
+class(ν, ω),
+(13)
+where T ⊆ {1, 2, . . . , T} is the set of timesteps. These
+additional noisy classifiers are schematically depicted in
+Figure 4(b) in which we highlight that the model is reused
+across various noisy images. It is important to mention that
+the noisy classifiers serve only for training purposes; they
+are not used for prediction. This procedure is similar to the
+data augmentation technique, where random noise is added
+to the input (Sietsma & Dow, 1991).
+5.3. Conditional sampling in joint diffusion models
+To improve the quality of samples generated by DDGM,
+(Dhariwal & Nichol, 2021) propose a classifier guidance
+approach, where an externally trained classifier can be used
+to guide the generation of the DDGM trained in an unsu-
+pervised way towards the desired class. In the standard
+DDGM, at each backward diffusion step, an image is sam-
+pled from the output of the diffusion model pθ according to
+the following formula:
+µ, Σ ←µθ (xt) , Σθ (xt)
+xt−1 ← sample from N (µ, Σ)
+(14)
+(Dhariwal & Nichol, 2021) proposed to change the second
+line of this equation and add a scaled gradient with respect
+to the target class from an externally trained classifier c(·)
+directly to the output of the denoising model:
+xt−1 ← sample from N (µ + sΣ∇xtc(xt), Σ) ,
+(15)
+where s is a gradient scale.
+With the joint training of a classifier and diffusion model
+introduced in this work, we propose to simplify the clas-
+sifier guidance technique. Using the alternative training
+introduced in the previous section, Section 5.2, we can use
+noisy classifiers to formulate conditional sampling. The
+encoder model eν encodes input data xt into the represen-
+tation vectors Zt that are used to both denoise an example
+into the previous diffusion timestep xt−1 ∼ dψ (Zt) as well
+as to predict the target label with a classifier ˆy = gω (Zt).
+Therefore, to guide the model towards a target label during
+sampling, we propose optimizing the representations Zt ac-
+cording to the gradient calculated through the classifier with
+respect to the desired class. The overview of this procedure
+is presented in the algorithm Algorithm 1.
+For the reformulation of the diffusion model proposed
+by (Ho et al., 2020) where instead of predicting the previous
+timestep xt−1 denoising model is optimized to predict noise
+ϵ that is subtracted from the image at the current timestep
+xt, we adequately change the optimization objective. In-
+stead of optimizing the noise to be specific to the target
+class, we optimize it to be anything except for the target
+class, which we implement by changing the optimization
+direction: Z′
+t ← Zt + α∇Zt log gω(y|Zt).
+
+Diffusion Models Learn Data Representations
+Table 1. An evaluation of generative capabilities by measuring the FID score, Precision and Recall of generations from various diffusion-
+based models, including our joint diffusion model.
+Model
+FashionMNIST
+CIFAR-10
+CIFAR-100
+CelebA
+FID ↓
+Prec ↑
+Rec ↑
+FID ↓
+Prec ↑
+Rec ↑
+FID ↓
+Prec ↑
+Rec ↑
+FID ↓
+Prec ↑
+Rec ↑
+DDGM
+7.8
+71.5
+65.3
+7.2
+64.8
+61.2
+29.7
+70.0
+47.8
+5.6
+66.5
+58.7
+DDGM (classifier guidance)
+7.9
+66.6
+59.5
+8.1
+63.2
+63.3
+22.1
+69.3
+46.9
+4.9
+66.0
+57.8
+Ours
+8.7
+71.1
+61.1
+10
+66.4
+56.3
+17.4
+63.2
+54
+7.5
+66.7
+50.7
+Ours (conditional sampling)
+5.9
+63.1
+63.2
+9.4
+66.8
+54.7
+16.8
+63.5
+54.1
+6.6
+64.2
+54.5
+Algorithm 1 Sampling with optimized representations
+given a diffusion model (an encoder eν(Zt|xt), a decoder
+dφ(xt−1|Zt)), a classifier gω(y|Zt), and a step size α.
+Input: class label y, step size α
+xT ← sample from N(0, I)
+for all t from T to 1 do
+Zt ← eν(xt)
+Z′
+t ← Zt − α∇Zt log gω(y|Zt)
+µ, Σ ← dψ(Z′
+t)
+xt−1 ← sample from N(µ, Σ)
+end for
+return x0
+6. Experiments
+In the experiments, we aim for observing the benefits of the
+proposed joint diffusion model over a stand-alone classifier
+or a marginal diffusion model. To that end, we run a series
+of experiments to verify various properties, namely:
+• We measure the quality of a classifier to evaluate
+whether training together with a diffusion model im-
+proves the robustness of the classifier.
+• We measure the generative capability of our model to
+check if representations optimized by the classifier can
+lead to more accurate conditional generations.
+• We train our model in a semi-supervised setup to see
+if shared representations between the classifier and the
+diffusion model can positively influence the classifica-
+tion accuracy for a limited number of labeled data.
+• We use a domain-adaptation task to check if optimizing
+the representations using our approach helps to adapt
+to new data compared to a stand-alone classifier.
+• We show that our joint model learns abstract features
+that can be used for the counterfactual explanation.
+We use a UNet-based model with a depth level of three in
+all experiments. We pool its latent features with average
+pooling into a single vector, on top of which we add a clas-
+sifier with two linear layers and the LeakyReLU activation.
+All metrics are reported for the standard training with the
+objective in (11), except for the conditional sampling where
+we additionally train the classifier on noisy samples, i.e.,
+additional losses as in (13). Hyperparameters and training
+details are included in the appendix and code repository1.
+6.1. Predictive performance of joint diffusion models
+In the first experiment, we evaluate the predictive perfor-
+mance of our method. To that end, we report the accu-
+racy of our model on four datasets: FashionMNIST, SVHN,
+CIFAR-10, and CIFAR-100. We compare our method with
+a baseline classifier trained with a standard cross-entropy
+loss and the MLP classifier trained on top of representations
+extracted from the pre-trained DDGM as in Section 4. The
+results of this experiment are presented in Table 2.
+As noticed before, a classifier trained on features extracted
+from the UNet of a DDGM pre-trained in an unsupervised
+manner achieves reasonable performance. However, it is
+always outperformed by a stand-alone classifier. Interest-
+ingly, the proposed joint diffusion model achieves the best
+performance on all four datasets. The reason for that could
+be two-fold. First, training a partially shared neural network
+(i.e., the encoder in the UNet architecture) benefits from the
+unsupervised training, similarly to how the pre-training us-
+ing Boltzmann machines benefited finetuning of deep neural
+networks (Hinton et al., 2006). Second, the shared encoder
+part is more robust since it is used in the backward diffusion
+for images with various levels of noise.
+Table 2. The classification accuracy calculated on the test sets. For
+each training, we used exactly the same architecture.
+Model
+F-MNIST
+SVHN
+CIFAR-10
+CIFAR-100
+Classifier
+92.0%
+95.1%
+81%
+60.8%
+Ours (pre-trained DDGM)
+60.6%
+79.6%
+80.9%
+43.8%
+Ours
+93.3%
+95.4%
+89.9%
+63.6%
+6.2. Generative performance of joint diffusion models
+In the second experiment, we check how adding a classifier
+in our joint diffusion models influences the generative per-
+formance. We use the FID score to quantify the quality of
+data synthesis. Additionally, we use distributed Precision
+1Anonymized due to the double-blind policy.
+
+Diffusion Models Learn Data Representations
+(Prec), and Recall (Rec) for assessing the exactness and
+diversity of generated samples (Sajjadi et al., 2018). For our
+joint diffusion model, we consider samples from the prior
+let through the backward diffusion. We also use the second
+sampling scheme in which we use conditional sampling,
+namely, the optimization procedure as described in Section
+5.3. We compare our approach with a vanilla DDGM, and a
+DDGM with classifier guidance (Dhariwal & Nichol, 2021).
+Overall, all methods performed similarly. In some cases, the
+vanilla DDGM and the DDGM with the classifier guidance
+obtain better results in terms of the FID score (CIFAR-10,
+CelebA) and Precision (FashionMNIST, CIFAR-100). How-
+ever, our joint diffusion model with conditional sampling
+outperforms the DDGMs in terms of the FID score (Fash-
+ionMNIST, CIFAR-100), Precision (CIFAR-10), and Recall
+(FashionMNIST, CIFAR-100). Hence, we conclude that
+all models perform well on the data synthesis task. Con-
+ditional sampling is beneficial in the joint diffusion model
+and yields better-quality generations. This could result from
+the fact that the optimization procedure drives Zt to a mode.
+Eventually, the backward diffusion generates better samples.
+0
+500
+1000
+Step size α
+40
+60
+Precision
+CIFAR10
+FashionMNIST
+0
+500
+1000
+Step size α
+40
+50
+60
+70
+Recall
+CIFAR10
+FashionMNIST
+Precision
+Recall
+Figure 5. The dependency between the value of the step size α
+and the value of Precision and Recall for the joint diffusion with
+conditional sampling.
+To get further insight into the role of conditional sampling,
+we carried out an additional study for the varying value of
+α (the step size in Algorithm 1). In Figure 5, we present
+how Precision and Recall change for different values of
+this parameter. Apparently, increasing the step size value α
+leads to more precise but less diverse samples. This is rather
+intuitive behavior because larger steps result in features Zt
+closer to modes. There seems to be a sweet spot around
+α ∈ [100, 250] for which both measures are high.
+𝛼 = 0
+𝛼 = 100
+𝛼 = 500
+𝛼 = 1000
+Figure 6. Samples from our joint diffusion model optimized to-
+wards a specific class (here: plane) with different step size α.
+We visualize this effect in Figure 6. For a chosen class, e.g.,
+plane, we observe that the larger α, the more precise the
+samples are but with limited diversity (i.e., the background
+is almost the same). For more samples, see Appendix B.
+6.3. A comparison to state-of-the-art approaches
+To get a better overview of the performance of our joint
+diffusion model, we present a comparison with other joint
+models and SOTA discriminative and generative models in
+Table 3. Importantly, we present the discriminative model
+and the generative model as the bounds of the performance.
+Importantly, within the class of the joint models, our joint
+diffusion performs on par with Joint Energy-based Model
+(JEM) (Grathwohl et al., 2019a) in terms of classification
+accuracy, but it visibly outperforms JEM in terms of the FID
+score. Moreover, our approach performs significantly better
+than flow-based methods (Residual Flows, Glow).
+Table 3. A comparison of our joint diffusion model with other
+joint models, and the SOTA discriminative model, and the SOTA
+generative model on the CIFAR-10 test set.
+Class
+Model
+Accuracy% ↑
+FID↓
+Joint
+Residual Flows (Chen et al., 2019)
+70.3
+46.4
+Glow (Kingma & Dhariwal, 2018)
+67.6
+48.9
+JEM (Grathwohl et al., 2019a)
+92.9
+38.4
+Ours
+89.9
+9.4
+Disc.
+VIT-H (Dosovitskiy et al., 2020)
+99.5
+N/A
+Gen.
+DDGM (our implementation)
+N/A
+7.2
+LSGM (Vahdat et al., 2021)
+N/A
+2.1
+6.4. Semi-supervised learning of joint diffusion models
+With satisfactory performance, we further evaluate other
+setups where one part of the model can benefit from another.
+In particular, we propose to assess our approach in the semi-
+supervised setup, where we artificially limit the amount of
+labeled data to 10%, 5% or 1% in three datasets SVHN,
+CIFAR-10, and CIFAR-100. We compare joint diffusion
+models to a deep neural network-based classifier and a deep
+neural network-based classifier on top of the pre-trained
+UNet encoder. The results are presented in Table 4.
+In the case of the stand-alone classifier, we observe that
+classification accuracy drastically drops with the number of
+labeled data. However, in our joint diffusion model, we can
+train the classifier on the smaller dataset while still optimiz-
+ing the generator part in an unsupervised manner, with all
+available unlabelled data. This approach significantly im-
+proves the classifier’s performance thanks to the improved
+quality of data representations. For CIFAR-10, we observe
+that the joint diffusion model with only 5% of labeled data
+(250 examples per class) performs almost as well as the
+stand-alone classifier trained with the fully labeled train-
+ing dataset. In more extreme scenarios, e.g., labeled data
+limited to 50, 25, or 5 examples per class, it seems to be
+slightly more beneficial to first learn the data representation
+in an unsupervised way and then add the classifier on top
+of them. However, overall, the joint diffusion model per-
+forms extremely well and greatly benefits from available
+unlabeled data in terms of classification accuracy. Our ex-
+
+Diffusion Models Learn Data Representations
+Table 4. The accuracy of the classifier trained in the semi-supervised setup, for each dataset we train the classifier with the fully labeled
+data or a limited amount of labeled examples and the remaining unlabelled examples. We compare standard classifier with classifier
+trained on a pre-trained DDGM as presented in Sec 4 and our joint diffusion method.
+SVHN
+CIFAR-10
+CIFAR-100
+Labelled data
+100%
+5%
+1%
+100%
+5%
+1%
+100%
+10%
+5%
+1%
+Images per class
+10000
+500
+100
+5000
+250
+50
+500
+50
+25
+5
+Classifier
+95.1
+87.8
+75.15
+81
+46.4
+31.5
+60.8
+22.2
+16.6
+6.9
+Ours (pre-trained DDGM)
+79.6
+51.7
+66.0
+80.9
+75.1
+65.3
+43.8
+33.9
+28.8
+15.4
+Ours
+95.4
+90.2
+76.7
+89.9
+78.2
+64.7
+63.6
+38.6
+21.5
+11.5
+periments align with the observation by (Baranchuk et al.,
+2021), where DDGMs were used to improve the perfor-
+mance in semi-supervised image segmentation.
+6.5. Domain adaptation with diffusion-based
+fine-tuning
+In the previous section, we evaluate whether the classifier
+can benefit from the generative part of our model when
+trained with limited access to labeled data. Now, we further
+extend those experiments and check if joint diffusion can
+adapt to the new data domain using only the generative part
+– in a fully unsupervised way. For this purpose, we run an
+experiment in which we first train the model on the source
+labeled data to retrain it on the target dataset without access
+to the labels. We compare our approach to a standalone
+deep neural network-based classifier, see Table 5.
+Table 5. The classification accuracy of the classifier trained in a
+domain adaption task. We first train the joint model on the source
+dataset, which we adapt to the target domain by retraining it using
+only the diffusion loss for the examples in the target one.
+SVHN → MNIST
+USPS → MNIST
+MNIST → USPS
+Classifier
+78.8
+54.7
+72.2
+Ours
+85.5
+90.5
+92.7
+Classifier on target
+(upper bound)
+96.1
+96.8
+99.4
+As expected, in all three scenarios, the classification accu-
+racy of the stand-alone classifier degrades on a target do-
+main.2 However, having access to unlabeled data from the
+target domain allows our joint diffusion model to adapt sur-
+prisingly well. Our approach outperforms the stand-alone
+classifier in all three cases by a significant margin. This
+result indicates that learning low-level features is essential
+for obtaining good predictive power while it is enough to
+transfer the classification head unchanged.
+6.6. Visual Counterfactual Explanations
+In the last experiment, we apply our joint diffusion model to
+real-world medical data, the MALARIA dataset (Rajaraman
+et al., 2018), that includes 27,558 cell images that are either
+2The classification accuracy does not drop to a random level
+because all datasets share the same task, i.e., digits classification.
+Negative examples
+Positive examples
+Figure 7. Data samples from the Malaria dataset classified as nega-
+tive examples (left) or parasitized cells (right). (top row) original
+data examples, (2nd row) data noised with 20% of forward diffu-
+sion steps, (3rd row) denoised images with conditional sampling,
+(bottom row) the difference between the 3rd and 4th rows.
+infected by the malaria parasite or not (a classification task).
+The cells have various shapes and different staining (i.e.,
+colors) and contain or not the parasite (visually apparent as
+a purple dot).
+After training our joint diffusion model, we obtain high
+classification accuracy (98%) on the test set. On top of
+this, we introduce an adaptation of visual counterfactual
+explanations (VCE) method (Augustin et al., 2022) that
+provides an answer to the question: What is the minimal
+change to the input image x0 to change the decision of the
+classifier. In our setup, we answer this question with a
+conditional sampling algorithm that we use to generate the
+counterfactual explanations. In Figure 7, we show a few
+examples from the negative (left) or positive (right) classes.
+We add 20% of noise to these images and run conditional
+sampling with the opposite class (i.e., changing negative
+examples to positive ones and vice versa). In both cases,
+the joint diffusion model with conditional sampling can
+either remove the parasite from the image (for the positive
+examples) or add the parasite to the image (for the negative
+ones). All presented images are not cherry-picked.
+This experiment shows that not only we can use our pro-
+posed approach to obtain a powerful classifier but also to
+visualize some regions of interest. In the considered case,
+calculating the difference between the original example and
+the image with a changed class label indicates the malaria
+plasmodium (see the last row in Figure 7). We provide more
+examples from the CelebA data in the Appendix C.
+
+Diffusion Models Learn Data Representations
+7. Conclusion
+In this work, we introduced a joint model that combines a
+diffusion model and a classifier through shared parameteri-
+zation. We first experimentally demonstrated that DDGMs
+learn semantically meaningful data representations that
+could be used for classification.
+Next, we showed that our approach improves the perfor-
+mance of the classification task while retaining the high
+quality of generations and enables conditional generations
+with built-in classifier guidance. Additionally, we show that
+the joint diffusion model can be used in semi-supervised
+learning, domain adaptation, and for counterfactual expla-
+nations, without any changes to the original setup.
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+
+Diffusion Models Learn Data Representations
+Appendix
+A. Training details and hyperparameters
+A.1. Pooling of the UNet features
+As discussed in Section 4, we pool the UNet features encoded to different UNet levels with the average pooling function.
+Precisely speaking, we take an average convolutional filter activation for a given filter across the whole image. This approach
+seems to result in the loss of information, such as the location of particular features extracted by the convolutional filter, but
+it allows us to create image representation with reasonable dimensionality. Depending on the dimensionality of input, with
+our method, we extract 1856 features for 28 × 28 Grayscale images (e.g. MNIST), 3712 features for 32 × 32 images with 3
+color channels (e.g. CIFAR), and 5248 features for 64 × 64 images with 3 color channels (CelebA).
+In all of our experiments, we use average pooling. Although other options such as max or min pooling might be used, our
+approach ensures that all of the features across the whole image are shared between the classifier and the generative models.
+A.2. Semi-supervised learning
+In our semi-supervised learning, we train our joint diffusion model on datasets with limited access to labeled samples. The
+simplest approach for this problem is to calculate the loss function on the diffusion using the whole batch of data while
+using only the labeled examples for the classifier loss. However, in some scenarios, we artificially omit up to 99% of labeled
+data. In practice, this would lead to a situation where for batch size equal to 128 or 254 examples, the classifier loss would
+be practically calculated on 1 or 2 samples. Therefore, to stabilize the training we propose to create a buffer where we put
+labeled examples from each batch. When the buffer reaches its capacity equal to the batch size, we calculate the classifier
+loss using the examples from the buffer and add it to the generative loss according to Equation 11.
+A.3. Domain adaptation
+In the experiments on the domain adaptation task, we propose the simplest setup, where we first train the joint model on the
+source task using the joint loss function (Eq. 11), and then we retrain the model on the target domain using only the DDGM
+loss in Equation 7. We show that without any alteration to our basic setup, we can observe a significant performance boost
+compared to the baseline classifier. We believe that we can further improve those results if we focus directly on the domain
+adaptation task and take advantage of the recent advantages in this field. Further experiments in this direction should for
+example include simultaneous training on examples from two domains. To improve the alignment, we can also benefit from
+adversarial training as introduced by (Ganin et al., 2016) in DANN.
+
+Diffusion Models Learn Data Representations
+B. Additional results: Conditional generations with optimized representations
+In Figure 8 we present how the decision of the classifier changes for sampling with the optimized generations. With a higher
+α step size value, optimization converges faster towards target classes. Interestingly, for the CIFAR10 dataset, there are
+certain classes (e.g., class 3) that converge later in the backward diffusion process than the others. In Figure 8 we also
+present associated samples from our model. Once more they depict that the higher value of the α parameter leads to more
+precise but less diverse samples.
+0
+200
+400
+600
+800
+1000
+Timestep
+0
+2
+4
+6
+8
+Avg. prediction
+9
+7
+6
+4
+3
+1
+0
+(a) α = 200
+0
+200
+400
+600
+800
+1000
+Timestep
+0
+2
+4
+6
+8
+Avg. prediction
+9
+7
+6
+4
+3
+1
+0
+(b) α = 1000
+Figure 8. CIFAR10: Classifier decisions at different diffusion steps, for conditional sampling with different values of step size α and
+associated conditional samples
+
+Diffusion Models Learn Data Representations
+Fashion MNIST
+CIFAR-100
+Figure 9. Conditional samples from our joint diffusion model for Fashion MNIST dataset (left) and first 10 classes of CIFAR100 dataset
+(right). Each row represents samples from one class.
+Fashion MNIST
+CIFAR-100
+CelebA
+Figure 10. Generated examples from our joint diffusion model without conditional sampling for CIFAR-10, CIFAR-100, and CelebA
+dataset.
+
+oDiffusion Models Learn Data Representations
+C. Additional results: Counterfactual image generation
+In the experiment described in Section 6.6, we presented how we can use our joint diffusion model to generate the
+counterfactual explanations to the classifier using the medical dataset. In Figure 11, we present more examples of this
+approach by perturbing original examples from the CelebA dataset. We select 3 attributes from the CelebA dataset namely:
+young, smiling, and moustache. For each attribute, we select 5 positive examples and 5 negative examples which we alter
+using our conditional sampling procedure with the classifier-based optimization. We present original examples (first row)
+noised with 20% of noise (second row) and generated towards counterfactual class (third row). In the last row, we show the
+differences between the original and modified examples.
+(a) Young to old (left), old to young (right)
+(b) Smiling to no-smiling (left), no-smiling to smiling (right)
+(c) Moustache to no-moustache (left), no-moustache to moustache (right)
+Figure 11. Counterfactual image generation for the CelebA dataset using three different attributes on random original examples. For each
+attribute, we select 5 positive examples that we change to negative ones and 5 negative ones that we change to positive ones.
+

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+1 
+ 
+The Real Number n-Degree Pythagorean Theorem 
+ 
+Jeffrey S. Lee1,2,4 
+Gerald B. Cleaver1,3 
+ 
+1Early Universe Cosmology and Strings Group 
+2Space Physics Numerical Modeling Group 
+Center for Astrophysics, Space Physics, and Engineering Research 
+3Department of Physics 
+4Department of Geosciences 
+Baylor University 
+One Bear Place 
+Waco, TX 76706 
+ 
+Jeff_Lee@Baylor.edu 
+Gerald_Cleaver@Baylor.edu 
+ 
+ 
+Key Words: ∞-degree Pythagorean Theorem, Law of Cosines, maximum triangle area, minimum triangle 
+area 
+ 
+Word Count: 4203 
+ 
+Abstract 
+This paper extends the Pythagorean Theorem to positive and negative real exponents to take the form 
+n
+n
+n
+a
+b
+c
++
+=
+ and makes use of the definition 
+1
+b
+a
+ =
+ . For the case of n
++
+
+, 
+1
+n   is necessary for 
+the vertex angle to be real, and there are no restrictions on γ beyond its definition. However, for n
+−
+
+, 
+two significant restrictions that are necessary for 
+n
+n
+n
+a
+b
+c
++
+=
+ to yield real vertex angles have been 
+discovered by this work: 1
+2
+
+
+
+, and n cannot exceed a critical value which is γ-dependent. 
+Additionally, the areas of the associated triangles have been determined as well as the conditions for those 
+areas to be maxima or minima.  
+1. Introduction 
+ 
+For centuries, the Pythagorean Theorem has been significant in the foundation of mathematics. 
+Although the finally proven Fermat’s Last Theorem [1] definitively establishes the non-existence of 
+Pythagorean Triples for 
+n
+n
+n
+a
+b
+c
++
+=
+ with 
+|
+2
+n
+n
+
+
+, the Pythagorean Theorem can be extended to 
+higher degrees which are not required to be positive integers. However, only positive integers possess the 
+physical representation of dimension for 
+n
+n
+n
+a
+b
+c
++
+=
+ (i.e., 
+1
+n =  defines a scalar sum of straight line 
+segments; 
+2
+n =
+ defines a scalar sum of areas; 
+3
+n =
+ defines a scalar sum of volumes; and 
+4
+n 
+ defines 
+a scalar sum of hypervolumes; 
+0
+n =
+ is mathematically meaningless because it results in 
+0
+0
+0
+2
+1
+a
+b
+c
++
+=
+→
+= ). 
+ 
+ 
+
+2 
+ 
+The extension of the Pythagorean Theorem to higher dimensions using a variety of methods has been 
+extensively addressed in the literature [2-15], most frequently by considering a formulation such as 
+2
+2
+Total
+1
+n
+k
+k
+a
+a
+=
+=
+. Rather, the n-degree Pythagorean Theorem contains a relationship between the ratio of the 
+adjacent side lengths (i.e., γ) and the vertex angle (θ). n
+ indicates that the triangles to which 
+n
+n
+n
+a
+b
+c
++
+=
+ is being applied are not necessarily right angled. 
+1
+   is defined as a precondition with no 
+loss of generality. If 0
+1
+
+
+ , the 
+1
+   triangle has merely been rotated within its plane, and 
+0
+ 
+ 
+implies the geometrically uninteresting imposition upon the triangle of non-physical side lengths. 
+ 
+The vertex angle θ is a function of γ and n and is therefore denoted (
+)
+,n
+ 
+. For 0
+1
+n
+
+ , the 
+vertex angle is complex (not addressed in this paper). For 1
+2
+n
+
+
+, the triangle is obtuse with 
+(
+)
+90
+,
+180
+n
+ 
+
+
+, and for 
+2
+n 
+, the triangle is acute with (
+)
+,
+90
+n
+ 
+
+. In each instance of 
+|
+1
+n
+n
++
+
+ , 
+(
+)
+0
+,
+180
+n
+ 
+
+
+, and there are no restrictions on the ratio of the adjacent side lengths 
+(other than 
+1
+  ). 
+ 
+When n
+−
+
+, the situation changes significantly. In order that (
+)
+,n
+ 
+
+, the restriction 
+1
+2
+
+
+
+ must be imposed. However, even with 1
+2
+
+
+
+, there is one additional compulsory restriction 
+for (
+)
+,n
+ 
+
+: if 
+1
+  , the degree of the negative real exponent Pythagorean Theorem must not 
+exceed a critical value which is dependent on γ 
+(
+)
+(
+)
+(
+)
+crit
+crit
+i.e., 
+1 , and thus, 
+1
+n
+n
+n
+n
+
+
+
+
+
+
+. If 
+(
+)
+crit
+1
+n
+n
+
+
+
+, the vertex angle is complex (also not addressed in this paper). 
+ 
+For 
+|
+1
+n
+n
++
+
+ , the areas of the associated triangles, with fixed a and γ values, reach a maximum 
+which occurs when the triangle is right isosceles (
+)
+i.e., 
+1 and 
+90
+
+
+=
+=
+; the areas increase as n → . 
+Conversely, if the perimeter of a triangle is fixed, the triangle area approaches a maximum value for 
+increasing n and approaches 0 for decreasing γ.  
+ 
+For 
+( )
+crit
+|
+ if 
+1
+n
+n
+n
+
+
+−
+
+
+ , the areas of the associated triangles with a fixed perimeter are 
+maximized for 
+2
+ =
+ and as n → − . For finite n, the maximum area occurs when (
+)
+,n
+ 
+ is a 
+maximum.  
+2. The n-Degree Pythagorean Theorem with Positive Real Exponents 
+ 
+For the general scalene triangle in Figure 1, the n-degree Pythagorean Theorem can be written as  
+ 
+n
+n
+n
+a
+b
+c
++
+=
+ 
+(1) 
+ 
+which must also conform to the standard Law of Cosines, 
+ 
+2
+2
+2
+2
+cos
+c
+a
+b
+ab
+
+=
++
+−
+, 
+(2) 
+
+3 
+ 
+ 
+with n
++
+
+.  
+ 
+In this work, the extension of the Pythagorean exponents to numbers other than n
++
+
+ does not 
+extend with it the Law of Cosines because unlike in [10, 12] because an n-dimensional simplex does not 
+arise. Here, the objective is the solution for the vertex angle θ such that eq. (1) conforms to eq. (2).  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+ 
+Figure 1: A scalene triangle. 
+ 
+Using 
+1
+b
+a
+ =
+ , rewriting eq. (1) as  
+ 
+(
+)
+2
+2
+n
+n
+n
+c
+a
+b
+=
++
+, 
+(3) 
+ 
+and equating eqs. (2) and (3) yields 
+ 
+(
+)
+2
+2
+1
+1
+1
+cos
+2
+n
+n
+
+
+
+
++
+−
+
+
++ −
++
+
+
+=
+
+
+
+
+
+
+, 
+(4) 
+where 
++  denotes that the vertex angle arises from a version of the n-degree Pythagorean Theorem in 
+which n
++
+
+. 
+ 
+If 
+0
+n =
+, as stated above, no triangle exists; there is no 0-degree (or 0-dimensional) Pythagorean 
+Theorem. If 0
+1
+n
+
+ ,   is a complex angle for all side ratios γ and is not applicable here. However, if 
+1
+n  ,   is always a real angle, and therefore, a corresponding real triangle does exist. As expected, for 
+the 
+1
+n =  case, eq. (4) becomes 
+(
+)
+(
+)
+1
+1
+cos
+1
+180o
+n
+
++
+−
+=
+=
+−
+=
+ which is sensible because the triangle 
+has collapsed into a straight line which is independent of the side ratio. For the 
+2
+n =
+ case, eq. (4) 
+θb 
+c 
+a 
+b 
+θa 
+θ 
+
+4 
+ 
+becomes 
+(
+)
+( )
+1
+2
+cos
+0
+90o
+n
+
++
+−
+=
+=
+=
+, and the traditional Pythagorean Theorem with a 90o vertex 
+angle, also independent of the side ratio, is recovered. 
+ 
+It is important to note that the vertex angle exists only for combinations of n and γ such that the 
+argument of the inverse cosine function in eq. (4) is not greater than 1 or less than -1, 
+(
+)
+2
+2
+1
+1
+i.e., 
+1
+2
+n
+n
+
+
+
+
+
++ −
++
+
+
+
+
+
+
+
+
+
+. However, the stipulation that 
+1
+   ensures that eq. (4) will always result 
+in real angles. Therefore, there are no forbidden side ratios, and eq. (4) applies without restriction for 
+1
+n  . 
+ 
+2.1 The 1 ≤ n ≤ 2 Case  
+ 
+If 1
+2
+n
+
+
+, the range of vertex angles is between 90o and 180o (as shown above). Plots of eq. (4) are 
+shown in Figure 2 and Figure 3.  
+ 
+Figure 2: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean 
+exponent for 1 ≤ n ≤ 2. 
+
+Vertex Angle (deg.)
+2
+180
+1.9
+170
+1.8
+160
+1.7
+150
+Pythagorean Exponent
+1.6
+140
+1.5
+130
+1.4
+120
+1.3
+110
+1.2
+1.1
+100
+06
+2
+3
+4
+5
+6
+8
+6
+10
+Side Ratio5 
+ 
+ 
+Figure 3: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean 
+exponent for 1 ≤ n ≤ 2. 
+For any given value of n, there is a corresponding value of γ that results in a maximum vertex angle 
+which is seen in Figure 3. That vertex angle and the side ratio that gives rise to it are found by equating 
+the first derivative of eq. (4) to zero.  
+ 
+(
+)(
+)
+2
+2
+0
+1
+1
+1
+0
+n
+n
+n
+n
+d
+d
+
+
+
+
+
++
+−
+=
+ −
++
+−
++
+=
+ 
+(5) 
+ 
+The solution to eq. (5) for ( )
+n
+
+ is not analytic, and the total number of complex solutions grows 
+rapidly with increasing n. However, it is clear that for all values of n, 
+1
+ =  is a solution – it is the only 
+positive real solution. Thus, the largest vertex angle occurs in an isosceles triangle. Therefore, by 
+substituting 
+1
+ =  into eq. (4), the maximum vertex angle can be found (eq. (6)).  
+ 
+2
+1
+max
+cos
+1
+2
+n
+n
+
++
+−
+− 
+
+=
+−
+
+
+
+
+, 
+(6) 
+where 
+max
+
++  denotes the maximum value of 
++ for any degree n. 
+ 
+The second derivative of eq. (4) is extremely unruly, and therefore, the confirmation that 
+max
+
++  is a 
+maximum angle was performed numerically with Maple®. A plot of eq. (6) is shown in Figure 4.  
+
+6 
+ 
+ 
+Figure 4: Maximum vertex angle versus Pythagorean exponent for 1 ≤ n ≤ 2. 
+ 
+2.2 The n ≥ 2 Case  
+ 
+For 
+2
+n 
+, plots of eq. (4) are shown in Figure 5 and Figure 6. In this case, the vertex angles do not 
+exceed 90o, and there is a minimum vertex angle for which eq. (4) is valid. That vertex angle and the side 
+length that gives rise to it are also found by equating the first derivative of eq. (4) to zero (as was done 
+above). Given that for all values of n, 
+1
+ =  is once again the only positive real solution to eq. (5), and by 
+substituting 
+1
+ =  into eq. (4), the minimum vertex angle can be found (eq. (7)). Thus, as was the case in 
+Section 2.1 for the largest vertex angle for 1
+2
+n
+
+
+, the smallest vertex angle for 
+2
+n 
+ occurs when the 
+triangle is isosceles. 
+ 
+2
+1
+min
+cos
+1 2
+n
+n
+
++
+−
+− 
+
+=
+−
+
+
+
+
+, 
+(7) 
+where 
+min
+
++  denotes the minimum value of 
++  for any degree n. 
+ 
+
+180
+170
+(deg.)
+160
+Maximum Vertex Angle
+150
+140
+130
+120
+110
+100
+90
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+1.7
+1.8
+1.9
+2
+Pythagorean Exponent7 
+ 
+ 
+Figure 5: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean 
+exponent for n ≥ 2. 
+ 
+ 
+Figure 6: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean 
+exponent for n ≥ 2. The side ratio is extended to the excluded regime of γ < 1.  
+ 
+The confirmation that 
+min
+
++  is a minimum angle was also performed numerically with Maple®. A plot of 
+eq. (7) is shown in Figure 7.  
+ 
+
+Vertex Angle (deg.)
+10
+06
+6
+85
+Pythagorean Exponent
+80
+6
+75
+70 
+4
+65
+3
+2
+09
+1
+2
+3
+4
+5
+7
+8
+6
+10
+Side Ratio8 
+ 
+ 
+ 
+Figure 7: Minimum vertex angle versus Pythagorean exponent. 
+  
+In the limit that the Pythagorean exponent becomes infinite,  
+ 
+(
+)
+2
+2
+1
+1
+1
+lim
+lim cos
+2
+n
+n
+n
+n
+
+
+
+
+
+−
+
+→
+→
+
+
+
+
++ −
++
+
+
+
+
+=
+=
+
+
+
+
+
+
+
+
+
+
+
+
+, 
+ (8) 
+which must be evaluated as a piecewise function. 
+Case 1 (
+)
+γ < 1
+i 
+(
+)
+(
+)
+2
+2
+1
+1
+1
+0 1
+1
+lim cos
+cos
+2
+2
+n
+n
+
+
+
+
+
+−
+−
+
+→
+
+
+
+
++ −
++
+
+
+
+
+
+
+
+=
+=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+ 
+ 
+Case 2 (
+)
+=
+γ
+1  
+(
+)
+( )
+2
+1 1 1 2
+1
+1
+lim cos
+60
+2
+n
+o
+n
+
+
+−
+
+→
+
+
+
+
++ −
+
+
+
+
+=
+=
+=
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+ 
+Case 3 (
+)
+
+γ
+1  
+(
+)
+(
+)
+2
+2
+1
+1
+1
+1
+1
+lim cos
+cos
+2
+2
+n
+n
+n
+
+
+
+
+
+
+−
+−
+
+→
+
+
+
+
++ −
+
+
+
+
+
+
+
+=
+=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+ 
+i This case is included for completeness, even though it was specified that γ ≥ 1. 
+
+06
+85
+(deg.)
+Minimum Vertex Angle 
+80
+75
+70
+65
+60
+2
+3
+4
+5
+6
+7
+6
+10
+11
+12
+13
+14
+15
+Pythagorean Exponent9 
+ 
+Case 2, illustrated graphically by 
+100
+n =
+ in Figure 6, indicates that the infinite degree Pythagorean 
+Theorem generates an equilateral triangle, and expectedly, it is the only case that does. For case 3 with an 
+infinite side ratio, 
+(
+)
+lim lim
+1
+90o
+n
+
+
+
+
+→
+→
+
+
+
+=
+
+
+. Thus, the infinite degree Pythagorean Theorem for a 
+triangle with an infinite side ratio (a straight line) requires the same vertex angle as the standard (
+)
+2
+n =
+ 
+Pythagorean Theorem. A plot of the behavior of the three cases of eq. (8) is shown in Figure 8.  
+ 
+ 
+ 
+Figure 8: Plot of the infinite degree Pythagorean Theorem which was produced with n = 106. 
+3. The n-Degree Pythagorean Theorem with Negative Real Exponents 
+ 
+Section 2 can be adapted to examine real triangles that conform to the negative exponent n-degree 
+Pythagorean Theorem, and a very different picture emerges. Once again, eqs. (2) and (3) can be equated 
+to form  
+ 
+(
+)
+2
+2
+1
+1
+1
+cos
+2
+n
+n
+
+
+
+
+−
+−
+
+
++ −
++
+
+
+=
+
+
+
+
+
+
+, 
+(9) 
+where 
+−  denotes that the vertex angle arises from a version of the n-degree Pythagorean Theorem in 
+which n
+−
+
+. 
+ 
+Unlike the positive exponents case, there is an infinite set of (
+)
+,n
+
+ for which the always positive 
+argument of the inverse cosine function exceeds 1. If 
+1
+ = , then eq. (9) yields 
+ 
+(
+)
+2
+2
+2
+1
+1
+1
+1
+1
+1 1
+cos
+cos
+1 2
+2
+n
+n
+
+−
+
+
+−
+
+
+−
+−
+
+
+
+
+
+
++ −
++
+
+
+=
+=
+−
+
+
+
+
+
+
+
+
+
+
+
+
+. 
+(10) 
+
+10 
+ 
+2 1
+1 2
+1
+n
+
+
+−
+
+
+
+
+−
+  for all n, and isosceles triangles are permitted for all degrees of the negative real exponent 
+Pythagorean Theorem. 
+ 
+However, as γ increases, 
+(
+)
+2
+2
+1
+1
+2
+n
+n
+
+
+
++ −
++
+ (the argument of the inverse cosine function in eq. (9)) 
+also increases and can exceed 1, and the vertex angle becomes complex. Therefore, there exists a γ-
+dependent critical value of the Pythagorean degree, 
+( )
+crit
+n
+
+, which is the largest exponent for a given 
+side ratio that will produce a real vertex angle. This occurs when 
+(
+)
+crit
+crit
+2
+2
+1
+1
+1
+2
+n
+n
+
+
+
++ −
++
+= , which can be 
+simplified as  
+ 
+(
+)
+crit
+crit
+1
+1
+1
+n
+n
+
+
++
+=
+− . 
+(11) 
+ 
+Equation (11) must be solved numerically, and its solution, 
+( )
+crit
+n
+
+, is shown in Figure 9. 
+ 
+ 
+ 
+Figure 9: The Pythagorean critical degree as a function of the side ratio. 
+ 
+Thus, if 
+( )
+crit
+n
+n
+
+
+, then   is a complex angle. If, on the other hand, 
+( )
+crit
+n
+n
+
+
+,   will be real, 
+and a corresponding real triangle exists. The upper limit of γ is not immediately apparent, but consider the 
+argument of the inverse cosine function in eq. (9). For 
+−  to be a real angle, 
+(
+)
+2
+2
+1
+1
+1
+2
+n
+n
+
+
+
++ −
++
+ . As 
+such, (
+)
+(
+)
+2
+2
+1
+1
+n
+n
+
+
+−
+
++
+ . However, the term (
+)
+2
+1
+1
+n
+n
+ +
+  for all γ and all n including n → − .    
+
+Side Ratio
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+1.7
+1.8
+1.9
+2
+0
+-1
+-2 
+Critical Degree
+-3
+-4
+5
+-6
+-7
+-811 
+ 
+This requires that 
+(
+)
+2
+min
+1
+1
+
+
+
+−
+
+
+
+ (for n  −), and therefore, 
+2
+ 
+. Consequently, the n-degree 
+Pythagorean Theorem with negative real exponents cannot be applied to real triangles with side ratios 
+greater than or equal to 2. As indicated by Figure 10 and Figure 11, 0.5
+2
+
+
+
+ is required to produce a 
+real vertex angle for 
+( )
+crit
+n
+n
+
+
+. However, by definition
+1
+  , and therefore, 1
+2
+
+
+
+.  
+ 
+Also different from the 
+1
+n =  and 
+2
+n =
+ cases is that the 
+1
+n = −  and 
+2
+n = −  cases produce non-
+constant but equal γ-dependent angles. From eq. (9): when 
+1
+n = −  or 
+2
+n = − , 
+(
+)
+2
+1
+2
+1
+cos
+2
+2
+1
+
+
+
+
+
+−
+− 
+
++
+=
+−
+
+
++
+
+
+
+
+. 
+ 
+ 
+3.1 The n ≤ ncrit(γ) Case  
+ 
+For 
+( )
+crit
+n
+n
+
+
+, plots of eq. (9) are shown in Figure 10 and Figure 11. Additionally, a maximum 
+value of the vertex angle exists for a given side ratio. As in Section 2.1, that vertex angle and the side 
+ratio that gives rise to it are found by equating the first derivative of eq. (9) (instead of eq. (4)) to zero.  
+ 
+ 
+Figure 10: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean 
+exponent for n ≤ ncrit(γ). The missing section at the top and on the right side of the figure are due to n > ncrit(γ) for a 
+given value of γ; the border of this section has the equation ncrit(γ) as shown in Figure 9. The side ratio is extended to 
+the excluded regime of γ < 1. 
+ 
+
+Vertex Angle (deg.)
+0
+60
+-1
+50
+Pythagorean Exponent
+40
+30
+20
+8
+10
+-9
+-10
+1.6
+1.2
+1.4
+1.8
+2
+Side Ratio12 
+ 
+ 
+Figure 11: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio. The gaps between 
+the plots and the horizontal axis are due to n > ncrit(γ) at those values of γ. The side ratio is extended to the excluded 
+regime of γ < 1. 
+ 
+As was the case with eq. (5), the solution to eq. (12) for ( )
+n
+
+ is not analytic. However, it is clear that 
+for all values of n, 
+1
+ =  is once again a solution. By substituting 
+1
+ =  into eq. (9), the vertex angle can 
+be found (eq. (13)) which expectedly has the same mathematical form as eq. (7)).  
+ 
+(
+)(
+)
+2
+2
+0
+1
+1
+1
+0
+n
+n
+n
+n
+d
+d
+
+
+
+
+
+−
+−
+=
+ −
++
+−
++
+=
+ 
+(12) 
+ 
+2
+1
+max
+cos
+1
+2
+n
+n
+
+−
+−
+− 
+
+=
+−
+
+
+
+
+, 
+(13) 
+where 
+max
+
+−  denotes the maximum value of 
+−  for a given degree n. 
+ 
+The second derivative of eq. (9) is the same as the second derivative of eq. (4), and therefore, the 
+mathematical confirmation that 
+max
+
+−  is a maximum angle was also performed numerically with Maple®. 
+A plot of eq. (13) is shown in Figure 12.  
+ 
+
+60 
+n=-1
+n=-2
+50
+n=-3
+n = -100
+40
+30
+20
+10
+0
+0
+0.5
+1.5
+2
+2.5
+Side Ratio13 
+ 
+ 
+Figure 12: Maximum vertex angle versus Pythagorean exponent. 
+  
+In the limit that the Pythagorean exponent becomes infinitely negative,  
+ 
+(
+)
+2
+2
+1
+1
+1
+lim
+lim
+cos
+2
+n
+n
+n
+n
+
+
+
+
+
+−
+−
+→−
+→−
+
+
+
+
++ −
++
+
+
+
+
+=
+=
+
+
+
+
+
+
+
+
+
+
+
+
+, 
+ (14) 
+which is like the case with positive exponents. Even with the imposed condition that 1
+2
+
+
+
+, eq. (14) 
+must be evaluated as a piecewise function.  
+ 
+ 
+ 
+Case 1 (
+)
+=
+γ
+1  
+(
+)
+1
+60o
+
+
+−
+=
+=
+ 
+ 
+ 
+Case 2 (
+)
+1 < γ < 2  
+(
+)
+1
+1
+2
+cos
+2
+
+
+
+−
+−
+
+
+
+
+=
+
+
+
+
+ 
+ 
+Case 3 (
+)
+−
+→
+γ
+2
+ 
+(
+)
+2
+0
+
+
+−
+−
+→
+=
+ 
+ 
+ 
+ 
+
+60
+50
+ (deg.)
+Maximum Vertex Angle
+40
+30
+20
+10
+0
+-15
+-12
+6-
+-6
+-3
+0
+Pythagorean Exponent14 
+ 
+Case 1 produces the same result as did case 2 for positive exponents (
+)
+i.e., 
+1
+ =
+ – a maximum 
+vertex angle of 60o and thus, an equilateral triangle. Also, like the positive exponents case, the equilateral 
+triangle is never actually realized because although γ can equal 1, n cannot be infinite. As illustrated in 
+Figure 13, the negative infinite degree Pythagorean Theorem can produce real triangles, and the 
+requirement that 1
+2
+
+
+
+ is retained. Furthermore, a straight line will result when  
+ 
+( )
+( )
+(
+)
+crit
+crit
+2
+2
+1
+1
+1
+lim
+lim
+cos
+0
+2
+n
+n
+n
+n
+n
+n
+
+
+
+
+
+
+−
+→
+→
+
+
+
+
++ −
++
+
+
+
+
+=
+=
+
+
+
+
+
+
+
+
+
+
+
+
+.  
+ 
+ 
+ 
+Figure 13: Plot of the negative infinite degree Pythagorean Theorem. 
+ 
+4. The Areas of the Associated Triangles 
+ 
+The area of the scalene triangle in Figure 1, with a real vertex angle, is 
+1
+sin
+2
+A
+ab
+
+=
+. Its side 
+lengths are a, b
+a
+
+=
+, and from eq. (1), 
+(
+)
+1
+1
+n
+n
+c
+a 
+=
++
+. Therefore, its area is  
+ 
+2
+1
+sin
+2
+A
+a 
+
+=
+. 
+ (15) 
+ 
+Applying 
+2
+2
+cos
+1 sin
+
+
+= −
+ and eq. (4) or eq. (9) depending on whether n
++
+
+ or n
+−
+
+, 
+respectively, eq. (15) can be written in terms of n and γ. 
+ 
+
+15 
+ 
+(
+)
+2
+2
+2
+2
+2
+4
+1
+1
+2
+n
+n
+a
+A
+
+
+
+
+
+
+
+=
+−
++ −
++
+
+
+
+
+
+
+
+
+. 
+ (16) 
+ 
+The restrictions on γ remain: 
+1
+   for n
++
+
+, and 1
+2
+
+
+
+ for n
+−
+
+. However, there are no 
+restrictions on a (other than 
+0
+a 
+). 
+ 
+4.1 The Maximum Areas of Triangles for the n-Degree Pythagorean Theorem with Positive 
+Real Exponents 
+ 
+A plot of eq. (16) is shown in Figure 14. It is clear as if a is constant and γ increases, the area of the 
+associated triangle increases without bound. Also, if a is constant, increasing n increases the area, but this 
+is a negligible effect compared to increasing γ. However, the effect that n has on the area increases as γ 
+increases. 
+ 
+ 
+Figure 14: Triangle area (in dimensionless units) with unit side length (a = 1). In the left-side figure, the effect of n 
+on the area is difficult to distinguish. However, in the right-side figure, the effect of n on the area is clearer. 
+Determining the conditions that maximize the area of a triangle can be done with an “angular” 
+approach. From eq. (15), the maximum area of a triangle with given values of a and γ clearly occurs when 
+90
+ =
+ (corresponding to 
+2
+n =
+) and is 
+2
+1
+2
+A
+a 
+=
+. 
+In summary, 
+• 
+If γ and n are fixed, increasing a will increase the triangle’s area without bound.  
+• 
+If a and n are fixed, increasing γ will increase the triangle’s area without bound.  
+ 
+Therefore, no absolute maximum area triangle exists because the triangle will experience infinite 
+dilation. The only minimum area is the trivial solution (
+)
+0
+A =
+. Most significant from this analysis is 
+that a right triangle, with fixed values of a and γ, has the maximum area.  
+
+Triangle Area
+Triangle Area
+10
+5
+3
+4.5
+2.8
+0.9
+8
+4
+2.6
+0.8
+Pythagorean Exponent
+7
+3.5
+Pythag orean Exponent
+2.4
+0.7
+3
+2.2
+0.6
+2.5
+2
+0.5
+5
+2
+0.4
+1.8
+1.5
+0.3
+1.6
+3
+1
+0.2
+1.4
+2
+0.5 
+0.1
+1.2
+0
+2
+3
+4
+5
+6
+7
+8
+9
+10
+1.1
+1.2 
+1.3
+1.4
+1.5
+1.6
+1.7
+1.8
+1.9
+Side Ratio
+Side Ratio16 
+ 
+The above result can, of course, be determined from the dependence of 
++  on n and γ. From eq. (4), 
+(
+)
+2
+2
+1
+1
+1
+cos
+90
+2
+n
+n
+
+
+
+
++
+−
+
+
++ −
++
+
+
+=
+=
+
+
+
+
+
+
+ which gives 
+(
+)
+2
+2
+1
+1
+n
+n
+
+
++
+=
++  
+(17) 
+because 
+(
+)
+2
+2
+1
+1
+0
+2
+n
+n
+
+
+
++ −
++
+=
+. The only real solution to eq. (17) for n is 
+2
+n =
+.  
+ 
+An alternative and more rigorous approach to finding the degree which gives rise to the maximum 
+area case for a given value of γ is from eq. (16); 
+0
+A
+n
+
+=
+
+ gives 
+ 
+(
+)
+(
+)
+(
+)
+2
+2
+2
+1
+1
+1
+1
+ln
+ln
+1
+0
+1
+n
+n
+n
+n
+n
+n
+n
+n
+
+
+
+
+
+
+
+
+
+
+ 
+
++
++ −
++
+−
++
+=
+
+
+
+
+
+
++
+
+
+
+ 
+
+. 
+ (18) 
+ 
+But (
+)
+2
+1
+0
+n
+n
+ +
+
+ for any value of γ or n, and therefore,  
+ 
+(
+)
+(
+)
+2
+2
+1
+1
+1
+ln
+ln
+1
+0
+1
+n
+n
+n
+n
+n
+n
+
+
+
+
+
+
+
+
+
+ 
+
++ −
++
+−
++
+=
+
+
+
+
+
+
++
+
+
+
+ 
+
+. 
+ (19) 
+ 
+(
+)
+2
+2
+1
+1
+0
+n
+n
+
+
+
+
++ −
++
+=
+
+
+
+
+ iff 
+2
+n =
+ (as shown with eq. (17)). 
+(
+)
+1
+ln
+ln
+1
+0
+1
+n
+n
+n
+n
+
+
+
+
+
+
+
+
+−
++
+
+
+
+
+
++
+
+
+
+
+, 
+regardless of the value of γ, although it asymptotically approaches zero for increasing n. As expected, this 
+somewhat more complex approach gives the same result as the “angular approach” used above – the 
+maximum area triangle for positive real exponents occurs when 
+2
+n =
+.  
+ 
+Thus, the n-degree Pythagorean Theorem that produces triangles with the largest area for a given a 
+and γ is the standard Pythagorean Theorem, and the triangles it produces are right angled. Even though 
+the triangles that result from 1
+2
+n
+
+
+ have vertex angles which are larger than 90o, they have smaller 
+areas than 
+2
+1
+2
+A
+a 
+=
+ because sin
+1
+   for obtuse  . 
+ 
+ 
+ 
+ 
+
+17 
+ 
+For a given a and a given n (not necessarily 
+2
+n =
+), the value of γ that produces the triangle with the 
+maximum area would be calculated from the derivative of eq. (16) i.e., 
+0
+A
+
+
+
+
+=
+
+
+
+
+
+ which yields 
+ 
+(
+)
+(
+)
+2
+2 1
+2
+2
+1
+1
+1
+1
+4
+n
+n
+n
+n
+n
+
+
+
+
+
+
+−
+
+
+−
+
+
+
+
+
+
++ −
++
+−
++
+=
+
+
+
+
+
+ 
+
+. 
+ (20) 
+ 
+However, equation (20) has no solution for 
+0
+n 
+. This is physically reasonable because increasing γ 
+for a given value of a continuously dilates the triangle; therefore, there is no finite maximum area. The 
+smallest of all of the areas of these continuously enlarging triangles (with a fixed side length a) occurs 
+when 
+1
+ =  and is 
+ 
+3
+2
+1
+2
+2
+min
+2
+1 4
+n
+n
+n
+n
+A
+a
+−
+−
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+=
+−
+
+
+
+
+
+
+. 
+ (21) 
+ 
+Equation (21) (a plot of which is shown in Figure 15) represents the areas of the members of an 
+infinite set of isosceles triangles. Each of these triangles has the minimum possible area among all of the 
+various triangles of the same degree. The triangle in this infinite set with the largest area occurs from the 
+infinite degree case, is denoted by 
+max
+min
+A
+ , and is given by the limit of eq. (21). 
+ 
+max
+2
+min
+min
+lim
+6
+n
+A
+A
+a
+
+→
+=
+=
+, 
+ (22) 
+ 
+Figure 15: The minimum area (in dimensionless units) of an isosceles triangle with unit side length (a = 1) for the  
+n-degree Pythagorean Theorem with positive real exponents. The red line indicates 
+max
+min
+A
+ . 
+ 
+ 
+ 
+
+18 
+ 
+4.1.1 
+The Area of a Triangle with a Fixed Perimeter (n > 0) 
+ 
+To further examine these ideas, it is sensible to consider the area of a triangle with a fixed perimeter 
+P. Then, determine the values of γ and n that produce the maximum area for the n
++
+
+ case (and then in 
+Section 4.2, for the n
+−
+
+ case).  
+ 
+The triangle perimeter is trivially 
+ 
+(
+)
+1
+1
+1
+n
+n
+P
+a
+b
+c
+a 
+
+
+
+=
++
++
+=
++ +
++
+
+
+
+
+. 
+ (23) 
+ 
+The area, in terms of the fixed perimeter, is denoted 
+P
+A , and it results from combining eqs. (16) and (23). 
+ 
+(
+)
+(
+)
+2
+2
+2
+2
+2
+1
+4
+1
+1
+2
+1
+1
+P
+n
+n
+n
+n
+P
+A
+
+
+
+
+
+
+
+
+
+
+
+
+
+=
+−
++ −
++
+
+
+
+
+
+
+
+
+
+
++ +
++
+
+
+
+
+
+
+
+
+ 
+ (24) 
+ 
+From eq. (24): for a fixed P, and as a function of n, the area of the triangle is a maximum when  
+1
+ = . This area is denoted by 
+max
+P
+A
+ and is given by  
+ 
+1
+1
+2
+max
+2
+1
+4
+16
+16
+1 2
+n
+n
+n
+P
+n
+n
+P
+A
++
+
+
+
+
+
+
+
+
+
+
+
+
+−
+
+
+
+
+
+
+
+
+
+
+
+
+−
+=
+
+
+ 
+
+
++
+
+
+
+
+
+
+
+
+
+
+
+
+, 
+ (25) 
+ 
+Equation (25) represents the areas of an infinite set of maximum area isosceles triangles with a fixed 
+perimeter. The triangle in that set with the largest area is the infinite degree member which is denoted by 
+max
+P
+A
+ , and its area is given by  
+ 
+1
+1
+2
+2
+max
+2
+1
+4
+16
+3
+lim
+16
+36
+1 2
+n
+n
+n
+P
+n
+n
+n
+P
+A
+P
+
++
+
+
+
+
+
+
+
+
+
+
+
+
+→
+−
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−
+=
+=
+
+
+
+
+
+
+ 
+
+
+
+
++
+
+
+
+
+
+
+
+
+
+
+
+
+. 
+ (26) 
+ 
+ 
+
+19 
+ 
+From eq. (24), as  →  , the areas of the triangles approach zero (as shown in Figure 17), as they 
+take the form of a straight line. Also, from eq. (24) (and as shown in Figure 16), triangles with a fixed P 
+will see their areas (as a function of γ) asymptotically increase as n → . Those areas are denoted 
+P
+A  
+and are given by 
+ 
+(
+)
+2
+2
+2
+4
+1
+1
+4
+2
+1
+P
+A
+P
+
+
+
+
+
+−
+=
+
+
++
+
+
+
+
+, 
+ (27) 
+ 
+ 
+ 
+Figure 16: Area (in dimensionless units) of Pythagorean triangles with unit perimeter. 
+ 
+ 
+Figure 17: Area (in dimensionless units) of Pythagorean triangles with unit perimeter. 
+ 
+ 
+ 
+ 
+
+20 
+ 
+4.2 The Maximum Areas of Triangles for the n-Degree Pythagorean Theorem with Negative 
+Real Exponents 
+ 
+For n
+−
+
+, a plot of eq. (16) is shown in Figure 18. The area relationship for positive real exponents 
+also applies to the case of negative real exponents. However, for all 
+( )
+crit
+n
+n
+
+
+, the vertex angles are 
+real and acute, and if a and γ are fixed, the maximum area triangle has the largest vertex angle.  
+ 
+ 
+ 
+Figure 18: Triangle area (in dimensionless units) with unit side length (a = 1). The missing section at the top right 
+and on the right side of figure are due to  n > ncrit(γ) for a given value of γ; the border of this section has the equation 
+ncrit(γ) as shown in Figure 9. 
+ 
+To determine the triangle with the maximum area, consider eqs. (15) and (9). 
+ 
+
+
+(
+)
+
+
+(
+)
+(
+)
+(
+)
+1
+2
+2
+2
+1
+2
+2
+2
+2
+2
+2
+max
+1
+1
+1
+1
+1
+max sin
+1
+min cos
+1
+min
+2
+2
+2
+2
+n
+n
+A
+a
+a
+a
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++ −
++
+
+
+
+
+
+
+=
+=
+−
+=
+−
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+. 
+ (28) 
+ 
+In the special case of an isosceles triangle (
+)
+1
+ =
+, 
+(
+)
+2
+2
+2
+1
+1
+min
+1 2
+2
+n
+n
+n
+n
+
+
+
+−
+
+
+
+
+
+
+
+
++ −
++
+
+ = −
+
+
+
+
+
+
+ for 
+1
+n  −  
+because 
+(
+)
+crit
+1
+n
+ =
+ does not exist.  
+ 
+ 
+
+Triangle Area
+0.5
+2
+0.45
+3
+0.4
+Pythagorean Exponent
+0.35
+4
+0.3 
+4
+0.25
+0.2 
+0.15
+8
+0.1
+-9
+0.05
+-10
+1.2
+1.4
+1.6
+1.8
+2
+Side Ratio21 
+ 
+The maximum triangle area is then 
+ 
+1
+1
+1
+2
+1
+2
+2
+max
+1
+sin
+2
+1 4
+2
+n
+n
+n
+n
+A
+a
+a
+
+
+
+−
+−
+
+
+
+
+
+
+
+
+=
+
+
+
+
+
+
+
+=
+=
+−
+
+
+
+
+
+
+
+
+
+. 
+ (29) 
+ 
+If the negative infinite degree case of eq. (28) is considered, the following results. 
+ 
+(
+)
+2
+2
+1
+1
+lim
+min
+2
+2
+n
+n
+n
+
+
+
+
+→−
+
+
+
+
++ −
++
+
+
+
+ =
+
+
+
+
+
+
+
+
+
+
+
+
+ for any side ratio that conforms to 1
+2
+
+
+
+. For this negative 
+infinite degree case, the maximum triangle area is 
+ 
+2
+2
+2
+max
+1
+1
+sin
+4
+2
+4
+A
+a
+a
+
+
+
+
+− =
+=
+−
+. 
+ (30) 
+ 
+Equation (30) necessarily complies with the requirement that 1
+2
+
+
+
+. Additionally, three interesting 
+results emerge about this infinite set of negative infinite degree triangles: 
+1. In this set, there are two values of γ that produce triangles with equal areas: 
+1
+ =  and 
+3
+ =
+, 
+and that maximum area is 
+1, 
+3
+2
+2
+max
+1
+3
+sin
+2
+4
+A
+a
+a
+
+
+
+
+−
+=
+=
+=
+=
+. 
+2. 
+2
+ =
+ necessarily gives zero area because the triangle has collapsed into a straight line. 
+3. In this infinite set, the triangle with the largest area is the member for which 
+2
+ =
+, and its area 
+is 
+max
+2
+2
+max
+1
+1
+sin
+2
+2
+A
+a
+a
+
+
+− =
+=
+. 
+ 
+ 
+4.2.1 
+The Area of a Triangle with a Fixed Perimeter (n < 0) 
+ 
+If, on the other hand, the perimeter is fixed, eq. (24) applies for 1
+2
+
+
+
+, and the largest area 
+triangle will be isosceles, and it will have an area of 
+2
+3
+36 P . This area is expected because the conditions 
+that 
+1
+ =  and n → −  result in equilateral triangle in which 
+3
+P
+a
+=
+, and therefore, 
+2
+2
+3
+3
+36
+4
+P
+a
+=
+.  
+ 
+From eq. (24), if lim
+P
+n
+A
+→−
+ is taken, eq. (31) results. As seen in Figure 19 (a plot of eq. (31)), the area 
+will asymptotically increase as n → −  and will asymptotically decrease toward zero as 
+2
+ →
+. Eq. 
+(31) determines the area for a fixed perimeter, arbitrary side ratio, and infinite degree triangle.  
+ 
+
+22 
+ 
+(
+)
+2
+2
+2
+1
+4
+4
+2
+P
+A
+P
+
+
+
+−
+
+
+=
+−
+
+
++
+
+
+
+
+, 
+ (31) 
+where 
+P
+A−  is the asymptotic area of the triangles with a fixed perimeter and for which n → − . 
+ 
+Among the infinite set of triangles whose areas are given by eq. (31), the triangle with the largest area 
+is isosceles (as stated above), and its area is 
+2
+3
+36 P . 
+ 
+ 
+Figure 19: Area of Pythagorean triangles with unit perimeter. 
+ 
+ 
+Figure 20: Area of Pythagorean triangles with unit perimeter. 
+ 
+
+23 
+ 
+5. Summary 
+ 
+The Pythagorean Theorem has been extended to positive and negative real exponents unfettered by 
+the physical requirement of dimension. The relationship between the ratio of the adjacent sides and the 
+vertex angle was determined for a given degree. It was found that for positive exponents, the stipulation 
+that 
+1
+   can be applied to all degrees, and no complex vertex angles arose. For 1
+2
+n
+
+
+, an obtuse 
+triangle results, and if 
+2
+n 
+, the triangle is acute. However, for negative exponents to produce real 
+vertex angles, the restriction 1
+2
+
+
+
+ is necessary but not sufficient. The additional requirement that if 
+1
+2
+
+
+
+, then 
+( )
+crit
+n
+n
+
+
+ needs to be imposed.  
+ 
+The areas of the associated triangles for positive and negative real exponents were explored. With 
+fixed a and γ values, the areas for 
+|
+1
+n
+n
++
+
+  are maximized when the triangle is right isosceles 
+requiring 
+1
+ =  and 
+2
+n =
+. Additionally, triangle areas increase as n → . Alternatively, if the 
+perimeter of a triangle is kept constant, the triangle area approaches a maximum value with increasing n 
+and approaches 0 for decreasing γ.  
+ 
+For the n
+−
+
+ case, as n → − , the triangle with a fixed perimeter and the maximum area has a 
+side ratio of 
+2
+ =
+. In contrast, if the degree is finite and 
+( )
+crit
+n
+n
+
+
+ (if 
+1
+  ), the maximum area 
+occurs when the vertex angle (
+)
+,n
+ 
+ is a maximum. 
+ 
+ 
+ 
+ 
+ 
+
+24 
+ 
+Works Cited 
+1. Faltings, G., The proof of Fermat’s last theorem by R. Taylor and A. Wiles. Notices of the AMS, 1995. 
+42(7): p. 743-746. 
+2. Agarwal, R.P., Pythagorean theorem before and after Pythagoras. Adv. Stud. Contemp. Math, 2020. 
+30: p. 357-389. 
+3. Amir-Moéz, A.R., R.E. Byerly, and R.R. Byerly, Pythagorean theorem in unitary spaces. Publikacije 
+Elektrotehničkog fakulteta. Serija Matematika, 1996: p. 85-89. 
+4. Atzema, E.J., Beyond Monge's Theorem: A Generalization of the Pythagorean Theorem. Mathematics 
+Magazine, 2000. 73(4): p. 293-296. 
+5. Conant, D. and W. Beyer, Generalized pythagorean theorem. The American Mathematical Monthly, 
+1974. 81(3): p. 262-265. 
+6. Cook, W.J., An n-dimensional Pythagorean theorem. The College Mathematics Journal, 2013. 44(2): 
+p. 98-101. 
+7. Czyzewska, K., Generalization of the Pythagorean theorem. Demonstratio Mathematica, 1991. 24(1-
+2): p. 305-310. 
+8. Donchian, P.S. and H. Coxeter, 1142. An n-dimensional extension of Pythagoras' Theorem. The 
+Mathematical Gazette, 1935. 19(234): p. 206-206. 
+9. Drucker, D., A comprehensive Pythagorean theorem for all dimensions. The American Mathematical 
+Monthly, 2015. 122(2): p. 164-168. 
+10. Eifler, L. and N.H. Rhee, The n-dimensional Pythagorean theorem via the divergence theorem. The 
+American Mathematical Monthly, 2008. 115(5): p. 456-457. 
+11. Kadison, R.V., The Pythagorean theorem: I. The finite case. Proceedings of the National Academy of 
+Sciences, 2002. 99(7): p. 4178-4184. 
+12. Lee, J.-R., The law of cosines in a tetrahedron. The Pure and Applied Mathematics, 1997. 4(1): p. 1-6. 
+13. Quadrat, J.-P., J.B. Lasserre, and J.-B. Hiriart-Urruty, Pythagoras' theorem for areas. The American 
+Mathematical Monthly, 2001. 108(6): p. 549-551. 
+14. Veljan, D., The 2500-year-old Pythagorean theorem. Mathematics Magazine, 2000. 73(4): p. 259-272. 
+15. Yeng, S., T. Lin, and Y.-F. Lin, The n-dimensional Pythagorean theorem. Linear and multilinear 
+algebra, 1990. 26(1-2): p. 9-13. 
+ 
+

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@@ -0,0 +1,1506 @@
+ 
+ 
+1 
+ 
+Mid-Infrared spectroscopy of 
+impactites from the Nördlinger Ries impact crater  
+Corresponding Author: Andreas Morlok, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 
+Münster, Germany. Email: morlokan@uni-muenster.de, Tel. +49-251-83-39069 
+Aleksandra Stojic, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany. Email: 
+a.stojic@uni-muenster.de;  
+Isabelle Dittmar, Hochschule Emden/Leer, Constantiaplatz 4, 26723 Emden, Germany, Email: 
+isabelle.dittmar@hs-emden-leer.de 
+Harald Hiesinger, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany. Email: 
+hiesinger@uni-muenster.de 
+Manuel Ahmedi, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany. Email: 
+ahmedi@uni-muenster.de 
+Martin Sohn, Hochschule Emden/Leer, Constantiaplatz 4, 26723 Emden, Germany, Email: 
+martin.sohn@hs-emden-leer.de 
+Iris Weber, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany. Email: 
+sonderm@uni-muenster.de 
+Joern Helbert, Institute for Planetary Research, DLR, Rutherfordstrasse 2, 12489 Berlin, Germany, Email: 
+joern.helbert@dlr.de 
+© 2016 This manuscript version is made available under the CC-BY-NC-ND 4.0 
+ 
+
+ 
+ 
+2 
+ 
+ABSTRACT 
+This study is part of an effort to build a mid-infrared database (7-14µm) of spectra for MERTIS (Mercury 
+Radiometer and Thermal Infrared Spectrometer), an instrument onboard of the ESA/JAXA BepiColombo 
+space probe to be launched to Mercury in 2017. 
+Mercury was exposed to abundant impacts throughout its history. This study of terrestrial impactites can 
+provide estimates of the effects of shock metamorphism on the mid-infrared spectral properties of 
+planetary materials.   
+In this study, we focus on the Nördlinger Ries crater in Southern Germany, a well preserved and easily 
+accessible impact crater with abundant suevite impactites. Suevite and melt glass bulk samples from 
+Otting and Aumühle, as well as red suevite from Polsingen were characterized and their reflectance 
+spectra in mid-infrared range obtained. In addition, in-situ mid-infrared spectra were made from glasses 
+and matrix areas in thin sections. The results show similar, but distinguishable spectra for both bulk 
+suevite and melt glass samples, as well as in-situ measurements.    
+Impact melt glass from Aumühle and Otting have spectra dominated by a Reststrahlen band at 9.3-9.6 
+µm. Bulk melt rock from Polsingen and bulk suevite and fine-grained matrix have their strongest band 
+between 9.4 to 9.6 µm. There are also features between 8.5 and 9 µm, and 12.5 - 12.8 µm associated 
+with crystalline phases. There is evidence of weathering products in the fine-grained matrix, such as 
+smectites. Mercury endured many impacts with impactors of all sizes over its history. So spectral 
+characteristics observed for impactites formed only in a single impact like in the Ries impact event can be 
+expected to be very common on planetary bodies exposed to many more impacts in their past. We 
+conclude that in mid-infrared remote sensing data the surface of Mercury can be expected to be 
+dominated by features of amorphous materials. 
+ 
+
+ 
+ 
+3 
+ 
+Keywords: Spectroscopy, Impact processes, Infrared observations, Instrumentation 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+4 
+ 
+1. Introduction   
+The aim of this study is to provide mid-infrared reflectance spectra with special emphasis on the region 
+from 7µm – 14µm for a range of impactites for the application in planetary remote sensing. We generate 
+these spectra for a database for the ESA/JAXA BepiColombo mission to Mercury (Benkhoff et al., 2010). 
+Onboard is a mid-infrared spectrometer (MERTIS-Mercury Radiometer and Thermal Infrared 
+Spectrometer). This unique device allows mapping spectral features in the 7-14 µm range, with a spatial 
+resolution of ~500 m (Helbert et al., 2009; Benkhoff et al., 2010; Hiesinger et al., 2010).  
+Infrared spectroscopy provides a means to characterize the mineralogy of rocks via remote 
+sensing, in contrast to Gamma-Ray, Neutron and X-ray spectrometers, which determine elemental 
+compositions (Pieters and Englert, 1993). Thus, IR spectroscopy provides a central analytical tool to 
+determine the mineralogy of remote planetary surfaces.  In order to correctly interpret the remote 
+sensing data, laboratory spectra of natural, synthetic rocks and minerals have to be collected to compare 
+them to the spectra that will be obtained from MERTIS, once BepiColombo enters the Hermean orbit 
+(Maturilli et al., 2008; Hiesinger et al., 2010). Comparable earlier instruments were the Thermal Emission 
+Spectrometer (TES) on the Mars Global Surveyor (Christensen et al., 2005) and the Thermal Emission 
+Imaging System (THEMIS) on the Mars Odyssey orbiter (Christensen et al., 2004). The lunar surface was 
+mapped in the mid-infrared with the DIVINER Lunar Radiometer Instrument on the Lunar 
+Reconnaissance Orbiter (Paige et al., 2010). The OSIRIS-REx Thermal Emission Spectrometer (OTES) will 
+map asteroid Bennu (1999 RQ36) (Hamilton and Christensen, 2014), and the Thermal Infrared Imager 
+(TIR) onboard Hayabusa 2 will map asteroid 1999JU3 (Okada et al., 2015).  
+The Ries crater in southern Germany (Fig.1) provides well preserved layers of impactites and is 
+one of the best studied impact sites in the world (von Engelhardt, 1995). The 14.6 Myr old crater with a 
+diameter of 24 kilometers (Buchner et al., 2010) offers the whole range of impact-associated rocks and 
+
+ 
+ 
+5 
+ 
+minerals (von Engelhardt, 1990). Large impact events play a key role in surface modification processes in 
+effect on most terrestrial planets and their moons (Hörz and Cintala, 1997). Thus, the investigation of 
+terrestrial impact processed rocks and understanding how these processes affect the spectral properties 
+of the resulting impact generated rocks and melt glass is important for the interpretation of infrared data 
+from the surfaces of other planetary bodies.  Given the characteristics of surface regolith, we need 
+spectral data from different grain size fractions. This is necessary because grain size variations affect the 
+corresponding spectra immensely by reducing the spectral contrast of features and creating additional 
+features at small grain sizes (e.g., Salisbury and Eastes, 1985; Salisbury and Wald, 1992; Mustard and 
+Hayes, 1997; Ruff and Christensen, 2002).  
+Reflectance and emission studies about spectral features in the mid-infrared in minerals 
+undergoing high shock metamorphism including formation of melt glass were made on experimentally 
+shocked samples, e.g., on anorthosite, pyroxenite, basalt, and feldspar (Johnson et al. , 2002, 2003, 2007, 
+2012; and Jaret et al. ,2015). In these studies, especially feldspar-rich material showed loss and 
+degradation of features, as well as band shifts with increasing structural disorder resulting from 
+increasing shock pressure. Moroz et al. (2010) analyzed impact glasses from laser pulse experiments with 
+Martian soil analogue JSC Mars-1. Similar, Basilevsky et al. (2000) and Morris et al. (2000) spectrally 
+studied melt glass also made from Martian soil analogs. Byrnes et al. (2007), and Lee et al.,(2010) 
+measured synthetic quartzofeldspathic glasses and Dufresne et al. (2009), Minitti et al. (2002), Minitti 
+and Hamilton (2010) measured synthetic glass with basaltic to intermediate composition. In general, the 
+resulting mid-infrared reflectance or emission spectra display a dominant feature in the 9.2-10.5 µm 
+range. Pollack et al. (1973), Crisp et al., 1990; Nash and Salisbury (1991) and Wyatt et al. (2001), studied 
+obsidian or basalt. Further infrared reflectance and emission spectra of natural impact glass and tektites 
+were made, e.g., by Thomson and Schultz (2002), Gucsik et al. (2004), Faulques et al. (2005) and Palomba 
+et al. (2006). Wright et al. (2011), Basavaiah and Chavan (2013), and Jaret et al. (2013) investigated 
+
+ 
+ 
+6 
+ 
+shocked bulk material from the Lonar Crater, India, showing mainly a simple spectrum with a dominating 
+feature in the ~9.4 - 10.2 µm range. 
+Complementary infrared transmission and absorbance spectra of related materials were also 
+made, e.g, of experimentally shocked feldspar materials (Stöffler and Hornemann, 1972; Ostertag, 1983) 
+and of CM2 chondrite Murchison (Morlok et al., 2010). Glasses with feldspathic composition produced in 
+static high pressure experiments were measured by, e.g., Iiishi et al. (1971), Velde et al. (1987), and 
+Williams and Jeanloz (1988, 1989). In addition, transmission infrared analyses of natural impact glass and 
+tektites for studying water contents were made by Beran and Koeberl (1997), for identification purposes 
+(Fröhlich et al., 2013) or for astrophysical studies of circumstellar dust (Morlok et al., 2014). King et al. 
+(2004) provided an overview of silicate glass analysis with various mid-infrared techniques. 
+Furthermore, there are also numerous studies how impact shock affects the spectral properties 
+in the visible and near-infrared, such as Johnson and Hörz (2003), Adams et al. (1979), and Bruckenthal 
+and Pieters (1984) for experimentally shocked feldspars or enstatite. Moroz et al. (2009) studied 
+synthetic glasses with Martian soil composition as impact melt analogs, and Cannon and Mustard (2015) 
+identified glass-rich impactites on Mars. Bell et al. (1976) and Stockstill-Cahill et al. (2014) analyzed lunar 
+glass analogues, while Keppler (1992) studied synthetic silicate glasses with albite and diopside 
+composition. Schulz and Mustard (2004) studied terrestrial impact melt rocks. 
+Shock metamorphism begins with fracturing and brecciation of the rocks at lower pressures; 
+above 2 GPa the mineral phases start to change. Shatter cones and conical fracturing patterns form. 
+Between 8 and 25 GPa, planar deformation features appear along with microscopic changes in the 
+crystal structure due to impact shock in, e.g., quartz and feldspar. At pressures >25 GPa, shock 
+metamorphism continues with the solid-state transformation of minerals into diaplectic glasses, which 
+are amorphous materials generated at pressures up to 40 GPa. At pressures over 35 GPa, partial melting 
+of phases begins and at over 60 GPa rocks completely melt, followed by vaporization at pressures above 
+100 GPa (e.g., Stöffler, 1966, 1971, 1984; Chao, 1967; von Engelhardt and Stöffler, 1968; Stöffler and 
+
+ 
+ 
+7 
+ 
+Langenhorst, 1994; French, 1998). Also, high pressure mineral polymorphs form; stishovite and coesite 
+form from quartz at pressures from 12-15 GPa and over 30 GPa, respectively. Carbon is transformed into 
+diamond at 13 GPa (Stöffler and Langenhorst, 1994; French, 1998).   
+In the impactite suevite, shock metamorphism can be divided into stages 0 – IV.  Stage 0 ranges 
+from 0-10 GPa, stage I (10-35 GPa) and II (35-60 GPa) cover changes in crystal structure and partial 
+melting; stage IV  pressures over 60 GPa result in melt glasses (Stöffler, 1971 and 1984; French, 1998; 
+Stöffler and Grieve, 2007; Stöffler et al., 2013). Suevite consists of materials of all five shock stages 
+(Stöffler et al., 2013). Thus the infrared spectra obtained from the suevites can be expected to contain a 
+mixture of spectral features depending on the level of shock metamorphism the material underwent.  
+In this study, we focus on impactites from the Ries crater, since the crater site allows easy access 
+to a range of naturally shocked rocks and minerals (von Engelhardt, 1990, 1995). Prior to the impact 
+event, the Ries area was covered with sediments mainly consisting of limestones. The underlying 
+crystalline basement is dominated by granites and gneisses, with significant amounts of amphibolite (von 
+Engelhardt and Graup, 1984; von Engelhardt, 1997). The average chemical composition of very 
+homogeneous impact melts (shock stage IV; Stöffler et al, 2013) is similar to the composition of the 
+modelled crystalline basement rock clasts based on the fallout suevite, identifying these rocks as the 
+main source of the homogenized melt (Staehle, 1972; von Engelhardt et al., 1984 , 1995, 1997; 
+Vennemann et al., 2001).  
+We focus on the suevite, the top layer of ejecta in the Ries that was deposited on all other 
+impact ejecta, Bunte Breccia (e.g. Abadian, 1972; Hörz et al., 1983), megablocks (e.g. Sturm et al., 2015) 
+and polymict crystalline breccia (e.g. von Engelhardt, 1997). The most voluminous ejecta is the Bunte 
+Breccia, consisting mainly of sedimentary material. Megablocks are rocks in the size range from 25 m to 
+kilometers, consisting of sedimentary (limestone) and crystalline materials. Both Bunte Breccia and 
+
+ 
+ 
+8 
+ 
+megablocks show low degrees of shock metamorphism. Polymict crystalline breccias are mixtures of 
+basement rocks with shock stages up to stage II. Suevite is the only ejecta layer to contain material in all 
+stages of shock metamorphism (e.g., von Engelhardt, 1969, 1990, 1997). Moldavites, a type of tektites 
+formed in the Ries impact are an additional type of ejecta that was deposited 350 km east of the impact 
+site (e.g., von Engelhardt, 1987).  
+The mostly granitic and felsic petrology of the crystalline basement of the Ries, which controlled 
+the composition of the suevite (von Engelhardt et al., 1990, 1997), is quite different compared to that of 
+Mercury and other  planetary surfaces in the Solar System.  Also, granite in general is very rare outside 
+Earth (Bonin, 2012). Based on MESSENGER data, the best terrestrial analogues for the surface of Mercury 
+are basalts or ultramafic komatiites (Nittler et al., 2011; Stockstill-Cahill et al., 2012; Charlier et al., 2013; 
+Maturilli et al., 2014), but natural shocked forms are rare on Earth (e.g. Wright et al., 2011). However, a 
+spectral study of naturally shocked granitic material is of interest for our remote sensing purposes 
+because it gives insight into the spectral characteristics of impactites and their components, while 
+naturally shocked basalts are rare (Wright et al., 2011). Furthermore, the easy access to the Ries site 
+gives access to larger, representative amounts of sample material. Although rare, granitic materials have 
+been recognized on other planetary bodies. Granitic material has been found as fragments and clasts in 
+lunar samples (e.g., Warren et al., 1983; Jolliff et al., 1999; Shervais and McGee, 1999; Seddio et al., 
+2015). Granitoid or felsic materials may also occur on the mostly basaltic surface of Mars (e.g., 
+Christensen et al., 2005; Bandfield, 2006, Ehlmann and Edwards, 2014; Sautter et al., 2014, 2015). There 
+are also indications for felsic, granitoid material on Venus (Müller et al., 2008; Gilmore, 2015).  
+Suevites are divided into crater or fallback suevite, occurring inside the inner ring of the Ries, and 
+outer suevite, encompassing suevite found outside of the inner part of the crater (Pohl et al., 1977, 
+Stöffler, 1977, 2013). Normal suevite in this study is outer suevite that originates from the Otting and 
+Aumühle quarries (Fig.1, 2a,b). It mainly consists of three components: a porous matrix of fine-grained 
+
+ 
+ 
+9 
+ 
+rock, melt glass, and crystalline basement rocks displaying all stages of shock metamorphism ( Stöffler, 
+1966; von Engelhardt and Graup, 1984; Stöffler et al., 2013). Red suevite (Fig. 2c) from the Polsingen 
+quarry is a rare variation of the outer suevite. Its red color is a characteristic owing to high hematite 
+content (Stöffler et al., 2013). In the red suevite, the actual groundmass consists of impact melt, in which 
+fragments are embedded (Reimold et al., 2010). Also, chemical differences between normal suevite and 
+red suevite (Na2O, K2O) were reported by Reimold et al. (2013) and Stöffler (2013). In addition to the 
+bulk materials, we are also interested in the components of the suevite. In particular, we studied the 
+melt glasses (Fig.2a, b), i.e., completely shock-melted and quenched rock material. Furthermore, we 
+looked at the fine-grained matrix itself, which was affected by weathering processes (von Engelhardt, 
+1995; Stöffler et al., 2013). 
+ 
+2. Samples and Techniques 
+2.1 Sample Selection 
+We selected three bulk suevite samples from three different Ries localities of the suevite layer 
+(von Engelhardt and Stöffler, 1974; von Engelhardt et al., 1995, 1997; Bayerisches Geologisches 
+Landesamt, 2004): normal crater suevite from the Otting (samples Otting Bulk 1-3) and Aumühle 
+(Aumühle Bulk4, Bulk 18, Bulk 18 Matrix) quarries, and red suevite (samples Polsingen 2-4) from a small 
+outcrop in Polsingen (outer suevite) (von Engelhardt, 1997; Reimold et al., 2010).  In addition, we used 
+material from three separated impact melt glasses: from a large, 30 cm sized  glass ‘bomb’ (Otting 
+Glasbombe) and a separated glass ‘Flädle’ from Otting (Otting Glas 1), and another glass separated from 
+the Aumühle suevite (Aumühle 13).  
+In addition, for quantitative chemical and additional IR-microscopy analyses, polished thin 
+sections of Aumühle, Otting and Polsingen were produced from the bulk sample rocks.  The vitreous 
+
+ 
+ 
+10 
+ 
+state of the glass was confirmed using polarized light microscopy by the total extinction of the glass 
+under crossed polarizers (Fig.2a-c). However, the two Otting samples show brownish halos or ‘Schlieren’, 
+signs of incipient devitrification by the formation of small crystallites of pyroxenes, feldspar, oxides and 
+possibly incipient signs of alteration phases like clays (see discussion)(von Engelhardt et al., 1995).   
+ 
+2.2 Sample Preparation  
+We used aliquots of larger amounts of material to avoid bias due to larger 
+fragments/components thus ensuring a homogeneous and representative sample. The original sample 
+masses were greater than 100 grams for each sample. Bulk samples were first powdered in steel and 
+agate mortars. Subsequently, the powders were cleaned in acetone and dry sieved into four size 
+fractions: 0-25 µm, 25-63 µm, 63-125 µm and 125-250 µm. This was done using an automatic Retsch Tap 
+Sieve; each size fraction was dry sieved for at least one hour. In order to remove clinging fines, the larger 
+two fractions were cleaned with acetone.  
+For additional in situ measurements involving optical microscopy, micro-FTIR, and Scanning 
+Electron Microscopy (SEM) measurements, we used thin sections of representative blocks of the 
+samples. The thin sections were polished to ~30 µm using standard procedures for petrological thin 
+sections, ensuring a high specular reflectance from a flat surface from which especially microscopic IR 
+investigations benefit greatly.  
+ 
+2.3 Optical Microscopy 
+Overview images of the samples in normal light and between crossed polarizers were obtained with the 
+KEYENCE Digital Microscope VHX-500F. Light microscopy allows rapid assessment of the general 
+
+ 
+ 
+11 
+ 
+homogeneity, as well as first mineral identification in the samples. Images were made using a 
+magnification of 10.  
+ 
+2.4 Infrared Analyses 
+2.4.1 Diffuse Reflectance Powder Analyses 
+To ensure diffuse reflectance, the size fractions were gently placed in aluminum sample cups (1 
+cm diameter), and the surface flattened with a spatula following a similar procedure described by 
+Mustard and Hayes (1997). For mid-infrared analyses from 2-20 µm, we used a Bruker Vertex 70 infrared 
+system with a MCT detector at the Infrared and Raman for Interplanetary Spectroscopy (IRIS) laboratory 
+at the Institut für Planetologie in Münster.  To avoid sample surface disturbance due to pore collapse 
+during evacuation, we avoided analyses under near vacuum and measured under normal atmosphere. As 
+a consequence, water and atmosphere related features slightly affect the spectral range of interest near 
+7 µm. To ensure a high signal-to-noise ratio, we accumulated 512 scans for each size fraction. For 
+background calibration a diffuse gold standard was applied. For the MERTIS database, we obtained 
+analyses in a variable geometry stage (Bruker A513) in order to emulate various observational 
+geometries of the orbiter. The data presented here were obtained at 30° incidence (i) and 30° 
+emergence angle (e). 
+The spectra of this study are intended to be compared with remote sensing data in the thermal 
+infrared. Emission and reflectance spectra can be compared using Kirchhoff’s law:  ε = 1 – R 
+(R=Reflectance, ε = Emission) (Nicodemus, 1965).  This relation works very well for the comparison of 
+directional emissivity and directional hemispherical reflectance (Hapke, 1993, Salisbury et al., 1994). 
+
+ 
+ 
+12 
+ 
+However, in order to directly relate directional emissivity with reflectance by using Kirchhoff’s law, the 
+reflected light in all directions has to be collected (Thomson and Salisbury, 1993). A bi-directional, 
+variable mirror set-up was used for this study without a hemisphere integrating all radiation. This has to 
+be kept in mind when comparing the results in a quantitative manner with emission data (Salisbury et al., 
+1991; Christensen et al., 2000). In the case of the spectra obtained with the micro-FTIR in specular 
+reflectance mode, the diffusely scattered part of the light is very small, since highly polished glass was 
+analyzed. Although specular reflectance is not entirely relatable to the directed emission, as required by 
+Kirchhoff’s law, the differences will be mainly in spectral contrast, otherwise (e.g. band shape and 
+positions) the data will be comparable (Ramsey and Fink, 1999; Byrnes et al, 2007; Lee et al., 2010). 
+Although the spectral range of interest for the database is from 7-14 µm, we measured our 
+powder from 7-20 µm (Fig. 4a-c), since features of interest can appear at longer wavelengths. The signal 
+of the detector used becomes weak at wavelengths above 18 µm, resulting in a low signal to noise ratio. 
+Above 19 µm, the spectral features are mainly random noise. The analyses were conducted under 
+ambient pressure, which possibly affected the water bands at ~3 and ~6 µm. We therefore only present 
+representative features from 2-7 µm for each sample in Fig.4d. The impactites analyzed in this study are 
+often mixtures of various mineral phases. So in order to identify specific mineral bands in our laboratory 
+spectra, we used spectra from the Arizona State University Thermal Emission laboratory (Christensen et 
+al., 2000) and the Johns Hopkins ASTER laboratory (Baldridge et al. 2009).  
+ 
+2.4.2 In-situ FTIR Microspectroscopy 
+For in situ analyses, we used a Bruker Hyperion 2000 IR microscope attached to the external port 
+of a Bruker Vertex 70v at the Hochschule Emden/Leer. Here a 1000×1000 µm2 sized aperture was 
+applied to obtain analyses of interesting features with in situ reflectance spectroscopy on polished thin 
+
+ 
+ 
+13 
+ 
+sections. For each spectrum, 128 scans were added. A gold mirror was used for background calibration. 
+The analyses were made in the range from 2-15 µm. Since all features below 7 µm are very weak, we 
+present results in the range of interest, 7-14 µm (Fig.5).  
+The small shift between powder and microscope analyses observed (see 3. Results) is probably 
+due to the inherent differences among the samples (polished thin sections compared to powders) and 
+the different optical set-ups of the techniques. Differences in the positions between the CF in powdered 
+and solid samples were also observed by Cooper et al. (2002).  
+ 
+2.5 SEM EDX Analyses 
+In order to document the suevites and their components, we used a JEOL 6610-LV Scanning 
+electron microscope equipped with a silicon drift Oxford EDX (Energy Dispersive X-Ray Spectroscopy) 
+system to obtain micrographs of particular areas of interest and perform quantitative chemical analyses. 
+Each chemical analysis was quantified with an ASTIMEX™ standard set for major elements prior to the 
+measurement. Beam current stability was controlled and measured before each analysis using a Faraday 
+cup. The calibration was confirmed by re-analyzing the standards after the calibration procedure. For 
+analyses of the chemical composition of melt glass and fine-grained matrix areas, we analyzed areas of 
+100 x 100 µm2 using 90 seconds integration times.  The rather small area was chosen to maintain 
+comparability between amorphous parts and fine-grained matrix areas. Otherwise it would be difficult to 
+obtain representative measurements of larger areas due to abundant cracks, gaps, holes, and veins, 
+which are often re-filled by secondary alteration phases. A broad beam and shorter integration times are 
+helpful to measure volatile elements correctly.  
+ 
+
+ 
+ 
+14 
+ 
+3. Results 
+3.1 Optical Images 
+Optical images (Fig2a-c) give an overview of the thin sections for the three investigated samples. 
+The Otting (Fig.2a) and Aumühle (Fig.2b) samples in the transmitted light show the components of the 
+normal suevite, the darker melt glasses, embedded in a brighter matrix, which consists of fine fragments 
+of glass, rock and also secondary alteration products. Under crossed polarizers, the glasses are 
+recognizable by their nearly black appearance due to complete extinction, while the fine mixture of the 
+matrix appears brighter.  The Polsingen sample (Fig.2c), being a coherent melt rock, consists entirely of a 
+red groundmass with abundant larger and smaller fragments in transmitted light.  Brownish rims around 
+minerals and holes indicate significant weathering.  In the image under crossed polarizers, the 
+groundmass appears dark, a few crystalline fragments are visible by their brighter appearance.   
+ 
+3.2. SEM/EDX Analyses 
+Melt glass and fine-grained matrix area measurements in the suevites are shown in Table 1. Due 
+to porosity and volatile contents (water) the total analysis of matrix material in weight (wt)% is lower 
+than 100 wt%. The melt glasses also show lower totals, which is due to crystallized or oxidized inclusions 
+as well as inevitable cracks and veins.  
+For a better comparison with earlier data, we plot characteristic oxide ratios for MgO, FeO, CaO, 
+Na2O, K2O of our SEM/EDX results with those from the literature (Fig. 3) (Stöffler et al., 2013). Results for 
+melt glass in Aumühle and Otting are very similar to each other and to earlier studies of glass from outer 
+suevite. The results for the fine-grained matrix analyses cluster in an area overlapping with the 
+composition of outer suevite (Fig. 3) (Stöffler et al., 2013). The analyses plotting outside the area are 
+
+ 
+ 
+15 
+ 
+explained by the lack of larger rock fragments within the analyzed areas. This resulted in a higher 
+content of fine grained secondary phases typical for the matrix, such as clay minerals, which is also  
+indicated by elevated Al2O3 contents (Tab. 1) (Stöffler et al., 2013). Otherwise, the compositions of the 
+fine-grained matrices can be explained as a mixture of the endmember glass and various basement rock 
+fragments. Melt rock from the red suevite in Polsingen plots slightly below the studies for bulk red 
+suevite (Stöffler et al., 2013). This is probably due to the high degree of weathering of the samples from 
+the only accessible outcrops, resulting in increased alkali (Na2O, K2O, CaO) concentrations (Reimold et al., 
+2013; Stöffler, 2013). 
+ 
+3.3 Mid-Infrared Analyses of Powders  
+The powder spectra of the bulk normal suevite from Otting and Aumühle (Fig.4a, Tab. 2a) show 
+very similar band shapes and intensities for all samples, reflecting the high chemical homogeneity 
+already indicated by SEM/EDX. The Christiansen Features (CF) of the bulk Otting and Aumühle bulk 
+suevites are between 7.4 and 8.1 µm, respectively, for all size fractions within one sample group. In the 
+finest fraction from 0-25 µm, the CF is usually at longer wavelengths, from 7.9-8.1 µm indicating a 
+mineralogical heterogeneity among the finer fraction. Clays are extremely fine-grained and likely remain 
+in the smallest grain size fraction, which could account for the slight shift in the CF (see discussion). The 
+Transparency features (TF) appear at 10.4 - 11.7 µm usually in the smallest size fractions of the bulk 
+suevites. The strongest feature in this region is between 11.6 and 11.7µm. The strong Reststrahlen band 
+(RB) at 9.4 µm again indicates a highly amorphous and homogeneous internal structure of the minerals 
+comprising the sample. Minor bands or shoulders are found on the slope of the RB at 8.5-8.6 µm and 
+8.8-8.9 µm, indicating crystalline species (see Discussion). Further RB or potential TF are observed in the 
+bulk suevites at 10.4-10.5 µm and 11.1-11.2 µm (overlapping with the transparency feature in the 
+
+ 
+ 
+16 
+ 
+smallest size fractions) and 12.5-12.9 µm. Broad features in the 18-19 µm region are also typical. All bulk 
+suevite spectra show very strong water features at 2.8 µm and 6.1 µm (Fig. 4d). These volatile bands 
+probably result from clay minerals, but analysis under atmospheric conditions could also have affected 
+the spectra. A spectrum of separated fine-grained matrix from Aumühle 18 (Fig. 4a, Tab. 2a) is very 
+similar in band shape and peak positions to the bulk spectrum Aumühle 18, indicating a high abundance 
+of melt and crystalline clasts in the fine-grained matrix. 
+Melt glasses from Aumühle and Otting (Fig. 4b; Tab. 2b) are also very homogeneous. The 
+Christiansen Features (CF) are at 7.6-7.9 µm, the Transparency Features (TF) at 11.7-11.8 µm in the finest 
+size fraction. The strongest Reststrahlen band is at 9.3-9.4 µm, with a weak shoulder in the finest size 
+fraction at 8.6 µm. This all confirms the amorphous nature of the material. Only weak bands are found in 
+the 18-19 µm region. Water features occur at 6.1 µm and 2.8-2.9 µm (Fig. 4d).  
+In the red suevite (Polsingen) (Fig. 4c; Tab. 2c), the CF occurs between 7.6 and 7.9 µm. A strong 
+TF band is located at 11.8-12.0 µm, and the strongest Reststrahlen band is at 9.4-9.5 µm. Further, weak 
+Reststrahlen bands or shoulders are between 8.2 and 8.8 µm in red suevite samples. Also, the slopes 
+between 10 and 12 µm in Polsingen 3 and 4 are less steep than in the normal suevite.  Broad bands at 
+~17 µm, and between 18-19 µm are also typical. All red suevites show strong water features at 6.1 and 
+2.7-3.0 µm, which are probably caused by weathering phases (Fig.4d).  
+ 
+3.4 In Situ Analyses 
+Fine-grained matrix analyses for Aumühle (Fig.5, Table 3) show CFs between 7.6 and 8.3 µm, those for 
+Otting are located at 7.6-8 µm. The only powder analysis of separated matrix from Aumühle 18 overlaps 
+with these ranges (7.4-7.9µm).  The strongest Reststrahlen band is at 9.5-9.6 µm in the Aumühle matrix, 
+at slightly longer wavelengths  than in the powdered sample (9.4 µm). In the fine-grained Otting matrix, 
+
+ 
+ 
+17 
+ 
+the Reststrahlen band is very similar (between 9.4 and 9.5 µm) for both in situ and powder 
+measurements. There are several additional Reststrahlen bands for fine grained matrix in the in situ 
+analyses:  at 8.5-8.6 µm, 8.9-9.0 µm, and from 12.5 to 12.9 µm. The intensities vary from shoulders to 
+clear bands. Here the band positions are very similar to the powdered matrix from Aumühle 18. 
+The in-situ analyses of glasses in the polished sections of Aumühle and Otting (Fig.5, Tab.3) have 
+Christiansen Features between 7.7 and 7.9 µm, identical to the powder measurements. The dominating 
+Reststrahlen bands are at 9.4 to 9.6 µm, thus slightly shifted to longer wavelengths compared to the 
+powder data. The Polsingen samples where the ‘matrix’ consists of melt glass, show a high homogeneity: 
+the CF is at 7.9-8 µm, the Reststrahlen band at 9.6 µm. Characteristic is a shoulder/feature at 8.8 µm. 
+Again, a slight shift for the position of the Reststrahlen band was observed compared to the powder 
+analyses, whereas the other features occur at similar wavelength positions.  
+ 
+4. Discussion  
+The investigated impact melt glasses in Aumühle and Otting do not have significant bands of 
+crystalline features, and are dominated by the Reststrahlen band at 9.3-9.4 µm in the powdered samples 
+and 9.4-9.6 µm in the micro-FTIR analyses. This range is similar to quartzofeldspathic glasses or glass 
+with granitic/rhyolitic composition (9.1-9.8 µm) with a mafic component (9.4-10.5 µm) (Wyatt et al., 
+2001; Byrnes et al, 2007; Johnson et al., 2007; DuFresne et al., 2009; Lee et al., 2010; Minitti and 
+Hamilton, 2010; Wright et al., 2011). This mixture reflects the starting composition of the Ries impact 
+site, which was dominated by gneiss, granite and about 13% mafic rock (amphibolite; von Engelhardt, 
+1997). The CF of the melt glasses (7.6-7.9 µm) also falls into the region for acidic and intermediate rocks, 
+while mafic minerals and rocks tend to have the CF greater than 8 µm (Pieters and Englert, 1993; 
+Salisbury and Walter, 1989; Cooper et al., 2002). 
+
+ 
+ 
+18 
+ 
+Bulk melt rock, i.e. the red suevite from Polsingen, shows the main RB band slightly shifted by about 0.1 
+µm to longer wavelengths. The similarity of the various samples even on millimeter scale in in-situ 
+analyses confirms that the red suevite is a homogeneous melt rock. Compared to the melt glasses in 
+normal suevite, the spectra from Polsingen show weak, but clear RB or shoulders of crystalline materials. 
+These indicate higher contents of crystalline components from granite and gneiss fragments (e.g., Hecker 
+et al., 2012).  
+Bulk suevites also form a homogeneous group, only the relative intensity of the mostly weak 
+crystalline bands hints at varying contents of partially shocked fragments.  Separated matrix is spectrally 
+similar to the bulk material, indicating a high amount of fine-grained rock fragments in the matrix. This is 
+confirmed by the in-situ analyses of the fine-grained matrix, which have the same/similar band positions, 
+but varying intensities of the Restrahlen bands. The Reststrahlen bands in bulk powder suevite and fine-
+grained matrix are a mixture of features characteristic for glasses and crystalline components.  
+The dominating band in the bulk samples between 9.4 µm (powders) and 9.4 to 9.6 µm (micro-
+FTIR) is similar to the melt glasses (Byrnes et al, 2007; Lee et al., 2010; Wright et al., 2011). However, 
+crystalline features between 8.5-8.6 µm, 8.8-9.0 µm, and from 12.5 to 12.9 µm in the bulk and fine-
+grained matrices are identical to that of quartz (e.g., Wenrich and Christensen, 1996; Christensen et al., 
+2000; Michalski et al., 2003; Baldridge et al., 2009), a major component of biotite-rich granite and gneiss 
+as found in the basement of the Ries Crater (von Engelhardt, 1997; Hecker et al., 2012).  
+The ranges for the position of the Christiansen Features in glass and bulk suevite overlap, but 
+some of the bulk samples have their CF at significantly shorter wavelengths compared to the glass. This 
+probably reflects the increased content of a crystalline quartz component. The lowest CF are between  
+7.4 and 7.5 µm, close to that of quartz with 7.35-7.4 µm (Tab. 2a)(e.g., Christensen et al., 2000; Michalski 
+et al., 2003; Baldridge et al., 2009; Tappert et al., 2013). This in turn would point to a significant 
+
+ 
+ 
+19 
+ 
+crystalline component at least in some of the bulk suevites.  A comparison with CF positions of terrestrial 
+rocks (Salisbury and Walter, 1989; Cooper et al., 2002) also indicates that the bulk powder suevites (7.4-
+8.1µm) cover the range for acidic rocks (7.6-7.8 µm) and intermediate rocks (7.8-8.2 µm).  
+The variations in the positions of the CF are probably due to slight compositional variations as a 
+result of sieving. A higher content of quartz drives the CF to shorter wavelengths (e.g., Christensen et al., 
+2000; Michalski et al., 2003; Baldridge et al., 2009; Tappert et al., 2013), while an increase of clay 
+minerals or a mafic component (amphibolite) in the finest fraction during sieving could shift the position 
+of the CF towards longer wavelengths. Hornblende, a major phase of amphibolite, which comprises the 
+mafic component in the suevite (von Engelhardt, 1997), has a high CF from 8.2-8.5 µm (Salisbury, 1992; 
+1993; Baldridge et al., 2009). The CF of alteration phase montmorillionite falls between 7.9 and 8.2 µm 
+and would be difficult to identify directly in a mixture with rocks of intermediate composition 
+(Christiansen et al., 2000; Cooper et al., 2002; Koeppen et al., 2005; Michalski et al., 2005; Baldridge, 
+2009).The CF shift is also pronounced in the micro-FTIR spectra of the Aumühle samples, where the 
+intensity of the crystalline quartz features also varied in the analyzed areas (Fig.5). 
+In summary, in the mid-infrared range, suevites can be identified by their highly amorphous 
+band shape and few crystalline features. This could help to identify impactites like suevite in remote 
+sensing data, but also provides information about the shocked basement rock. However, glassy, 
+amorphous spectral features can also be expected from volcanic glasses like obsidian, so additional 
+geologic information about an observed area is necessary to distinguish the sources of glassy material 
+(Hamilton et al., 2001; Moroz et al., 2009; Wright et al., 2011).  
+While the starting composition of Mercury was probably not like the granitic basement of the 
+impact site in the Ries, i.e., in terms of mineralogy, it is still possible to draw conclusions for future 
+observations of the Mercurian surface. Mercury is a planet that underwent massive impact cratering in 
+
+ 
+ 
+20 
+ 
+its early history (e.g., Hiesinger et al. 2010; Strom et al., 2011; Fassett et al., 2012). As a consequence, the 
+resulting future mid-infrared observations that will be made of Mercury by MERTIS on BepiColombo 
+might show only a few strong crystalline features. While the Ries crater represents only the effects of 
+one impact, surface material on Mercury underwent many impacts resulting in regolith gardening 
+(Domingue et al., 2014). The low degree of crystallinity of all involved rocks in suevite after only one 
+impact event indicates that the degree of amorphization can be expected to be much higher on the 
+surface of Mercury.  
+ It has to be taken into account that the suevite is not the only type of ejecta observed in the 
+Ries. By volume, it is only a relatively small component – the Bunte Breccia has an estimated volume of 
+95 km3, the megablocks about 47 km3, and up to 22 km3 of suevite (Stöffler et al., 2013; Sturm et al, 
+2015). However, the suevite is the uppermost ejecta layer (e.g., von Engelhardt, 1990), and so is the 
+material most likely to be observed in remote sensing. Also, in a continuing impact gardening of a 
+planetary surface, the less shocked rocks will also increasingly experience higher degrees of shock 
+metamorphism.  This might be even more enhanced by higher impact velocities on Mercury compared to 
+Earth, which result in the production of larger amounts of impact melt (e.g., Fassett et al., 2012). 
+In addition, the effects of space weathering will damage the crystalline structure of the remaining 
+material even more (Domingue et al., 2014). Observations in the ultraviolet, visible and near-infrared 
+range by MESSENGER do not show much variation over the surface, which could also point towards a 
+complete amorphization of the surface minerals (Izenberg et al., 2014). However, constant reprocessing 
+and gardening of the surface may also allow formation of new crystallites, resulting in a mixture of 
+crystallized and amorphous material. In addition volcanic activity (Thomas et al., 2014) will produce 
+crystalline phases. Consequently, ground based spectroscopic observations of larger areas on Mercury 
+(e.g., Sprague et al., 2000) show features of crystalline species such as pyroxene and feldspar.  
+
+ 
+ 
+21 
+ 
+Furthermore, for our studied samples, alteration of pristine impact material by weathering has 
+to be taken into account. The fine-grained parts of the matrix contain clay minerals, mainly 
+montmorillonite, but also minor illite and halloysite (Stöffler et al, 2013). A strong band of 
+montmorillonite is found at 9.4-9.5 µm, overlapping with the position for the strong feature of 
+amorphous silicates. The strong feature of illite and halloysite is at slightly longer wavelengths, 9.4-9.7 
+µm (Christensen et al., 2000; Koeppen et al., 2005; Michalski et al., 2006; Baldridge et al., 2009). A 
+further feature that could indicate montmorillonite is a weak band at about 8.8 µm (Christensen et al., 
+2000; Michalski et al., 2006; Baldridge et al., 2009), occurring in most bulk suevites and the Polsingen 
+sample (Tab.2). However, this band is also characteristic for quartz (e.g., Christensen et al., 2000; 
+Michalski et al., 2003; Tappert et al., 2013).  
+Another way to identify clay minerals are spectral features at longer wavelengths, in the 17-19 
+µm regions. Clay minerals have features in the 18-19 µm region, montmorillionite from 18.7-19 µm, illite 
+and halloysite from 18-19 µm (Christensen et al., 2000; Baldridge et al., 2006; Michalski et al., 2006). 
+Here the Aumühle and Otting glasses show only a few, weak features. The suevite bulk samples show 
+more features in the 18.0-19 µm region, especially from 18.4-19 µm. This points towards a clay 
+component, but quartz can also have a feature at 18.3 µm (Baldridge et al., 2009). In contrast to the 
+Aumühle and Otting glasses, the Polsingen melt rocks show clear bands in the 18-19 µm region, 
+indicating a stronger degree of weathering compared to the glasses. However, this part is at the limit of 
+the spectral range of the spectrometer used, so exact band positions are difficult to obtain. 
+The strongest transparency feature of the bulk suevites (11.6-11.7µm) and red suevite (12.0µm) 
+is similar to that of the Aumühle and Otting glasses (at between 11.7 – 11.8 µm), and is also typical for 
+acidic and intermediate rocks (Salisbury and Walter, 1989; Cooper et al., 2002). A potential TF of 
+montmorillionite and illite at 12.1-12.7 µm is not visible in the smallest size fractions, except possibly the 
+Red Suevite from Polsingen. Illite has another TF at 11.5-11.7, which would overlap with the glass TF 
+
+ 
+ 
+22 
+ 
+(Baldridge et al., 2006). A potential TF in bulk suevite at 11.1-11.2 µm is similar to one of the 
+montmorillonite TF found at 11.2-11.5 µm (Baldridge et al., 2006). However, quartz has its TF also in this 
+region (10.9-11.1µm) (Salisbury 1992; 1993; Baldridge et al., 2009).  
+Water bands could also potentially allow identifying secondary phases, but analyses were 
+conducted under ambient air pressure. The water bands for the samples in this study are all very similar 
+to each other, which indicate that adsorbed water could have influenced the spectra.  This would render 
+the features difficult to use for comparison.  
+Further hints for the occurrence of smectites are elevated aluminum contents in the matrices and the 
+Polsingen samples (Tab.1). Certainly, aqueous alteration due to volatiles appears implausible on Mercury 
+at large scales. But application of the data from this study for remote sensing observations of bodies 
+where impactites were affected by alteration, like Earth or Mars, is still valid, as (minor to moderate) 
+alteration features do not overprint the spectral signature of the impactites. 
+ 
+5. Conclusion 
+Despite the general similarity of their bulk spectra, bulk suevite, red suevite melt rock, and impact melts 
+are distinguishable by their mid-infrared features, although they are all dominated by amorphous 
+materials. Suevite glass from Aumühle and Otting shows simple spectra, dominated by an amorphous 
+feature. Suevites have clear bands of crystalline materials, while the Polsingen impact melt falls between 
+the two groups. 
+Secondary phases like clay minerals have features overlapping with other components (lithic rock clasts, 
+amorphous material) and are difficult to identify in the laboratory bulk spectra. This shows that low to 
+moderate amounts of alteration may not significantly affect the study of impactites on remote bodies. 
+
+ 
+ 
+23 
+ 
+The samples of Polsingen impact melt also show signs of weathering, in contrast to the melt glasses from 
+Aumühle and Otting.   
+On the basis of our observations, we conclude that in mid-infrared remote sensing data, the surface 
+layer of Mercury will be dominated by features of amorphous materials. Because of the high degree of 
+amorphization that occurs after only one impact, as inferred from the Ries event, and following impact 
+gardening, even remaining crystalline materials will undergo high degrees of shock metamorphism. 
+However, the suevite represents only the uppermost impact layer in the Ries, while the underlying, more 
+voluminous ejecta show higher degrees of crystallinity.  
+ 
+Acknowledgements 
+Many thanks to Prof. Alexander Deutsch (Münster) and Gisela Pösges (Rieskrater Museum, Nördlingen) 
+for precious help with the samples. We also thank the editor, Will M. Grundy, Steve Ruff and another 
+anonymous reviewer for helping to improve the manuscript.  
+This work is supported by the DLR funding 50 QW 1302 in the framework of the BepiColombo mission. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+24 
+ 
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+ 
+ 
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+ 
+Figure Captions 
+Figure 1: Map of the Ries area (adapted from Stöffler, 2013). Samples for this study were taken in the 
+Otting, Aumühle and Polsingen locations.  
+Figure 2a-c: Micrographs of representative areas of locations sampled for this study under normal 
+transmitted light (top) and crossed polarizers (bottom). The Otting (a) and Aumühle (b) samples show 
+the melt glasses embedded in the fine grained matrix. Spots (1 mm2 each) analyzed in situ with a FTIR-
+microscope are marked with black boxes. Red suevite from Polsingen (c) does not clearly show the 
+distinction between amorphous material and matrix, it is probably a coherent melt rock. 
+Figure 3: Comparison of SEM/EDAX data for melt glass and matrix. Data from this study: squares = 
+Aumühle, circles = Otting, triangles = Polsingen. Empty symbols = glass; filled symbols = fine-grained 
+matrix.  Literature data (Stöffler et al., 2013) are marked with encircled areas (see legend).  Chemical 
+data for impact melt glass in suevite from our study fall into an area for earlier analyses. Analyses of fine-
+grained matrix overlap with bulk–suevite data, but also fall outside the range, due to higher contents of 
+secondary phases.    
+Figure 4a-d. Mid-infrared reflectance spectra of samples in the size fractions 0-25µm (blue), 25-63µm 
+(pink), 63-125µm (red) and 125-250µm (brown) of (a) Suevite bulk and matrix analyses from Aumühle 
+and Otting, (b) Spectra of separated melt glasses from Aumühle and Otting, (c) Samples of red suevite 
+melt glass from Polsingen. Vertical lines mark characteristic Christiansen, Reststrahlen and Transparency 
+features. (d) Water features in the range from 2-7 µm for selected samples. Spectra are off-set for 
+clarity. Blue: melt glasses, Red: red suevite from Polsingen, Purple: suevite bulk samples. 
+Figure 5: Infrared and Raman for Interplanetary Spectroscopy (IRIS) laboratory In situ mid-infrared 
+spectra obtained from polished sections of suevite using a FTIR microscope. Each area was 1 mm2 in size; 
+the locations are shown in Figure2 a-c.   
+
+Figure 1
+
+Drilling
+& Outcrop
+Structural
+IN
+Aumuhle
+Inner ring
+Polsingen
+Centeri
+i
+Potting
+4
+-
+-
+Enkingen
+Itzing 0
+@Altenburg
+Mauren
+Seelbronn
+Bollstadt
+5 km
+Amerdingen3 mm
+Glass2
+Glass3
+Glass1
+Matrix1
+Matrix2
+Matrix4
+Matrix3
+Matrix5
+Otting
+Glass2
+Glass3
+Glass1
+Matrix1
+Matrix2
+Matrix4
+Matrix3
+Matrix5
+Transmitted Light
+Cross Polarized
+Figure 2a
+
+工
+1.00 mm2 mm3 mm
+Matrix4
+Matrix3
+Matrix1
+Matrix2
+Glass1
+Glass2
+Glass3
+Aumühle
+Matrix4
+Matrix3
+Matrix1
+Matrix2
+Glass1
+Glass2
+Glass3
+Transmitted Light
+Cross Polarized
+Figure 2b
+
+2 mm
+Polsingen1
+Polsingen2
+Polsingen3
+Polsingen
+Polsingen1
+Polsingen2
+Polsingen3
+Transmitted Light
+Cross Polarized
+Figure 2c
+
+TI
+1.00mm2mmMatrix 
+Glass
+Impact Melt Rock 
+Basement Rocks 
+Melt/Suevite Crater
+Melt Outer Suevite
+Outer Suevite
+Polsingen (Melt rock)
+Aumühle
+Otting
+FeO+MgO
+CaO+Na2O
+K2O
+Figure 3
+
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+0.00
+0.05
+0.10
+Reflectance
+Micron (µm)
+Aumühle 18
+Matrix
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+0.00
+0.05
+0.10
+Reflectance
+Micron (µm)
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+0.00
+0.05
+0.10
+Reflectance
+Micron (µm)
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+0.00
+0.05
+0.10
+Reflectance
+Micron (µm)
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+0.00
+0.05
+0.10
+Reflectance
+Micron (µm)
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+0.00
+0.05
+0.10
+Reflectance
+Micron (µm)
+Otting
+Bulk 1
+Aumühle 4
+Bulk
+Aumühle 18
+Bulk
+Otting
+Bulk 2
+Otting
+Bulk 3
+Christiansen 
+Feature
+Transparency
+Feature
+Reststrahlen 
+Bands
+125-250 µm
+25-63 µm
+0-25 µm
+63-125 µm
+Figure 4a
+
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+0.00
+0.05
+0.10
+Reflectance
+Micron (µm)
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+0.00
+0.05
+0.10
+Reflectance
+Micron (µm)
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+0.00
+0.05
+0.10
+Reflectance
+Micron (µm)
+Otting 
+Glass1
+Otting
+Glasbombe
+Aumühle 13
+Glass
+Figure 4b
+
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+Reflectance
+Micron (µm)
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+Reflectance
+Micron (µm)
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+Reflectance
+Micron (µm)
+Polsingen
+Bulk 3
+Polsingen
+Bulk 4
+Polsingen
+Bulk 2
+Figure 4c
+
+Otting Glasbombe
+Aumühle 13 Glass
+Polsingen Bulk 2
+Aumühle 18 
+Matrix
+Aumühle 18 
+Bulk
+Otting Bulk 1
+0.00
+0.10
+0.20
+0.30
+0.40
+Reflectance (off-set)
+2
+3
+4
+5
+6
+7
+Micron (µm)
+0.50
+0.60
+0.70
+0.80
+0.90
+1.00
+Figure 4d
+
+7
+8
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+Glass
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+Matrix
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+Figure 5
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+An AMM minimizing user-level extractable
+value and loss-versus-rebalancing
+Conor McMenamin1 and Vanesa Daza1,2
+1 Department of Information and Communication Technologies, Universitat Pompeu
+Fabra, Barcelona, Spain
+2 CYBERCAT - Center for Cybersecurity Research of Catalonia
+Abstract. We present V0LVER, an AMM protocol which solves an in-
+centivization trilemma between users, passive liquidity providers, and
+block producers. V0LVER enables users and passive liquidity providers
+to interact without paying MEV or incurring uncontrolled loss-versus-
+rebalancing to the block producer. V0LVER is an AMM protocol built
+on an encrypted transaction mempool, where transactions are decrypted
+after being allocated liquidity by the AMM. V0LVER ensures this liq-
+uidity, given some external market price, is provided at that price in ex-
+pectancy. This is done by providing just enough loss-versus-rebalancing
+profits to the block producer, incentivizing the block producer to move
+the pool price to the external market price. With this, users transact
+in expectancy at the external market price in exchange for a fee, with
+AMMs providing liquidity in expectancy at the external market price.
+Under block producer and liquidity provider competition, all of the fees
+in V0LVER approach zero. Without block producer arbitrage, V0LVER
+guarantees fall back to those of an AMM capable of processing encrypted
+transactions, free from loss-versus-rebalancing or user-level MEV.
+Keywords: Extractable Value · Decentralized Exchange · Incentives · Blockchain
+1
+Introduction
+In this paper we introduce V0LVER 3, an AMM which provides arbitrarily high
+protection against user-level MEV and LVR. V0LVER is the first AMM to align
+the incentives of the three, typically competing, entities in AMMs; the user, the
+pool, and the block producer. This is done by ensuring that at all times, a block
+producer is incentivized to move the pool to the price maximizing LVR. When
+the block producer chooses a price, the block producer is forced to assert this is
+correct, a technique introduced in [9]. Unfortunately, the protocol in [9] gives the
+This Technical Report is part of a project that has received funding from the
+European Union’s Horizon 2020 research and innovation programme under
+grant agreement number 814284
+3 near-0 Extractable Value and Loss-Versus-Rebalancing ⇝ V0LVER
+arXiv:2301.13599v1  [cs.GT]  31 Jan 2023
+
+2
+McMenamin and Daza
+block producer total power to extract value from users, due to order information
+being revealed to the block producer before it is allocated a trading price in the
+blockchain. To address this, V0LVER is built on encrypted mempools.
+Modern cryptographic tools allow us to encrypt the mempool using zero-
+knowledge based collateralized commit-reveal protocols [7,2,8,13], delay encryp-
+tion [4,5] and/or threshold encryption [1]. If a block producer adding an order to
+an AMM is forced to replicate the payoff of the AMM, we demonstrate the block
+producer maximizes her own utility by showing liquidity centred around the ex-
+ternal market price.4 Providing users with an AMM where the expected trade
+price is the external market price, excluding fees, is a significant advancement
+and the main contribution of this paper. Although batching orders against AMM
+liquidity has been proposed as a defense against LVR [12], naively batching or-
+ders against an AMM still allows a block producer to extract LVR by censoring
+user orders. In V0LVER, block producers are effectively forced to immediately
+repay LVR, while always being incentivized to include order commitments in the
+blockchain, and eventually allocate AMM liquidity to these orders.
+2
+Related Work
+As the phenomenon of LVR has only recently been identified, there are only
+two academic papers on the subject of LVR protection [6,9] to the best of our
+knowledge, with no work protecting against both LVR and user-level MEV.
+In [6], the AMM must receive the price of a swap from a trusted oracle before
+users can interact with the pool. Such sub-block time price data requires central-
+ized sources which are prone to manipulation, or require the active participation
+of AMM representatives, a contradiction of the passive nature of AMMs and
+their liquidity providers. We see this as an unsatisfactory dependency for DeFi
+protocols.
+Our work is based on some of the techniques of the Diamond protocol as
+introduced in [9]. The Diamond protocol requires block producers to effectively
+attest to the final price of the block given the orders that are to be proposed to
+the AMM within the block. This technique requires the block producer to know
+exactly what orders are going to be added to the blockchain. This unfortunately
+gives the block producer total freedom to extract value from users submitting
+orders to the AMM. With V0LVER, we address this issue while keeping the LVR
+protection guarantees of Diamond.
+Encrypting the transaction mempool using threshold encryption controlled
+by a committee has been proposed in [1] and applied in [11]. In [11], a DEX
+involving an AMM and based on frequent batch auctions [3] is proposed. This
+DEX does not provide LVR resistance, and incentivizes transaction censorship
+when a large LVR opportunity arises on the DEX. This is protected against in
+V0LVER.
+4 This holds true in many CFMMs, including the famous Uniswap V2 protocol.
+
+V0LVER
+3
+3
+Preliminaries
+This section introduces the key terminology and definitions needed to understand
+LVR, and the proceeding analysis. In this work we are concerned with a single
+swap between token x and token y. We use x and y subscripts when referring
+to quantities of the respective tokens. The external market price of a swap is
+denoted by ϵ, with the price of a swap quoted as the quantity of token x per
+token y.
+3.1
+Constant Function Market Makers
+A CFMM is characterized by reserves (Rx, Ry) ∈ R2
++ which describes the total
+amount of each token in the pool. The price of the pool is given by pool price
+function P : R2
++ → R taking as input pool reserves (Rx, Ry). P() has the
+following properties:
+(a) P() is everywhere differentiable, with ∂P
+∂Rx
+> 0,
+∂P
+∂Ry
+< 0.
+(b)
+lim
+Rx→0 P = 0,
+lim
+Rx→∞ P = ∞,
+lim
+Ry→0 P = ∞,
+lim
+Ry→∞ P = 0.
+(c) If P(Rx, Ry) = p, then P(Rx + cp, Ry + c) = p, ∀c > 0.
+(1)
+For a CFMM, the feasible set of reserves C is described by:
+C = {(Rx, Ry) ∈ R2
++ : f(Rx, Ry) = k}
+(2)
+where f : R2
++ → R is the pool invariant and k ∈ R is a constant. The pool is
+defined by a smart contract which allows any player to move the pool reserves
+from the current reserves (Rx,0, Ry,0) ∈ C to any other reserves (Rx,1, Ry,1) ∈ C
+if and only if the player provides the difference (Rx,1 − Rx,0, Ry,1 − Ry,0). To
+reason about V0LVER, we assume the existence of an external market price
+between x and y, which we define as follows, based on the definition of [3].
+Whenever an arbitrageur interacts with an AMM pool, say at time t with
+reserves (Rx,t, Ry,t), we assume as in [10] that the arbitrageur always moves the
+pool reserves to a point which maximizes arbitrageur profits, exploiting the dif-
+ference between P(Rx,t, Ry,t) and the external market price at time t, denoted ϵt.
+In this paper, we consider only the subset of CFMMs in which, given the LVR ex-
+tracted in block Bt+1 corresponds to reserves (Rx,t+1, Ry,t+1), P(Rx,t+1, Ry,t+1)
+= ϵt+1. This holds for Uniswap V2 pools, among others.
+3.2
+LVR-resistant AMM
+We provide here an overview of the most important features of Diamond [9], an
+AMM protocol which is proved to provide arbitrarily high LVR protection under
+competition to capture LVR among block producers. In V0LVER, we adapt these
+features for use on an encrypted transaction mempool.
+
+4
+McMenamin and Daza
+A Diamond pool Φ is described by reserves (Rx, Ry), a pricing function P(), a
+pool invariant function f(), an LVR-rebate parameter β ∈ (0, 1), and conversion
+frequency T ∈ N. The authors also define a corresponding CFMM pool of Φ,
+denoted CFMM(Φ). CFMM(Φ) is the CFMM pool with reserves (Rx, Ry) whose
+feasible set is described by pool invariant function f() and pool constant k =
+f(Rx, Ry). Conversely, Φ is the corresponding V0LVER pool of CFMM(Φ). The
+authors note that CFMM(Φ) changes every time the Φ pool reserves change. The
+protocol progresses in blocks, with one reserve update possible per block.
+For an arbitrageur wishing to move the price of CFMM(Φ) to p from starting
+reserves (Rx,0, Ry,0), let this require ∆y > 0 tokens to be added to CFMM(Φ),
+and ∆x > 0 tokens to be removed from CFMM(Φ). The same price in Φ is
+achieved by the following process:
+1. Adding (1 − β)∆y tokens to Φ and removing (1 − β)∆x tokens.
+2. Removing δx > 0 tokens such that:
+P(Rx,0 − (1 − β)∆x − δx, Ry,0 + (1 − β)∆y) = p.
+(3)
+These δx tokens are added to the vault of Φ.
+Vault tokens are periodically re-entered into Φ through what is effectively an
+auction process, where the tokens being re-added are in a ratio which approxi-
+mates the external market price at the time. The main result of [9] is the proving
+that given a block producer interacts with Φ when the LVR parameter is β, and
+there is an LVR opportunity of LV R in CFMM(Φ), the maximum LVR in Φ is
+(1 − β)LV R. This results is stated formally therein as follows:
+Theorem 1. For a CFMM pool CFMM(Φ) with LVR of L > 0, the LVR of Φ,
+the corresponding pool in Diamond, has expectancy of at most (1 − β)L.
+In this paper we use the same base functionality of Diamond to restrict the
+LVR of block producers. Given a block producer wants to move the price of
+CFMM(Φ) to some price p to extract maximal LVR LV R, the maximal LVR
+in Φ of (1 − β)LV R is also achieved by moving the price to p. An important
+point to note about applying LVR rebates as done in [9], is that directly after
+tokens are placed in the vault, the pool constant drops. This must be considered
+when calculating the profitability of an arbitrageur extracting LVR from a Dia-
+mond pool. We do this when analyzing the profitability of V0LVER in Section
+5. Importantly, tokens are eventually re-added to the pool, and over time the
+expected value of the pool constant is increasing, as demonstrated in [9].
+4
+Our Protocol
+We now outline the model in which we construct V0LVER, followed by a detailed
+description of V0LVER.
+
+V0LVER
+5
+4.1
+Model
+In this paper we consider a blockchain in which all transactions are attempting
+to interact with a single V0LVER pool between tokens x and y.
+1. A transaction submitted by a player for addition to the blockchain while
+observing blockchain height H, is finalized in a block of height at most
+H + T, for some known T > 0.
+2. The token swap has an external market price ϵ, which follows a Martingale
+process.
+3. There exists a population of arbitrageurs able to frictionlessly trade at exter-
+nal market prices, who continuously monitor and interact with the blockchain.
+4. Encrypted orders are equally likely to buy or sell tokens at ϵ, distributed
+symmetrically around ϵ.
+4.2
+Protocol Framework
+This section outlines the terminology and functionalities used in V0LVER. It is
+intended as a reference point to understand the core V0LVER protocol. Specifi-
+cally, we describe the possible transactions in V0LVER, the possible states that
+V0LVER orders/order commitments can be in, and the possible actions of block
+producers.
+As in the protocol of Section 3.2, a V0LVER pool Φ with reserves (Rx, Ry)
+is defined with respect to a CFMM pool, denoted CFMM(Φ), with reserves
+(Rx, Ry), a pricing function P(), under the restrictions of Section 3.1, and a
+pool invariant function f().
+Orders in V0LVER are intended to interact with the AMM pool with some
+delay due to commit-reveal. Therefore, we need to introduce the concept of
+allocated funds to be used when orders eventually get revealed. To do this, we
+define an allocated pool Φa. For a player committing to an order of size either
+sizex or sizey known to be of maximum size maxx or maxy, the allocation pool
+consists of (∆x, ∆y) tokens such that:
+f(Rx, Ry) = f(Rx + maxx, Ry − ∆y) = f(Rx − ∆x, Ry + maxy).
+(4)
+Moreover, if f(Rx, Ry) = f(Rx+sizex, Ry−δy), meaning the CFMM pool would
+sell δy tokens in return for sizex, the allocation pool also sells δy for sizex.
+There are three types of transaction in our protocol. To define these trans-
+actions, we need an LVR rebate function β : [0, 1, ..., Z, Z +1] → [0, 1]. It suffices
+to consider β() as a strictly decreasing function with β(z) = 0 ∀z ≥ Z.
+1. Order. These are straightforward buy or sell orders indicating a limit price,
+size and direction to be traded. WLOG, we assume all orders in our system
+are executable.
+2. Order commitment transaction (OCT). These are encrypted orders
+known to be collateralized by either maxx or maxy tokens. The exact size,
+direction, price, and sender of an OCT sent from player Pi is hidden from all
+
+6
+McMenamin and Daza
+other players. This is possible using anonymous ZK proofs of collateral such
+as used in [8,13,7]), which can be implemented on a blockchain in conjunction
+with a user-lead commit-reveal protocol, delay encryption scheme [4,5] or
+threshold encryption scheme [1,11].
+3. Update transaction. These transactions are executed in a block before
+any OCT is allowed to interact with the protocol pool. Let the current block
+height be H. Update transactions take as input an allocation block height
+Ha ≤ H, some number of transactions to allocate Ta ∈ Z≥0, and pool price
+p. Given an allocation block height of H′
+a in the previous update transaction,
+valid update transactions require:
+(a) Ha > H′
+a.
+(b) The number of OCTs in blocks [H′
+a + 1, ..., Ha] equals Ta.
+Given inputs (Ha, Ta, p), the block producer moves the price of the pool to
+p. The producer receives (1−β()) of the implied change in reserves from this
+price move, as is done in [9].
+The producer then deposits (Taβ(H − Ha)maxx, Taβ(H − Ha) maxx
+p
+) to an
+allocation pool denoted ΦHa, with (Ta(1 − β(H − Ha))maxx, Ta(1 − β(H −
+Ha)) maxx
+p
+) being added to ΦHa from the AMM reserves.
+In other words, if an allocation pool requires up to (Tamaxx, Ta maxx
+p
+) tokens
+to trade with orders corresponding to the Ta allocated orders, the block producer
+is forced to provide β(H − Ha) of the tokens in the pool, with starting bid and
+offer prices equal to the pool price set by the block producer. This is used to
+incentivize the block producer to always choose a pool price equal to the external
+market price.
+Every block, a block producer has four possible actions to perform on OCTs
+and their orders. Orders in our system are batch-settled with other orders allo-
+cated at the same time, and the liquidity in the respective allocation pool.
+1. Insert OCTs to the blockchain.
+2. Allocate inserted OCTs. For a block producer adding a block at height H
+to allocate any number (including 0) inserted OCTs with inserted height of
+at most Hi, the block producer must:
+(a) Allocate all unallocated OCTs with inserted height less than or equal to
+Hi. Let there be x such inserted OCTs to allocate.
+(b) Submit an update transaction with inputs (Ha = Hi, Ta = x, p), for
+some p > 0.
+3. Reveal allocated order. When a decrypted order corresponding to an OCT
+at height Ha is finalized on the blockchain within T blocks after the corre-
+sponding OCT is allocated, it is considered revealed. For allocation pool Φa
+Ha
+with reserves (Rx,Ha, Ry,Ha), the initial clearing price for orders allocated at
+height Ha is equal to Rx,Ha
+Ry,Ha . Upon order reveal, the clearing price for orders
+is updated. This clearing price maximizes trade volume, as is done in [8].
+4. Execute revealed orders. T blocks after OCTs are allocated, any correspond-
+ing revealed orders are executed at the last-updated clearing price for orders
+
+V0LVER
+7
+allocated at the same time. The final tokens in the allocation pool are re-
+distributed proportionally to the allocating block producer and V0LVER
+reserves.
+4.3
+Protocol Outline
+Our protocol can be considered as two sub-protocols, an update protocol and an
+execution protocol. The update protocol is analogous to the protocol in Section
+3.2, while the execution protocol, and its combination with the update protocol
+are novel and specific to this paper. Every round a block-producer can submit
+an update transaction. There are two scenarios for an update transaction with
+inputs (Ha, Ta, p) and block height of the previous update transaction H′
+a. Either
+Ta = 0 or Ta > 0. If Ta = 0. the update transaction is equivalent to an arbi-
+trageur operation on a Diamond pool with LVR-rebate parameter of β(Ha−H′
+a)
+(see Section 3.2).
+If Ta > 0, the arbitrageur must also deposit (Ta(1−β(H −Ha))maxx, Ta(1−
+β(H−Ha)) maxx
+p
+) to the Ha allocation pool ΦHa, with (Taβ(H−Ha)maxx, Taβ(H−
+Ha) maxx
+p
+) being added to ΦHa from the AMM reserves.
+After an allocation pool is created for allocated OCTs {oct1, ..., octn}, the
+orders corresponding to {oct1, ..., octn} can be revealed for up to T blocks. This
+is sufficient time for any user whose OCT is contained in that set to reveal the
+order corresponding to the OCT. To enforce revelation, tokens corresponding to
+unrevealed orders are burned. After all orders have been revealed, or T blocks
+have passed, any block producer can execute revealed orders against the alloca-
+tion pool at a clearing price which maximizes volume traded. Specifically, given
+an array of orders ordered by price, a basic smart-contract can verify that a
+proposed clearing price maximizes volume traded, as is done in [8].
+The final tokens in the allocation pool are redistributed to the allocating
+block producer and V0LVER reserves. Adding these tokens directly to the pool
+(and not the vault as in the protocol from Section 3.2) allows the pool to update
+its price to reflect the information of the revealed orders.
+5
+Protocol Properties
+The goal of this section is to show that the expected execution price of any user
+order is the external market price when the order is allocated, excluding at most
+impact and fees. Firstly, note that an update transaction prior to allocation
+moves the pool reserves of a V0LVER pool identically to an LVR arbitrage
+transaction in Section 3.2. If Ta = 0, from [9] we know the block producer moves
+the pool price to the max LVR price which is the external market price, and the
+result follows trivially.
+Now instead, assume Ta > 0. Let the reserves of a V0LVER pool Φ before
+the update transaction be (Rx,0, Ry,0). Given an external market price of ϵ, from
+Section 3.1 we know the max LVR occurs by moving the pool reserves to some
+(Rx,m, Ry,m) with Rx,m
+Ry,m = ϵ. WLOG, let Rx,0
+Ry,0 < Rx,m
+Ry,m . Let the block producer
+
+8
+McMenamin and Daza
+move the pool price to p corresponding to reserves in the corresponding CFMM
+pool of (Rx,p, Ry,p). For a non-zero β(), this means the tokens in Φ not in the
+vault (as per the protocol in Section 3.2) are (R′
+x,p, R′
+y,p) = (bRx,p, bRy,p) for
+some b < 1. This is because some tokens in Φ are removed from the pool and
+placed in the vault, while maintaining
+R′
+x,p
+R′y,p = p.
+There are three payoffs of interest here. For these, recall that by definition of
+the external market price, the expected imbalance of an encrypted order in our
+system is 0 at the external market price.
+1. Payoff of block producer vs. AMM pool: (1−β())(Rx,0−Rx,p+(Ry,0−
+Ry,p)ϵ).
+2. Payoff of block producer vs. users: Against a block producer’s own or-
+ders, the block producer has 0 expectancy. Against other player orders, the
+block producer strictly maximizes her own expectancy when (Rx,p, Ry,p) =
+(Rx,m, Ry,m). Otherwise the block producer is offering below ϵ against ex-
+pected buyers, or bidding above ϵ to expected sellers.
+3. Payoff of users vs. AMM pool: Consider a set of allocated orders ex-
+ecuted against the allocation pool, corresponding to the pool receiving δx
+and paying δy tokens. By definition of the allocation pool, this (δx, δy) is
+the same token vector that would be applied to the CFMM pool with re-
+serves (bRx,p, bRy,p) if those orders were batch executed directly against the
+CFMM. Let these new reserves be (bRx,1, bRy,1). Thus the profit of these
+orders is b(1 − β())(Rx,p − Rx,1 + (Ry,p − Ry,1)ϵ).
+Optimal strategy for block producer Let the block producer account for
+α ∈ [0, 1] of the orders executed against the allocation pool. The maximum
+payoff of the block producer against the AMM pool is the maximum of the sum
+of arbitrage profits and profits of block producer orders executed against the
+pool. Thus, the functions to be maximized is:
+(1−β())(Rx,0−Rx,p+(Ry,0−Ry,p)ϵ)+α
+�
+b(1−β())(Rx,p−Rx,1+(Ry,p−Ry,1)ϵ)
+�
+.
+(5)
+This is equal to
+(1−αb)(1−β())
+�
+Rx,0−Rx,p+(Ry,0−Ry,p)ϵ
+�
++αb(1−β())
+�
+Rx,0−Rx,1+(Rx,0−Ry,1)ϵ
+�
+.
+(6)
+We know the second term is maximized for (Rx,1, Ry,1) = (Rx,m, Ry,m), as
+this corresponds to LVR. Similarly, the first term is maximized for (Rx,p, Ry,p) =
+(Rx,m, Ry,m).
+Now consider the payoff for the block producer against user orders (Payoff 2.).
+We have already argued that this is maximized with (Rx,p, Ry,p) = (Rx,m, Ry,m).
+As such, moving the pool price to ϵ is a dominant strategy for the block producer.
+Given this, we can see that the expected execution price for a client is ϵ
+excluding impact and fees, with impact decreasing in expectancy in the number
+of orders allocated. The payoff for the AMM against the block producer via the
+
+V0LVER
+9
+update transaction is (1 − β())LV R, while the payoff against other orders is at
+least 0.
+5.1
+Minimal LVR
+In the previous section, it is demonstrated that user-level MEV is prevented in
+V0LVER, with users trading at the external market price in expectancy, exclud-
+ing fees. However, we have thus far only proved that LVR in a V0LVER pool is
+(1 − β()) of the corresponding CFMM pool. As in [9], under sufficient competi-
+tion among block producers the Nash Equilibrium for the LVR rebate function is
+β(0). This is under the assumption that block producers take all non-negligible
+extractable value opportunities, which we consider LVR to be.
+However, in reality frictionless arbitrage against the external market price
+in blockchain-based protocols is likely not possible, and so LVR extraction has
+some cost. As such, the expected value for β() may be less than β(0). Deploying
+V0LVER, and analyzing β() across different token pairs, and under varying costs
+for block producers makes for interesting future work.
+6
+Discussion
+If a V0LVER pool allows an OCT to be allocated with β() = 0, V0LVER effec-
+tively reverts to the corresponding CFMM pool, with MEV-proof batch settle-
+ment for all simultaneously allocated OCTs, albeit without LVR protection for
+the pool. To see this, note that as β() = 0, the block producer can fully extract
+any existing LVR opportunity, without requiring a deposit to the allocation pool.
+As such, the expected price of the allocation pool is the external market price,
+with orders executed directly against the V0LVER reserves at the external mar-
+ket price, excluding fees and impact. Importantly, there is never any way for the
+block producer to extract any value from allocated orders. This is because the
+settlement price for an OCT is effectively set when it allocated, before any price
+or directional information is revealed.
+Allocation of tokens to the allocation pool has an opportunity cost for both
+the V0LVER pool and the block producer. Given the informational superiority of
+the block producer, allocating tokens from the pool requires the upfront payment
+of a fee to the pool. Doing this anonymously is important to avoid MEV-leakage
+to the block producer. One possibility is providing an on-chain verifiable proof
+of membership to set of players who have bought pool credits, where a valid
+proof releases tokens to cover specific fees, as in [13,8]. Another possibility is
+providing a proof to the block-producer that the user has funds to pay the fee,
+with the block-producer paying the fee on behalf of the user. A final option
+based on threshold encryption [11] is creating a state directly after allocation
+before any more allocations are possible, in which allocated funds are either used
+or de-allocated. All of these proposals have merits and limitations, but further
+analysis of these are beyond the scope of this work.
+
+10
+McMenamin and Daza
+7
+Conclusion
+V0LVER is an AMM based on an encrypted transaction mempool in which LVR
+and MEV are protected against. V0LVER aligns the incentives of users, passive
+liquidity providers and block producers. This is done by ensuring the optimal
+block producer strategy under competition among block producers simultane-
+ously minimizes LVR against passive liquidity providers and MEV against users.
+Interestingly, the exact strategy equilibria of V0LVER depend on factors be-
+yond instantaneous token maximization for block producers. This is due to risks
+associated with liquidity provision and arbitrage costs. On one hand, allocating
+OCTs after setting the pool price to the external market price, and providing
+some liquidity to OCTs around this price should be positive expectancy for block
+producers. Similarly, increasing the number of OCTs should also reduce the vari-
+ance of block producer payoffs. On the other hand, there are caveats in which all
+OCTs are informed and uni-directional. Analyzing these trade-offs for various
+risk profiles and trading scenarios makes for further interesting future work.
+References
+1. Asayag, A., Cohen, G., Grayevsky, I., Leshkowitz, M., Rottenstreich, O., Tamari,
+R., Yakira, D.: Helix: A Fair Blockchain Consensus Protocol Resistant to Ordering
+Manipulation. IEEE Transactions on Network and Service Management 18(2),
+1584–1597 (2021). https://doi.org/10.1109/TNSM.2021.3052038
+2. Ben-Sasson, E., Chiesa, A., Garman, C., Green, M., Miers, I., Tromer, E., Virza,
+M.: Zerocash: Decentralized Anonymous Payments from Bitcoin. In: 2014 IEEE
+Symposium on Security and Privacy. pp. 459–474. IEEE Computer Society, New
+York, NY, USA (2014)
+3. Budish, E., Cramton, P., Shim, J.:
+The High-Frequency Trading Arms Race:
+Frequent Batch Auctions as a Market Design Response *. The Quarterly Journal
+of Economics 130(4), 1547–1621 (07 2015). https://doi.org/10.1093/qje/qjv027,
+https://doi.org/10.1093/qje/qjv027
+4. Burdges, J., De Feo, L.: Delay encryption. In: Canteaut, A., Standaert, F.X. (eds.)
+Advances in Cryptology – EUROCRYPT 2021. pp. 302–326. Springer International
+Publishing, Cham (2021)
+5. Chiang, J.H., David, B., Eyal, I., Gong, T.: Fairpos: Input fairness in proof-of-
+stake with adaptive security. https://eprint.iacr.org/2022/1442 (2022), ac-
+cessed: 23/01/2023
+6. Krishnamachari, B., Feng, Q., Grippo, E.: Dynamic Automated Market Mak-
+ers
+for
+Decentralized
+Cryptocurrency
+Exchange.
+In:
+2021
+IEEE
+Interna-
+tional Conference on Blockchain and Cryptocurrency (ICBC). pp. 1–2 (2021).
+https://doi.org/10.1109/ICBC51069.2021.9461100
+7. McMenamin,
+C.,
+Daza,
+V.:
+Dynamic,
+private,
+anonymous,
+collateraliz-
+able
+commitments
+vs.
+mev.
+https://arxiv.org/abs/2301.12818
+(2022).
+https://doi.org/10.48550/ARXIV.2301.12818, accessed: 31/01/2023
+8. McMenamin, C., Daza, V., Fitzi, M., O’Donoghue, P.: FairTraDEX: A De-
+centralised Exchange Preventing Value Extraction. In: Proceedings of the
+2022 ACM CCS Workshop on Decentralized Finance and Security. p. 39–46.
+DeFi’22, Association for Computing Machinery, New York, NY, USA (2022).
+
+V0LVER
+11
+https://doi.org/10.1145/3560832.3563439,
+https://doi.org/10.1145/3560832.
+3563439
+9. McMenamin,
+C.,
+Daza,
+V.,
+Mazorra,
+B.:
+Diamonds
+are
+Forever,
+Loss-
+Versus-Rebalancing
+is
+Not.
+https://arxiv.org/abs/2210.10601
+(2022).
+https://doi.org/10.48550/ARXIV.2210.10601, accessed: 04/01/2023
+10. Milionis, J., Moallemi, C.C., Roughgarden, T., Zhang, A.L.: Quantifying Loss in
+Automated Market Makers. In: Zhang, F., McCorry, P. (eds.) Proceedings of the
+2022 ACM CCS Workshop on Decentralized Finance and Security. ACM (2022)
+11. Penumbra: https://penumbra.zone/, accessed: 23/01/2023
+12. Ramseyer, G., Goyal, M., Goel, A., Mazi`eres, D.: Batch exchanges with constant
+function market makers: Axioms, equilibria, and computation. https://arxiv.
+org/abs/2210.04929 (2022). https://doi.org/10.48550/ARXIV.2210.04929, ac-
+cessed: 26/01/2023
+13. Tornado Cash: https://tornadocash.eth.link/, accessed: 31/01/2023
+

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@@ -0,0 +1,381 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf,len=380
+page_content='An AMM minimizing user-level extractable  value and loss-versus-rebalancing  Conor McMenamin1 and Vanesa Daza1,2  1 Department of Information and Communication Technologies, Universitat Pompeu  Fabra, Barcelona, Spain  2 CYBERCAT - Center for Cybersecurity Research of Catalonia  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' We present V0LVER, an AMM protocol which solves an in-  centivization trilemma between users, passive liquidity providers, and  block producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' V0LVER enables users and passive liquidity providers  to interact without paying MEV or incurring uncontrolled loss-versus-  rebalancing to the block producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' V0LVER is an AMM protocol built  on an encrypted transaction mempool, where transactions are decrypted  after being allocated liquidity by the AMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' V0LVER ensures this liq-  uidity, given some external market price, is provided at that price in ex-  pectancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This is done by providing just enough loss-versus-rebalancing  profits to the block producer, incentivizing the block producer to move  the pool price to the external market price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' With this, users transact  in expectancy at the external market price in exchange for a fee, with  AMMs providing liquidity in expectancy at the external market price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Under block producer and liquidity provider competition, all of the fees  in V0LVER approach zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Without block producer arbitrage, V0LVER  guarantees fall back to those of an AMM capable of processing encrypted  transactions, free from loss-versus-rebalancing or user-level MEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Keywords: Extractable Value · Decentralized Exchange · Incentives · Blockchain  1  Introduction  In this paper we introduce V0LVER 3, an AMM which provides arbitrarily high  protection against user-level MEV and LVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' V0LVER is the first AMM to align  the incentives of the three, typically competing, entities in AMMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' the user, the  pool, and the block producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This is done by ensuring that at all times, a block  producer is incentivized to move the pool to the price maximizing LVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' When  the block producer chooses a price, the block producer is forced to assert this is  correct, a technique introduced in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Unfortunately, the protocol in [9] gives the  This Technical Report is part of a project that has received funding from the  European Union’s Horizon 2020 research and innovation programme under  grant agreement number 814284  3 near-0 Extractable Value and Loss-Versus-Rebalancing ⇝ V0LVER  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='13599v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='GT]  31 Jan 2023  2  McMenamin and Daza  block producer total power to extract value from users, due to order information  being revealed to the block producer before it is allocated a trading price in the  blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' To address this, V0LVER is built on encrypted mempools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Modern cryptographic tools allow us to encrypt the mempool using zero-  knowledge based collateralized commit-reveal protocols [7,2,8,13], delay encryp-  tion [4,5] and/or threshold encryption [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' If a block producer adding an order to  an AMM is forced to replicate the payoff of the AMM, we demonstrate the block  producer maximizes her own utility by showing liquidity centred around the ex-  ternal market price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='4 Providing users with an AMM where the expected trade  price is the external market price, excluding fees, is a significant advancement  and the main contribution of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Although batching orders against AMM  liquidity has been proposed as a defense against LVR [12], naively batching or-  ders against an AMM still allows a block producer to extract LVR by censoring  user orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' In V0LVER, block producers are effectively forced to immediately  repay LVR, while always being incentivized to include order commitments in the  blockchain, and eventually allocate AMM liquidity to these orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  2  Related Work  As the phenomenon of LVR has only recently been identified, there are only  two academic papers on the subject of LVR protection [6,9] to the best of our  knowledge, with no work protecting against both LVR and user-level MEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  In [6], the AMM must receive the price of a swap from a trusted oracle before  users can interact with the pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Such sub-block time price data requires central-  ized sources which are prone to manipulation, or require the active participation  of AMM representatives, a contradiction of the passive nature of AMMs and  their liquidity providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' We see this as an unsatisfactory dependency for DeFi  protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Our work is based on some of the techniques of the Diamond protocol as  introduced in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The Diamond protocol requires block producers to effectively  attest to the final price of the block given the orders that are to be proposed to  the AMM within the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This technique requires the block producer to know  exactly what orders are going to be added to the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This unfortunately  gives the block producer total freedom to extract value from users submitting  orders to the AMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' With V0LVER, we address this issue while keeping the LVR  protection guarantees of Diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Encrypting the transaction mempool using threshold encryption controlled  by a committee has been proposed in [1] and applied in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' In [11], a DEX  involving an AMM and based on frequent batch auctions [3] is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This  DEX does not provide LVR resistance, and incentivizes transaction censorship  when a large LVR opportunity arises on the DEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This is protected against in  V0LVER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  4 This holds true in many CFMMs, including the famous Uniswap V2 protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  V0LVER  3  3  Preliminaries  This section introduces the key terminology and definitions needed to understand  LVR, and the proceeding analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' In this work we are concerned with a single  swap between token x and token y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' We use x and y subscripts when referring  to quantities of the respective tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The external market price of a swap is  denoted by ϵ, with the price of a swap quoted as the quantity of token x per  token y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='1  Constant Function Market Makers  A CFMM is characterized by reserves (Rx, Ry) ∈ R2  + which describes the total  amount of each token in the pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The price of the pool is given by pool price  function P : R2  + → R taking as input pool reserves (Rx, Ry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' P() has the  following properties:  (a) P() is everywhere differentiable, with ∂P  ∂Rx  > 0,  ∂P  ∂Ry  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  (b)  lim  Rx→0 P = 0,  lim  Rx→∞ P = ∞,  lim  Ry→0 P = ∞,  lim  Ry→∞ P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  (c) If P(Rx, Ry) = p, then P(Rx + cp, Ry + c) = p, ∀c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  (1)  For a CFMM, the feasible set of reserves C is described by:  C = {(Rx, Ry) ∈ R2  + : f(Rx, Ry) = k}  (2)  where f : R2  + → R is the pool invariant and k ∈ R is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The pool is  defined by a smart contract which allows any player to move the pool reserves  from the current reserves (Rx,0, Ry,0) ∈ C to any other reserves (Rx,1, Ry,1) ∈ C  if and only if the player provides the difference (Rx,1 − Rx,0, Ry,1 − Ry,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' To  reason about V0LVER, we assume the existence of an external market price  between x and y, which we define as follows, based on the definition of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Whenever an arbitrageur interacts with an AMM pool, say at time t with  reserves (Rx,t, Ry,t), we assume as in [10] that the arbitrageur always moves the  pool reserves to a point which maximizes arbitrageur profits, exploiting the dif-  ference between P(Rx,t, Ry,t) and the external market price at time t, denoted ϵt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  In this paper, we consider only the subset of CFMMs in which, given the LVR ex-  tracted in block Bt+1 corresponds to reserves (Rx,t+1, Ry,t+1), P(Rx,t+1, Ry,t+1)  = ϵt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This holds for Uniswap V2 pools, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='2  LVR-resistant AMM  We provide here an overview of the most important features of Diamond [9], an  AMM protocol which is proved to provide arbitrarily high LVR protection under  competition to capture LVR among block producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' In V0LVER, we adapt these  features for use on an encrypted transaction mempool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  4  McMenamin and Daza  A Diamond pool Φ is described by reserves (Rx, Ry), a pricing function P(), a  pool invariant function f(), an LVR-rebate parameter β ∈ (0, 1), and conversion  frequency T ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The authors also define a corresponding CFMM pool of Φ,  denoted CFMM(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' CFMM(Φ) is the CFMM pool with reserves (Rx, Ry) whose  feasible set is described by pool invariant function f() and pool constant k =  f(Rx, Ry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Conversely, Φ is the corresponding V0LVER pool of CFMM(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The  authors note that CFMM(Φ) changes every time the Φ pool reserves change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The  protocol progresses in blocks, with one reserve update possible per block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  For an arbitrageur wishing to move the price of CFMM(Φ) to p from starting  reserves (Rx,0, Ry,0), let this require ∆y > 0 tokens to be added to CFMM(Φ),  and ∆x > 0 tokens to be removed from CFMM(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The same price in Φ is  achieved by the following process:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Adding (1 − β)∆y tokens to Φ and removing (1 − β)∆x tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Removing δx > 0 tokens such that:  P(Rx,0 − (1 − β)∆x − δx, Ry,0 + (1 − β)∆y) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  (3)  These δx tokens are added to the vault of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Vault tokens are periodically re-entered into Φ through what is effectively an  auction process, where the tokens being re-added are in a ratio which approxi-  mates the external market price at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The main result of [9] is the proving  that given a block producer interacts with Φ when the LVR parameter is β, and  there is an LVR opportunity of LV R in CFMM(Φ), the maximum LVR in Φ is  (1 − β)LV R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This results is stated formally therein as follows:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' For a CFMM pool CFMM(Φ) with LVR of L > 0, the LVR of Φ,  the corresponding pool in Diamond, has expectancy of at most (1 − β)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  In this paper we use the same base functionality of Diamond to restrict the  LVR of block producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Given a block producer wants to move the price of  CFMM(Φ) to some price p to extract maximal LVR LV R, the maximal LVR  in Φ of (1 − β)LV R is also achieved by moving the price to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' An important  point to note about applying LVR rebates as done in [9], is that directly after  tokens are placed in the vault, the pool constant drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This must be considered  when calculating the profitability of an arbitrageur extracting LVR from a Dia-  mond pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' We do this when analyzing the profitability of V0LVER in Section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Importantly, tokens are eventually re-added to the pool, and over time the  expected value of the pool constant is increasing, as demonstrated in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  4  Our Protocol  We now outline the model in which we construct V0LVER, followed by a detailed  description of V0LVER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  V0LVER  5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='1  Model  In this paper we consider a blockchain in which all transactions are attempting  to interact with a single V0LVER pool between tokens x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' A transaction submitted by a player for addition to the blockchain while  observing blockchain height H, is finalized in a block of height at most  H + T, for some known T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The token swap has an external market price ϵ, which follows a Martingale  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' There exists a population of arbitrageurs able to frictionlessly trade at exter-  nal market prices, who continuously monitor and interact with the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Encrypted orders are equally likely to buy or sell tokens at ϵ, distributed  symmetrically around ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='2  Protocol Framework  This section outlines the terminology and functionalities used in V0LVER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' It is  intended as a reference point to understand the core V0LVER protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Specifi-  cally, we describe the possible transactions in V0LVER, the possible states that  V0LVER orders/order commitments can be in, and the possible actions of block  producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  As in the protocol of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='2, a V0LVER pool Φ with reserves (Rx, Ry)  is defined with respect to a CFMM pool, denoted CFMM(Φ), with reserves  (Rx, Ry), a pricing function P(), under the restrictions of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='1, and a  pool invariant function f().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Orders in V0LVER are intended to interact with the AMM pool with some  delay due to commit-reveal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Therefore, we need to introduce the concept of  allocated funds to be used when orders eventually get revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' To do this, we  define an allocated pool Φa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' For a player committing to an order of size either  sizex or sizey known to be of maximum size maxx or maxy, the allocation pool  consists of (∆x, ∆y) tokens such that:  f(Rx, Ry) = f(Rx + maxx, Ry − ∆y) = f(Rx − ∆x, Ry + maxy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  (4)  Moreover, if f(Rx, Ry) = f(Rx+sizex, Ry−δy), meaning the CFMM pool would  sell δy tokens in return for sizex, the allocation pool also sells δy for sizex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  There are three types of transaction in our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' To define these trans-  actions, we need an LVR rebate function β : [0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=', Z, Z +1] → [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' It suffices  to consider β() as a strictly decreasing function with β(z) = 0 ∀z ≥ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' These are straightforward buy or sell orders indicating a limit price,  size and direction to be traded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' WLOG, we assume all orders in our system  are executable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Order commitment transaction (OCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' These are encrypted orders  known to be collateralized by either maxx or maxy tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The exact size,  direction, price, and sender of an OCT sent from player Pi is hidden from all  6  McMenamin and Daza  other players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This is possible using anonymous ZK proofs of collateral such  as used in [8,13,7]), which can be implemented on a blockchain in conjunction  with a user-lead commit-reveal protocol, delay encryption scheme [4,5] or  threshold encryption scheme [1,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Update transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' These transactions are executed in a block before  any OCT is allowed to interact with the protocol pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Let the current block  height be H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Update transactions take as input an allocation block height  Ha ≤ H, some number of transactions to allocate Ta ∈ Z≥0, and pool price  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Given an allocation block height of H′  a in the previous update transaction,  valid update transactions require:  (a) Ha > H′  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  (b) The number of OCTs in blocks [H′  a + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=', Ha] equals Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Given inputs (Ha, Ta, p), the block producer moves the price of the pool to  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The producer receives (1−β()) of the implied change in reserves from this  price move, as is done in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  The producer then deposits (Taβ(H − Ha)maxx, Taβ(H − Ha) maxx  p  ) to an  allocation pool denoted ΦHa, with (Ta(1 − β(H − Ha))maxx, Ta(1 − β(H −  Ha)) maxx  p  ) being added to ΦHa from the AMM reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  In other words, if an allocation pool requires up to (Tamaxx, Ta maxx  p  ) tokens  to trade with orders corresponding to the Ta allocated orders, the block producer  is forced to provide β(H − Ha) of the tokens in the pool, with starting bid and  offer prices equal to the pool price set by the block producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This is used to  incentivize the block producer to always choose a pool price equal to the external  market price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Every block, a block producer has four possible actions to perform on OCTs  and their orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Orders in our system are batch-settled with other orders allo-  cated at the same time, and the liquidity in the respective allocation pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Insert OCTs to the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Allocate inserted OCTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' For a block producer adding a block at height H  to allocate any number (including 0) inserted OCTs with inserted height of  at most Hi, the block producer must:  (a) Allocate all unallocated OCTs with inserted height less than or equal to  Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Let there be x such inserted OCTs to allocate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  (b) Submit an update transaction with inputs (Ha = Hi, Ta = x, p), for  some p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Reveal allocated order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' When a decrypted order corresponding to an OCT  at height Ha is finalized on the blockchain within T blocks after the corre-  sponding OCT is allocated, it is considered revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' For allocation pool Φa  Ha  with reserves (Rx,Ha, Ry,Ha), the initial clearing price for orders allocated at  height Ha is equal to Rx,Ha  Ry,Ha .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Upon order reveal, the clearing price for orders  is updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This clearing price maximizes trade volume, as is done in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Execute revealed orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' T blocks after OCTs are allocated, any correspond-  ing revealed orders are executed at the last-updated clearing price for orders  V0LVER  7  allocated at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The final tokens in the allocation pool are re-  distributed proportionally to the allocating block producer and V0LVER  reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='3  Protocol Outline  Our protocol can be considered as two sub-protocols, an update protocol and an  execution protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The update protocol is analogous to the protocol in Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='2, while the execution protocol, and its combination with the update protocol  are novel and specific to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Every round a block-producer can submit  an update transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' There are two scenarios for an update transaction with  inputs (Ha, Ta, p) and block height of the previous update transaction H′  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Either  Ta = 0 or Ta > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' If Ta = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' the update transaction is equivalent to an arbi-  trageur operation on a Diamond pool with LVR-rebate parameter of β(Ha−H′  a)  (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  If Ta > 0, the arbitrageur must also deposit (Ta(1−β(H −Ha))maxx, Ta(1−  β(H−Ha)) maxx  p  ) to the Ha allocation pool ΦHa, with (Taβ(H−Ha)maxx, Taβ(H−  Ha) maxx  p  ) being added to ΦHa from the AMM reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  After an allocation pool is created for allocated OCTs {oct1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=', octn}, the  orders corresponding to {oct1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=', octn} can be revealed for up to T blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This  is sufficient time for any user whose OCT is contained in that set to reveal the  order corresponding to the OCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' To enforce revelation, tokens corresponding to  unrevealed orders are burned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' After all orders have been revealed, or T blocks  have passed, any block producer can execute revealed orders against the alloca-  tion pool at a clearing price which maximizes volume traded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Specifically, given  an array of orders ordered by price, a basic smart-contract can verify that a  proposed clearing price maximizes volume traded, as is done in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  The final tokens in the allocation pool are redistributed to the allocating  block producer and V0LVER reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Adding these tokens directly to the pool  (and not the vault as in the protocol from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='2) allows the pool to update  its price to reflect the information of the revealed orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  5  Protocol Properties  The goal of this section is to show that the expected execution price of any user  order is the external market price when the order is allocated, excluding at most  impact and fees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Firstly, note that an update transaction prior to allocation  moves the pool reserves of a V0LVER pool identically to an LVR arbitrage  transaction in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' If Ta = 0, from [9] we know the block producer moves  the pool price to the max LVR price which is the external market price, and the  result follows trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Now instead, assume Ta > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Let the reserves of a V0LVER pool Φ before  the update transaction be (Rx,0, Ry,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Given an external market price of ϵ, from  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='1 we know the max LVR occurs by moving the pool reserves to some  (Rx,m, Ry,m) with Rx,m  Ry,m = ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' WLOG, let Rx,0  Ry,0 < Rx,m  Ry,m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Let the block producer  8  McMenamin and Daza  move the pool price to p corresponding to reserves in the corresponding CFMM  pool of (Rx,p, Ry,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' For a non-zero β(), this means the tokens in Φ not in the  vault (as per the protocol in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='2) are (R′  x,p, R′  y,p) = (bRx,p, bRy,p) for  some b < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This is because some tokens in Φ are removed from the pool and  placed in the vault, while maintaining  R′  x,p  R′y,p = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  There are three payoffs of interest here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' For these, recall that by definition of  the external market price, the expected imbalance of an encrypted order in our  system is 0 at the external market price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Payoff of block producer vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' AMM pool: (1−β())(Rx,0−Rx,p+(Ry,0−  Ry,p)ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Payoff of block producer vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' users: Against a block producer’s own or-  ders, the block producer has 0 expectancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Against other player orders, the  block producer strictly maximizes her own expectancy when (Rx,p, Ry,p) =  (Rx,m, Ry,m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Otherwise the block producer is offering below ϵ against ex-  pected buyers, or bidding above ϵ to expected sellers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Payoff of users vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' AMM pool: Consider a set of allocated orders ex-  ecuted against the allocation pool, corresponding to the pool receiving δx  and paying δy tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' By definition of the allocation pool, this (δx, δy) is  the same token vector that would be applied to the CFMM pool with re-  serves (bRx,p, bRy,p) if those orders were batch executed directly against the  CFMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Let these new reserves be (bRx,1, bRy,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Thus the profit of these  orders is b(1 − β())(Rx,p − Rx,1 + (Ry,p − Ry,1)ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Optimal strategy for block producer Let the block producer account for  α ∈ [0, 1] of the orders executed against the allocation pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The maximum  payoff of the block producer against the AMM pool is the maximum of the sum  of arbitrage profits and profits of block producer orders executed against the  pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Thus, the functions to be maximized is:  (1−β())(Rx,0−Rx,p+(Ry,0−Ry,p)ϵ)+α  �  b(1−β())(Rx,p−Rx,1+(Ry,p−Ry,1)ϵ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  (5)  This is equal to  (1−αb)(1−β())  �  Rx,0−Rx,p+(Ry,0−Ry,p)ϵ  �  +αb(1−β())  �  Rx,0−Rx,1+(Rx,0−Ry,1)ϵ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  (6)  We know the second term is maximized for (Rx,1, Ry,1) = (Rx,m, Ry,m), as  this corresponds to LVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Similarly, the first term is maximized for (Rx,p, Ry,p) =  (Rx,m, Ry,m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Now consider the payoff for the block producer against user orders (Payoff 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  We have already argued that this is maximized with (Rx,p, Ry,p) = (Rx,m, Ry,m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  As such, moving the pool price to ϵ is a dominant strategy for the block producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Given this, we can see that the expected execution price for a client is ϵ  excluding impact and fees, with impact decreasing in expectancy in the number  of orders allocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' The payoff for the AMM against the block producer via the  V0LVER  9  update transaction is (1 − β())LV R, while the payoff against other orders is at  least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='1  Minimal LVR  In the previous section, it is demonstrated that user-level MEV is prevented in  V0LVER, with users trading at the external market price in expectancy, exclud-  ing fees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' However, we have thus far only proved that LVR in a V0LVER pool is  (1 − β()) of the corresponding CFMM pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' As in [9], under sufficient competi-  tion among block producers the Nash Equilibrium for the LVR rebate function is  β(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This is under the assumption that block producers take all non-negligible  extractable value opportunities, which we consider LVR to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  However, in reality frictionless arbitrage against the external market price  in blockchain-based protocols is likely not possible, and so LVR extraction has  some cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' As such, the expected value for β() may be less than β(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Deploying  V0LVER, and analyzing β() across different token pairs, and under varying costs  for block producers makes for interesting future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  6  Discussion  If a V0LVER pool allows an OCT to be allocated with β() = 0, V0LVER effec-  tively reverts to the corresponding CFMM pool, with MEV-proof batch settle-  ment for all simultaneously allocated OCTs, albeit without LVR protection for  the pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' To see this, note that as β() = 0, the block producer can fully extract  any existing LVR opportunity, without requiring a deposit to the allocation pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  As such, the expected price of the allocation pool is the external market price,  with orders executed directly against the V0LVER reserves at the external mar-  ket price, excluding fees and impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Importantly, there is never any way for the  block producer to extract any value from allocated orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This is because the  settlement price for an OCT is effectively set when it allocated, before any price  or directional information is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Allocation of tokens to the allocation pool has an opportunity cost for both  the V0LVER pool and the block producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Given the informational superiority of  the block producer, allocating tokens from the pool requires the upfront payment  of a fee to the pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Doing this anonymously is important to avoid MEV-leakage  to the block producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' One possibility is providing an on-chain verifiable proof  of membership to set of players who have bought pool credits, where a valid  proof releases tokens to cover specific fees, as in [13,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Another possibility is  providing a proof to the block-producer that the user has funds to pay the fee,  with the block-producer paying the fee on behalf of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' A final option  based on threshold encryption [11] is creating a state directly after allocation  before any more allocations are possible, in which allocated funds are either used  or de-allocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' All of these proposals have merits and limitations, but further  analysis of these are beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  10  McMenamin and Daza  7  Conclusion  V0LVER is an AMM based on an encrypted transaction mempool in which LVR  and MEV are protected against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' V0LVER aligns the incentives of users, passive  liquidity providers and block producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This is done by ensuring the optimal  block producer strategy under competition among block producers simultane-  ously minimizes LVR against passive liquidity providers and MEV against users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content='  Interestingly, the exact strategy equilibria of V0LVER depend on factors be-  yond instantaneous token maximization for block producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' This is due to risks  associated with liquidity provision and arbitrage costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' On one hand, allocating  OCTs after setting the pool price to the external market price, and providing  some liquidity to OCTs around this price should be positive expectancy for block  producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Similarly, increasing the number of OCTs should also reduce the vari-  ance of block producer payoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' On the other hand, there are caveats in which all  OCTs are informed and uni-directional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
+page_content=' Analyzing these trade-offs for various  risk profiles and trading scenarios makes for further interesting future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'}
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+arXiv:2301.11782v1  [math.NT]  27 Jan 2023
+A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS
+ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT
+Abstract. Given a sequence of frequencies {λn}n≥1, a corresponding generalized Dirichlet
+series is of the form f(s) = �
+n≥1 ane−λns. We are interested in multiplicatively generated
+systems, where each number eλn arises as a finite product of some given numbers {qn}n≥1,
+1 < qn → ∞, referred to as Beurling primes. In the classical case, where λn = log n, Bohr’s
+theorem holds: if f converges somewhere and has an analytic extension which is bounded in a
+half-plane {Re s > θ}, then it actually converges uniformly in every half-plane {Re s > θ +ε},
+ε > 0. We prove, under very mild conditions, that given a sequence of Beurling primes, a small
+perturbation yields another sequence of primes such that the corresponding Beurling integers
+satisfy Bohr’s condition, and therefore the theorem. Applying our result in conjunction with
+work of Diamond–Montgomery–Vorhauer and Zhang, we find a system of Beurling primes for
+which both Bohr’s theorem and the Riemann hypothesis are valid. We discuss the connections
+between our work and Diophantine approximation with Beurling integers, as well as with a
+conjecture of Helson concerning outer functions in Hardy spaces of generalized Dirichlet series.
+1. Introduction
+For an increasing sequence of positive frequencies λ = {λn}n≥1, and a generalized Dirichlet
+series
+f(s) =
+�
+n≥1
+ane−λns,
+the abscissas σc, σu, and σa of point-wise, uniform, and absolute convergence are defined as in
+the classical theory of Dirichlet series [10]. In this article we wish to find sets of frequencies such
+that the analogue of a theorem of Bohr [4] holds: if σc(f) < ∞ and f has a bounded analytic
+extension to a half-plane {Re s > θ}, then σu(f) ≤ θ. The problem of finding frequencies for
+which the abscissas of bounded and uniform convergence always coincide, which originated with
+Bohr and Landau [17], has recently been revisited [2, 19] with the context of Hardy spaces of
+Dirichlet series in mind.
+Indeed, Bohr’s theorem is essentially a necessity for a satisfactory
+Hardy space theory, see [18, Ch. 6].
+An important class of frequencies were introduced by Beurling [3]. Given an arbitrary in-
+creasing sequence q = {qn}n≥1, 1 < qn → ∞, such that {log qn}n≥1 is linearly independent over
+Q, we will denote by Nq = {νn}n≥1 the set of numbers that can be written (uniquely) as finite
+products with factors from q, ordered in an increasing manner. The numbers qn are known
+as Beurling primes, and the numbers νn are Beurling integers. The corresponding generalized
+Dirichlet series are of the form
+f(s) =
+�
+n≥1
+anν−s
+n .
+There are a number of criteria to guarantee the validity of Bohr’s theorem for frequencies
+{λn}n≥1. Bohr’s original condition asks for the existence of c1, c2 > 0 such that
+(1)
+λn+1 − λn ≥ c1e−c2λn+1,
+n ∈ N.
+Landau relaxed the condition somewhat: for every δ > 0 there should be a c > 0 such that
+(2)
+λn+1 − λn ≥ ce−eδλn+1 ,
+n ∈ N.
+1
+
+2
+ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT
+Landau’s condition was recently relaxed further by Bayart [2]: for every δ > 0 there should be
+a C > 0 such that
+(3)
+log
+�λm + λn
+λm − λn
+�
++ (m − n) ≤ Ceδλn,
+m > n ∈ N.
+All of these conditions are usually very difficult to check for any given Beurling system, since
+they involve the distances between the corresponding Beurling integers. Furthermore, while it
+is easy to construct Beurling integers for which conditions (1)-(3) fail, it appears to be rather
+difficult to construct Beurling systems which do satisfy the Bohr condition. This is especially
+true if one wants to retain properties of the ordinary integers, such as the asympotic behaviour
+of the counting function Nq(x) = �
+νn≤x 1, see for example [9].
+One motivation for considering Beurling integers is to investigate the properties of the q-zeta
+function
+ζq(s) =
+�
+n≥1
+ν−s
+n
+=
+�
+n≥1
+1
+1 − q−s
+n
+,
+and their interplay with the counting functions
+Nq(x) =
+�
+νn≤x
+1,
+πq(x) =
+�
+qn≤x
+1.
+As an example, Beurling [3] himself showed that the condition
+(4)
+Nq(x) = ax + O(
+x
+(log x)γ ),
+for some γ > 3
+2,
+implies the analogue of the prime number theorem,
+(5)
+πq(x) :=
+�
+qn≤x
+1 ∼
+x
+log x.
+We refer to [15] for an overview of further developments.
+In Section 2 we begin with a preparatory result which is interesting in its own right. It states
+that starting with the classical set of primes numbers we can add almost any finite sequence of
+Beurling primes while retaining the validity of Bohr’s theorem.
+Theorem 1.1. Let {pn}n≥1 be the sequence of ordinary prime numbers and let N ≥ 1. Then
+Bohr’s condition (1) holds for the Beurling integers generated by the primes
+q = {pn}n≥1
+�
+{qj}N
+j=1,
+for almost every choice (q1, . . . , qN) ∈ (1, ∞)N.
+Sequences of Beurling primes of the type considered in Theorem 1.1 previously appeared in
+[16].
+Our main result requires more careful analysis.
+Theorem 1.2. Let q = {qn}n≥1 be an increasing sequence of primes such that q1 > 1 and
+σc(ζq) < ∞. Then, for every A > 0 there exists a sequence of Beurling primes ˜q = {˜qn}n≥1 for
+which Bohr’s condition (1) holds and
+|qn − ˜qn| ≤ q−A
+n
+,
+n ∈ N.
+Theorem 1.2 and the work of Diamond, Montgomery, Vorhauer [8] and Zhang [21], allows
+us to construct a system of Beurling primes that satisfies Bohr’s theorem and the “Riemann
+Hypothesis”.
+Corollary 1.3. There exists a system of Beurling primes q = {qn}n≥1 such that:
+
+A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS
+3
+(i) The Riemann zeta function ζq(s) has an analytic extension to Re s > 1
+2, except for a simple
+pole at s = 1.
+(ii) The Riemann zeta function has no zeros in the half–plane C 1
+2 = {Re s > 1/2}.
+(iii) The prime counting πq(x) satisfies
+πq(x) = li(x) + O
+�√x
+�
+,
+where li(x) =
+x´
+2
+(log u)−1 du.
+(iv) Bohr’s condition holds for the associated class of generalized Dirichlet series.
+We remark that as a direct consequence of (iii), we have the prime number theorem (5) as
+well as Nq(x) ∼ ax, where a is the residue of ζq(s) at s = 1, see for example [7, 15].
+The proofs of our results investigate how well “irrational numbers” may be approximated by
+fractions of Beurling integers. We will comment further on this kind of Diophantine approxi-
+mation problems in Section 3.
+In Section 3 we will also return to the original motivation for our work. There has been an
+interest in studying Hardy spaces of generalized Dirichlet series since the 60s [6, 11, 12]. However,
+other than the ordinary integers, no examples of Beurling integers exist which simultaneously
+satisfy the prime number theorem (iii) and Bohr’s theorem, despite the fact that Bohr’s theorem
+is crucial. Furthermore, since many aspects of the function theory of the Hardy space do not
+depend on the choice q of Beurling primes, the idea behind Corollary 1.3 was to find a Beurling
+system Nq in which we can assume the Riemann hypothesis. As a function theoretic application,
+we construct an outer function f which has a zero,
+(6)
+f(s) =
+1
+ζq(s + 1/2 + ε),
+0 < ε < 1/2.
+Originally, we intended to make use of (6) in order to disprove a conjecture of Helson [13]
+about outer functions in Hardy spaces formed from frequencies satisfying the Bohr condition.
+Unfortunately, we did not quite succeed, since we are unable to demonstrate the convergence
+of (6) for Res > 0; the typical proof of this relies on the Lindel¨of hypothesis. Finding a system
+which in addition to the properties of Corollary 1.3 satisfies the Lindel¨of hypothesis seems to
+demand a deeper interplay between our tools and the probabilistic methods of [8].
+Notation. Throughout the article, we will be using the convention that C denotes a positive
+constant which may vary from line to line. We will write that C = C(Ω) when the constant
+depends on the parameter Ω.
+2. Proof of the main results
+Lemma 2.1. Suppose that {qn}n≥1 is a Beurling system such that dn := νn+1 − νn ≫ ν−C
+n+1.
+Then, for almost every q′ > 1 and every ε > 0, the Beurling system {qn}n≥1∪{q′} has a distance
+function satisfying
+(7)
+d′
+n = ν′
+n+1 − ν′
+n ≫ ν−C
+′
+n+1 ,
+n ∈ N,
+where C′(q′, q) = max (C, 2σc(ζq) − 1 + ε).
+Proof. Let x0 > 1. First we will prove that the set M of all numbers q′ ≥ x0 such that there
+exist infinitely many triples (j, n, m) ∈ N3 with
+����(q′)j − νn
+νm
+���� ≤ ν−C0
+n
+ν−C0
+m
+,
+C0 = σ(ζq) + ε,
+
+4
+ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT
+has measure zero. Since
+�����q′ −
+� νn
+νm
+� 1
+j ����� ≤ C(x0)x−j
+0
+����(q′)j − νn
+νm
+���� .
+we have that M ⊂ lim supm,n,j Ωm,n,j, where
+Ωm,n,j =
+�� νn
+νm
+� 1
+j
+− C(x0)x−j
+0 ν−C0
+n
+ν−C0
+m
+,
+� νn
+νm
+� 1
+j
++ C(x0)x−j
+0 ν−C0
+n
+ν−C0
+m
+�
+,
+j, n, m ≥ 1.
+The Borel–Cantelli lemma thus shows that |M| = 0, since
+�
+m≥1
+�
+n≥1
+�
+j≥1
+|Ωm,n,j| ≤ C(x0)ζq (C0)2 < ∞.
+Fix a number q′ ∈ [x0, ∞) \ M such that log q′ is not in the (countable) set spanQ{log qn}.
+Note that the set of such numbers has full measure in [x0, ∞), and that x0 > 1 is arbitrary. By
+construction, there are finitely many triples (j, n, m) such that
+(8)
+����(q′)j − νn
+νm
+���� ≤ ν−C0
+n
+ν−C0
+m
+.
+For these exceptional triples, the left-hand side is at least positive, since log q′ /∈ spanQ{log qn}.
+Therefore
+����(q′)j − νn
+νm
+���� ≫ ν−C0
+n
+ν−C0
+m
+for all (j, n, m) ∈ N3.
+Now we consider two arbitrary consecutive Beurling integers generated by the prime system
+{qn}n≥1 ∪ {q′},
+ν′
+n+1 = (q′)aνm,
+ν′
+n = (q′)bνl.
+If a = b, then l = m − 1 and
+ν′
+n+1 − ν′
+n ≫ ν−C
+m
+≥
+�
+ν′
+n+1
+�−C ,
+by the hypothesis on the distances dn for the original Beurling system. Otherwise, if, say, b < a,
+then
+��ν′
+n+1 − ν′
+n
+�� = (q′)bνm
+����(q′)a−b − νl
+νm
+���� ≫ ν−C0
+l
+ν−C0+1
+m
+(q′)b ≫
+�
+ν′
+n+1
+�−C
+′
+.
+where C
+′ = 2σc(ζq) − 1 + ε.
+□
+Proof of Theorem 1.1. The proof is a direct consequence of Lemma 2.1.
+□
+In order to prove Bohr’s theorem for more general Beurling systems, we need to control the
+constant in the distance estimate (7), which comes from the exceptional triples satisfying (8).
+Proof of Theorem 1.2. Fix a small ε > 0 and x0 ∈ (1+ε/2, 1+ε). Consider first any Beurling
+system Nρ = {νn}n≥1 generated by Beurling primes such that ρ1 > 1 + ε and σc(ζρ) < ∞. For
+a number σ∞ > max(2, A) to be chosen in a moment, let
+N =
+�
+m≥2
+�
+n≥2
+�
+j≥1
+Ωm,n,j,
+where Ωm,n,j is defined as in the proof of Lemma 2.1,
+Ωm,n,j =
+�� νn
+νm
+� 1
+j
+− C(x0)x−j
+0 ν−σ∞
+n
+ν−σ∞
+m
+,
+� νn
+νm
+� 1
+j
++ C(x0)x−j
+0 ν−σ∞
+n
+ν−σ∞
+m
+�
+.
+
+A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS
+5
+Then |N| ≤ C(ε) (ζρ(σ∞) − 1)2 . Furthermore, for x > 2, let
+Ix = [x − x− σ∞
+2 , x + x− σ∞
+2 ].
+Note that if σ∞ is sufficiently large, σ∞ ≥ C(ε), then Ix∩Ωm,n,j ̸= ∅ only if νn ≥ x/2. Therefore
+|Ix ∩ N| ≤
+�
+m≥2
+j≥1
+νn≥ x
+2
+|Ωm,n,j| ≤ C(ε) (ζρ(σ∞) − 1)
+�
+νn≥ x
+2
+ν−σ∞
+n
+≤ C(ε) (ζρ(σ∞) − 1) ζρ
+�σ∞
+4
+�
+x− 3σ∞
+4 .
+We will construct a sequence of Beurling systems such that
+(9)
+(ζρ(σ∞) − 1) ζρ
+�σ∞
+4
+�
+≤ 1
+for the number σ∞ > 0, still to be chosen later. Therefore
+(10)
+|Ix ∩ N| ≤ C(ε)x− σ∞
+4 |Ix|,
+We conclude that whenever x is sufficiently large, Ix ̸⊂ N.
+To include triples where νn or νm equals one in our considerations, we increase the power
+σ∞. The inequality
+(11)
+����xj − νn
+νm
+���� ≤ ν−3σ∞
+n
+ν−3σ∞
+m
+implies, whenever x ≥ x0, that
+�����x −
+� νn
+νm
+� 1
+j ����� ≤ C(x0)x−j
+0
+����xj − ν2
+nνm
+ν2mνn
+���� ≤ C(x0)x−j
+0
+�
+ν2
+mνn
+�−σ∞ �
+ν2
+nνm
+�−σ∞ .
+Therefore M ⊂ N, where M this time denotes the set of all x ≥ x0 for which there exists an
+exceptional triple (j, n, m) ∈ N3 such that (11) holds.
+Now let q be a sequence of primes in the statement of Theorem 1.2, assuming that ε < q1 −1.
+As described, we will only be able to effectively apply (10) when x is sufficiently large, say,
+x ≥ B = B(ε) = C(ε)4 + 2, where C(ε) in this instance refers to the same constant as in (10).
+Let N be such that {q1, . . . , qN} = (1, B) ∩ q. Then, as a corollary of Theorem 1.1, we already
+know that there exists an increasing finite sequence of primes {˜q1, . . . ˜qN}, ˜q1 > 1, such that
+|qj − ˜qj| ≤ q−A
+j
+, j = 1, . . . , N, and such that Bohr’s condition holds for {ν(N)
+n
+}n≥1 = N{˜q1,...˜qN }.
+Further, we choose σ∞ so large that
+���ν(N)
+n+1 − ν(N)
+n
+��� ≥
+�
+ν(N)
+n+1
+�−6σ∞
+,
+n ∈ N,
+and
+(12)
+(ζq′(σ∞) − 1) ζq′
+�σ∞
+4
+�
+≤ 1,
+q′ = {˜q1, . . . ˜qN, qN+1 − 1, qN+2 − 1, qN+3 − 1, . . .}.
+This is made possible by the hypothesis that σc(ζq) < ∞, since
+ζq′(σ) ≤
+�
+j≥1
+1
+1 − (q′
+j)−σ ≤ ζq
+� σ
+C
+�
+,
+σ > 0,
+C ≥ sup
+n≥1
+log(qn)
+log(q′n).
+From here we proceed by induction. Suppose that ˜q1, . . . ˜qk have been chosen, where k ≥ N,
+with corresponding Beurling integers {ν(k)
+n }n≥1 = N{˜qn}k
+n=1 satisfying that
+���ν(k)
+n+1 − ν(k)
+n
+��� ≥
+�
+ν(N)
+n+1
+�−6σ∞ .
+
+6
+ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT
+We apply the preceding discussion to the Beurling primes ρ = {˜q1, . . . ˜qk} and x = qk+1, con-
+cluding that there exists a number ˜qk+1 ∈ Iqk+1 such that
+�����˜qj
+k+1 − ν(k)
+n
+ν(k)
+m
+����� ≥
+�
+ν(k)
+n
+�−3σ∞ �
+ν(k)
+m
+�−3σ∞
+,
+(j, n, m) ∈ N3.
+By the same argument as in the last paragraph of the proof of Theorem 1.1 the Beurling system
+{ν(k+1)
+n
+}n≥1 = N{˜qn}k+1
+n=1, then satisfies that
+���ν(k+1)
+n+1
+− ν(k+1)
+n
+��� ≥
+�
+ν(k+1)
+n+1
+�−6σ∞
+,
+n ∈ N.
+At each step of the construction, (12) ensures that (9) holds. We hence obtain a sequence
+˜q = {˜qn}n≥1, satisfying that |˜qn − qn| ≤ q
+− σ∞
+2
+n
+as well as Bohr’s condition (1), specifically,
+|˜νn+1 − ˜νn| ≥ (˜νn+1)−6σ∞ ,
+n ∈ N.
+where {˜νn}n≥1 = N˜q.
+□
+Proof of Corollary 1.3. By [21, Theorem 1] there exists a Beurling system qRH that satisfies
+(i)–(iii). Theorem 1.2 then implies that there exist primes q, asymptotically approaching qRH,
+satisfying Bohr’s theorem, and such that
+(13)
+πq(x) = li(x) + O
+��√x
+�
+.
+We observe that
+log ζq(s) = −
+∞
+ˆ
+1
+log
+�
+1 − u−s�
+dπq(u),
+Re s > 1,
+see [8, Lemma 10]. With f(u) =
+u−1
+u log u, u > 1, consider the function
+Z(s) = exp
+
+
+∞
+ˆ
+1
+u−sf(u) du
+
+ =
+s
+s − 1,
+Re s > 1.
+We then have that
+(14)
+log ζq(s)
+Z(s) = −
+∞
+ˆ
+1
+�
+log
+�
+1 − u−s�
++ u−s�
+dπq(u) +
+∞
+ˆ
+1
+u−s (dπq(u) − f(u) du) .
+The quantity under the first integral sign satisfies
+log
+�
+1 − u−s�
++ u−s = O
+�
+u−2σ�
+.
+Thus, the first integral in (14) is analytic and uniformly bounded for Re s > 1
+2 + ε, ε > 0. By
+(13), the second integral in (14) is also analytic in the half-plane C 1
+2 . Therefore, ζq has an
+analytic continuation to the half–plane C 1
+2 , except for a simple pole at s = 1. Furthermore, ζq
+cannot have any zeros in this half-plane.
+□
+3. Further discussion
+Diophantine approximation and Beurling integers. Using the Borel–Cantelli theorem to
+study the irrationality of real numbers is a standard technique of Diophantine approximation.
+The irrationality measure µ(x) of a real number x ∈ R is defined as the infimum of the set
+Rx =
+�
+r > 0 :
+���x − m
+n
+��� < 1
+nr for at most finitely many pairs (m, n) ∈ N × N
+�
+.
+
+A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS
+7
+For a Beurling system Nq = {νn}n≥1, we may also introduce the irrationality measure µq(x) of
+a real number x ∈ R as the infimum of the set
+Rx =
+�
+r > 0 :
+����x − νm
+νn
+���� < 1
+νrn
+for at most finitely many pairs (m, n) ∈ N × N
+�
+.
+Then, by slightly modifying the proof of Lemma 2.1, we obtain the following proposition.
+Proposition 3.1. Let q = {qn}n≥1 be a sequence of Beurling primes with σc(ζq) < ∞. Then,
+for almost every x ∈ R, it holds that
+µq(x) ≤ 2σc(ζq).
+In the classical case, Dirichlet’s approximation theorem therefore implies that µ(x) = 2 for
+almost every x ∈ R. We also recall Roth’s theorem [5], which states that µ(x) = 2 for every
+algebraic irrational number. It would be very interesting to develop corresponding results in the
+context of Beurling integers.
+Hardy spaces of Dirichlet series and a conjecture of Helson. For a sequence q of Beurling
+primes, we introduce the Hardy space H2
+q as
+H2
+q =
+
+
+f(s) =
+�
+n≥1
+anν−s
+n
+: ∥f∥2
+H2q =
+�
+n≥1
+|an|2 < ∞
+
+
+ .
+More generally, for 1 ≤ p < ∞, we define Hp
+q as the completion of polynomials (finite sums
+� anν−s
+n ) under the Besicovitch norm
+∥P∥Hp
+q :=
+
+ lim
+T →∞
+1
+2T
+T
+ˆ
+−T
+|P(it)|p dt
+
+
+1
+p
+.
+The function theory of these spaces originated with Helson [12], and was, in the distuingished
+case where q is the sequence of ordinary primes, continued in very influential papers of Bayart
+[1] and Hedenmalm, Lindqvist, and Seip [11]. More generally, there is a developing theory of
+Hardy spaces of Dirichlet series � ane−λns whose frequencies are related to other groups than
+T∞, but we shall restrict our attention to frequencies given by Beurling primes. A cornerstone
+of the theory is that there is a natural multiplicative linear isometric isomorphism between Hp
+q
+and the Hardy space Hp
+q (T∞) of the infinite torus [6, 13]. However, more is needed in order to
+identify H∞(T∞) with H∞
+q , the space of Dirichlet series � anν−s
+n
+which converge to a bounded
+function in C0. In fact, Bohr’s condition is typically used in order to establish this isomorphism
+[19].
+In identifying Hp
+q with Hp
+q (T∞) one is naturally led to consider twisted Dirichlet series
+fχ(s) =
+�
+n≥1
+anχ(νn)ν−s
+n ,
+where a point χ ∈ T∞ is interpreted as the completely multiplicative character χ: Nq → T
+such that χ(qn) = χn. Helson [13] proved that if f ∈ H2
+q and the associated frequencies satisfy
+Bohr’s condition, then fχ(s) converges in C0 for almost every χ ∈ T∞. Helson went on to
+make a conjecture, which we state only in the special case that the frequencies correspond to a
+Beurling system.
+Recall that f ∈ H2
+q is said to be outer if
+�
+fg : g ∈ H∞
+q
+�
+is dense in H2
+q.
+Conjecture. If Nq is a Beurling system that satisfies Bohr’s condition and f is outer in H2
+q,
+then fχ never has any zeros in its half-plane of convergence.
+
+8
+ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT
+Suppose now that the Beurling primes q are chosen as in Corollary 1.3, so that we have the
+“Riemann hypothesis” at our disposal, and consider the Dirichlet series
+f(s) =
+1
+ζq(s + 1/2 + ε),
+for some 0 < ε < 1/2.
+Through a routine calculation with coefficients, one checks that
+f, f 2, 1/f, 1/f 2 ∈ H2
+q. Therefore, there are polynomials pn which converge to 1/f in H4
+q, so
+that
+∥1 − pnf∥H2q ≤ ∥f∥H4q∥1/f − pn∥H4q → 0,
+n → ∞.
+Thus f is outer. On the other hand, it has a zero at s = 1/2 − ε.
+Corollary 1.3 ensures that f has an analytic extension to C0. The problem is that we do not
+know if f actually converges there, as required by Helson’s conjecture. Since we have Bohr’s
+condition, a standard argument [20, Section 14.25] with the Perron formula shows that the
+Lindel¨of hypothesis for ζq implies the convergence of f. However, in the Beurling prime setting,
+“Riemann implies Lindel¨of” is only true when ζq has finite order in C 1
+2 .
+This crux further
+highlights one of the main issues of [14, 15]: how can we ensure that ζq is zero-free and has finite
+order in some half-plane Re s > 1 − η? Or, equivalently, when do we have that
+πq(x) = li(x) + O(xθ1),
+Nq(x) = kx + O(xθ2),
+for some θ1, θ2 < 1? The best estimate we are able to add to Corollary 1.3 comes from the
+general result [15, Theorem 2.2]: there exists a number c > 0 such that
+Nq(x) = ax + O
+�
+x exp(−c
+�
+log x log log x)
+�
+.
+References
+[1] Fr´ed´eric Bayart, Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math. 136
+(2002), no. 3, 203–236.
+[2] Fr´ed´eric Bayart, Convergence and almost sure properties in Hardy spaces of Dirichlet series, Math. Ann.
+382 (2022), no. 3-4, 1485–1515.
+[3] Arne Beurling, Analyse de la loi asymptotique de la distribution des nombres premiers g´en´eralis´es. I, Acta
+Math. 68 (1937), no. 1, 255–291.
+[4] Harald Bohr, ¨Uber die gleichm¨aßige Konvergenz Dirichletscher Reihen, J. Reine Angew. Math. 143 (1913),
+203–211.
+[5] H. Davenport and K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 160–
+167.
+[6] Andreas Defant and Ingo Schoolmann, Hp-theory of general Dirichlet series, J. Fourier Anal. Appl. 25
+(2019), no. 6, 3220–3258.
+[7] Harold G. Diamond, When do Beurling generalized integers have a density?, J. Reine Angew. Math. 295
+(1977), 22–39.
+[8] Harold G. Diamond, Hugh L. Montgomery, and Ulrike M. A. Vorhauer, Beurling primes with large oscilla-
+tion, Math. Ann. 334 (2006), no. 1, 1–36.
+[9] Andrew Granville, The lattice points of an n-dimensional tetrahedron, Aequationes Math. 41 (1991), no. 2-3,
+234–241.
+[10] G. H. Hardy and M. Riesz, The general theory of Dirichlet’s series, Cambridge Tracts in Mathematics and
+Mathematical Physics, No. 18, Stechert-Hafner, Inc., New York, 1964.
+[11] H˚akan Hedenmalm, Peter Lindqvist, and Kristian Seip, A Hilbert space of Dirichlet series and systems of
+dilated functions in L2(0, 1), Duke Math. J. 86 (1997), no. 1, 1–37.
+[12] Henry Helson, Compact groups with ordered duals, Proc. London Math. Soc. (3) 14a (1965), 144–156.
+[13] Henry Helson, Compact groups and Dirichlet series, Ark. Mat. 8 (1969), 139–143.
+[14] Titus W. Hilberdink, Well-behaved Beurling primes and integers, J. Number Theory 112 (2005), no. 2,
+332–344.
+[15] Titus W. Hilberdink and Michel L. Lapidus, Beurling zeta functions, generalised primes, and fractal mem-
+branes, Acta Appl. Math. 94 (2006), no. 1, 21–48.
+[16] Athanasios Kouroupis, Composition operators and generalized primes, Proc. Amer. Math. Soc., to appear,
+https://doi.org/10.1090/proc/16395.
+
+A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS
+9
+[17] Edmund Landau, ¨Uber die gleichm¨aßige Konvergenz Dirichletscher Reihen, Math. Z. 11 (1921), no. 3-4,
+317–318.
+[18] Herv´e Queffelec and Martine Queffelec, Diophantine approximation and Dirichlet series, Texts and Readings
+in Mathematics, vol. 80, Hindustan Book Agency, New Delhi; Springer, Singapore, [2020] ©2020, Second
+edition [of 3099268].
+[19] I. Schoolmann, On Bohr’s theorem for general Dirichlet series, Math. Nachr. 293 (2020), no. 8, 1591–1612.
+[20] E. C. Titchmarsh, The theory of the Riemann zeta-function, second ed., The Clarendon Press, Oxford
+University Press, New York, 1986, Edited and with a preface by D. R. Heath-Brown.
+[21] Wen-Bin Zhang, Beurling primes with RH and Beurling primes with large oscillation, Math. Ann. 337
+(2007), no. 3, 671–704.
+Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU),
+7491 Trondheim, Norway
+Email address: athanasios.kouroupis@ntnu.no
+Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU),
+7491 Trondheim, Norway
+Email address: karl-mikael.perfekt@ntnu.no
+

69FKT4oBgHgl3EQfUC2Z/content/tmp_files/load_file.txt

@@ -0,0 +1,311 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf,len=310
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='11782v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='NT]  27 Jan 2023  A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS  ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Given a sequence of frequencies {λn}n≥1, a corresponding generalized Dirichlet  series is of the form f(s) = �  n≥1 ane−λns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' We are interested in multiplicatively generated  systems, where each number eλn arises as a finite product of some given numbers {qn}n≥1,  1 < qn → ∞, referred to as Beurling primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' In the classical case, where λn = log n, Bohr’s  theorem holds: if f converges somewhere and has an analytic extension which is bounded in a  half-plane {Re s > θ}, then it actually converges uniformly in every half-plane {Re s > θ +ε},  ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' We prove, under very mild conditions, that given a sequence of Beurling primes, a small  perturbation yields another sequence of primes such that the corresponding Beurling integers  satisfy Bohr’s condition, and therefore the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Applying our result in conjunction with  work of Diamond–Montgomery–Vorhauer and Zhang, we find a system of Beurling primes for  which both Bohr’s theorem and the Riemann hypothesis are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' We discuss the connections  between our work and Diophantine approximation with Beurling integers, as well as with a  conjecture of Helson concerning outer functions in Hardy spaces of generalized Dirichlet series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Introduction  For an increasing sequence of positive frequencies λ = {λn}n≥1, and a generalized Dirichlet  series  f(s) =  �  n≥1  ane−λns,  the abscissas σc, σu, and σa of point-wise, uniform, and absolute convergence are defined as in  the classical theory of Dirichlet series [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' In this article we wish to find sets of frequencies such  that the analogue of a theorem of Bohr [4] holds: if σc(f) < ∞ and f has a bounded analytic  extension to a half-plane {Re s > θ}, then σu(f) ≤ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' The problem of finding frequencies for  which the abscissas of bounded and uniform convergence always coincide, which originated with  Bohr and Landau [17], has recently been revisited [2, 19] with the context of Hardy spaces of  Dirichlet series in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Indeed, Bohr’s theorem is essentially a necessity for a satisfactory  Hardy space theory, see [18, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  An important class of frequencies were introduced by Beurling [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Given an arbitrary in-  creasing sequence q = {qn}n≥1, 1 < qn → ∞, such that {log qn}n≥1 is linearly independent over  Q, we will denote by Nq = {νn}n≥1 the set of numbers that can be written (uniquely) as finite  products with factors from q, ordered in an increasing manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' The numbers qn are known  as Beurling primes, and the numbers νn are Beurling integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' The corresponding generalized  Dirichlet series are of the form  f(s) =  �  n≥1  anν−s  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  There are a number of criteria to guarantee the validity of Bohr’s theorem for frequencies  {λn}n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Bohr’s original condition asks for the existence of c1, c2 > 0 such that  (1)  λn+1 − λn ≥ c1e−c2λn+1,  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Landau relaxed the condition somewhat: for every δ > 0 there should be a c > 0 such that  (2)  λn+1 − λn ≥ ce−eδλn+1 ,  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  1  2  ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT  Landau’s condition was recently relaxed further by Bayart [2]: for every δ > 0 there should be  a C > 0 such that  (3)  log  �λm + λn  λm − λn  �  + (m − n) ≤ Ceδλn,  m > n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  All of these conditions are usually very difficult to check for any given Beurling system, since  they involve the distances between the corresponding Beurling integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Furthermore, while it  is easy to construct Beurling integers for which conditions (1)-(3) fail, it appears to be rather  difficult to construct Beurling systems which do satisfy the Bohr condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' This is especially  true if one wants to retain properties of the ordinary integers, such as the asympotic behaviour  of the counting function Nq(x) = �  νn≤x 1, see for example [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  One motivation for considering Beurling integers is to investigate the properties of the q-zeta  function  ζq(s) =  �  n≥1  ν−s  n  =  �  n≥1  1  1 − q−s  n  ,  and their interplay with the counting functions  Nq(x) =  �  νn≤x  1,  πq(x) =  �  qn≤x  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  As an example, Beurling [3] himself showed that the condition  (4)  Nq(x) = ax + O(  x  (log x)γ ),  for some γ > 3  2,  implies the analogue of the prime number theorem,  (5)  πq(x) :=  �  qn≤x  1 ∼  x  log x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  We refer to [15] for an overview of further developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  In Section 2 we begin with a preparatory result which is interesting in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' It states  that starting with the classical set of primes numbers we can add almost any finite sequence of  Beurling primes while retaining the validity of Bohr’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Let {pn}n≥1 be the sequence of ordinary prime numbers and let N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Then  Bohr’s condition (1) holds for the Beurling integers generated by the primes  q = {pn}n≥1  �  {qj}N  j=1,  for almost every choice (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' , qN) ∈ (1, ∞)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Sequences of Beurling primes of the type considered in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='1 previously appeared in  [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Our main result requires more careful analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Let q = {qn}n≥1 be an increasing sequence of primes such that q1 > 1 and  σc(ζq) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Then, for every A > 0 there exists a sequence of Beurling primes ˜q = {˜qn}n≥1 for  which Bohr’s condition (1) holds and  |qn − ˜qn| ≤ q−A  n  ,  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='2 and the work of Diamond, Montgomery, Vorhauer [8] and Zhang [21], allows  us to construct a system of Beurling primes that satisfies Bohr’s theorem and the “Riemann  Hypothesis”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' There exists a system of Beurling primes q = {qn}n≥1 such that:  A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS  3  (i) The Riemann zeta function ζq(s) has an analytic extension to Re s > 1  2, except for a simple  pole at s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  (ii) The Riemann zeta function has no zeros in the half–plane C 1  2 = {Re s > 1/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  (iii) The prime counting πq(x) satisfies  πq(x) = li(x) + O  �√x  �  ,  where li(x) =  x´  2  (log u)−1 du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  (iv) Bohr’s condition holds for the associated class of generalized Dirichlet series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  We remark that as a direct consequence of (iii), we have the prime number theorem (5) as  well as Nq(x) ∼ ax, where a is the residue of ζq(s) at s = 1, see for example [7, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  The proofs of our results investigate how well “irrational numbers” may be approximated by  fractions of Beurling integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' We will comment further on this kind of Diophantine approxi-  mation problems in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  In Section 3 we will also return to the original motivation for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' There has been an  interest in studying Hardy spaces of generalized Dirichlet series since the 60s [6, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' However,  other than the ordinary integers, no examples of Beurling integers exist which simultaneously  satisfy the prime number theorem (iii) and Bohr’s theorem, despite the fact that Bohr’s theorem  is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Furthermore, since many aspects of the function theory of the Hardy space do not  depend on the choice q of Beurling primes, the idea behind Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='3 was to find a Beurling  system Nq in which we can assume the Riemann hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' As a function theoretic application,  we construct an outer function f which has a zero,  (6)  f(s) =  1  ζq(s + 1/2 + ε),  0 < ε < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Originally, we intended to make use of (6) in order to disprove a conjecture of Helson [13]  about outer functions in Hardy spaces formed from frequencies satisfying the Bohr condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Unfortunately, we did not quite succeed, since we are unable to demonstrate the convergence  of (6) for Res > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' the typical proof of this relies on the Lindel¨of hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Finding a system  which in addition to the properties of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='3 satisfies the Lindel¨of hypothesis seems to  demand a deeper interplay between our tools and the probabilistic methods of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Throughout the article, we will be using the convention that C denotes a positive  constant which may vary from line to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' We will write that C = C(Ω) when the constant  depends on the parameter Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Proof of the main results  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Suppose that {qn}n≥1 is a Beurling system such that dn := νn+1 − νn ≫ ν−C  n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Then, for almost every q′ > 1 and every ε > 0, the Beurling system {qn}n≥1∪{q′} has a distance  function satisfying  (7)  d′  n = ν′  n+1 − ν′  n ≫ ν−C  ′  n+1 ,  n ∈ N,  where C′(q′, q) = max (C, 2σc(ζq) − 1 + ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Let x0 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' First we will prove that the set M of all numbers q′ ≥ x0 such that there  exist infinitely many triples (j, n, m) ∈ N3 with  ����(q′)j − νn  νm  ���� ≤ ν−C0  n  ν−C0  m  ,  C0 = σ(ζq) + ε,  4  ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT  has measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Since  �����q′ −  � νn  νm  � 1  j ����� ≤ C(x0)x−j  0  ����(q′)j − νn  νm  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  we have that M ⊂ lim supm,n,j Ωm,n,j, where  Ωm,n,j =  �� νn  νm  � 1  j  − C(x0)x−j  0 ν−C0  n  ν−C0  m  ,  � νn  νm  � 1  j  + C(x0)x−j  0 ν−C0  n  ν−C0  m  �  ,  j, n, m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  The Borel–Cantelli lemma thus shows that |M| = 0, since  �  m≥1  �  n≥1  �  j≥1  |Ωm,n,j| ≤ C(x0)ζq (C0)2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Fix a number q′ ∈ [x0, ∞) \\ M such that log q′ is not in the (countable) set spanQ{log qn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Note that the set of such numbers has full measure in [x0, ∞), and that x0 > 1 is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' By  construction, there are finitely many triples (j, n, m) such that  (8)  ����(q′)j − νn  νm  ���� ≤ ν−C0  n  ν−C0  m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  For these exceptional triples, the left-hand side is at least positive, since log q′ /∈ spanQ{log qn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Therefore  ����(q′)j − νn  νm  ���� ≫ ν−C0  n  ν−C0  m  for all (j, n, m) ∈ N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Now we consider two arbitrary consecutive Beurling integers generated by the prime system  {qn}n≥1 ∪ {q′},  ν′  n+1 = (q′)aνm,  ν′  n = (q′)bνl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  If a = b, then l = m − 1 and  ν′  n+1 − ν′  n ≫ ν−C  m  ≥  �  ν′  n+1  �−C ,  by the hypothesis on the distances dn for the original Beurling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Otherwise, if, say, b < a,  then  ��ν′  n+1 − ν′  n  �� = (q′)bνm  ����(q′)a−b − νl  νm  ���� ≫ ν−C0  l  ν−C0+1  m  (q′)b ≫  �  ν′  n+1  �−C  ′  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  where C  ′ = 2σc(ζq) − 1 + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  □  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' The proof is a direct consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  □  In order to prove Bohr’s theorem for more general Beurling systems, we need to control the  constant in the distance estimate (7), which comes from the exceptional triples satisfying (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Fix a small ε > 0 and x0 ∈ (1+ε/2, 1+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Consider first any Beurling  system Nρ = {νn}n≥1 generated by Beurling primes such that ρ1 > 1 + ε and σc(ζρ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' For  a number σ∞ > max(2, A) to be chosen in a moment, let  N =  �  m≥2  �  n≥2  �  j≥1  Ωm,n,j,  where Ωm,n,j is defined as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='1,  Ωm,n,j =  �� νn  νm  � 1  j  − C(x0)x−j  0 ν−σ∞  n  ν−σ∞  m  ,  � νn  νm  � 1  j  + C(x0)x−j  0 ν−σ∞  n  ν−σ∞  m  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS  5  Then |N| ≤ C(ε) (ζρ(σ∞) − 1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Furthermore, for x > 2, let  Ix = [x − x− σ∞  2 , x + x− σ∞  2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Note that if σ∞ is sufficiently large, σ∞ ≥ C(ε), then Ix∩Ωm,n,j ̸= ∅ only if νn ≥ x/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Therefore  |Ix ∩ N| ≤  �  m≥2  j≥1  νn≥ x  2  |Ωm,n,j| ≤ C(ε) (ζρ(σ∞) − 1)  �  νn≥ x  2  ν−σ∞  n  ≤ C(ε) (ζρ(σ∞) − 1) ζρ  �σ∞  4  �  x− 3σ∞  4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  We will construct a sequence of Beurling systems such that  (9)  (ζρ(σ∞) − 1) ζρ  �σ∞  4  �  ≤ 1  for the number σ∞ > 0, still to be chosen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Therefore  (10)  |Ix ∩ N| ≤ C(ε)x− σ∞  4 |Ix|,  We conclude that whenever x is sufficiently large, Ix ̸⊂ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  To include triples where νn or νm equals one in our considerations, we increase the power  σ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' The inequality  (11)  ����xj − νn  νm  ���� ≤ ν−3σ∞  n  ν−3σ∞  m  implies, whenever x ≥ x0, that  �����x −  � νn  νm  � 1  j ����� ≤ C(x0)x−j  0  ����xj − ν2  nνm  ν2mνn  ���� ≤ C(x0)x−j  0  �  ν2  mνn  �−σ∞ �  ν2  nνm  �−σ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Therefore M ⊂ N, where M this time denotes the set of all x ≥ x0 for which there exists an  exceptional triple (j, n, m) ∈ N3 such that (11) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Now let q be a sequence of primes in the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='2, assuming that ε < q1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  As described, we will only be able to effectively apply (10) when x is sufficiently large, say,  x ≥ B = B(ε) = C(ε)4 + 2, where C(ε) in this instance refers to the same constant as in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Let N be such that {q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' , qN} = (1, B) ∩ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Then, as a corollary of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='1, we already  know that there exists an increasing finite sequence of primes {˜q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' ˜qN}, ˜q1 > 1, such that  |qj − ˜qj| ≤ q−A  j  , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' , N, and such that Bohr’s condition holds for {ν(N)  n  }n≥1 = N{˜q1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='˜qN }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Further, we choose σ∞ so large that  ���ν(N)  n+1 − ν(N)  n  ��� ≥  �  ν(N)  n+1  �−6σ∞  ,  n ∈ N,  and  (12)  (ζq′(σ∞) − 1) ζq′  �σ∞  4  �  ≤ 1,  q′ = {˜q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' ˜qN, qN+1 − 1, qN+2 − 1, qN+3 − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  This is made possible by the hypothesis that σc(ζq) < ∞, since  ζq′(σ) ≤  �  j≥1  1  1 − (q′  j)−σ ≤ ζq  � σ  C  �  ,  σ > 0,  C ≥ sup  n≥1  log(qn)  log(q′n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  From here we proceed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Suppose that ˜q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' ˜qk have been chosen, where k ≥ N,  with corresponding Beurling integers {ν(k)  n }n≥1 = N{˜qn}k  n=1 satisfying that  ���ν(k)  n+1 − ν(k)  n  ��� ≥  �  ν(N)  n+1  �−6σ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  6  ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT  We apply the preceding discussion to the Beurling primes ρ = {˜q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' ˜qk} and x = qk+1, con-  cluding that there exists a number ˜qk+1 ∈ Iqk+1 such that  �����˜qj  k+1 − ν(k)  n  ν(k)  m  ����� ≥  �  ν(k)  n  �−3σ∞ �  ν(k)  m  �−3σ∞  ,  (j, n, m) ∈ N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  By the same argument as in the last paragraph of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='1 the Beurling system  {ν(k+1)  n  }n≥1 = N{˜qn}k+1  n=1, then satisfies that  ���ν(k+1)  n+1  − ν(k+1)  n  ��� ≥  �  ν(k+1)  n+1  �−6σ∞  ,  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  At each step of the construction, (12) ensures that (9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' We hence obtain a sequence  ˜q = {˜qn}n≥1, satisfying that |˜qn − qn| ≤ q  − σ∞  2  n  as well as Bohr’s condition (1), specifically,  |˜νn+1 − ˜νn| ≥ (˜νn+1)−6σ∞ ,  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  where {˜νn}n≥1 = N˜q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  □  Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' By [21, Theorem 1] there exists a Beurling system qRH that satisfies  (i)–(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='2 then implies that there exist primes q, asymptotically approaching qRH,  satisfying Bohr’s theorem, and such that  (13)  πq(x) = li(x) + O  �√x  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  We observe that  log ζq(s) = −  ∞  ˆ  1  log  �  1 − u−s�  dπq(u),  Re s > 1,  see [8, Lemma 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' With f(u) =  u−1  u log u, u > 1, consider the function  Z(s) = exp  \uf8eb  \uf8ed  ∞  ˆ  1  u−sf(u) du  \uf8f6  \uf8f8 =  s  s − 1,  Re s > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  We then have that  (14)  log ζq(s)  Z(s) = −  ∞  ˆ  1  �  log  �  1 − u−s�  + u−s�  dπq(u) +  ∞  ˆ  1  u−s (dπq(u) − f(u) du) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  The quantity under the first integral sign satisfies  log  �  1 − u−s�  + u−s = O  �  u−2σ�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Thus, the first integral in (14) is analytic and uniformly bounded for Re s > 1  2 + ε, ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' By  (13), the second integral in (14) is also analytic in the half-plane C 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Therefore, ζq has an  analytic continuation to the half–plane C 1  2 , except for a simple pole at s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Furthermore, ζq  cannot have any zeros in this half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Further discussion  Diophantine approximation and Beurling integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Using the Borel–Cantelli theorem to  study the irrationality of real numbers is a standard technique of Diophantine approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  The irrationality measure µ(x) of a real number x ∈ R is defined as the infimum of the set  Rx =  �  r > 0 :  ���x − m  n  ��� < 1  nr for at most finitely many pairs (m, n) ∈ N × N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS  7  For a Beurling system Nq = {νn}n≥1, we may also introduce the irrationality measure µq(x) of  a real number x ∈ R as the infimum of the set  Rx =  �  r > 0 :  ����x − νm  νn  ���� < 1  νrn  for at most finitely many pairs (m, n) ∈ N × N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Then, by slightly modifying the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='1, we obtain the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Let q = {qn}n≥1 be a sequence of Beurling primes with σc(ζq) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Then,  for almost every x ∈ R, it holds that  µq(x) ≤ 2σc(ζq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  In the classical case, Dirichlet’s approximation theorem therefore implies that µ(x) = 2 for  almost every x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' We also recall Roth’s theorem [5], which states that µ(x) = 2 for every  algebraic irrational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' It would be very interesting to develop corresponding results in the  context of Beurling integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Hardy spaces of Dirichlet series and a conjecture of Helson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' For a sequence q of Beurling  primes, we introduce the Hardy space H2  q as  H2  q =  \uf8f1  \uf8f2  \uf8f3f(s) =  �  n≥1  anν−s  n  : ∥f∥2  H2q =  �  n≥1  |an|2 < ∞  \uf8fc  \uf8fd  \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  More generally, for 1 ≤ p < ∞, we define Hp  q as the completion of polynomials (finite sums  � anν−s  n ) under the Besicovitch norm  ∥P∥Hp  q :=  \uf8eb  \uf8ed lim  T →∞  1  2T  T  ˆ  −T  |P(it)|p dt  \uf8f6  \uf8f8  1  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  The function theory of these spaces originated with Helson [12], and was, in the distuingished  case where q is the sequence of ordinary primes, continued in very influential papers of Bayart  [1] and Hedenmalm, Lindqvist, and Seip [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' More generally, there is a developing theory of  Hardy spaces of Dirichlet series � ane−λns whose frequencies are related to other groups than  T∞, but we shall restrict our attention to frequencies given by Beurling primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' A cornerstone  of the theory is that there is a natural multiplicative linear isometric isomorphism between Hp  q  and the Hardy space Hp  q (T∞) of the infinite torus [6, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' However, more is needed in order to  identify H∞(T∞) with H∞  q , the space of Dirichlet series � anν−s  n  which converge to a bounded  function in C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' In fact, Bohr’s condition is typically used in order to establish this isomorphism  [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  In identifying Hp  q with Hp  q (T∞) one is naturally led to consider twisted Dirichlet series  fχ(s) =  �  n≥1  anχ(νn)ν−s  n ,  where a point χ ∈ T∞ is interpreted as the completely multiplicative character χ: Nq → T  such that χ(qn) = χn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Helson [13] proved that if f ∈ H2  q and the associated frequencies satisfy  Bohr’s condition, then fχ(s) converges in C0 for almost every χ ∈ T∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Helson went on to  make a conjecture, which we state only in the special case that the frequencies correspond to a  Beurling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Recall that f ∈ H2  q is said to be outer if  �  fg : g ∈ H∞  q  �  is dense in H2  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' If Nq is a Beurling system that satisfies Bohr’s condition and f is outer in H2  q,  then fχ never has any zeros in its half-plane of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  8  ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT  Suppose now that the Beurling primes q are chosen as in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='3, so that we have the  “Riemann hypothesis” at our disposal, and consider the Dirichlet series  f(s) =  1  ζq(s + 1/2 + ε),  for some 0 < ε < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Through a routine calculation with coefficients, one checks that  f, f 2, 1/f, 1/f 2 ∈ H2  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Therefore, there are polynomials pn which converge to 1/f in H4  q, so  that  ∥1 − pnf∥H2q ≤ ∥f∥H4q∥1/f − pn∥H4q → 0,  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Thus f is outer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' On the other hand, it has a zero at s = 1/2 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='3 ensures that f has an analytic extension to C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' The problem is that we do not  know if f actually converges there, as required by Helson’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Since we have Bohr’s  condition, a standard argument [20, Section 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='25] with the Perron formula shows that the  Lindel¨of hypothesis for ζq implies the convergence of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' However, in the Beurling prime setting,  “Riemann implies Lindel¨of” is only true when ζq has finite order in C 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  This crux further  highlights one of the main issues of [14, 15]: how can we ensure that ζq is zero-free and has finite  order in some half-plane Re s > 1 − η?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Or, equivalently, when do we have that  πq(x) = li(x) + O(xθ1),  Nq(x) = kx + O(xθ2),  for some θ1, θ2 < 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' The best estimate we are able to add to Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='3 comes from the  general result [15, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='2]: there exists a number c > 0 such that  Nq(x) = ax + O  �  x exp(−c  �  log x log log x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  References  [1] Fr´ed´eric Bayart, Hardy spaces of Dirichlet series and their composition operators, Monatsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
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+page_content='  [9] Andrew Granville, The lattice points of an n-dimensional tetrahedron, Aequationes Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' 41 (1991), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
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+page_content='  [10] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
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+page_content='  [16] Athanasios Kouroupis, Composition operators and generalized primes, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=', to appear,  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
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+page_content=' 80, Hindustan Book Agency, New Delhi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Springer, Singapore, [2020] ©2020, Second  edition [of 3099268].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
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+page_content=', The Clarendon Press, Oxford  University Press, New York, 1986, Edited and with a preface by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Heath-Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  [21] Wen-Bin Zhang, Beurling primes with RH and Beurling primes with large oscillation, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' 337  (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content=' 3, 671–704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='  Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU),  7491 Trondheim, Norway  Email address: athanasios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='kouroupis@ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='no  Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU),  7491 Trondheim, Norway  Email address: karl-mikael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='perfekt@ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}
+page_content='no' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'}

6NE2T4oBgHgl3EQfOwb7/content/tmp_files/2301.03753v1.pdf.txt

@@ -0,0 +1,776 @@
+arXiv:2301.03753v1  [math.NA]  10 Jan 2023
+ENFORCING NEUMANN BOUNDARY CONDITIONS WITH POLYNOMIAL
+EXTENSION OPERATORS TO ACHEIVE OPTIMAL CONVERGENCE RATES ON
+POLYTOPIAL MESHES IN THE FINITE ELEMENT METHOD
+JAMES CHEUNG
+Millennium Space Systems, A Boeing Company. 2265 E. El Segundo Blvd, El Segundo, CA.
+Abstract. In [4], the authors presented two finite element methods for approximating second order bound-
+ary value problems on polytopial meshes with optimal accuracy without having to utilize curvilinear map-
+pings. This was done by enforcing the boundary conditions through judiciously chosen polynomial extension
+operators. The H1 error estimates were proven to be optimal for the solutions of both the Dirichlet and
+Neumann boundary value problems. It was also proven that the Dirichlet problem approximation converges
+optimally in L2. However, optimality of the Neumann approximation in the L2 norm was left as an open
+problem. In this work, we seek to close this problem by presenting new analysis that proves optimal error
+estimates for the Neumann approximation in the W 1
+∞ and L2 norms.
+1. Introduction
+The purpose of this note is to derive optimal error estimates for the polynomial extension finite element
+method (PE-FEM) described in [4] for approximating elliptic boundary value problems with Neumann
+conditions. This manuscript is very much an appendix to our previous work. As such, we highly suggest
+that the reader refer to that work, especially since we will not redefine our notation here for the sake of
+brevity. Our analysis will be structured in the following manner: We present new technical lemmas involving
+the Averaged Taylor series in §2, then in §3 we move on to prove well-posedness and derive optimal error
+estimates for the numerical solution in the W 1
+∞(Ω) norm, using this new result we are finally able to derive
+optimal error estimates for the numerical solution in the L2(Ω) norm in §4. We then discuss our results and
+propose additional future research in §5.
+2. Additional Results for Averaged Taylor Polynomials
+We begin our analysis by deriving some technical results for the Averaged Taylor polynomials in the
+W m
+∞(Ωh) setting.
+Lemma 1. Let v ∈ L∞(Ωh), then ���T k
+h (v)
+��
+η(ξ)
+���
+L∞(Ωh) ≤ C∥v∥L∞(Ωh).
+Proof. Using a scaling argument on [2, Corollary 4.1.15], we immediately have that
+���T k
+h (v)
+��
+η(ξ)
+���
+L∞(Ei∩Si,ℓ) ≤ Cδ−d
+h ∥v∥L1(σi,ℓ)
+≤ C∥v∥L∞(σi,ℓ)
+We conclude by seeing that���T k
+h (v)
+��
+η(ξ)
+���
+L∞(Γh) = max
+i,ℓ
+���T k
+h (v)
+��
+η(ξ)
+���
+L∞(Ei∩Si,ℓ) .
+□
+Lemma 2. Let v ∈ W k+1
+∞
+(Ωh). Then for an integer m < k + 1, the following is satisfied
+���T k
+h (v)
+��
+η(ξ) − v
+���
+W m
+∞(Γh) ≤ Cδk+1−m
+h
+|v|W k+1
+∞
+(Ωh).
+1
+
+Proof. Using [2, Proposition 4.3.2], we have that
+���T k
+h (v)
+��
+η(ξ) − v
+���
+W m
+∞(Ei∩Si,ℓ) =
+��T k
+h (v) − v
+��
+W m
+∞(η(Ei)∩Si,ℓ)
+≤
+��T k
+h (v) − v
+��
+W m
+∞(Si,ℓ)
+≤ Cδk+1−m
+h
+|v|W k+1
+∞
+(Si,ℓ).
+Taking the maximum over all i, ℓ ∈ N concludes the proof.
+□
+Lemma 3. Let v ∈ V
+k
+h, then
+����T k′,k
+h
+(v)
+���
+η(ξ)
+����
+L∞(Γh)
+≤ C
+k
+�
+α|=1
+h−|α|δ|α|
+h ∥v∥L∞(Ωh)
+Proof. From the definition of T k′,k
+h
+(v), we see directly that
+����T k′,k
+h
+(v)
+���
+η(ξ)
+����
+L∞(Γh)
+≤
+k
+�
+|α|=1
+δ|α|
+h
+|α|! ∥Dαv∥L∞(Γh)
+≤
+k
+�
+|α|=1
+δ|α|
+h
+|α|! ∥Dαv∥L∞(Ωh)
+≤
+k
+�
+|α|=1
+δ|α|
+h h−|α|
+|α|!
+∥v∥L∞(Ωh) ,
+after applying the inverse inequality.
+□
+With these technical lemmas derived, we are now ready to prove that the solution of the Neumann
+approximation presented in [4] is well-posed and optimal in W 1
+∞(Ω).
+3. Well-Posedness and Error Estimates in W 1
+∞(Ωh)
+In this section, we determine that the solution uh ∈ V k
+h is bounded in W 1
+∞(Ωh).
+Additionally, we
+demonstrate that the error estimate is optimal in the same norm. The analysis presented here is remarkably
+standard since the perturbations incurred by the extensions in the discrete bilinear form are not large enough
+to cause stability loss. Additionally, Strang’s Lemma arguments are also used to handle the error incurred
+by the nonconforming perturbations in the bilinear form.
+Theorem 1. Assume that δh ∼ O(h2) and that h ∈ R+ is sufficiently small. Further assume that �p, �q ∈
+Ck+1(Ωh). Then there exist positive constants γ, M ∈ R+ such that
+(1)
+sup
+vh∈V k
+h ∩W 1
+1 (Ωh)
+Bh,N(wh, vh)
+∥vh∥W 1
+1 (Ωh)
+≥ γ∥wh∥W 1
+∞(Ωh)
+and
+(2)
+Bh,N(wh, vh) ≤ M∥wh∥W 1
+∞(Ωh)∥vh∥W 1
+1 (Ωh)
+for all uh, vh ∈ V k
+h .
+Proof. We begin by recalling that
+Bh,N(wh, vh) =
+�
+Ωh
+(�p∇wh · ∇vh + ˜qwhvh) dx +
+�
+�p ◦ η(ξ) Tk−1
+h
+(∇wh)
+��
+η(ξ) · n − �p(ξ)∇wh · nh, vh
+�
+Γh
+.
+We then see that
+Bh,N(wh, vh) ≥
+�
+Ωh
+(�p∇wh · ∇vh + �qwhvh) dx − Ch|u|W 1
+∞(Ωh)∥vh∥W 1
+1 (Ωh),
+2
+
+after applying Lemma 3 and the trace inequality [2, Theorem 1.6.6]. Dividing both sides with ∥vh∥W 1
+1 (Ωh)
+allows us to see that
+sup
+vh∈V k
+h ∩W 1
+1 (Ωh)
+Bh,N(wh, vh)
+∥vh∥W 1
+1 (Ωh)
+≥
+sup
+vh∈V k
+h ∩W 1
+1 (Ωh)
+�
+Ωh (�p∇wh · ∇vh + �qwhvh) dx
+∥vh∥W 1
+1 (Ωh)
+− Ch|u|W 1
+∞(Ωh).
+Now, choosing vh = wh allows us to see that supvh∈V k
+h ∩W 1
+1 (Ωh)
+�
+Ωh(�p∇wh·∇vh+�qwhvh)dx
+∥vh∥W 1
+1 (Ωh)
+> 0. As such, we can
+choose a constant Cp,q > 0 such that
+sup
+vh∈V k
+h ∩W 1
+1 (Ωh)
+�
+Ωh (�p∇wh · ∇vh + �qwhvh) dx
+∥vh∥W 1
+1 (Ωh)
+≥ Cp,q
+sup
+vh∈V k
+h ∩W 1
+1 (Ωh)
+�
+Ωh (∇wh · ∇vh + whvh) dx
+∥vh∥W 1
+1 (Ωh)
+= Cp,q∥wh∥W 1
+∞(Ωh).
+Therefore,
+sup
+vh∈V k
+h ∩W 1
+1 (Ωh)
+Bh,N(wh, vh)
+∥vh∥W 1
+1 (Ωh)
+≥ Cp,q∥wh∥W 1
+∞Ωh − Ch∥uh∥W 1
+∞(Ωh).
+And thus, we have that (1) is satisfied.
+Using Lemma 1, H¨older’s inequality, and the trace inequality allows us to derive (2).
+□
+Theorem 2. Let uh ∈ V k
+h satisfy [4, Equation 26]. Assume that u ∈ W k+1
+∞
+(Ω) and that f ∈ W k−1
+∞
+(Ω).
+Furthermore, let �u ∈ W k+1
+∞
+(Ωh) and �f, �f ∈ W k−1
+∞
+(Ωh) be extensions of u and f respectively from Ω to Ωh.
+We then have that
+∥�u − uh∥W 1
+∞(Ωh) ≤ Chk �
+|u|W k+1
+∞
+(Ω) + ∥f∥W k−1
+∞
+(Ω)
+�
+under the conditions specified in Theorem 1.
+Proof. Let uI ∈ V k
+h be the piecewise polynomial interpolant of �u ∈ W k+1
+∞
+(Ωh) defined on Ωh. From [2,
+Theorem 4.2.20] and the Stein extension theorem [1], we have that
+(3)
+∥�u − uI∥W 1
+∞(Ωh) ≤ Chk|u|W k+1
+∞
+(Ω).
+We begin the analysis by seeing that
+∥uI − uh∥W 1
+∞(Ωh) ≤
+sup
+vh∈V k
+h ∩W 1
+1 (Ωh)
+Bh,N(uI − uh, vh)
+∥vh∥W 1
+1 (Ωh)
+=
+sup
+vh∈V k
+h ∩W 1
+1 (Ωh)
+Bh,N(uI − �u, vh) + Bh,N(�u − uh, vh)
+∥vh∥W 1
+1 (Ωh)
+≤ M∥�u − uh∥W 1
+∞(Ωh) +
+sup
+vh∈V k
+h ∩W 1
+1 (Ωh)
+�
+�f − �f, vh
+�
+Ωh
+−
+�
+�p ◦ η(ξ)Rk−1 (∇�u)|η(ξ) · n, vh
+�
+Γh
+∥vh∥W 1
+1 (Ωh)
+≤ Chk|u|W k+1
+∞
+(Ω) + Cδk−1
+h
+|f|W k−1
+∞
+(Ω) + Cδk−1
+h
+|u|W k+1
+∞
+(Ω)
+≤ Chk|u|W k+1
+∞
+(Ω)
+after utilizing (3), Lemma 2, and seeing that
+Bh,N(�u, vh) =
+�
+�f, vh
+�
+Ωh
++
+�
+gN ◦ η(ξ) − �p ◦ η(ξ)Rk−1 (∇u)|η(ξ) · n, vh
+�
+Γh
+.
+The proof is completed by seeing that
+∥�u − uh∥W 1
+∞(Ωh) ≤ ∥u − uI∥W 1
+∞(Ωh) + ∥uI − uh∥W 1
+∞(Ωh),
+and applying the above bound along with (3).
+□
+3
+
+4. Error Estimates in L2(Ωh)
+We are now ready to derive the optimal error estimates for the solution of the Neumann approximation
+in the L2(Ω) norm. The analysis begins by estimating the nonconformity error. This nonconformity error
+will then be used in the following duality argument to bound the terms in the discrete problem that are not
+orthogonal in the Galerkin sense with respect to the continuous bilinear form.
+4.1. Nonconformity Error. Let us begin the derivation of the L2(Ωh) error bound by analyzing the
+nonconformity induced by Bh,N(·, ·) := H1(Ωh) × V k
+h → R+.
+Lemma 4. Assume that all the conditions in Theorem 2 hold, then the following is satisfied
+Nh(�u − uh, vh) ≤ Chk+1 �
+|u|W k+1
+∞
+(Ω) + |f|W k−1
+∞
+(Ω)
+�
+∥vh∥1,Ωh
+for all V k
+h ∩ W 1
+1 (Ωh).
+Proof. Notice that
+Bh,N(�u, vh) =
+�
+�f, vh
+�
+Ωh
++
+�
+gN ◦ η(ξ) − �p ◦ η(ξ) Rk−1
+h
+(∇�u)
+��
+η(ξ) · n, vh
+�
+Γh
+∀v ∈ H1(Ωh).
+Taking the difference with [4, Equation (24)] yields
+Bh,N(�u − uh, vh) =
+�
+�f − �f, vh
+�
+Ωh
+−
+�
+�p ◦ η(ξ) Rk−1
+h
+(∇�u)
+��
+η(ξ) · n, vh
+�
+Γh
+∀vh ∈ V k
+h .
+Let us define eh := �u − uh, then from the definition of Bh,N(·, ·) (See [4, Equation (25)]), we have that
+(4)
+Nh(eh, vh) =
+�
+�f − �f, vh
+�
+Ωh
+−
+�
+�p ◦ η(ξ) Rk−1
+h
+(∇�u)
+��
+η(ξ) · n, vh
+�
+Γh
+−
+�
+�p ◦ η(ξ) Tk−1
+h
+(∇eh)
+��
+η(ξ) · n, vh
+�
+Γh
++ ⟨�p(ξ)∇eh · nh, vh⟩Γh
+=
+�
+�f − �f, vh
+�
+Ωh
+−
+�
+�p ◦ η(ξ) Rk−1
+h
+(∇�u)
+��
+η(ξ) · n, vh
+�
+Γh
++ ⟨�p ◦ η(ξ)∇eh · (n − nh), vh⟩Γh +
+�
+�p(ξ) T1,k−1
+h
+(∇eh)
+���
+η(ξ) · n, vh
+�
+Γh
++
+�
+(�p(ξ) − �p ◦ η(ξ)) Tk−1
+h
+(∇eh)
+��
+η(ξ) · n, vh
+�
+Γh
+.
+We will now analyze each of the terms on the right hand side of (4) seperately.
+Let Ωh
+diff := Ωh \ (Ω ∩ Ωh), then we have by the definition of the extension that
+�
+�f − �f, vh
+�
+Ωh
+=
+�
+�f − �f, vh
+��
+Ωh
+diff
+≤
+�
+max
+i,ℓ ∥ �f − T k−1
+h
+f∥L∞(Si,ℓ) + max
+i,ℓ ∥ �f − T k−1
+h
+f∥L∞(Si,ℓ)
+� �
+Ωh
+diff
+1 · |v|dx
+≤ Cδk−1
+h
+|Ωh
+diff|
+1
+2 |f|W k−1
+∞
+(Ωh)∥vh∥0,Ωh
+≤ Cδ
+k− 1
+2
+h
+|f|W k−1
+∞
+(Ωh)∥vh∥1,Ωh,
+after applying [2, Lemma 4.3.8], H¨older’s inequality, and seeing that |Ωh
+diff| ∼ O(δh). Since we have assumed
+that δh ∼ O(h2), we have that
+(5)
+�
+�f − �f, vh
+�
+Ωh
+≤ Ch2k−1|f|W k−1
+∞
+(Ωh)∥vh∥1,Ωh.
+Next, using Lemma 2, we have that
+(6)
+�
+�p ◦ η(ξ) Rk−1
+h
+(∇�u)
+��
+η(ξ) · n, vh
+�
+Γh
+≤ p
+���Rk−1
+h
+∇�u
+��
+η(ξ)
+���
+L∞(Γh) ∥vh∥L1(Γh)
+≤ Cδk
+h|∇�u|W k
+∞(Ω)∥vh∥0,Γh
+≤ Ch2k|u|W k+1
+∞
+(Ω)∥vh∥1,Ωh,
+4
+
+after applying the trace theorem and the assumption that δh ∼ O(h2).
+Additionally, we have that
+(7)
+⟨�p ◦ η(ξ)∇eh · (n − nh), vh⟩Γh ≤ p
+�
+max
+ξ∈Γh ∥n ◦ η(ξ) − nh(ξ)∥Rd
+�
+∥∇eh∥L∞(Γh)∥vh∥L1(Γh)
+≤ Ch∥eh∥W 1
+∞(Ωh)∥vh∥1,Ωh
+≤ Chk+1 �
+|u|W k+1
+∞
+(Ω) + |f|W k−1
+∞
+(Ω)
+�
+∥vh∥1,Ωh,
+after applying Theorem 2.
+Then, we have that
+(8)
+�
+�p(ξ) T1,k−1
+h
+(∇eh)
+���
+η(ξ) · n, vh
+�
+Γh
+≤ Cp∥vh∥1,1Ωh
+k−1
+�
+|α|=1
+δ|α|
+h ∥Dα∇eh∥L∞(Γh)
+≤ Cp∥vh∥1,1Ωh
+k−1
+�
+|α|=1
+δ|α|
+h
+�
+∥Dα∇(�u − uI)∥L∞(Ωh) + ∥Dα∇(uI − uh)∥L∞(Ωh)
+�
+≤ Cp∥vh∥1,1Ωh
+k−1
+�
+|α|=1
+δ|α|
+h
+�
+hk−|α||u|W k+1
+∞
+(Ω) + Ch−|α| �
+∥∇(eh)∥L∞(Ωh) − ∥∇(�u − uI)∥L∞(Ωh)
+��
+≤ Cp∥vh∥1,1Ωh
+k−1
+�
+|α|=1
+δ|α|
+h hk−|α| �
+|u|W k+1
+∞
+(Ω) + |f|W k−1
+∞
+(Ω)
+�
+≤ Chk+1 �
+|u|W k+1
+∞
+(Ω) + |f|W k−1
+∞
+(Ω)
+�
+∥vh∥1,Ωh
+after applying Theorem 2 and the interpolation estimate [2, Theorem 4.2.20].
+We finally move on to the last term, where we see that
+(9)
+�
+(�p(ξ) − �p ◦ η(ξ)) Tk−1
+h
+(∇eh)
+��
+η(ξ) · n, vh
+�
+Γh
+≤ Cδh
+���Tk−1
+h
+(∇eh)
+��
+η(ξ)
+���
+L∞(Γh) ∥vh∥L1(Γh)
+≤ Cδh∥∇eh∥L∞(Ωh)∥vh∥1,Ωh
+≤ Chk+2 �
+|u|W k+1
+∞
+(Ω) + |f|W k−1
+∞
+(Ω)
+�
+∥vh∥1,Ωh,
+where we used Lemma 1 and Theorem 2.
+Inserting (5), (6), (7), (8), and (9) into (4) gives us
+Nh(�u − uh, vh) ≤ Chk+1 �
+|u|W k+1
+∞
+(Ω) + |f|W k−1
+∞
+(Ω)
+�
+∥vh∥1,Ωh
+∀vh ∈ V k
+h
+This concludes this proof.
+□
+Now that we have established that the nonconformity in the bilinear form is bounded above by O(hk+1),
+we are now ready to analyze the L2(Ω) error of the numerical solution.
+4.2. The Dual Problem. The dual problem we are interested in utilizing is to seek a ψ ∈ H1(Ωh) such
+that
+Nh(v, ψ) = ⟨�u − uh, v⟩Ωh
+∀v ∈ H1(Ωh).
+The corresponding finite element approximation to the dual problem is to seek a ψh ∈ V k
+h such that
+Nh(v, ψh) = ⟨�u − uh, v⟩Ωh
+∀vh ∈ V k
+h .
+In general, φ ∈ H1(Ωh) does not full H2(Ωh) regularity due to the presence of interior angles in Γh. From
+the literature, [2, 5] it is known that
+(10)
+∥ψ∥1+s,Ωh ≤ C∥u − uh∥0,Ωh
+5
+
+and
+(11)
+∥φ∥1,Ωh ≤ C∥u − uh∥−1,Ωh ≤ C∥u − uh∥0,Ωh.
+Furthermore, we have that
+(12)
+∥ψ − ψh∥1,Ωh ≤ Chs|φ|1+s,Ωh,
+where s ∈
+� 1
+2, 1
+�
+depends on the magnitude of the largest interior angle of Γh. See [4, Remark 7].
+4.3. Analysis of L2(Ωh) Convergence. With the previous results proven, we are now ready to derive op-
+timal L2(Ωh) error estimates for the PE-FEM approximation of elliptic Neumann boundary value problems.
+Theorem 3. Assume that all the conditions in Theorem 2 are satisfied. Then we have that
+∥�u − uh∥0,Ωh ≤ Chk+s �
+|u|W k+1
+∞
+(Ω) + |f|W k−1
+∞
+Ω
+�
+,
+where s ∈
+� 1
+2, 1
+�
+depends on the largest interior angle of Γh. If Γh is convex, then s = 1.
+Proof. From the definition of the dual problem, we have that
+∥�u − uh∥2
+0,Ωh = Nh(�u − uh, φ)
+= Nh(�u − uh, φ − φh) + Nh(�u − uh, φh)
+≤ C∥�u − uh∥1,Ωh∥φ − φh∥1,Ωh + Nh(�u − uh, φh)
+([4, Theorem 2])
+≤ Chk+s (|u|k+1,Ωh + |f|k−1,Ωh) |φ|2,Ωh + Nh(�u − uh, φh)
+(12) and [4, Theorem 5]
+≤ Chk+s (|u|k+1,Ωh + |f|k−1,Ωh) |φ|2,Ωh
++ Chk+1 �
+|u|W k+1
+∞
+(Ω) + |f|W k−1
+∞
+Ω
+�
+∥φh∥1,Ωh
+Lemma 4
+≤ Chk+s �
+|u|W k+1
+∞
+(Ω) + |f|W k−1
+∞
+Ω
+�
+∥�u − uh∥0,Ωh
+(10) and (11).
+This concludes this proof.
+□
+5. Conclusion
+In this manuscript, we have presented new results that indicate that the solution of the Neumann ap-
+proximation presented in [4] is optimal in W 1
+∞(Ω) and L2(Ω). Our analysis implies hat the solution to the
+continuous Neumann problem must be pointwise bounded a.e. in its (k + 1)-th derivative in order for us to
+achieve additional accuracy in the L2(Ω) norm, otherwise the L2(Ω) error is bounded by the O(hk) H1(Ω)
+error.
+In our future work, we plan on utilizing these results in the derivation of optimal L2(Ω) × L2(Ω) error
+estimates for the interface coupling method presented in [3]. We will then aim to establish a generalized
+theory for the well-posedness and approximation properties of extension boundary methods. We believe that
+this approach can be generalized across many types of partial differential equations since the work presented
+in this manuscript and in [4] indicates that the averaged Taylor series approximation only generates an O(h)
+perturbation to the discretized variational operator that vanishes as h → 0.
+References
+[1] Robert A Adams and John JF Fournier. Sobolev spaces. Elsevier, 2003.
+[2] Susanne C Brenner, L Ridgway Scott, and L Ridgway Scott. The mathematical theory of finite element methods, volume 3.
+Springer, 2008.
+[3] James Cheung, Max Gunzburger, Pavel Bochev, and Mauro Perego. An optimally convergent higher-order finite element
+coupling method for interface and domain decomposition problems. Results in Applied Mathematics, 6:100094, 2020.
+[4] James Cheung, Mauro Perego, Pavel Bochev, and Max Gunzburger. Optimally accurate higher-order finite element methods
+for polytopial approximations of domains with smooth boundaries. Mathematics of Computation, 88(319):2187–2219, 2019.
+[5] Philippe G Ciarlet. The finite element method for elliptic problems. SIAM, 2002.
+6
+

6NE2T4oBgHgl3EQfOwb7/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf,len=183
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='03753v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='NA]  10 Jan 2023  ENFORCING NEUMANN BOUNDARY CONDITIONS WITH POLYNOMIAL  EXTENSION OPERATORS TO ACHEIVE OPTIMAL CONVERGENCE RATES ON  POLYTOPIAL MESHES IN THE FINITE ELEMENT METHOD  JAMES CHEUNG  Millennium Space Systems, A Boeing Company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' 2265 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' El Segundo Blvd, El Segundo, CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' In [4], the authors presented two finite element methods for approximating second order bound-  ary value problems on polytopial meshes with optimal accuracy without having to utilize curvilinear map-  pings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' This was done by enforcing the boundary conditions through judiciously chosen polynomial extension  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' The H1 error estimates were proven to be optimal for the solutions of both the Dirichlet and  Neumann boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' It was also proven that the Dirichlet problem approximation converges  optimally in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' However, optimality of the Neumann approximation in the L2 norm was left as an open  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' In this work, we seek to close this problem by presenting new analysis that proves optimal error  estimates for the Neumann approximation in the W 1  ∞ and L2 norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Introduction  The purpose of this note is to derive optimal error estimates for the polynomial extension finite element  method (PE-FEM) described in [4] for approximating elliptic boundary value problems with Neumann  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' This manuscript is very much an appendix to our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' As such, we highly suggest  that the reader refer to that work, especially since we will not redefine our notation here for the sake of  brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Our analysis will be structured in the following manner: We present new technical lemmas involving  the Averaged Taylor series in §2, then in §3 we move on to prove well-posedness and derive optimal error  estimates for the numerical solution in the W 1  ∞(Ω) norm, using this new result we are finally able to derive  optimal error estimates for the numerical solution in the L2(Ω) norm in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' We then discuss our results and  propose additional future research in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Additional Results for Averaged Taylor Polynomials  We begin our analysis by deriving some technical results for the Averaged Taylor polynomials in the  W m  ∞(Ωh) setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Let v ∈ L∞(Ωh), then ���T k  h (v)  ��  η(ξ)  ���  L∞(Ωh) ≤ C∥v∥L∞(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Using a scaling argument on [2, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='15], we immediately have that  ���T k  h (v)  ��  η(ξ)  ���  L∞(Ei∩Si,ℓ) ≤ Cδ−d  h ∥v∥L1(σi,ℓ)  ≤ C∥v∥L∞(σi,ℓ)  We conclude by seeing that���T k  h (v)  ��  η(ξ)  ���  L∞(Γh) = max  i,ℓ  ���T k  h (v)  ��  η(ξ)  ���  L∞(Ei∩Si,ℓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Let v ∈ W k+1  ∞  (Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Then for an integer m < k + 1, the following is satisfied  ���T k  h (v)  ��  η(ξ) − v  ���  W m  ∞(Γh) ≤ Cδk+1−m  h  |v|W k+1  ∞  (Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Using [2, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='2], we have that  ���T k  h (v)  ��  η(ξ) − v  ���  W m  ∞(Ei∩Si,ℓ) =  ��T k  h (v) − v  ��  W m  ∞(η(Ei)∩Si,ℓ)  ≤  ��T k  h (v) − v  ��  W m  ∞(Si,ℓ)  ≤ Cδk+1−m  h  |v|W k+1  ∞  (Si,ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Taking the maximum over all i, ℓ ∈ N concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Let v ∈ V  k  h, then  ����T k′,k  h  (v)  ���  η(ξ)  ����  L∞(Γh)  ≤ C  k  �  α|=1  h−|α|δ|α|  h ∥v∥L∞(Ωh)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' From the definition of T k′,k  h  (v), we see directly that  ����T k′,k  h  (v)  ���  η(ξ)  ����  L∞(Γh)  ≤  k  �  |α|=1  δ|α|  h  |α|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' ∥Dαv∥L∞(Γh)  ≤  k  �  |α|=1  δ|α|  h  |α|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' ∥Dαv∥L∞(Ωh)  ≤  k  �  |α|=1  δ|α|  h h−|α|  |α|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  ∥v∥L∞(Ωh) ,  after applying the inverse inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  □  With these technical lemmas derived, we are now ready to prove that the solution of the Neumann  approximation presented in [4] is well-posed and optimal in W 1  ∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Well-Posedness and Error Estimates in W 1  ∞(Ωh)  In this section, we determine that the solution uh ∈ V k  h is bounded in W 1  ∞(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Additionally, we  demonstrate that the error estimate is optimal in the same norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' The analysis presented here is remarkably  standard since the perturbations incurred by the extensions in the discrete bilinear form are not large enough  to cause stability loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Additionally, Strang’s Lemma arguments are also used to handle the error incurred  by the nonconforming perturbations in the bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Assume that δh ∼ O(h2) and that h ∈ R+ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Further assume that �p, �q ∈  Ck+1(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Then there exist positive constants γ, M ∈ R+ such that  (1)  sup  vh∈V k  h ∩W 1  1 (Ωh)  Bh,N(wh, vh)  ∥vh∥W 1  1 (Ωh)  ≥ γ∥wh∥W 1  ∞(Ωh)  and  (2)  Bh,N(wh, vh) ≤ M∥wh∥W 1  ∞(Ωh)∥vh∥W 1  1 (Ωh)  for all uh, vh ∈ V k  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' We begin by recalling that  Bh,N(wh, vh) =  �  Ωh  (�p∇wh · ∇vh + ˜qwhvh) dx +  �  �p ◦ η(ξ) Tk−1  h  (∇wh)  ��  η(ξ) · n − �p(ξ)∇wh · nh, vh  �  Γh  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  We then see that  Bh,N(wh, vh) ≥  �  Ωh  (�p∇wh · ∇vh + �qwhvh) dx − Ch|u|W 1  ∞(Ωh)∥vh∥W 1  1 (Ωh),  2  after applying Lemma 3 and the trace inequality [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Dividing both sides with ∥vh∥W 1  1 (Ωh)  allows us to see that  sup  vh∈V k  h ∩W 1  1 (Ωh)  Bh,N(wh, vh)  ∥vh∥W 1  1 (Ωh)  ≥  sup  vh∈V k  h ∩W 1  1 (Ωh)  �  Ωh (�p∇wh · ∇vh + �qwhvh) dx  ∥vh∥W 1  1 (Ωh)  − Ch|u|W 1  ∞(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Now, choosing vh = wh allows us to see that supvh∈V k  h ∩W 1  1 (Ωh)  �  Ωh(�p∇wh·∇vh+�qwhvh)dx  ∥vh∥W 1  1 (Ωh)  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' As such, we can  choose a constant Cp,q > 0 such that  sup  vh∈V k  h ∩W 1  1 (Ωh)  �  Ωh (�p∇wh · ∇vh + �qwhvh) dx  ∥vh∥W 1  1 (Ωh)  ≥ Cp,q  sup  vh∈V k  h ∩W 1  1 (Ωh)  �  Ωh (∇wh · ∇vh + whvh) dx  ∥vh∥W 1  1 (Ωh)  = Cp,q∥wh∥W 1  ∞(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Therefore,  sup  vh∈V k  h ∩W 1  1 (Ωh)  Bh,N(wh, vh)  ∥vh∥W 1  1 (Ωh)  ≥ Cp,q∥wh∥W 1  ∞Ωh − Ch∥uh∥W 1  ∞(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  And thus, we have that (1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Using Lemma 1, H¨older’s inequality, and the trace inequality allows us to derive (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  □  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Let uh ∈ V k  h satisfy [4, Equation 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Assume that u ∈ W k+1  ∞  (Ω) and that f ∈ W k−1  ∞  (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Furthermore, let �u ∈ W k+1  ∞  (Ωh) and �f, �f ∈ W k−1  ∞  (Ωh) be extensions of u and f respectively from Ω to Ωh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  We then have that  ∥�u − uh∥W 1  ∞(Ωh) ≤ Chk �  |u|W k+1  ∞  (Ω) + ∥f∥W k−1  ∞  (Ω)  �  under the conditions specified in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Let uI ∈ V k  h be the piecewise polynomial interpolant of �u ∈ W k+1  ∞  (Ωh) defined on Ωh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' From [2,  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='20] and the Stein extension theorem [1], we have that  (3)  ∥�u − uI∥W 1  ∞(Ωh) ≤ Chk|u|W k+1  ∞  (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  We begin the analysis by seeing that  ∥uI − uh∥W 1  ∞(Ωh) ≤  sup  vh∈V k  h ∩W 1  1 (Ωh)  Bh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='N(uI − uh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh)  ∥vh∥W 1  1 (Ωh)  =  sup  vh∈V k  h ∩W 1  1 (Ωh)  Bh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='N(uI − �u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh) + Bh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='N(�u − uh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh)  ∥vh∥W 1  1 (Ωh)  ≤ M∥�u − uh∥W 1  ∞(Ωh) +  sup  vh∈V k  h ∩W 1  1 (Ωh)  �  �f − �f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh  �  Ωh  −  �  �p ◦ η(ξ)Rk−1 (∇�u)|η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh  �  Γh  ∥vh∥W 1  1 (Ωh)  ≤ Chk|u|W k+1  ∞  (Ω) + Cδk−1  h  |f|W k−1  ∞  (Ω) + Cδk−1  h  |u|W k+1  ∞  (Ω)  ≤ Chk|u|W k+1  ∞  (Ω)  after utilizing (3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Lemma 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' and seeing that  Bh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='N(�u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh) =  �  �f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh  �  Ωh  +  �  gN ◦ η(ξ) − �p ◦ η(ξ)Rk−1 (∇u)|η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh  �  Γh  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  The proof is completed by seeing that  ∥�u − uh∥W 1  ∞(Ωh) ≤ ∥u − uI∥W 1  ∞(Ωh) + ∥uI − uh∥W 1  ∞(Ωh),  and applying the above bound along with (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  □  3  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Error Estimates in L2(Ωh)  We are now ready to derive the optimal error estimates for the solution of the Neumann approximation  in the L2(Ω) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' The analysis begins by estimating the nonconformity error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' This nonconformity error  will then be used in the following duality argument to bound the terms in the discrete problem that are not  orthogonal in the Galerkin sense with respect to the continuous bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Nonconformity Error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Let us begin the derivation of the L2(Ωh) error bound by analyzing the  nonconformity induced by Bh,N(·, ·) := H1(Ωh) × V k  h → R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Assume that all the conditions in Theorem 2 hold, then the following is satisfied  Nh(�u − uh, vh) ≤ Chk+1 �  |u|W k+1  ∞  (Ω) + |f|W k−1  ∞  (Ω)  �  ∥vh∥1,Ωh  for all V k  h ∩ W 1  1 (Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Notice that  Bh,N(�u, vh) =  �  �f, vh  �  Ωh  +  �  gN ◦ η(ξ) − �p ◦ η(ξ) Rk−1  h  (∇�u)  ��  η(ξ) · n, vh  �  Γh  ∀v ∈ H1(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Taking the difference with [4, Equation (24)] yields  Bh,N(�u − uh, vh) =  �  �f − �f, vh  �  Ωh  −  �  �p ◦ η(ξ) Rk−1  h  (∇�u)  ��  η(ξ) · n, vh  �  Γh  ∀vh ∈ V k  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Let us define eh := �u − uh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' then from the definition of Bh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='N(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' ·) (See [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Equation (25)]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' we have that  (4)  Nh(eh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh) =  �  �f − �f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh  �  Ωh  −  �  �p ◦ η(ξ) Rk−1  h  (∇�u)  ��  η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh  �  Γh  −  �  �p ◦ η(ξ) Tk−1  h  (∇eh)  ��  η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh  �  Γh  + ⟨�p(ξ)∇eh · nh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh⟩Γh  =  �  �f − �f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh  �  Ωh  −  �  �p ◦ η(ξ) Rk−1  h  (∇�u)  ��  η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh  �  Γh  + ⟨�p ◦ η(ξ)∇eh · (n − nh),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh⟩Γh +  �  �p(ξ) T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='k−1  h  (∇eh)  ���  η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh  �  Γh  +  �  (�p(ξ) − �p ◦ η(ξ)) Tk−1  h  (∇eh)  ��  η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh  �  Γh  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  We will now analyze each of the terms on the right hand side of (4) seperately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Let Ωh  diff := Ωh \\ (Ω ∩ Ωh), then we have by the definition of the extension that  �  �f − �f, vh  �  Ωh  =  �  �f − �f, vh  �  Ωh  diff  ≤  �  max  i,ℓ ∥ �f − T k−1  h  f∥L∞(Si,ℓ) + max  i,ℓ ∥ �f − T k−1  h  f∥L∞(Si,ℓ)  � �  Ωh  diff  1 · |v|dx  ≤ Cδk−1  h  |Ωh  diff|  1  2 |f|W k−1  ∞  (Ωh)∥vh∥0,Ωh  ≤ Cδ  k− 1  2  h  |f|W k−1  ∞  (Ωh)∥vh∥1,Ωh,  after applying [2, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='8], H¨older’s inequality, and seeing that |Ωh  diff| ∼ O(δh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Since we have assumed  that δh ∼ O(h2), we have that  (5)  �  �f − �f, vh  �  Ωh  ≤ Ch2k−1|f|W k−1  ∞  (Ωh)∥vh∥1,Ωh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Next, using Lemma 2, we have that  (6)  �  �p ◦ η(ξ) Rk−1  h  (∇�u)  ��  η(ξ) · n, vh  �  Γh  ≤ p  ���Rk−1  h  ∇�u  ��  η(ξ)  ���  L∞(Γh) ∥vh∥L1(Γh)  ≤ Cδk  h|∇�u|W k  ∞(Ω)∥vh∥0,Γh  ≤ Ch2k|u|W k+1  ∞  (Ω)∥vh∥1,Ωh,  4  after applying the trace theorem and the assumption that δh ∼ O(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Additionally, we have that  (7)  ⟨�p ◦ η(ξ)∇eh · (n − nh), vh⟩Γh ≤ p  �  max  ξ∈Γh ∥n ◦ η(ξ) − nh(ξ)∥Rd  �  ∥∇eh∥L∞(Γh)∥vh∥L1(Γh)  ≤ Ch∥eh∥W 1  ∞(Ωh)∥vh∥1,Ωh  ≤ Chk+1 �  |u|W k+1  ∞  (Ω) + |f|W k−1  ∞  (Ω)  �  ∥vh∥1,Ωh,  after applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' we have that  (8)  �  �p(ξ) T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='k−1  h  (∇eh)  ���  η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' vh  �  Γh  ≤ Cp∥vh∥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='1Ωh  k−1  �  |α|=1  δ|α|  h ∥Dα∇eh∥L∞(Γh)  ≤ Cp∥vh∥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='1Ωh  k−1  �  |α|=1  δ|α|  h  �  ∥Dα∇(�u − uI)∥L∞(Ωh) + ∥Dα∇(uI − uh)∥L∞(Ωh)  �  ≤ Cp∥vh∥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='1Ωh  k−1  �  |α|=1  δ|α|  h  �  hk−|α||u|W k+1  ∞  (Ω) + Ch−|α| �  ∥∇(eh)∥L∞(Ωh) − ∥∇(�u − uI)∥L∞(Ωh)  ��  ≤ Cp∥vh∥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='1Ωh  k−1  �  |α|=1  δ|α|  h hk−|α| �  |u|W k+1  ∞  (Ω) + |f|W k−1  ∞  (Ω)  �  ≤ Chk+1 �  |u|W k+1  ∞  (Ω) + |f|W k−1  ∞  (Ω)  �  ∥vh∥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='Ωh  after applying Theorem 2 and the interpolation estimate [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  We finally move on to the last term, where we see that  (9)  �  (�p(ξ) − �p ◦ η(ξ)) Tk−1  h  (∇eh)  ��  η(ξ) · n, vh  �  Γh  ≤ Cδh  ���Tk−1  h  (∇eh)  ��  η(ξ)  ���  L∞(Γh) ∥vh∥L1(Γh)  ≤ Cδh∥∇eh∥L∞(Ωh)∥vh∥1,Ωh  ≤ Chk+2 �  |u|W k+1  ∞  (Ω) + |f|W k−1  ∞  (Ω)  �  ∥vh∥1,Ωh,  where we used Lemma 1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Inserting (5), (6), (7), (8), and (9) into (4) gives us  Nh(�u − uh, vh) ≤ Chk+1 �  |u|W k+1  ∞  (Ω) + |f|W k−1  ∞  (Ω)  �  ∥vh∥1,Ωh  ∀vh ∈ V k  h  This concludes this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  □  Now that we have established that the nonconformity in the bilinear form is bounded above by O(hk+1),  we are now ready to analyze the L2(Ω) error of the numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' The Dual Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' The dual problem we are interested in utilizing is to seek a ψ ∈ H1(Ωh) such  that  Nh(v, ψ) = ⟨�u − uh, v⟩Ωh  ∀v ∈ H1(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  The corresponding finite element approximation to the dual problem is to seek a ψh ∈ V k  h such that  Nh(v, ψh) = ⟨�u − uh, v⟩Ωh  ∀vh ∈ V k  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  In general, φ ∈ H1(Ωh) does not full H2(Ωh) regularity due to the presence of interior angles in Γh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' From  the literature, [2, 5] it is known that  (10)  ∥ψ∥1+s,Ωh ≤ C∥u − uh∥0,Ωh  5  and  (11)  ∥φ∥1,Ωh ≤ C∥u − uh∥−1,Ωh ≤ C∥u − uh∥0,Ωh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Furthermore, we have that  (12)  ∥ψ − ψh∥1,Ωh ≤ Chs|φ|1+s,Ωh,  where s ∈  � 1  2, 1  �  depends on the magnitude of the largest interior angle of Γh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' See [4, Remark 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Analysis of L2(Ωh) Convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' With the previous results proven, we are now ready to derive op-  timal L2(Ωh) error estimates for the PE-FEM approximation of elliptic Neumann boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Assume that all the conditions in Theorem 2 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Then we have that  ∥�u − uh∥0,Ωh ≤ Chk+s �  |u|W k+1  ∞  (Ω) + |f|W k−1  ∞  Ω  �  ,  where s ∈  � 1  2, 1  �  depends on the largest interior angle of Γh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' If Γh is convex, then s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' From the definition of the dual problem, we have that  ∥�u − uh∥2  0,Ωh = Nh(�u − uh, φ)  = Nh(�u − uh, φ − φh) + Nh(�u − uh, φh)  ≤ C∥�u − uh∥1,Ωh∥φ − φh∥1,Ωh + Nh(�u − uh, φh)  ([4, Theorem 2])  ≤ Chk+s (|u|k+1,Ωh + |f|k−1,Ωh) |φ|2,Ωh + Nh(�u − uh, φh)  (12) and [4, Theorem 5]  ≤ Chk+s (|u|k+1,Ωh + |f|k−1,Ωh) |φ|2,Ωh  + Chk+1 �  |u|W k+1  ∞  (Ω) + |f|W k−1  ∞  Ω  �  ∥φh∥1,Ωh  Lemma 4  ≤ Chk+s �  |u|W k+1  ∞  (Ω) + |f|W k−1  ∞  Ω  �  ∥�u − uh∥0,Ωh  (10) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  This concludes this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Conclusion  In this manuscript, we have presented new results that indicate that the solution of the Neumann ap-  proximation presented in [4] is optimal in W 1  ∞(Ω) and L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Our analysis implies hat the solution to the  continuous Neumann problem must be pointwise bounded a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' in its (k + 1)-th derivative in order for us to  achieve additional accuracy in the L2(Ω) norm, otherwise the L2(Ω) error is bounded by the O(hk) H1(Ω)  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  In our future work, we plan on utilizing these results in the derivation of optimal L2(Ω) × L2(Ω) error  estimates for the interface coupling method presented in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' We will then aim to establish a generalized  theory for the well-posedness and approximation properties of extension boundary methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' We believe that  this approach can be generalized across many types of partial differential equations since the work presented  in this manuscript and in [4] indicates that the averaged Taylor series approximation only generates an O(h)  perturbation to the discretized variational operator that vanishes as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  References  [1] Robert A Adams and John JF Fournier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Elsevier, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  [2] Susanne C Brenner, L Ridgway Scott, and L Ridgway Scott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' The mathematical theory of finite element methods, volume 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  Springer, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  [3] James Cheung, Max Gunzburger, Pavel Bochev, and Mauro Perego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' An optimally convergent higher-order finite element  coupling method for interface and domain decomposition problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Results in Applied Mathematics, 6:100094, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  [4] James Cheung, Mauro Perego, Pavel Bochev, and Max Gunzburger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Optimally accurate higher-order finite element methods  for polytopial approximations of domains with smooth boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' Mathematics of Computation, 88(319):2187–2219, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  [5] Philippe G Ciarlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' The finite element method for elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content=' SIAM, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}
+page_content='  6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'}

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@@ -0,0 +1,1816 @@
+Draft version January 4, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+Deep Synoptic Array science: a 50 Mpc fast radio burst constrains the mass of the Milky Way
+circumgalactic medium
+Vikram Ravi,1, 2 Morgan Catha,2 Ge Chen,1 Liam Connor,1 James M. Cordes,3, 4 Jakob T. Faber,1
+James W. Lamb,2 Gregg Hallinan,1, 2 Charlie Harnach,2 Greg Hellbourg,1, 2 Rick Hobbs,2 David Hodge,1
+Mark Hodges,2 Casey Law,1, 2 Paul Rasmussen,2 Kritti Sharma,1 Myles B. Sherman,1 Jun Shi,1 Dana Simard,1
+Jean J. Somalwar,1 Reynier Squillace,1 Sander Weinreb,1 David P. Woody,2 Nitika Yadlapalli,1
+(The Deep Synoptic Array team)
+1Cahill Center for Astronomy and Astrophysics, MC 249-17 California Institute of Technology, Pasadena CA 91125, USA.
+2Owens Valley Radio Observatory, California Institute of Technology, Big Pine CA 93513, USA.
+3Department of Astronomy, Cornell University, Ithaca, NY 14853, USA
+4Cornell Center for Astrophysics and Planetary Science and Department of Astronomy, Cornell University, Ithaca, NY 14853, USA
+ABSTRACT
+We present the Deep Synoptic Array (DSA-110) discovery and interferometric localization of the so
+far non-repeating FRB 20220319D. The FRB originates in a young, rapidly star-forming barred spiral
+galaxy, IRAS 02044+7048, at just 50 Mpc. Although the NE2001 and YMW16 models for the Galactic
+interstellar-medium (ISM) contribution to the DM of FRB 20220319D exceed its total observed DM,
+we show that uncertainties in these models accommodate an extragalactic origin for the burst. We
+derive a conservative upper limit on the DM contributed by the circumgalactic medium (CGM) of
+the Milky Way: the limit is either 28.7 pc cm−3 and 47.3 pc cm−3, depending on which of two pulsars
+nearby on the sky to FRB 20220319D is used to estimate the ISM DM. These limits both imply that
+the total Galactic CGM mass is < 1011M⊙, and that the baryonic mass of the Milky Way is ≲ 60% of
+the cosmological average given the total halo mass. More stringent albeit less conservative constraints
+are possible when the DMs of pulsars in the distant globular cluster M53 are additionally considered.
+Although our constraints are sensitive to possible anisotropy in the CGM and to the assumed form of
+the radial-density profile, they are not subject to uncertainties in the chemical and thermal properties of
+the CGM. Our results strongly support scenarios commonly predicted by galaxy-formation simulations
+wherein feedback processes expel baryonic matter from the halos of galaxies like the Milky Way.
+Keywords: Barred spiral galaxies — circumgalactic medium — radio interferometers — radio transient
+sources — star formation — warm ionized medium
+1. INTRODUCTION
+Galaxies like the Milky Way are embedded in a multi-
+phase (∼ 104 − 107 K), highly ionized (hydrogen neu-
+tral fractions ≪ 0.01%), kinematically complex, spa-
+tially clumpy and anisotropic circumgalactic medium
+(CGM; Tumlinson et al. 2017). Likely extending beyond
+dark-matter halo virial radii rvir, the CGM may repre-
+sent the dominant baryon component by mass within
+Corresponding author: Vikram Ravi
+vikram@caltech.edu
+halos.
+The physical properties of the CGM of exter-
+nal galaxies, including density, temperature, kinematic
+structure, and chemical composition have long been
+probed by absorption-line measurements towards back-
+ground objects.
+More recently, detections of thermal
+bremsstrahlung (e.g., Li et al. 2018) and the Sunyaev-
+Zeldovich effect around nearby galaxies (e.g., Bregman
+et al. 2022) provide independent constraints on the
+CGM density and temperature structure, potentially
+confirming the presence of extended coronae around
+galaxies.
+Evidence for extended gas reservoirs associ-
+ated with nearby galaxies, either in the CGM or the
+intra-group medium, has also been observed in the dis-
+arXiv:2301.01000v1  [astro-ph.GA]  3 Jan 2023
+
+2
+Ravi et al.
+persion measures (DMs) of background fast radio bursts
+(FRBs; Connor & Ravi 2022; Wu & McQuinn 2022).
+Galaxy-formation simulations highlight the dependence
+of CGM properties on feedback from stellar winds, su-
+pernovae, and AGN, as well as on the galaxy merger his-
+tories (e.g., Wijers et al. 2020; Hafen et al. 2020; Zheng
+et al. 2020; Fielding et al. 2020; Appleby et al. 2021;
+Ramesh et al. 2022). The simulations predict varying
+degrees of anisotropy in the CGM of individual galax-
+ies, and different total CGM masses for galaxies with
+comparable global characteristics. A ubiquitous predic-
+tion of simulations is that feedback processes determine
+the integrated CGM mass, MCGM.
+In general, CGM
+baryon fractions fCGM = MCGMΩM
+MtotΩb
+≲ 0.5 are predicted,
+where Mtot is the total (dark matter and baryonic) halo
+mass.
+A halo of ∼ 106 K gas at densities ≲ 10−3 cm−3 sur-
+rounding the Milky Way was proposed by Spitzer (1956)
+as a solution to the problem of confining distant cold-
+gas clouds at high Galactic latitudes.
+Tentative evi-
+dence for such coronal gas had previously been found
+in observations of the diffuse radio-synchrotron back-
+ground (e.g., Baldwin 1956).
+A theoretical basis for
+the origins of such a hot CGM around galaxies was de-
+veloped in the 1970s (Rees & Ostriker 1977; White &
+Rees 1978), with the gas infalling onto dark-matter ha-
+los heated to virial temperatures of ≳ 106 K, possibly
+in shocks at the halo virial radii (see also Birnboim &
+Dekel 2003). It would, however, be nearly two decades
+before further observational evidence for a hot Milky
+Way CGM was identified using observations of shadows
+in the soft X-ray background from high-latitude (Snow-
+den et al. 1991) and high-velocity (Herbstmeier et al.
+1995) cold clouds, and in the spectral decomposition of
+the background into absorbed and unabsorbed compo-
+nents (Garmire et al. 1992).
+Today the hot CGM of
+the Milky Way is best traced through X-ray observa-
+tions of OVII and OVIII diffuse emission, and absorp-
+tion towards bright background sources. Although high
+Galactic latitude data are well fit by a spheroidal com-
+ponent with a mass of a few ×1010M⊙ (e.g., Bregman
+& Lloyd-Davies 2007; Gupta et al. 2012), more recent
+models include a substantially less massive (∼ 0.1%),
+but denser disk-like component to explain the line ob-
+servations (Li & Bregman 2017; Nakashima et al. 2018;
+Kaaret et al. 2020; Ueda et al. 2022). Structure in the
+hot Milky Way CGM is also observed in the form of
+the Fermi (Su et al. 2010), Parkes (Carretti et al. 2013)
+and eROSITA (Predehl et al. 2020) bubbles. The neu-
+tral and warm (≲ 105 K) components of the Milky Way
+CGM, as primarily traced through observations of high-
+velocity clouds, are likely sub-dominant in mass to the
+hot component (few ×108M⊙; Putman et al. 2012), but
+highlight the complexities of the disk-halo interface (e.g.,
+Koo et al. 1992; Werk et al. 2019; Clark et al. 2022).
+The mass of the Galactic CGM is uncertain. Here we
+assume Mtot = 1.5 × 1012M⊙ (following Prochaska &
+Zheng 2019) and a total Galactic mass in stars and cold
+gas of 6 × 1010M⊙ (Draine 2011). Setting fCGM = 1
+and Ωb/ΩM = 0.188 (Planck Collaboration et al. 2020)
+implies MCGM = 2.82 × 1011M⊙. Recent models for the
+distribution of OVII and OVIII emission and absorp-
+tion result in MCGM values between 3 − 4 × 1010M⊙
+(Li & Bregman 2017), 5.5 − 8.6 × 1010M⊙ (Kaaret et
+al. 2020), and ∼ 1.2 × 1011M⊙ (Yamasaki & Totani
+2020). Faerman et al. (2017) argue that a portion of
+the hot CGM is also traced by OVI absorption, as is
+observed in external galaxies (Tumlinson et al. 2011),
+and find MCGM = 1.2 × 1011M⊙.
+Systematic effects
+that make the estimation of MCGM difficult include the
+model uncertainties in foregrounds and in the spatial
+distribution and clumpiness of the gas, and the poorly
+constrained gas metallicity. An alternate measurement
+of MCGM was found by Salem et al. (2015) by model-
+ing the ram-pressure stripping of the Large Magellanic
+Cloud (LMC); a low value of (2.7 ± 1.4) × 1010M⊙ was
+found.
+The DMs of FRBs and pulsars within the Milky Way
+halo offer a unique probe of the content of the Galactic
+CGM (e.g., Anderson & Bregman 2010; Platts et al.
+2020). FRBs are transient radio emissions from distant
+extragalactic sources that, like the emissions from radio
+pulsars, are so short in duration that the dispersion in
+intervening plasma is evident as an arrival-time delay at
+lower frequencies:
+∆t(ν) =
+q2
+e
+2πmecν2
+� D
+0
+ne(l)dl,
+(1)
+where qe is the electron charge, me is the electron mass,
+ν is the frequency, D is the source distance, and ne(l) is
+the electron number density along the sightline. Under
+physical conditions of CGM gas, DMs quantify the col-
+umn densities along the source sightlines. In all cases,
+however, the observed DMs of FRBs, DMobs, are com-
+posed of a selection of additive components:
+DMobs = DMISM + DMCGM + DMIGM + DMhost, (2)
+where in this formulation DMISM includes the warm ion-
+ized medium (WIM) in the Milky Way disk (e.g., Cordes
+& Lazio 2002; Yao et al. 2017), DMCGM is the DM con-
+tributed by the Galactic CGM, DMhost is the DM as-
+sociated with the FRB host-galaxy interstellar medium
+(ISM) and their halos, and DMIGM includes contribu-
+tions from the intergalactic medium (IGM) and any in-
+tervening galaxy systems. For pulsars within the Milky
+
+Mass of the Milky Way CGM
+3
+Way halo, such as in the Magellanic Clouds and in sev-
+eral globular clusters, DMIGM = 0 pc cm−3. These ex-
+pressions apply to the redshift ∼ 0 regime relevant to
+this paper. If the components of DMobs can be accu-
+rately modeled or bounded, constraints can be placed
+on DMCGM.
+Modeling of the DMs of pulsars in the
+LMC led Anderson & Bregman (2010) to find a very low
+MCGM ∼ 1.5×1010M⊙ assuming a Navarro et al. (1997)
+profile (an NFW profile); it is likely that an overly large
+DMISM was assumed, and the NFW profile is no longer
+favored for the Galactic CGM. Although most FRBs are
+too distant to allow for accurate modeling of the compo-
+nents of DMobs, FRBs from nearby galaxies (Bhardwaj
+et al. 2021; Kirsten et al. 2022) have begun to be used to
+place constraints on DMCGM, assuming specific models
+for DMISM, that are in tension with some models for the
+Milky Way CGM.
+In this paper we present observations with the
+Deep
+Synoptic
+Array
+(DSA-110)
+of
+a
+new
+FRB
+(FRB 20220319D) that motivates stringent new con-
+straints on the mass of the Milky Way CGM. Remark-
+ably, FRB 20220319D was observed to have DMobs <
+DMISM for leading models of the warm-gas distribution
+in the Galactic disk, despite the unambiguous associ-
+ation we find with a galaxy at a distance of 50 Mpc.
+We detail observations of the FRB in §2.
+We con-
+sider the association with its host galaxy in §3, wherein
+we show that the DM of FRB 20220319D is consis-
+tent with an extragalactic origin given uncertainties in
+models for DMISM, using observations of the DMs of
+pulsars in globular clusters with accurate parallax dis-
+tances.
+We present an analysis of the host environ-
+ment of FRB 20220319D in §4. We then place conser-
+vative constraints on DMCGM, and hence MCGM, using
+FRB 20220319D and the distant high-latitude globular
+cluster M53, in §5. We discuss the implications of our
+results in §6, and conclude in §7.
+2. DSA-110 OBSERVATION OF FRB 20220319D
+The DSA-1101 is a radio interferometer hosted at
+the Owens Valley Radio Observatory (OVRO), pur-
+pose built for the discovery and arcsecond-localization
+of FRBs. A full description of the instrument will be
+presented in Ravi et al. (in prep.). During the observa-
+tions discussed herein, the DSA-110 was configured as
+described in Ravi et al. (2022).
+Of particular note is
+that when FRB 20220319D was observed, early in DSA-
+110 commissioning, all candidates at DMs in excess of
+1 https://deepsynoptic.org
+100
+0
+100
+degrees
+Intrinsic PPA
+0
+10
+20
+30
+S/N
+Intensity (I)
+Linear Polarization (L)
+Circular Polarization (V)
+-0.8
+-0.5
+-0.2
+0.0
+0.2
+0.5
+0.8
+Time (ms)
+1325
+1350
+1375
+1400
+1425
+1450
+1475
+Frequency (MHz)
+Figure 1. Dedispersed temporal profile and dynamic spec-
+trum of FRB 20220319D. A value of DMobs = 110.95 pc cm−3
+was used. The top panel shows the estimated absolute po-
+sition angle based on an approximate parallactic angle, the
+∼ 20◦ error due to RM uncertainty is not included in the
+error bars. The middle panel shows temporal profiles in to-
+tal intensity, linearly polarized intensity, and Stokes V, as la-
+beled. The bottom panel shows the dynamic spectrum of the
+total-intensity data. The time resolution in the time-series
+and dynamic-spectrum plots is 32.768 µs, and the temporal
+profile is in units of signal to noise ratio. The reference time
+is MJD 59657.93275696795, which was the burst arrival time
+at OVRO at 1530 MHz. The polarized data on the burst was
+corrected for the measured RM of 50 rad m−2.
+80 pc cm−3 were saved for further inspection, regardless
+of the expected DM through the Galactic disk.
+FRB 20220319D was detected during standard com-
+missioning observations, with an arrival time at OVRO
+at 1530 MHz of MJD 59657.93275696795. During these
+observations, the DSA-110 was parked at a pointing-
+center declination of +71.6◦.
+The DSA-110 refer-
+ence position is −118.283◦ longitude, +37.2334◦ lat-
+itude.
+FRB 20220319D was detected with a DM of
+110.96 pc cm−3, and a signal to noise ratio (S/N) of 41.7.
+To derive optimized burst parameters and polarization
+properties, we coherently combined voltage data from
+
+4
+Ravi et al.
+the 48 core antennas towards the detection-beam direc-
+tion. The resulting temporal profile and calibrated dy-
+namic spectrum, produced with incoherent dedispersion
+at the native time and frequency resolutions, are shown
+in Figure 1, and optimized parameters are given in Ta-
+ble 1. The polarization analysis was done following pro-
+cedures described in Ravi et al. (2022). The matched-
+filter total-intensity S/N estimate is 79.
+The burst exhibits a narrow temporal profile with
+a strongly modulated spectrum.
+The DM smearing
+timescale ranges between 8–13 µs within the DSA-110
+band, and so the data resolve the temporal structure of
+the burst. Scaling the system-equivalent flux density of
+the DSA-110 by the primary-beam attenuation at the
+burst position, we derive a fluence of 11.2 ± 0.2 Jy ms.
+Following techniques described in Connor et al. (2020),
+we derive a spectral decorrelation bandwidth, νd =
+3.6 ± 0.2 MHz, defined as the 1/e scale of the autocor-
+relation function of the spectrum. The burst spectro-
+temporal morphology is consistent with the population
+of so far non-repeating FRBs (Pleunis et al. 2021). We
+derive a low fractional polarization of just 16% (Fig-
+ure 1), with no significant circular polarization. A Fara-
+day rotation measure (RM) of 50 ± 15 rad m−2 was de-
+tected. The error was estimated by simulating the re-
+covery of RMs with the same linear-polarization S/N as
+FRB 20220319D (Sherman et al., in prep.).
+The interferometric localization of FRB 20220319D
+was derived following procedures largely described in
+Ravi et al. (2022). Correlation products for all baselines
+formed from the 63 functioning antennas of the DSA-110
+at the time of detection were integrated over 262.144µs
+centered on the burst arrival time. Bandpass calibra-
+tion was done using a 10-min observation of 3C309.1 ob-
+served 11 hr prior to the burst detection. Complex gain
+calibration at the time of the burst detection was derived
+using an NRAO VLA Sky Survey (NVSS; Condon et al.
+1998) model and 5 min of visibility data. We show im-
+ages of the point-spread function (PSF; a flat spectrum
+was assumed), and the pre- and post-deconvolution im-
+ages of FRB 20220319D in Figure 2. Briggs weighting
+with a ‘robust’ parameter of 0.5 was used to suppress
+sidelobes, given the high S/N of FRB 20220319D, yield-
+ing a PSF FWHM of 26.9′′ × 15.6′′.
+The position of
+FRB 20220319D, given in Table 1, was derived by fit-
+ting an elliptical Gaussian to the deconvolved image of
+FRB 20220319D.
+We derived the uncertainty in the FRB position us-
+ing data on nine compact bright sources from the Radio
+Fundamental Catalog (RFC; rfc 2022c), obtained within
+2 hr of the burst observation. A fit to a linear trend in
+R.A. and Decl. was used to derive arcsecond-level astro-
+Table 1. Properties of FRB 20220319D.
+Parameter
+Value
+Arrival time (MJD)a
+59657.93275696795
+DM (pc cm−3)
+110.95(1)
+Full-width half-maximum (ms)
+0.16(1)
+Fluence (Jy ms)
+11.2(2)
+Spectral energy (erg Hz−1)
+3.3(1)×1028
+L/Ib
+0.16(3)
+|V/I|b
+0.04(3)
+RM (rad m−2)
+50(15)
+νd (MHz)c
+3.6(2)
+R.A. (J2000)
+02:08:42.7(1)
+Decl. (J2000)
++71:02:06.9(6)
+aArrival time at OVRO at 1530 MHz.
+b L/I is the fraction of linearly polarized fluence, and
+|V/I| is the absolute value of the fraction of circularly
+polarized fluence.
+c νd is the characteristic spectral decorrelation band-
+width.
+metric corrections, and the corresponding uncertainties.
+These uncertainties (0.52′′ in R.A. and 0.5′′ in Decl.)
+were then added in quadrature to the statistical uncer-
+tainty in the fitted burst position (0.29′′ in R.A. and
+0.17′′ in Decl.) to derive a final 90% confidence contain-
+ment ellipse with semi-axes of 1.25′′ in R.A. and 1.18′′ in
+Decl. This ellipse is shown in the bottom-left panel of
+Figure 2, together with the detrended position offsets of
+the RFC sources. We also assessed the quality of the
+in-field calibration by checking the positions of bright
+(> 50 mJy) compact (< 20′′) NVSS sources within the
+primary-beam FWHM against NVSS catalog positions.
+These offsets (with the RFC corrections applied) are also
+shown in Figure 2. A more detailed analysis of the DSA-
+110 localization accuracy will be presented in Ravi et al.
+(in prep.).
+3. ASSOCIATION WITH IRAS 02044+7048, AND
+UNCERTAINTIES IN DMISM
+The
+90%
+confidence
+localization
+region
+of
+FRB 20220319D is shown in Figure 3 overlaid on a
+Pan-STARRS1 (PS1) i-band image.
+The galaxy co-
+incident with the FRB location is cataloged in the
+NASA Extragalactic Database as IRAS 02044+7048,
+with a spectroscopic redshift of 0.011 indicating a dis-
+tance of approximately 50 Mpc.
+Using observations
+described below, we derive a spectroscopic redshift of
+
+Mass of the Milky Way CGM
+5
+100
+50
+0
+50
+100
+100
+50
+0
+50
+100
+PSF
+100
+50
+0
+50
+100
+100
+50
+0
+50
+100
+FRB 20220319 - dirty
+100
+50
+0
+50
+100
+100
+50
+0
+50
+100
+FRB 20220319 - clean
+2
+0
+2
+2
+1
+0
+1
+2
+Offsets
+NVSS offsets
+VLBI sources
+Right Ascension offset (arcsec)
+Declination offset (arcsec)
+Figure 2. DSA-110 localization of FRB 20220319D. Top left: Point-spread function (PSF) of the DSA-110 for the observation
+of FRB 20220319D, assuming a flat-spectrum source. Top right: Dirty image of FRB 20220319D, with no deconvolution applied.
+Bottom left: Deconvolved image of FRB 20220319D. The synthesized beam is represented by an ellipse with half-power diameters
+of 26.9′′×15.6′′, at a position angle of 80◦. Briggs weighting was used with a ‘robust’ parameter of 0.5, in order to partially
+suppress PSF sidelobes. Baselines shorter than 200λ were excluded from the analysis because they were affected by spurious
+cross-coupling. Contours in the preceding three panels are at −0.4, −0.2 (dashed), 0.2, 0.4, 0.6, 0.8 and 0.9 (solid) of the peak
+intensity. The images are centered on the coordinates (R.A. J2000, decl. J2000) = (02:08:42.7, +71:02:06.9). Bottom right:
+Offsets of known astronomical sources from their true positions as measured by the DSA-110. The blue disks show offsets
+of sources from cataloged positions in the NRAO VLA Sky Survey (NVSS; Condon et al. 1998) in a 5 min DSA-110 image
+formed from data taken at a time centered on the burst detection. Only sources with cataloged flux densities > 50 mJy and
+measured major-axis diameters < 20′′, within 2 deg of the pointing center, were considered. The symbol size is proportional to
+flux density. The red crosses show measured offsets of nine RFC calibrator sources (see text for details) observed immediately
+preceding and after the burst. The ellipse indicates the 90% confidence containment region for FRB 20220319D, derived from
+the RFC calibrators.
+
+6
+Ravi et al.
+2h08m45s
+42s
+39s
+36s
+71°02'30"
+15"
+00"
+01'45"
+Right Ascension (J2000)
+Declination (J2000)
+Figure 3. Pan-STARRS i-band image of IRAS 02044+7048,
+with the 3.7 × 2.2 arcsec 90% confidence localization region
+of FRB 20220319D indicated as a white ellipse.
+0.0111 ± 0.0004, indicating a luminosity distance of
+49.6 Mpc. An isophotal fit to the PS1 i-band image in-
+dicates an effective radius of 2.7 ± 0.2 kpc, and the FRB
+is offset by 2.3 ± 0.3 kpc. The isophotal analysis also
+indicates an inclination of 23 ± 3 deg. A barred spiral
+morphology, with a classification of SBa, is evident from
+the image.
+Under
+the
+assumption
+that
+FRB 20220319D
+is
+extragalactic,
+the
+spatial
+association
+between
+FRB 20220319D and IRAS 02044+7048 is secure. It is
+unlikely that FRB 20220319D lies significantly beyond
+IRAS 02044+7048; the morphology of the galaxy indi-
+cates the presence of a warm-ISM phase, and scattering
+in this ISM would likely cause temporal broadening
+of several hundred milliseconds at our observing fre-
+quencies (Cordes et al. 2022; Ocker et al. 2022). The
+Gravitational Wave Galaxy Catalog (GWGC; White
+et al. 2011) lists 15869 galaxies at distances less than
+50 Mpc, subtending a total of 4 deg2. Thus the chance-
+association probability of FRB 20220319D and a galaxy
+at < 50 Mpc is ≲ 10−4.
+The extragalactic nature of FRB 20220319D is called
+into question by its DM of 110.95 pc cm−3.
+The pre-
+dicted DM for extragalactic objects along its l = 129.2◦,
+b = +9.1◦ sightline is 132.9 pc cm−3 according to the
+NE2001 model for the Galactic ionized ISM distribu-
+tion (Cordes & Lazio 2002), and 187.7 pc cm−3 accord-
+ing to the YMW16 model (Yao et al. 2017). The RM
+detection for FRB 20220319D of 50±15 rad m−2 is, how-
+ever, marginally consistent with being in excess of the
+expected Milky Way contribution along the burst sight-
+line. A recent model for the Galactic RM foreground
+along this sightline (Hutschenreuter et al. 2022) indi-
+cates an expected Galactic RM of −14 ± 18 rad m−2
+towards FRB 20220319D. We cannot draw conclusions
+from the measured spectral decorrelation bandwidth of
+νd = 3.6±0.2 MHz, which is far in excess of the 0.2 MHz
+decorrelation bandwidth predicted by NE2001 due to
+scattering in the Milky Way ISM, because the origins of
+the spectral structure cannot easily be determined.
+We now analyze the performance of the NE2001 and
+YMW16 models with the aim of determining whether
+the model uncertainties can accommodate an extra-
+galactic origin for FRB 20220319D. Both model the dis-
+tribution of the WIM (Draine 2011) in the Galactic
+disk using observations of radio-wave propagation, with
+a dominant thick-disk component encapsulating several
+overdensities and voids motivated by multiwavelength
+data. The YMW16 model is fit to pulsars with indepen-
+dent DM and distance estimates, whereas the NE2001
+model is additionally fit to pulsar and extragalactic
+radio-source scattering measurements. For low-latitude
+extragalactic sightlines, the radial extent of the thick
+disk is critical in determining DMISM.
+In NE2001, a
+cutoff at a radius of 20 kpc is motivated by observations
+of HII regions in other galaxies and scattering of extra-
+galactic sources. Surveys of Galactic HII regions on the
+other hand motivate the cutoff of 15 kpc in YMW16.
+Neither cutoff is estimated in the NE2001 or YMW16
+fits; rather they are fixed model choices that do not im-
+pact the match between the models and the data un-
+der consideration. The functional forms for the WIM
+distribution in the thick disk also differ, with NE2001
+incorporating an oblate spheroid, and YMW16 incorpo-
+rating a plane-parallel slab that results in significant de-
+viations from NE2001 at low latitudes in the second and
+third quadrants (e.g., Price et al. 2021). High-latitude
+extragalactic sightlines are likely to be better modeled
+because the scale height of the thick disk is a critical fit-
+ted parameter. However, density fluctuations caused by
+turbulence in the WIM, together with clumps and voids,
+complicate our picture even at high latitudes (Ocker et
+al. 2020). Large-scale WIM inhomogeneities at the disk-
+halo interface such as Galactic chimneys (e.g., Koo et al.
+1992; Normandeau et al. 1996) can result in significant
+departures from spatially smooth models for DMISM.
+We first note that the recent PSRπ sample of pul-
+sar parallax measurements (Deller et al. 2019) reveals
+discrepancies between the NE2001 and YMW16 models
+and measured pulsar distances at low latitudes in the
+second quadrant, where FRB 20220319D appears on the
+sky. In general, known errors in NE2001 and YMW16
+
+Mass of the Milky Way CGM
+7
+exhibit significant spatial correlation (e.g., Price et al.
+2021), motivating the consideration of sources along
+sightlines similar to FRB 20220319D. Table 2 lists the
+DMs of the three nearest pulsars in the sample to
+FRB 20220319D, together with the predicted DMs given
+the measured distances.
+The YMW16 model signifi-
+cantly overpredicts all three DMs, and although NE2001
+is more accurate for the modestly distant pulsars likely
+in the Cygnus-Orion arm, it also overpredicts the DM
+of the more distant PSR J0406+6138.
+A more detailed analysis of the performance of the
+models for sightlines towards distant objects is enabled
+by the recent convergence in the distances to globu-
+lar clusters hosting pulsars.
+We obtain accurate dis-
+tance estimates (less than few-percent errors) to all
+21 globular clusters with associated pulsars (Manch-
+ester et al. 2005) at |b| > 5◦ from the compilation
+of Baumgardt & Vasiliev (2021), primarily based on
+Gaia parallaxes (Vasiliev & Baumgardt 2021).
+Clus-
+ters at lower latitudes are not considered because their
+DMs are influenced by the thin disk and other compo-
+nents in the Galactic plane. For each cluster we com-
+pare the difference between the DMs predicted by the
+NE2001 and YMW16 models for the cluster distances,
+and the true cluster DMs defined as the mean DMs of
+the associated pulsars. The results are shown in Fig-
+ure 4; FRB 20220319D is also included with the pre-
+dicted DM set to DMISM.
+The errors in NE2001 at
+low latitudes can clearly accommodate an extragalactic
+origin for FRB 20220319D. YMW16 generally performs
+better than NE2001 for the low-latitude clusters, but
+these are nearly all in the first and fourth quadrants
+(with NGC 1851 as the only exception), and so perhaps
+not representative of errors along sightlines closer to the
+Galactic anticenter.
+As an additional check of the Galactic sightline to-
+wards FRB 20220319D, we also consider independent
+measures of DMISM. First, we use the relation between
+the neutral hydrogen column density and pulsar DMs
+derived by He et al. (2013) to estimate the DMs to-
+wards FRB 20220319D and each globular cluster, as-
+suming no HI gas beyond the clusters. We obtain HI
+column densities from the HI4PI data (HI4PI Collabo-
+ration et al. 2016). The results are shown in Figure 4,
+including errors representing intrinsic scatter in the rela-
+tion. The HI column densities result in underestimates
+of the DMs towards low-latitude clusters, possibly re-
+flecting the increased ionization of HI gas towards the
+central regions of the Galaxy (e.g., Madsen & Reynolds
+2005).
+The HI-predicted DMISM for FRB 20220319D
+is consistent with an extragalactic origin, although the
+uncertainties are large. Second, we estimate the DMs
+towards FRB 20220319D and each globular cluster us-
+ing Hα emission measures (EMs) derived from the Wis-
+consin Hα Mapper (WHAM) all sky maps (Haffner et
+al. 2003).
+We follow the analysis of Berkhuijsen et
+al. (2006), wherein the photon fluxes, FHα (in units of
+Rayleighs), integrated over the range of Galactic veloci-
+ties are converted to DMs using the following relations:
+EM = 2.25FHαe2.2E(B−V ) cm−6 pc
+(3)
+DM = (EM × D × F)1/2.
+(4)
+F =
+f
+ζ(1 + ϵ2)
+(5)
+The expression for F (e.g., Ocker et al. 2020, and
+references therein) encodes an ionized cloudlet model,
+wherein the WIM is composed of cloudlets with a volume
+filling factor f, ζ = ⟨n2
+e⟩/⟨ne⟩/2 ∼ 2 captures the cloud
+to cloud variation, and ϵ2 ≲ 1 is the density variance
+internal to the cloudlets. Further, E(B − V ) is a mea-
+sure of the interstellar redenning along a given sightline,
+and D is the distance through the WIM. We have as-
+sumed a uniform WIM electron temperature of 8000 K,
+and equivalence between the line-of-sight and volume
+filling factors. We find that for the cluster sightlines the
+Hα-based DM estimates perform remarkably well for a
+nominal value F = 0.1 (i.e., f ≳ 0.4), comparably to
+YMW16 and better than NE2001 for low-latitude sight-
+lines. This is surprising given previous results for pulsar
+DMs (e.g., Schnitzeler 2012), but may be explained by
+globular clusters generally lying beyond the outer extent
+of the WIM, obviating the need for distance corrections
+to the WHAM EMs.
+For the FRB 20220319D sight-
+line, the Hα flux implies DMISM = 89 pc cm−3, which is
+22 pc cm−3 lower than the measured DM.
+We
+conclude
+that
+models
+for
+the
+DM
+towards
+FRB 20220319D contributed by the Galactic WIM are
+consistent with an extragalactic origin.
+This se-
+cures the association of the burst with the galaxy
+IRAS 02044+7048. Consistent with the original presen-
+tations of the NE2001 and YMW16 models for the WIM
+distribution, we confirm that there are significant model
+uncertainties at low Galactic latitudes. We used a large
+sample of globular-cluster parallax distances, which have
+DM estimates from associated pulsars, to demonstrate
+that an alternative predictor of DMISM based on the to-
+tal Galactic Hα flux can provide a useful check on the
+aforementioned models.
+In order to estimate DMISM
+towards FRB 20220319D we consider pulsars that are
+nearby on the sky and at a similar Galactic latitude. The
+two nearest pulsars within 1 deg of the burst in Galactic
+latitude, PSR J0231+7026 (total separation of 1.9 deg)
+
+8
+Ravi et al.
+Table 2. Measured and predicted DMs of PSRπ pulsars near FRB 20220319D.
+Pulsar
+l (deg.)
+b (deg.)
+FRB offset (deg)
+Distance (kpc)
+DMa
+NE2001a
+YMW16a
+PSR J0147+5922
+130.1
+−2.7
+12
+2.02+0.46
+−0.16
+40.1
+31.8
+79.0
+PSR J0157+6212
+130.6
+0.3
+9
+1.80+0.08
+−0.12
+30.2
+31.5
+59.3
+PSR J0406+6138
+144.0
+7.0
+15
+4.58+1.63
+−0.87
+65.3
+123.7
+141.7
+aAll DMs are given in units of pc cm−3.
+NGC6517
+NGC6539
+M62
+M22
+FRB20220319
+NGC6652
+NGC6397
+NGC5986
+OmegaCen
+M4
+M10
+NGC6752
+M3
+M15
+M14
+NGC1851
+M2
+M13
+47Tuc
+M5
+M30
+M53
+200
+150
+100
+50
+0
+50
+100
+150
+(Predicted DM) - (True DM) (pc/cc)
+6.7
+6.8
+7.3
+7.6
+9.8
+11.4
+12.0
+13.3
+14.9
+16.0
+23.1
+25.6
+26.1
+27.3
+27.7
+35.0
+35.8
+40.9
+44.9
+46.8
+46.8
+79.8
+|b| (deg)
+NE2001
+YMW16
+Halpha (ff=0.1)
+NH
+Figure 4. Difference between the predicted DM and true DM of all globular clusters that host pulsars, sorted by increasing
+|b|. Predictions are made for each globular cluster given its distance and location on the sky, using four different models for the
+warm ISM of the Milky Way. The two literature models are NE2001 (blue solid thick curve) and YMW16 (orange solid thin
+curve). We also convert the measured Hα emission measures (green dashed curve) and HI columns (red dots and shaded region)
+along each sightline to DM predictions using methods described in the text. We also include predictions for FRB 20220319D
+assuming a location beyond the Galactic disk.
+and PSR J0325+6744 (total separation of 7.4 deg), have
+DMs of 46.7 pc cm−3 and 65.3 pc cm−3, and we consider
+these as conservative and less conservative estimates re-
+spectively of DMISM along the burst sightline. We note
+that the predicted NE2001 and YMW16 DMISM values
+along the sightlines towards these pulsars are lower than
+along the burst sightline.
+4. THE HOST ENVIRONMENT OF FRB 20220319D
+We
+obtained
+long-slit
+optical
+spectroscopy
+of
+IRAS 02044+7048
+using
+the
+Double
+Spectrograph
+(DBSP; Oke & Gunn 1982) mounted on the Palo-
+mar 200-inch Hale telescope on 2022 June 01 (UT).
+The observations were undertaken under conditions of
+1.2′′ seeing, at an airmass of 1.9, and used a 1.5′′ slit
+positioned to include both the FRB location and the
+galaxy nucleus at a position angle of 104◦. Two 600 s
+exposures were obtained with the D55 dichroic and the
+316/7500 grating on the red arm; we only considered
+data from the red arm of the spectrograph given the
+significant extinction along this sightline (AV = 2.209;
+Schlafly & Finkbeiner 2011). The two-dimensional spec-
+tral data were reduced according to standard procedures
+using a PypeIt-based pipeline (Prochaska et al. 2020).
+We then defined the trace function using observations
+of the standard star Feige 34, and extracted spectra in
+1.5′′ windows along the slit.
+Feige 34 was also used
+
+Mass of the Milky Way CGM
+9
+Table 3. Observed and derived parame-
+ters of the host galaxy of FRB 20220319D,
+IRAS 02044+7048.
+1σ errors in the last
+significant figures are given in parentheses.
+Parameter
+Value
+Redshift
+0.0111(4)
+Luminosity distance (Mpc)
+49.6
+Effective radius (kpc)
+2.7(2)
+FRB projected offset (kpc)
+2.3(3)
+log M∗ (M⊙)
+9.93(7)
+log Za
+0.1(0.2)
+Internal AV b
+0.2+0.2
+−0.1
+SFR (M⊙ yr−1)c
+1.8(0.7)
+Inclination (deg)
+23(3)
+aMetallicity with respect to solar.
+b V -band extinction corresponding to a
+uniform dust slab.
+c Averaged over the past 100 Myr.
+6
+4
+2
+0
+2
+Offset along slit (kpc)
+0
+10
+20
+30
+40
+50
+H  flux (1040 erg s
+1)
+FRB
+Nucleus
+Figure
+5.
+Total Hα flux observed at different lo-
+cations within our longslit spectroscopic observations of
+IRAS 02044+7048. Measurements were obtained at 1.5′′ in-
+tervals along a 1.5′′ longslit oriented at a position angle of
+104◦, which covered both the FRB position and the cen-
+ter of the galaxy. The locations of the galaxy nucleus and
+FRB 20220319D are indicated in the figure.
+for flux calibration, and all spectra were corrected for
+extinction according to the Fitzpatrick & Massa (2007)
+extinction curve.
+We detected Hα emission at several locations along
+the slit, as shown in Figure 5. The strongest Hα emission
+is observed at the galaxy core, from which we also detect
+lines corresponding to [NII] and [SII]. These lines were
+together used to derive a redshift of 0.0111 ± 0.0004 for
+IRAS 02044+7048. Although we do not detect Hβ or
+[OIII], the ratio log [NII]/Hα = −0.23 ± 0.02 indicates a
+< 10% chance of the nuclear ionization being purely due
+to star formation activity (Ho et al. 1997; Kewley et al.
+2006). There is evidence of Hα emission from spiral-arm
+features on either side of the nucleus; these features are
+too extended to represent HII regions. The FRB appears
+associated with the eastern arm, although not at the
+center of its Hα radial profile. No additional evidence
+for or against an arm association can be gleaned from
+the residuals of a fit of the IRAS 02044+7048 image in
+Figure 3 to a two-dimensional Sersic profile.
+In
+order
+to
+determine
+global
+properties
+of
+IRAS 02044+7048, we derived and modeled its spec-
+tral energy distribution (SED). We executed aperture
+photometry on archival images from the Panoramic
+Survey Telescope and Rapid Response System (Pan-
+STARRS; Chambers et al. 2016), Two Micron All Sky
+Survey (2MASS; Skrutskie et al. 2006), and ALLWISE
+(Cutri et al. 2021) surveys. We identified an elliptical
+aperture that captured the i-band extent of the galaxy
+in Pan-STARRS data, and masked likely foreground
+objects. We then used this aperture, convolved with the
+PSF of each survey, to measure extinction-corrected AB
+magnitudes in all filters used by Pan-STARRS, 2MASS,
+and WISE. We modeled this SED using the Prospector
+stellar population synthesis modeling code (Johnson et
+al. 2021). We ran Prospector using recommended tech-
+niques and priors and a non-parametric star-formation
+history (SFH) model (Leja et al. 2019), and sampled
+from the posterior using emcee (Foreman-Mackey et al.
+2013).
+Non-parametric models for the SFH result in
+less bias in both stellar-mass (M∗) and star-formation
+rate (SFR) estimates, because specific SFH models are
+not excluded a priori. We included a model for dust re-
+radiation in the likelihood function. The spectral energy
+distribution of IRAS 02044+7048 is shown in Figure 6,
+together with the results from the Prospector analysis,
+and derived maximum aposteriori probability parame-
+ters are listed in Table 3. The SFR of 1.8 M⊙ yr−1 was
+derived by integrating the SFH over the past 100 Myr.
+IRAS 02044+7048,
+and
+the
+location
+of
+FRB 20220319D within it, are largely consistent with
+previous results on FRB host galaxies and environ-
+ments (Bhandari et al. 2020; Heintz et al. 2020; Boch-
+enek et al. 2021; Mannings et al. 2021; Bhandari et al.
+2022). Most FRB host galaxies have detectable ongoing
+star formation, and some exhibit signatures of addi-
+tional nuclear ionization sources.
+The stellar mass of
+IRAS 02044+7048 is entirely consistent with the distri-
+
+10
+Ravi et al.
+Figure 6.
+Left:
+Prospector fit to the spectral energy distribution (SED) of the host galaxy of FRB 20220319D,
+IRAS 02044+7048. The SED measurements from archival PanSTARRS, 2MASS, and WISE data are shown in red, together
+with representative 10% errors. Filter transmission curves are shown in grey. The maximum aposteriori probability (MAP)
+model photometry is shown in green, together with the MAP spectrum. 100 spectra generated using random draws from the
+posterior distributions of the model parameters are shown in black. Right: Measured SFH of IRAS 02044+7048 (green). The
+results from 100 random draws from the posterior distributions are shown in black.
+bution of masses found for both repeating and so far
+non-repeating FRB hosts. The potential association of
+FRB 20220319D with a spiral-arm feature is consistent
+with the results of Mannings et al. (2021) based on
+Hubble Space Telescope imaging of eight FRB hosts.
+In the absence of very long baseline interferometry, our
+data do not have sufficient angular resolution to deter-
+mine whether or not the FRB originates from within an
+association of young stars (e.g., Tendulkar et al. 2021).
+The location of FRB 20220319D at an offset of ∼ 85% of
+the effective radius from the galaxy nucleus is consistent
+with the range of previously observed FRB offsets.
+However, the SFR of IRAS 02044+7048 is rather high
+for its stellar mass, with respect to the observed sam-
+ple of hosts of so far non-repeating FRBs (Bhandari
+et al. 2022).
+Only one so far non-repeating FRB
+(20190102C, at z = 0.2912) has a comparably high
+specific SFR (sSFR), of log sSFR = −9.74. Although
+IRAS 02044+7048 is consistent with a location on the
+star-forming main sequence of galaxies (Speagle et al.
+2014), non-repeating FRB hosts as a population are con-
+sistent with originating from below this sequence.
+It
+is possible that FRBs from more actively star-forming
+hosts are selected against in existing radio surveys due to
+propagation effects in the host ISM (Seebeck et al. 2021),
+in which case it is not surprising that IRAS 02044+7048
+has a face-on orientation.
+5. THE MASS OF THE MILKY WAY CGM
+The constraints on the extragalactic DM contribution
+along the sightline towards FRB 20220319D are severe.
+Adopting the two estimates discussed above for DMISM
+from pulsars nearby to the FRB on the sky, we find
+possible upper limits on the extragalactic DM contribu-
+tion (DMCGM+DMIGM+DMhost) of either 45.7 pc cm−3
+or 64.3 pc cm−3. Even with our conservative estimates
+for DMISM, FRB 20220319D has the lowest extragalactic
+DM yet measured for an FRB localized to a host galaxy.
+We thus have the opportunity to stringently bound the
+heretofore poorly constrained value of DMCGM, and thus
+directly bound the mass of the Galactic CGM. The IGM
+likely contributes 7 pc cm−3 towards IRAS 02044+7048,
+assuming an IGM baryon fraction of 0.7 (e.g., Macquart
+et al. 2020).
+The detection of extended Hα emission
+from the galaxy itself, and in particular at the location
+of FRB 20220319D, indicates that the WIM component
+of the ISM is present, although we do not know where
+within the ISM column the FRB source is located.2 We
+simply assume a nominal value of DMhost = 10 pc cm−3
+for our analysis below.
+Thus, we find that for the
+FRB 20220319D sightline, two possible upper limits on
+DMCGM are 28.7 pc cm−3 and 47.3 pc cm−3, depending
+on the assumption for DMISM. These upper limits are
+comparable to previous results from the closest known
+FRB source (FRB 20200120E), in a globular cluster as-
+sociated with M81, which provide possible upper limits
+on DMCGM of 32 pc cm−3 and 42 pc cm−3] depending on
+the choice of NE2001 or YMW16 for DMISM (Kirsten et
+al. 2022). The constraints from FRB 20200120E are af-
+fected by a reliance on the NE2001/YMW16 models for
+2 With a measurement of scattering in the host ISM, it is possible
+to estimate DMhost (Cordes et al. 2022). However, upper limits
+on the temporal broadening of FRB 20220319D of ∼ 0.1 ms are
+not usefully constraining.
+
+Model spectrum (MAP)
+Model photometry (MAP)
+Observed photometry
+Flux Density [maggies]
+10
+10
+Wavelength [A]5
+4 -
+SFR [Mo/yr]
+3
+2
+0
+.6
+18
+0
+2
+4
+10
+12
+Lookback Time [Gyr]Mass of the Milky Way CGM
+11
+DMISM, and by stronger assumptions for the halo DM
+of M81.
+A recent synthesis of models for the Galactic CGM
+DM by Keating & Pen (2020) highlighted the wide range
+of extant predictions. In this work, we use a representa-
+tive set of models to demonstrate the implications of the
+constraint on DMCGM from FRB 20220319D. The mod-
+els are illustrated in Figure 7. We also consider measure-
+ments of the DM contributed by sightlines through the
+halo from pulsars in the LMC (Ridley et al. 2013) and
+in the distant (18.5 kpc; Baumgardt & Vasiliev 2021)
+globular cluster M53 (Kulkarni et al. 1991; Pan et al.
+2021). Although the LMC is at b ≈ −33◦ and M53 is at
+b ≈ +80◦, we conservatively subtract the lowest DMISM
+to be found in either the NE2001 or YMW16 models of
+18 pc cm−3 to estimate the halo DM contributions along
+their sightlines. In the top two panels of Figure 7, we
+show the resulting upper limits on DMCGM towards M53
+and the LMC, with ranges corresponding to the range of
+associated pulsar DMs. The bottom-left panel shows the
+upper limit on DMCGM from FRB 20220319D, placed at
+the virial radius of the Milky Way halo. In all cases we
+show model predictions for DMCGM assuming a spher-
+ically symmetric galactocentric baryon halo, but with
+the DM evaluated along sightlines from the position of
+the Earth (GRAVITY Collaboration et al. 2019). We
+evaluate the modified NFW (mNFW) and Maller & Bul-
+lock (2004) profiles following Prochaska & Zheng (2019),
+with a total Milky Way mass of Mtot = 1.5 × 1012M⊙,
+and a halo baryon fraction of fCGM = 0.75. The Pen
+(1999) model is evaluated for the same Mtot and with
+the core radius set to the halo virial radius. The semi-
+empirical Miller & Bregman (2013) and Faerman et
+al. (2017) models and the semi-analytic Faerman et al.
+(2020) model are all evaluated as given. Finally, in the
+bottom-right panel of Figure 7, we also show the model
+predictions for the halo baryon density along the LMC
+sightline, together with constraints from ram-pressure
+stripping analyses of dwarf galaxies (Salem et al. 2015;
+Putman et al. 2021) and modeling of OVII and OVIII
+emission measurements (Kaaret et al. 2020).
+The data are in favor of models that predict lower to-
+tal CGM DMs and lower inner densities (e.g., Pen 1999;
+Faerman et al. 2020; Miller & Bregman 2013). The DMs
+of M53 pulsars and FRB 20220319D deliver consistent
+constraints on the range of models, albeit on very dif-
+ferent radial distance scales. The constraints from LMC
+pulsars are less impactful, but nonetheless also exclude
+the three models (Maller & Bullock 2004; Faerman et
+al. 2017; Prochaska & Zheng 2019) considered here that
+predict larger values of DMCGM. We note that our treat-
+ment of the CGM contribution to the DMs of LMC pul-
+sars is far more conservative than that of Anderson &
+Bregman (2010); this is motivated by the uncertainties
+discussed above in estimating DMISM.
+The DM con-
+straints are consistent with the most robust density es-
+timates that we consider (Salem et al. 2015; Kaaret et
+al. 2020).
+The density estimates are however subject
+to the uncertainties discussed in §1, and we proceed by
+considering only the constraints on DMCGM.
+We now derive constraints on the mass of the Galac-
+tic CGM, MCGM, by finding β models that are consis-
+tent with the FRB 20220319D and M53 DMs.
+When
+converting the electron column density to a mass col-
+umn density, we assume 1.18 proton masses per elec-
+tron following Yamasaki & Totani (2020), which corre-
+sponds to roughly solar metallicity gas. The β profile is
+widely adopted in the field to convert (column-)density
+estimates to halo masses (e.g., Miller & Bregman 2013;
+Salem et al. 2015; Kaaret et al. 2020), as it is empirically
+motivated. The electron number density at a radius r is
+given by
+n(r) = n0
+�
+1 +
+� r
+rc
+�2�−3β/2
+,
+(6)
+where n0 is a central density, and rc is a core radius that
+we fix to 0.47 kpc following Miller & Bregman (2013).
+The exact value of the core radius is unimportant for
+our conclusions regarding MCGM.
+Typical values of
+β found using various CGM tracers are in the range
+0.4 ≲ β ≲ 0.5 (e.g., Miller & Bregman 2013, 2015; Salem
+et al. 2015; Kaaret et al. 2020). We consider three con-
+straints on DMCGM: the lower and upper constraints
+from FRB 20220319D, and the lower constraint from the
+M53 pulsars. For each constraint, we derive correspond-
+ing values of MCGM from an integrated β profile for dif-
+ferent values of β. A useful joint constraint on MCGM
+and β is not possible with the data in hand. The results
+are shown in Figure 8.
+For typical values of β of between 0.4 and 0.5,
+the DM of FRB 20220319D implies an upper limit on
+log MCGM of between 11.0 and 10.8.
+The M53 pul-
+sars are even more constraining, limiting log MCGM to
+below a value of between 10.7 to 10.5.
+We consider
+the FRB 20220319D constraints to be more conserva-
+tive because the M53 estimate of DMCGM is very sen-
+sitive to the assumed DMISM along its sightline. The
+mass constraints all favor values of fCGM < 0.5 (re-
+call fCGM = MCGMΩM
+MtotΩb ) for our fiducial value of Mtot =
+1.5×1012M⊙, with M53 potentially accommodating val-
+ues of fCGM ≲ 0.2 for reasonable values of β.
+
+12
+Ravi et al.
+0
+50
+100
+150
+200
+250
+Distance (kpc)
+100
+101
+102
+DM (pc cm
+3)
+M53
+mNFW
+MB04
+MB13
+Pen99
+Faerman1
+Faerman2
+0
+50
+100
+150
+200
+250
+Distance (kpc)
+100
+101
+102
+DM (pc cm
+3)
+LMC
+mNFW
+MB04
+MB13
+Pen99
+Faerman1
+Faerman2
+0
+50
+100
+150
+200
+250
+Distance (kpc)
+100
+101
+DM (pc cm
+3)
+FRB 20220319
+mNFW
+MB04
+MB13
+Pen99
+Faerman1
+Faerman2
+0
+50
+100
+150
+200
+250
+Distance (kpc)
+10
+5
+10
+4
+10
+3
+Baryon number
+ density (cm
+3)
+Density (LMC direction)
+mNFW
+MB04
+MB13
+Pen99
+Faerman1
+Faerman2
+Figure 7. Illustration of the constraining power of different measurements of the CGM DM and density on a range of models.
+In all panels, curves show predictions of the default modified NFW (mNFW) model of Prochaska & Zheng (2019) (blue), the
+model of Maller & Bullock (2004) (orange; MB04), the model of Miller & Bregman (2013) (green; MB13), a Pen (1999) model
+(Pen99) assuming a core radius set to the virial radius (red), and the models of Faerman et al. (2017) (purple; Faerman1) and
+Faerman et al. (2020) (brown; Faerman2). Top panels: upper limit on the CGM DM towards M53 (left) and the LMC (right).
+In both cases, the minimum DM (18 pc cm−3) across the sky in either the NE2001 or YMW16 models has been subtracted. The
+ranges indicate the range of DMs of pulsars associated with each object. Bottom left: upper limit on the CGM DM towards
+FRB 20220319D. The range indicates different assumptions for the ISM DM that correspond to the two closest pulsars on the
+sky. Bottom right: constraints on the CGM density compared with the models. The dark disks indicate measurements using
+ram-pressure stripping modeling of the LMC (Salem et al. 2015) and modeling of diffuse X-ray line emission (Kaaret et al.
+2020). The lower limits displayed as triangles indicate estimates using ram-pressure stripping modeling of a large selection of
+Milky Way satellites (Putman et al. 2021).
+6. DISCUSSION
+Our results demonstrate that the total Galactic
+baryon mass is likely significantly lower than the cos-
+mological average for a halo as massive as the Milky
+Way.
+The mass constraints we derive are consistent
+with some previous observational results from analyses
+of OVII and OVIII emission and absorption (e.g., Li
+& Bregman 2017; Kaaret et al. 2020) and ram-pressure
+stripping of the LMC (Salem et al. 2015), but inconsis-
+tent with other results that posit higher CGM masses
+(e.g., Faerman et al. 2017; Yamasaki & Totani 2020).
+Analyses of the dynamics of Milky Way satellites im-
+ply a total virial mass of Mtot = (1.4 ± 0.3) × 1012M⊙
+(Watkins et al. 2010), a recent Gaia-based analysis
+of the dynamics of Milky Way globular clusters yields
+Mtot = (1.3 ± 0.3) × 1012M⊙ (Posti & Helmi 2019), and
+an analysis of the dynamics of the Magellanic Stream
+yields Mtot = (1.5 ± 0.3) × 1012M⊙ (Craig et al. 2022).
+It is possible that a full accounting for the orbit of the
+LMC would reduce the estimates of Mtot by ∼ 15%
+(Correa Magnus & Vasiliev 2022).
+However, in all
+cases fCGM ≲ 0.4 is implied by FRB 20220319D, and
+fCGM ≲ 0.2 is implied by the M53 DM. Our constraints
+are affected by different systematic effects to previous
+results, and do not suffer from uncertainties in model-
+ing the chemical and thermal states of the CGM, nor
+from uncertainties in modeling the interaction of satel-
+lite galaxies with the CGM. These values are consistent
+with several simulations of the CGM contents of galaxies
+like the Milky Way that account for the effects of kinetic
+and thermal feedback on reducing the halo baryon con-
+tent.
+The observational constraints on MCGM are affected
+by uncertainties in identifying the CGM DM contri-
+bution along the sightlines of interest.
+For the low-
+latitude FRB 20220319D sightline, we adopted conser-
+vative estimates of DMISM, and there are uncertainties
+at the level of a few pc cm−3 in the DMIGM and DMhost
+
+Mass of the Milky Way CGM
+13
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+10.4
+10.6
+10.8
+11.0
+11.2
+11.4
+11.6
+log MCGM (M )
+fCGM = 0.5
+fCGM = 0.2
+fCGM = 0.1
+FRB lower
+FRB upper
+M53
+Figure 8.
+Upper limits on the CGM mass for different
+assumptions for the radial density profile.
+We adopt a β
+model (see text for details), and show limits corresponding
+to the upper and lower ends of the range of constraints from
+FRB 20220319D (dashed and solid lines respectively), and
+from the most constraining pulsar in M53 (red solid line).
+Typical values of β found in the literature are ≳ 0.4. For
+these values, the inferred fCGM values are typically below
+0.5, as indicated by blue horizontal lines.
+terms. We therefore consider the constraints on MCGM
+based on FRB 20220319D to be robust to uncertainties
+in DMCGM. The impact of uncertainty in DMISM for the
+M53 sightline is greater given the low values of DMCGM
+under consideration. Although we attempt to be con-
+servative, even a few-unit increase in DMCGM would sig-
+nificantly raise the derived upper limits on MCGM. Our
+analysis is also sensitive to how we model the distribu-
+tion of baryons in the CGM. First, we do not account
+for sightline to sightline variance, which simulations sug-
+gest may be at the level of ≳ 10% (e.g., Zheng et al.
+2020; Ramesh et al. 2022). Second, it may be that a β
+profile is not the correct model to adopt, and our con-
+straints are sensitive to the specific value of β. Guidance
+on these points from simulations for future analyses of
+DMCGM will be important. Finally, although we assume
+a specific helium abundance in translating the DM con-
+straints to a total CGM mass, our results are sensitive
+to variations in this assumption only at the < 10% level.
+The discovery of FRB 20220319D has major implica-
+tions for the search for FRBs in the local universe, and
+for future FRB studies of the CGM of the Milky Way
+and other galaxies. First, we have shown that for low
+Galactic latitudes (|b| ≲ 15◦; see Figure 4) there can
+be significant uncertainties in the widely used NE2001
+and YMW16 models for DMISM, which may lead to
+nearby extragalactic FRBs being missed in surveys. In
+all cases we recommend that FRB surveys save candi-
+dates at DMs below the model values of DMISM. The
+possibility of missing FRBs with low extragalactic DMs
+also needs to be considered when using FRB samples to
+directly infer a characteristic DMCGM (e.g., Platts et al.
+2020). These studies are also likely to be impacted by
+the uncertainties in deriving DMCGM that in most cases
+will be on the order of the values of DMCGM allowed
+by our analysis (see Figure 7). We recommend that in-
+ferences of DMCGM with FRBs focus on high-latitude
+sightlines towards FRBs interferometrically localized to
+nearby galaxies, wherein uncertainties in DMIGM and
+DMhost can be minimized. Nonetheless, if the charac-
+teristic DMCGM is indeed on the order of 10 pc cm−3,
+a very large number of local FRBs will be required to
+suppress variance between halo sightlines, and uncer-
+tainties in other DM contributions. These samples may
+be furnished by future coherent all-sky radio monitors
+(e.g., Lin et al. 2022). Finally, consistent with previ-
+ous studies (Keating & Pen 2020; Kirsten et al. 2022),
+our constraints on fCGM firmly exclude the fiducial pa-
+rameterizations of the widely used modified-NFW and
+Maller & Bullock (2004) models for the CGM DM as
+described by Prochaska & Zheng (2019). This impacts
+studies that have assumed the correspondingly large val-
+ues of DMCGM (e.g., Ravi 2019), as well as forecasts for
+the contributions of the CGM of intervening galaxies to
+FRB DMs.
+The host of FRB 20220319D, IRAS 02044+7048, ap-
+pears to have an unusually large specific SFR. This is
+despite FRB 20220319D being the closest so far non-
+repeating FRB yet discovered, and the strong evolution
+towards higher SFRs of the star-forming main sequence
+of galaxies with redshift (Speagle et al. 2014). Our anal-
+ysis of the SFH of the IRAS 02044+7048 host galaxy
+suggests that most stars were formed in a burst in the
+last ≲ 2 Gyr. Thus it is likely that the progenitor of
+FRB 20220319D does not have a delay time (i.e., age
+from its formation epoch) in excess of this timescale.
+Although several progenitor scenarios remain consistent
+with this constraint, the additional possible association
+of the FRB source with a spiral arm may imply a pro-
+genitor age that is a fraction of a ∼ 100 Myr dynamical
+timescale. We note that it is not straightforward to com-
+pare our inferred SFR, which is averaged over the past
+100 Myr of the SFH, to the Hα-based SFRs common in
+the FRB literature (e.g., Bhandari et al. 2022), which
+are sensitive to the last ∼ 10 Myr (Kennicutt 1998). A
+direct comparison of non-parametric SFH estimates may
+lead to a more complete view of the formation channels
+of FRB progenitors.
+7. SUMMARY AND CONCLUSIONS
+We present the DSA-110 discovery and interfero-
+metric localization of the nearest so far non-repeating
+
+14
+Ravi et al.
+FRB (20220319). The burst was observed with a high
+S/N of 79, a DM of 110.95 pc cm−3, and has a lin-
+ear polarization fraction of 16 ± 3% and an RM of
+50 ± 15 rad m−2 (Figure 1, and Table 1).
+The FRB
+originated from the position (R.A. J2000, decl. J2000)
+= (02:08:42.7(1), +71:02:06.9(6)), where uncertainties
+in the last significant figures are given in parenthe-
+ses (Figure 2).
+We associate FRB 20220319D with a
+spiral-arm feature of a face-on star-forming galaxy at a
+distance of 50 Mpc (IRAS 02044+7048; Figures 3 and
+5; Table 3).
+IRAS 02044+7048 is moderately mas-
+sive (log M∗ = 9.93 ± 0.07), with an approximately
+solar metallicity and an SFR averaged over the past
+100 Myr of 1.8 ± 0.7 M⊙ yr−1 (Figure 6). The low DM
+of FRB 20220319D motivates a sensitive new constraint
+on the baryon mass of the Milky Way CGM. Our con-
+clusions are summarized as follows.
+• The spatial coincidence of FRB 20220319D and
+IRAS 02044+7048
+(false
+alarm
+probability
+of
+<
+10−4),
+and a potential RM detection for
+FRB 20220319D that is in excess of the Galac-
+tic foreground, both imply a secure association.
+However, the DM of FRB 20220319D is somewhat
+lower than the predicted Galactic ISM DM along
+its sightline from leading models for the WIM dis-
+tribution (NE2001 and YMW16; Cordes & Lazio
+2002; Yao et al. 2017). We show through an anal-
+ysis of the DMs towards pulsar-hosting globular
+clusters with Gaia parallax distances (Figure 4)
+that there are significant uncertainties in NE2001
+predictions for DMISM for |b| ≲ 15◦, and that
+YMW16 is particularly uncertain at low latitudes
+in the second quadrant, where FRB 20220319D is
+located (see also Table 2). We find that estimates
+for DMISM based on Hα EMs perform compara-
+bly well to the models. We conclude that models
+for DMISM towards FRB 20220319D can accom-
+modate an extragalactic origin, and urge FRB sur-
+veys to consider candidate bursts at DMs below
+standard model predictions for DMISM.
+• The SFH of IRAS 02044+7048 is consistent with
+most stars being formed in a burst within the
+past two Gigayears. This, together with the pos-
+sible location of FRB 20220319D within a spi-
+ral arm of IRAS 02044+7048, suggests a mod-
+erately low delay time for the burst progenitor.
+Despite its nearby distance, the specific SFR of
+IRAS 02044+7048 derived from SED modeling is
+larger than all but one so far non-repeating FRB,
+although the comparison with literature samples
+of host galaxies would be aided by the widespread
+use of non-parametric SFH analyses.
+• We find conservative upper limits on the Galactic
+CGM contribution to the DM of FRB 20220319D
+of either 28.7 pc cm−3 or 47.3 pc cm−3: the two val-
+ues are based on DMISM-estimates from the DMs
+of two pulsars nearby on the sky. These limits ex-
+clude some literature models (Maller & Bullock
+2004; Faerman et al. 2017; Prochaska & Zheng
+2019) for the baryon distribution in the CGM (Fig-
+ure 7).
+We find that consistent results are ob-
+tained from estimates of the CGM DM contribu-
+tions towards pulsars in the LMC and in the dis-
+tant (18.5 kpc) high-latitude globular cluster M53.
+The M53 pulsars in particular provide interest-
+ing constraints in the inner regions of the Milky
+Way halo that complement the FRB constraint,
+although the M53 DMCGM estimate is sensitive to
+the assumed ISM contribution.
+• We derive an upper limit on the mass of baryons in
+the Galactic CGM of log MCGM ≲ 10.8 − 11 from
+FRB 20220319D, and log MCGM ≲ 10.5−10.7 from
+M53 (Figure 8).
+The range corresponds to the
+range of assumed indices (0.4–0.5) of the β pro-
+file for the baryon halo density distribution. For
+a fiducial total mass (baryons and dark matter)
+of the Milky Way of 1.5 × 1012M⊙, and an as-
+sumed baryon-disk mass of 6×1010M⊙, the Milky
+Way contains ≪ 60% of the cosmological-average
+baryon mass.
+This is consistent with a baryon
+census, predicted by galaxy-formation simulations,
+wherein these “missing” baryons are expelled from
+halos like the Milky Way into the IGM through ki-
+netic and thermal feedback from AGN, supernovae
+and massive stars.
+Although our analysis relies
+on just a few sightlines, the conservative upper
+bounds on MCGM assuming spherical symmetry
+are robust to variations in the density distribution
+of halo baryons, and to assumptions on the chem-
+ical and thermal state of the halo.
+Studies of unseen baryons with FRB propagation sig-
+natures, like DMs, promise to transform our understand-
+ing of the distribution of baryons around and in be-
+tween galaxies (Ravi et al. 2019). As more FRBs like
+FRB 20220319D are localized to nearby galaxies (see
+also Kirsten et al. 2022), constraints on the CGM con-
+tent of the Milky Way will continue to improve, and
+direct measurements of MCGM may be possible. These
+measurements are likely to be important in assembling
+an in-situ picture of the processes whereby a galaxy
+grows from and impacts its environment.
+
+Mass of the Milky Way CGM
+15
+The authors thank staff members of the Owens Valley
+Radio Observatory and the Caltech radio group, includ-
+ing Kristen Bernasconi, Stephanie Cha-Ramos, Sarah
+Harnach, Tom Klinefelter, Lori McGraw, Corey Posner,
+Andres Rizo, Michael Virgin, Scott White, and Thomas
+Zentmyer. Their tireless efforts were instrumental to the
+success of the DSA-110. The DSA-110 is supported by
+the National Science Foundation Mid-Scale Innovations
+Program in Astronomical Sciences (MSIP) under grant
+AST-1836018. We acknowledge use of the VLA calibra-
+tor manual and the radio fundamental catalog. Some of
+the data presented herein were obtained at the W. M.
+Keck Observatory, which is operated as a scientific part-
+nership among the California Institute of Technology,
+the University of California and the National Aeronau-
+tics and Space Administration. The Observatory was
+made possible by the generous financial support of the
+W. M. Keck Foundation.
+Facility: Hale
+Software: astropy, CASA, heimdall, pypeit, Prospec-
+tor, wsclean
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+

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+Selecting Models based on the Risk of Damage Caused by Adversarial Attacks
+Jona Klemenc * 1 Holger Trittenbach * 1
+Abstract
+Regulation, legal liabilities, and societal con-
+cerns challenge the adoption of AI in safety and
+security-critical applications. One of the key con-
+cerns is that adversaries can cause harm by manip-
+ulating model predictions without being detected.
+Regulation hence demands an assessment of the
+risk of damage caused by adversaries. Yet, there
+is no method to translate this high-level demand
+into actionable metrics that quantify the risk of
+damage.
+In this article, we propose a method to model and
+statistically estimate the probability of damage
+arising from adversarial attacks. We show that
+our proposed estimator is statistically consistent
+and unbiased. In experiments, we demonstrate
+that the estimation results of our method have
+a clear and actionable interpretation and outper-
+form conventional metrics. We then show how
+operators can use the estimation results to reliably
+select the model with the lowest risk.
+1. Introduction
+Adversarial perturbations are a security risk since an adver-
+sary can use them to alter machine learning model predic-
+tions without being noticed by a human (Yuan et al., 2019;
+Ren et al., 2020). For instance, think of an upload filter that
+uses a machine learning model to identify prohibited mate-
+rial on social media platforms, such as copyright-protected
+images or hate speech. By perturbing prohibited content,
+an adversary may bypass an upload filter even though the
+content appears identical to the original prohibited content
+to the human observer. When machine learning models
+are deployed in systems that take or enable action in phys-
+ical environments, the security risks can result in safety
+hazards (Deng et al., 2020). An example is autonomous
+driving, where a false classification, e.g., of a stop sign, can
+severely damage property and life. Given these potentially
+*Equal contribution
+1neurocat, Berlin, Germany. Correspon-
+dence to: Jona Klemenc <jona.klemenc@neurocat.ai>, Holger
+Trittenbach <holger.trittenbach@neurocat.ai>.
+severe consequences, regulators have made clear that one
+must treat risk from adversaries seriously. Broad regulation,
+such as the “EU AI Act” (European Commission, 2021),
+as well as domain-specific norms, such as safety standards
+for autonomous vehicles (e.g., ISO 21448 (SOTIF) (The
+British Standards Institution, 2022), UL 4600 (Underwriters
+Laboratories Inc., 2022), and ISO PAS 8800 (International
+Organization for Standardization, 2023 (forthcoming)), ex-
+plicitly require an assessment of the risk arising from adver-
+sarial attacks. Consequently, operators of machine learning
+models strive to manage the risk that stems from adversarial
+perturbations (Piorkowski et al., 2022). While the motiva-
+tion and intentions are clear, neither the regulation nor the
+academic literature currently provides sufficient technical
+guidelines on how to assess the “risk of adversarial attacks”.
+A risk assessment of adversarial attacks requires a reliable
+estimate that an adversary causes damage. Intuitively, this
+probability of damage describes how likely an adversary can
+find perturbations that go undetected and alter model predic-
+tions. The probability of damage hence depends on the capa-
+bilities of an adversary and the effectiveness of measures put
+in place to detect adversarial perturbations. Previous work
+has focused on evaluating adversaries by comparing adver-
+sarial attacks (Yuan et al., 2019; Ren et al., 2020) with each
+other, e.g., by a per-attack drop in accuracy (Brendel et al.,
+2020). These evaluations do not consider the likelihood that
+measures detect adversarial perturbations. Instead, evalua-
+tions assume a threshold (Croce et al., 2021; Maho et al.,
+2021) on the perturbation size beyond which perturbations
+are detected with certainty. This is a stark simplification.
+Practical experience shows that a suitable threshold is hard
+to find or may not exist. Other proposed evaluation metrics
+that compare perturbation sizes (Carlini et al., 2019) are
+not viable alternatives as they are either prone to outliers or
+suffer from statistical bias. It is an open question of how to
+reliably estimate the probability of damage and how to use
+the estimate to select machine learning models.
+In this article, we propose a statistical approach to assessing
+the risk caused by adversarial attacks. Our main contri-
+bution is a model-agnostic estimator for the probability of
+damage that is statistically unbiased and consistent. Our
+estimator explicitly considers the probability of detecting
+perturbations instead of assuming a hard threshold. As it
+turns out, calculating estimates for the probability of detec-
+arXiv:2301.12151v1  [cs.LG]  28 Jan 2023
+
+Selecting Models based on the Risk of Damage Caused by Adversarial Attacks
+tion efficiently is challenging for large sample sizes (see
+Section 4). Hence, we propose a strategy to make estimates
+efficient (see Section 4.2), even when querying the detector,
+e.g., a human in the loop, is out of reach (see Section 4.3).
+The estimates resulting from our method allow comparing
+different models with each other to select the model with the
+lowest risk of damage caused by adversarial attacks. Our
+experiments demonstrate that our estimator is more reliable
+than existing metrics in adversarial robustness benchmarks.
+2. Notation
+Let X be a data space, and X = ⟨x1, . . . , xI⟩ ⊆ X a sample
+of I observations. A machine learning model M : X → O
+is a function that maps the data space to a prediction space
+O. O differs depending on the model task, e.g., for image
+classification it might be the space of logit scores or class
+labels; for object detection it might be the space of bounding
+boxes and classification scores. When there are multiple
+models, we index them as M1, . . . , MJ.
+We further define a ground truth as a function g: X → O
+that assigns each observation a value of the prediction space
+that is considered “correct” by some gold standard. We say
+a prediction is “incorrect” if M(x) ̸= g(x). We say that
+two model predictions “disagree” for a pair of observations
+if M(x) ̸= M(x′), x, x′ ∈ X. Herein, we do not specify
+further what it means to disagree since the specifics usually
+depend on the model task and application. For instance, in
+one case, one would say two models disagree if the argmax
+of their logit outputs are different; in another case, they
+disagree if the ranking of their top-k logit scores differs.
+We use PS to denote probability distribution functions over
+some space S with the corresponding probability density
+function dPS. The hat notation indicates empirical esti-
+mates, e.g., an empirical distribution ˆP.
+We say an estimator of a probability function is unbiased
+if the mean of the sampling distribution of the estimator is
+equal to the true probability. An estimator that converges to
+the estimated value with increasing sample size is consis-
+tent (see Cram´er, 1999, p. 351).
+3. Fundamentals and Related Work
+We expect operators to assess the risk of using a machine
+learning model in a security-critical application as an ex-
+pected value of the total damage, i.e., the product of the
+occurrence probability of damage when operating a model
+P dam(M) and the expected size of the damage Cdam.
+Operational Risk(M) = P dam(M) × Cdam
+(1)
+We assume that Cdam is constant and independent of whether
+the operator relies on machine learning or any other system,
+e.g., a human in the loop. We focus on the risk of a malicious
+adversary seeking to intentionally manipulate predictions
+in a way that can inflict damage. We do not consider other
+categories of machine-learning-related security and privacy
+risks, such as the risks of model stealing (Tram`er et al.,
+2016) and model inversion (Fredrikson et al., 2015).
+Adversarial Risk. The adversarial machine learning com-
+munity often uses the term “risk” in the sense of Adver-
+sarial Risk (AR). AR is concerned with the empirical risk
+estimation of a model under small perturbations. Formally,
+the adversarial risk is the expected loss over perturbations
+within a neighborhood Nϵ(x)
+AR(M) = Ex∼X
+�
+sup
+x′∈N (x)
+l(M(x′), g(x))
+�
+(2)
+with loss function l (Uesato et al., 2018). With ϵ=0, Equa-
+tion 2 reduces to the standard empirical risk. Typically, the
+neighborhood N is constrained to an epsilon ball around
+x, i.e., N(x) ≡ Bϵ(x) for a fixed perturbation budget ϵ.
+Details of this definition differ across the literature (Pydi
+& Jog, 2021). For instance, instead of the loss against a
+ground truth, one may compute the loss of prediction change
+l(M(x), M(x′)) if the ground truth is unknown (Diochnos
+et al., 2018).
+Related research focuses on average risk under ran-
+dom (Levy & Katz, 2021; Rice et al., 2021) and natural
+perturbations (Pedraza et al., 2021; Hendrycks & Dietterich,
+2019; Schwerdtner et al., 2020). Here, one measure of inter-
+est is the Error-Region Risk (ERR) (Diochnos et al., 2018),
+i.e., the probability that a successful perturbation exists
+ERR(M) = Px∈X [∃x′ ∈ Bϵ(x) : M(x′) ̸= g(x′)]
+(3)
+However, estimations of ERR rely on random perturbations
+drawn from a uniform distribution (Diochnos et al., 2018;
+Webb et al., 2018). Thus, it does not account for “an explicit
+and effective adversary” (Webb et al., 2018).
+Benchmarks. Adversarial Risk has inspired several bench-
+marks that compare the effectiveness of individual adver-
+sarial attacks and defenses. The comparisons are based
+on either the success rate of attacks for a defined pertur-
+bation budget ϵ, see RobustBench (Croce et al., 2021)
+and RoBIC (Maho et al., 2021), or the average perturba-
+tion size attacks require to find successful perturbations,
+see RobustVision (Brendel et al., 2020). Such com-
+parisons are attack-centric, i.e., effective in evaluating at-
+tacks against each other. However, estimating P dam requires
+model-centric evaluations that measure how likely an adver-
+sary can find successful adversarial perturbations given a set
+of attacks and perturbation budgets (cf. Carlini et al., 2019).
+Instead of evaluating adversarial risk for a defined budget,
+one can also plot accuracy or attack success rates against a
+
+Selecting Models based on the Risk of Damage Caused by Adversarial Attacks
+perturbation budget (Dong et al., 2020; Carlini et al., 2019).
+Such plots are useful to investigate the effectiveness of at-
+tacks for varying perturbation budgets. However, visual
+inspection of plots does not scale beyond the comparison of
+a few attacks. It is an open question how one can use these
+curves to estimate the operational risk; we will come back
+to this question in Section 5.2.
+In summary, there are a variety of evaluation methods that
+seek to capture some element of adversarial risk. They pro-
+vide a set of measures for attack benchmarks but are not
+immediately applicable to estimate operational risk. The
+reason is that they (i) do not account for the probability of
+detecting adversarial attacks (see Section 4) and (ii) are ei-
+ther biased or inconsistent and hence not useful as statistical
+estimates (see Section 5.2).
+4. Estimating the Probability of Damage
+The operational risk of a machine learning model depends
+on the probability of damage P dam, see Equation 1. Intu-
+itively, P dam is the joint probability of finding a successful
+perturbation (Succ) from the space of adversarial perturba-
+tions AdvM(X) and of a detector, e.g., a human in the loop,
+not detecting it (¬ Det). Formally, we can express P dam as
+the joint probability
+P dam :=Px∼AdvM(X)(Succ(x), ¬Det(x))
+= Px∼AdvM(X)(Succ(x))
+�
+��
+�
+Probability of Attack Success
+× Px∼AdvM(X)(¬Det(x) | Succ(x))
+�
+��
+�
+Probability of Detection
+(4)
+Considering this equation, a natural solution to estimat-
+ing P dam seems to be Monte Carlo Simulation: One can
+draw a sample from the space of adversarial perturbations
+AdvM(X) to estimate the probability of attack success
+Px∼AdvM(X)(Succ(x)) and then pass the successful per-
+turbations on to a detector to estimate the probability of de-
+tection Px∼AdvM(X)(¬Det(x) | Succ(x)). However, one
+must be careful with designing this sampling process – a
+naive sampling of random perturbations is not sufficient here.
+An adversary is efficient and uses all means available to find
+successful perturbations. The adversary does not search
+through the infinite space of random perturbations but in-
+stead uses adversarial attacks, i.e., efficient optimization
+methods that make heuristic assumptions on the perturba-
+tion space. Thus, naive random sampling underestimates an
+efficient adversary. Valid Monte Carlo Simulation requires
+to simulate an adversary, i.e., to run attacks that are in line
+with the resources and capabilities of the adversary.
+While Monte Carlo Simulation is conceptually sound, it has
+two significant limitations in our context:
+Small Sample Size: If detection requires a human in the
+loop, collecting data on detection will inevitably be time-
+consuming: a detector must inspect each successful pertur-
+bation individually. This may significantly limit the feasible
+sample size and reduce the estimation quality.
+Model-Specific Estimates: Probability estimates are specific
+to the machine learning model and attacks used to search
+for perturbations. This is because the conditional probabil-
+ity of detection Px∼AdvM(X)(¬Det(x) | Succ(x)) relies
+on AdvM(X) which is specific to the model and attacks.
+Consequently, one must repeat the entire estimation of P dam
+for any change in the choice of models or attacks. Because
+detection is costly, the estimation becomes impractical for
+operators who have frequent model iterations and face a
+research field that frequently produces novel attacks.
+Both limitations stand in the way of obtaining an estimate
+of P dam in many practical settings.
+In this section, we propose a sampling method that over-
+comes both limitations. We first establish a rigorous formal
+framework for the probability of attack success (Section 4.1).
+We then turn to the estimation of the probability of detection
+(Section 4.2) and introduce a method that is not limited by
+Small Sample Size and Model-Specific Estimates. Lastly,
+we look at the special case of providing an estimation for
+P dam when there is no explicit detector – a setting that often
+occurs in academic benchmarks (Section 4.3). The result is
+a formal expression of the probability of damage that has
+both a clear interpretation and an unbiased and consistent
+estimator.
+4.1. Estimating the Probability of Attack Success
+So far, we have defined P dam as an estimate over the space
+of adversarial perturbations AdvM(X). However, one typi-
+cally only has access to a sample X ∼ X, used for training
+or testing of a machine learning model. The space of adver-
+sarial perturbations is defined implicitly by a pushforward
+of X along a function ΠM : x �→ x′
+Px∼AdvM(X)(Succ(x), ¬Det(x))
+= Px∼X (Succ(ΠM(x)), ¬Det(ΠM(x)))
+(5)
+We call ΠM the attack strategy.
+One can use a sample X and an attack strategy ΠM to
+simulate an adversary. To illustrate, think of an adversary
+that wants to evade an upload filter for copyright-protected
+material. The adversary would select a copyright-protected
+image and manipulate it via an attack strategy ΠM such
+that the classifier of the upload filter predicts it to be
+non-protected content. To estimate the probability that
+the adversary finds such a successful perturbation, one
+can simulate the adversary by sampling images from the
+data distribution of interest to the adversary, here the
+
+Selecting Models based on the Risk of Damage Caused by Adversarial Attacks
+copyright-protected images, and then manipulate the im-
+ages with the attack strategy the adversary is expected to use.
+An attack strategy depends on the choice of search algo-
+rithms that an efficient adversary has at their disposal to
+search for successful adversarial perturbations. The most
+effective search algorithms known today are adversarial at-
+tacks, i.e., heuristics to find small successful perturbations.
+Many adversarial attacks require access to the model, e.g.,
+to obtain inference results or gradients with respect to the
+model input. An adversary with restricted model access
+can only use a subset of adversarial attacks that do not rely
+on gradient calculations. We define the set of applicable
+adversarial attacks as
+A = {ak : (x, M) �→ x′}k∈K
+where each ak ∈ {a1, a2, . . . , aK} is an attack that has
+access to an input x and potentially restricted access to
+a model M. In practice, an adversary also may have a
+limited computational budget, which forces them to select a
+computationally feasible subset A′ ⊂ A; the selection of a
+good subset of adversarial attacks under budget restrictions,
+however, is a question orthogonal to our current article.
+An adversary executes an attack strategy as follows. First,
+the adversary uses A to generate candidate perturbations
+CandA(x, M) = {a(x, M) | a ∈ A}
+Out of the candidates, the adversary filters the ones x′ ∈
+CandA(x, m) where Succ(x′) = True∧¬Det(x′) = True.
+Filtering for Succ(x′) is straightforward if the adversary can
+access the model predictions. For instance, if the adversary
+is interested in an untargeted misclassification, e.g., chang-
+ing the prediction from “copyright-protected” images to any
+other class, the success filter is
+CandSucc
+A
+(x, M) = {x′ ∈CandA(x, M)|M(x′)̸=M(x)}
+where the sample X are images that initially are classified
+as M(x) = “copyright-protected”.
+However, filtering is not possible for an adversary because it
+requires access to ¬Det(x′), e.g., a human in the loop. If an
+adversary would have access to the detector, then there is no
+need to use adversarial attacks that minimize perturbation
+sizes. The adversary could instead directly optimize for
+evading the detector with adaptive attacks (Tramer et al.,
+2020). To minimize the chance of being detected without ac-
+cess to ¬Det(x′), the adversary thus makes an assumption:
+small perturbations are less likely to be detected than large
+ones. The adversary then uses the perturbation size as a
+proxy for detectability and selects the smallest perturbation
+per observation among all successful perturbations. This
+Algorithm 1: Monte Carlo Simulation of P dam
+Input
+:X, M, A, Det, d
+Output : �P dam
+1 r ← 0
+2 for x ∈ X do
+▷ Outer Loop
+3
+dmin ← ∞
+4
+for a ∈ A do
+▷ Inner Loop
+5
+x′ ← a(x, M)
+6
+if M(x′) ̸= M(x) ∧ d(x′, x) < dmin then
+7
+x′′ ← x′, dmin ← d(x′, x)
+8
+end
+9
+end
+10
+if x′′ ̸= x and ¬Det(x′′) then
+11
+r ← r + 1
+12
+end
+13 end
+14 return
+r
+|X|
+means that an operator has to assume the worst-case, i.e.,
+the smallest successful perturbation for each observation.
+A difficulty for the adversary, and in turn also for the simu-
+lation, is to select a distance metric to measure perturbation
+size that correlates well with the chance of detection. A
+common choice are Lp metrics, but there is an active debate
+about which metrics align well with human perception, (see
+for instance Zhang et al., 2018). With this in mind, we can
+now introduce a formal definition of the attack strategy:
+Definition 4.1 (Attack Strategy). An attack strategy is a
+function
+ΠM
+A (x)=
+�
+�
+�
+arg min
+x′∈CandSucc
+A
+(x,M)
+d(x′, x), if CandSucc
+A
+(x, M)̸=∅
+x
+otherwise,
+of type ΠM
+A : X → X that returns the smallest perturbation
+obtained by applying a set of applicable adversarial attacks
+A to an observation x given a distance metric d.
+We say an attack strategy is successful if ΠM
+A (x) ̸= x. Since
+an attack strategy relies on empirical evaluations, it yields
+an upper bound on the minimum perturbation required for
+any attack strategy to be successful.
+Definition 4.2 (Smallest Upper Bound on Perturbation Size).
+The smallest upper bound on the perturbation size obtained
+by a successful attack strategy is
+dA(x, M) =
+�
+d(ΠM
+A (x), x), if ΠM
+A (x) ̸= x
+∞
+otherwise,
+where d is a distance metric on X.
+We can now define an algorithm to find an estimate for P dam
+by Monte Carlo Simulation. Algorithm 1 illustrates the idea:
+
+Selecting Models based on the Risk of Damage Caused by Adversarial Attacks
+iterate over observations in a sample X (Line 2) and indi-
+vidual attacks of an attack strategy A (Line 4), measure the
+attack success (Line 6) and return the ratio of observations
+for which an attack strategy was successful, i.e., the ones
+that are not detected (Line 10). The result is an unbiased
+and consistent estimate of the probability of attack success
+for an attack strategy.
+4.2. Estimating the Probability of Detection
+Algorithm 1 reveals the two limitations that we introduced
+earlier. First, samples sizes |X| must be small if evaluating
+Det(x′) is costly (Small Sample Size). This, in turn, means
+that the sample size and the number of attacks to evaluate
+the success of an attack strategy must be small. The small
+sample sizes limit the quality of the estimate. Second, the
+evaluation of Det occurs after generating x′. Since x′ de-
+pends on M, the result of the estimation is model-dependent
+(Model-Specific Estimates). One must repeat the estimation
+of P dam for each model. We now show how one can over-
+come both limitations.
+4.2.1. OVERCOMING SMALL SAMPLE SIZE
+One way to overcome Small Sample Size is to substitute
+the detector with a surrogate function that is inexpensive to
+evaluate. Specifically, if there is a function F that substitutes
+¬Det, such that F(x) = ¬Det(ΠM
+A (x)), one can use F
+instead of ¬Det to evaluate the detector during a Monte
+Carlo Simulation. We can express this in commutative
+diagram notation as
+ΠM
+A (x)
+¬Det(ΠM
+A (x))
+¬Det
+F
+Finding a suitable F is difficult since the domain of F is
+the high-dimensional observation space. In particular, es-
+timating a function in a high-dimensional space may still
+require large sample sizes. However, under the assumption
+that the detection probability correlates well with the pertur-
+bation size, one can first map x to a distance and then use
+the distance as the domain of the detection function. With
+this assumption, the commutative diagram changes to
+dA(x)
+x
+ΠM
+A (x)
+¬Det(ΠM
+A (x))
+F ′
+¬Det
+where F ′ is a function of type F ′ : R → {0, 1}, and the
+image of dA the distance space induced by a metric d.
+We further say that such an F ′ has the point-wise detection
+commutation property if
+∀x ∈ X : Succ(ΠM
+A (x)) ⇒ F ′(dA(x)) = ¬Det(ΠM
+A (x)).
+Such an F ′ may not exist.
+Think of two observations
+x, ˜x ∈ X, x ̸= ˜x with x′ = ΠM
+A (x) and ˜x′ = ΠM
+A (˜x),
+where dA(x) = dA(˜x) but ¬Det(x′) ̸= ¬Det(˜x′), i.e.,
+both observations have the same upper bound on the per-
+turbation size. Then, there is no F ′ that has the detection
+commutation property.
+Fortunately, we do not require a point-wise detection com-
+mutation. Since �P dam is a probabilistic estimate, it suffices
+that F ′ is equivalent to the probability of detection.
+Definition 4.3 (Probabilistic Detection Commutation Prop-
+erty). We say a function F ′ : R → R has the probabilistic
+detection commutation property if
+Px∼X ′(¬Det(ΠM
+A (x))) = Ex∼X ′(F ′(dA(x))),
+(6)
+where X ′ = {x ∈ X : Succ(ΠM
+A (x))}.
+Given an F ′ that has the probabilistic detection commu-
+tation property, one can obtain P dam using the cumulative
+distribution function of dA, the Attack Success Distribution.
+Definition 4.4 (Attack Success Distribution).
+ASDA,M(τ) := Px∼AdvM(X)(dA(x, M)≤ τ), τ ≥ 0 (7)
+Formally, this leads to the following theorem.
+Theorem 4.5. Let F ′ : R → R be a function that fulfills
+the probabilistic detection commutation property. Then
+P dam =
+� ∞
+0
+F ′(τ) dASDA,M(τ) dτ.
+Proof. See Appendix A.
+Using Theorem 4.5, the key to an estimation of P dam is an
+estimation of ASDA,M. Given a sample X ∼ X, one can
+approach the Attack Success Distribution with the following
+empirical distribution function (cf. Dong et al., 2020).
+Definition 4.6 (Attack Success Ratio).
+ASRA,M(τ) = |{x ∈ X | dA(x, M) ≤ τ}|
+|X|
+Empirical distribution functions are unbiased and consis-
+tent estimators of their distribution functions. Applying
+this general property to ASRA,M allows us to formulate an
+unbiased and consistent estimator for P dam.
+Theorem 4.7. The estimator
+�P dam =
+� ∞
+0
+F ′(τ) dASRA,M(τ) dτ
+=
+1
+|X|
+�
+x∈X,dA(x)̸=∞
+F ′(dA(x))
+(�)
+is an unbiased, consistent estimator of P dam.
+Proof. See Appendix A.
+
+Selecting Models based on the Risk of Damage Caused by Adversarial Attacks
+4.2.2. OVERCOMING MODEL-SPECIFIC ESTIMATES
+What is left to discuss is how to construct an F ′ that fulfills
+the probabilistic commutation property. A natural choice
+is to use the probability of detection conditioned on τ. In-
+tuitively, this probability is the ratio of observations with
+dA(x) = τ that one expects to be detected. Formally, we
+define this as a probability function.
+Definition 4.8 (Detection Probability Function).
+ΨA,M(τ) = Px∼X (¬Det(ΠM
+A (x)) | dA(x) = τ)
+With a suitable distance measurement, we can assume that
+the perturbation size statistically determines the detection
+probability function. In this case, the detection probability
+does not depend on the choice of a model, i.e.,
+ΨA,M1(τ) = Px∼X (¬Det(ΠM1
+A (x)) | dA(x) = τ)
+= Px∼X (¬Det(ΠM2
+A (x)) | dA(x) = τ)
+= ΨA,M2(τ).
+where M1 ̸= M2. One can show that ΨA,M is indeed a
+suitable choice for F ′.
+Theorem 4.9. The Detection Probability Function has the
+probabilistic detection commutation property.
+Proof. See Appendix B.
+Combining Theorem 4.9 with Theorem 4.5 gives
+Corollary 4.10. The probability of damage P dam is
+P dam =
+� ∞
+0
+ΨA,M(τ) dASDA,M(τ) dτ
+Furthermore, the estimator
+�P dam =
+� ∞
+0
+ΨA,M(τ) dASRA,M(τ) dτ
+=
+1
+|X|
+�
+x∈X,dA(x)̸=∞
+ΨA,M(dA(x))
+is an unbiased, consistent estimator of P dam.
+A useful implication of Corollary 4.10 is that one can further
+refine ΨA,M to include prior knowledge and assumptions
+in the calculation of P dam. For instance, recall that an ad-
+versary uses an attack strategy ΠM
+A to find the smallest
+perturbation for an observation. The rationale is that the ad-
+versary expects that the chances of being detected increase
+with the perturbation size. With this assumption, one can
+simplify ΨA,M to monotonic non-decreasing functions or
+even a logistic curve. One must then only query the detector
+with a data sample and then fit a posterior for ˆΨA,M, e.g.,
+with Bayesian estimation, see Appendix C. A further take-
+away from this section is that the estimate ˆΨA,M depends
+on the application but not on a specific model. Think of our
+upload filter example. Upload filters are used in different ap-
+plications, e.g., to detect copyright infringement of portrait
+photographs (Application A) and violent content in pictures
+(Application B). For each application, one must estimate
+a detection probability function: ˆΨphoto
+A,M for Application A
+and ˆΨviolent
+A,M
+for Application B. However, within Applica-
+tion A, ˆΨphoto
+A,M can be used to estimate the risk for different
+copyright detection models, e.g., trained with different hy-
+perparameter settings. Likewise, ˆΨviolent
+A,M
+can be used to
+estimate the risk of different violence detection models. This
+reduces the number of detection probability functions one
+has to fit by querying a detector from one per model to one
+per application. Thus, the independent estimation of the
+detection probability function mitigates both Small Sample
+Size and Model-Specific Estimates, and hence reduces the
+effort for risk estimation.
+4.3. Estimations without a Detector
+In some cases, collecting a sample from a detector to fit
+ΨA,M is infeasible. For instance, think of academic bench-
+marks that compare adversarial attacks or defense methods.
+There, querying a human detector is often beyond the scope
+of the study. One may not even have an indication of which
+magnitude of perturbation size is actually required for a
+detection to be successful. In such a case, selecting and
+estimating a suitable detection probability function is not
+possible.
+However, one can still obtain a relative comparison of the
+operational risk between models. One way is to assume that
+the average sensitivity of models to adversarial perturbations
+is similar to the sensitivity of a potential detector. Formally,
+this gives an average detection function
+ˆΨavg
+A,M(τ) = 1 − 1
+J
+J
+�
+j=1
+ASRA,Mj(τ)
+(8)
+where M1, M2, . . . , Mj, . . . , MJ are the models to com-
+pare. Intuitively, using ˆΨavg
+A,M as a detection function means
+that models more sensitive to adversarial examples than the
+average will obtain a high operational risk.
+If all ASRA,MJ estimates are based on the same sample
+X, one can rearrange Equation 8 to make its computation
+efficient. We first define W(τ) to count the combination
+of observations and models where the adversarial example
+with the smallest perturbation size is further away than τ.
+W(τ) = |{(i, j) | i ∈ [I], j ∈ [J], dA(xi, Mj) > τ}| .
+We then have
+
+Selecting Models based on the Risk of Damage Caused by Adversarial Attacks
+0.0
+0.1
+0.2
+0.3
+8
+255
+2
+255
+Perturbation Size (L∞)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ASR
+Baseline
+Engstrom-Robust
+Rice-Overfit
+Carmon-Semi
+Calculated ΨA,M
+Figure 1. Solid, colored: ASR of the different models on 200
+observations. Dashed, black: ΨA,M estimated according to Sec-
+tion 4.3. Solid, black: Vertical lines marking L∞ ∈ {
+2
+255,
+8
+255}.
+ˆΨavg
+A,M(τ) = 1 −
+�
+j=1...J ASRA,Mj(τ)
+J
+= 1
+J
+�
+� �
+j=1...J
+(1 − ASRA,Mj(τ))
+�
+�
+=
+1
+|X| · J
+�
+� �
+j=1...J
+(|X| − |{x ∈ X | dA(x) ≤ τ}|)
+�
+�
+=
+1
+|X| · J
+�
+j=1...J
+|{x ∈ X | dA(x) > τ}|
+=
+1
+|X| · J W(τ)
+With Corollary 4.10, we have
+�P dam =
+1
+|X|2 · J
+�
+j=1...J
+W(dA(xi, Mj))
+(9)
+Algorithm 2 in Appendix D summarizes the estimation of
+�P dam using Equation 9. The algorithm helps identify the
+model with the lowest probability of damage even if collect-
+ing a sample from a detector is infeasible.
+5. Experiments
+This section demonstrates that �P dam provides a consistent
+and unbiased evaluation to compare the robustness of ma-
+chine learning models without the need to choose a threshold
+on the perturbation size. Our experimental setup is represen-
+tative of how academic benchmarks comparing adversarial
+robustness are typically constructed. Our goal is to under-
+line the usefulness of our metric in common setups using
+open-source models and attack implementations. Hence, our
+choice of models, attacks, and parametrization is arbitrary
+and can be replaced with any other use-case.1
+1Our
+implementations
+and
+results
+are
+available
+at
+https://github.com/duesenfranz/risk_scores_
+paper_code.
+Table 1. Summary statistics of the model robustness over 200
+observations. Smaller values are better for all metrics but MPS.
+The best values are highlighted in bold.
+Model
+�P dam
+ASR
+� 2
+255
+�
+ASR
+� 8
+255
+�
+MPS
+Baseline
+0.76
+0.70
+1.00
+0.00018
+Engstrom-Robust
+0.44
+0.16
+0.48
+0.00020
+Rice-Overfit
+0.43
+0.20
+0.42
+0.00119
+Carmon-Semi
+0.33
+0.13
+0.33
+0.00095
+5.1. Setup
+We compare the operational risk of publicly available mod-
+els on CIFAR-10 (Krizhevsky et al., 2009).
+Models. The list of models includes a baseline classifier and
+three other models trained to achieve high adversarial ro-
+bustness. We obtained all models from public repositories.2
+Baseline (Croce et al., 2021): A baseline model trained
+without a specific focus on robustness.
+Carmon-Semi (Carmon et al., 2019): A model trained
+for robustness with a semi-supervised learning method.
+Engstrom-Robust (Engstrom et al., 2019): A model
+trained for robustness by adversarial training.
+Rice-Overfit (Rice et al., 2020): A model trained for
+robustness by a combination of adversarial training and
+a focus on minimizing overfitting.
+Attacks. We define an attack strategy based on a set of
+attacks A based on Foolbox (Rauber et al., 2020), an open-
+source attack library. We use Projected Gradient Descent
+(PGD), PGD with Adam optimizer, and DeepFool.
+Parametrization. We use RobustBench (Croce et al., 2021)
+to run attacks on observations in the test set. We instantiate
+each attack with eight different values for epsilon, i.e., |A| =
+24, set d = d∞, and consider a perturbation successful if
+the model prediction does not agree with the ground truth.
+Metrics. Next to our metric �P dam, we compute three alter-
+native metrics. We compute ASR (τ), with two thresholds
+τ ∈ { 2
+255, 8
+255}: the fraction of observations with at least
+one successful adversarial example with perturbation size
+L∞ ≤ τ. ASR (τ) is the maximum likelihood estimator for
+the adversarial risk using the 0-1 loss (cf. Section 3). The
+threshold
+8
+255 is arbitrary but common (e.g., see Rice et al.,
+2020). Another common approach is to estimate the popu-
+lation parameters of the perturbation sizes dA(x), x ∈ X,
+(e.g., see Carlini et al., 2019). This approach is not reliable
+since estimates are either not robust to outliers (e.g., the
+average over dA(x), see Appendix E) or biased by nature
+(e.g., the median). To demonstrate the issue of relying on
+perturbation size as a metric, we compute the size of the
+smallest perturbation that alters the model prediction, the
+2All models can be downloaded using RobustBench (Croce
+et al., 2021).
+
+Selecting Models based on the Risk of Damage Caused by Adversarial Attacks
+25
+50
+75
+100
+125
+150
+175
+200
+Number of Observations
+0.0000
+0.0025
+0.0050
+0.0075
+0.0100
+0.0125
+0.0150
+0.0175
+MPS
+Baseline
+Engstrom-Robust
+Rice-Overfit
+Carmon-Semi
+50
+100
+150
+200
+Number of Observations
+0.00
+0.01
+0.02
+MPS
+(a) MPS with increasing sample size.
+50
+100
+150
+200
+Number of Observations
+0.00
+0.25
+0.50
+0.75
+�P dam
+(b) �P dam with increasing sample size.
+Figure 2. Convergence of robustness statistics. Each line plots a statistic measured on n ∈ [20, 200] observations. The error band is the
+5% and the 95% percentile, calculated by sampling 50 times with replacement.
+Minimal Perturbation Size (MPS). MPS is non-robust to
+outliers and biased.
+5.2. Results
+Figure 1 plots the attack success rate (ASR) against the per-
+turbation size. Since this is an academic benchmark, we do
+not have access to a human detector in the loop. Hence, we
+proceed as outlined in Section 4.3 to calculate ΨA,M. Based
+on Figure 1, an operator would choose Carmon-Semi, the
+model with the lowest ASR for all perturbation sizes. The
+second best is Rice-Overfit since the ASR (green line)
+is lower than the one of Engstrom-Robust (yellow line)
+on most of the perturbation spectrum. The Baseline per-
+forms poorly: an adversary can find successful perturbations
+for each observation even for a small perturbation budget.
+Table 1 summarizes the ASR plot with the metrics �P dam,
+ASR
+� 2
+255
+�
+, ASR
+� 8
+255
+�
+and MPS. The selection based on
+�P dam corresponds to the visual inspection: Carmon-Semi
+is the model with the lowest value ( �P dam = 0.33), followed
+by Rice-Overfit ( �P dam = 0.43). A benefit of �P dam is
+that it remains actionable even if there are too many models
+for a visual inspection.
+The ASR(τ) metrics, however, contradict each other.
+ASR
+� 8
+255
+�
+suggests that Rice-Overfit is more robust
+than Engstrom-Robust; ASR
+� 2
+255
+�
+suggests the oppo-
+site. Further, ASR (τ) = 1.0 for all models if the perturba-
+tion size is unconstrained, i.e., τ → ∞. Thus selecting any
+threshold on ASR is arbitrary, and results are volatile.
+MPS suggests that Rice-Overfit is the most robust
+model. However, MPS is biased: with increasing sample
+size, its value approaches the true size of the smallest ad-
+versarial example, which is close to 0 for all models, see
+Figure 2(a). Indeed, if adversaries can choose from an infi-
+nite number of observations, they likely find an adversarial
+example with a very small perturbation. MPS has high
+variance for small sample sizes n < 100, i.e., results are
+insignificant. On the other hand, �P dam is consistent and un-
+biased, i.e., the metric converges to its true expected value
+with increasing sample size, see Figure 2(b).
+In summary, our experimental results confirm our theoretical
+analyses. Neither ASR (τ) nor MPS are reliable metrics for
+selecting a robust model. �P dam allows for relative model
+robustness comparisons, even when no detector is available.
+6. Conclusions
+Estimating the damage caused by adversarial attacks is diffi-
+cult. Standard Monte Carlo Simulation is inefficient because
+it is limited to small sample sizes and model-specific esti-
+mates. A consequence is that resulting metrics do not give
+reliable estimates of model robustness. This prevents opera-
+tors of machine learning models from translating high-level
+regulations on AI safety and security into actionable techni-
+cal requirements.
+In this article, we put forward an original approach to quan-
+tifying the risk of adversarial attacks that overcomes current
+limitations. To this end, we first decompose the damage
+caused by adversarial attacks into the probability that an
+attacker is successful and the probability that an attack goes
+undetected. We then propose an unbiased and consistent
+estimator for both quantities. For cases where one does not
+have access to a detector, we provide an alternative method
+that allows comparing the risk between models. The results
+are interpretable statistical estimates that provide an empiri-
+cal basis for operators to select the model with the least risk
+of damage from adversarial attacks.
+Acknowledgments
+This work was partially funded by the German BMWI
+project KI-LOK.
+
+Selecting Models based on the Risk of Damage Caused by Adversarial Attacks
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+Selecting Models based on the Risk of Damage Caused by Adversarial Attacks
+Appendix
+A. Proof of Theorem 4.5 and Theorem 4.7
+Proof. Starting with Equation 5, we have
+P dam = Px∼D(Succ(ΠM
+A (x)), ¬Det(ΠM
+A (x)))
+= Px∼D(¬Det(ΠM
+A (x)) | Succ(ΠM
+A (x)))
+× Px∼D(Succ(ΠM
+A (x)))
+= Ex∼D(F ′(dA(x)) | Succ(ΠM
+A (x)))
+× Px∼D(Succ(ΠM
+A (x)))
+= Ex∼D(
+�
+F ′(dA(x)), if dA(x) < ∞
+0
+otherwise
+)
+(*)
+=
+� ∞
+0
+F ′(τ) dASDA,M(τ) dτ,
+where the last line is because ASD is the distribution func-
+tion of dA and because of the Law of the Unconscious
+Statistician (LOTUS). What remains to be proven is the
+unbiasedness and consistency of �P dam and the equality in
+Line �. The latter is a direct result of the equality
+ASRA,M =
+1
+|X|
+�
+x∈X,dA(x)̸=∞
+χ[dA(x),∞),
+where χ denotes the characteristic function. For unbiased-
+ness and consistency, note that �P dam in Line � is the sample
+mean of the expected value in Line *, and therefore an
+unbiased and consistent estimator of P dam.
+B. Proof of Theorem 4.9
+Proof. We have to prove that
+Px∼D′(¬Det(ΠM
+A (x))) = Ex∼D′(ΨA,M(dA(x))).
+With the Law of Total Probability, we have
+Px∼D′(¬Det(ΠM
+A (x)))
+= E˜x∼D′(Px∼D′(¬Det(ΠM
+A (x)) | dA(x) = dA(˜x)))
+= E˜x∼D′(ΨA,M(dA(˜x)))
+= Ex∼D′(ΨA,M(dA(x)))
+C. Sketch of Fitting a Detection Probability
+Function ΨA,M with Logistic Regression
+Figure 3 sketches how a detection probability function can
+be estimated with logistic regression using only 30 samples.
+D. Algorithm for �P dam without a Detector
+Algorithm 2 calculates P dam if no detector is available, see
+Equation 9. The algorithm consists of two parts:
+0
+2
+4
+6
+8
+10
+d(x′, x)
+0.0
+0.5
+1.0
+¬Det(x′)
+Figure 3. Sketch of fitting a detection probability function ΨA,M
+with logistic regression.
+Blue:
+Scatterplot for d(x′, x) =
+d(a(x, M), x), a ∈ A and ¬Det(x′).
+Red: Estimate ˆΨA,M
+by logistic regression.
+25
+50
+75
+100
+125
+150
+175
+200
+Number of Observations
+0.0000
+0.0025
+0.0050
+0.0075
+0.0100
+0.0125
+0.0150
+0.0175
+MPS
+Baseline
+Engstrom-Robust
+Rice-Overfit
+Carmon-Semi
+50
+100
+150
+200
+Number of Observations
+0.00
+0.05
+0.10
+mean(dA(x))
+Figure 4. Convergence of the average perturbation size. Each line
+plots the average perturbation size, measured on n ∈ [20, 200]
+observations. The error band is the 5% and the 95% percentile,
+calculated by sampling n observations 50 times with replacement.
+Part A calculates the smallest perturbation sizes dA(x) of
+the adversarial examples for all models and observa-
+tions x and collects them in two arrays:
+(a)
+A sorted array W which contains the smallest suc-
+cessful perturbations of all models.
+(b)
+A double array ASR, which contains the smallest
+successful perturbations organized into subarrays
+per model.
+Part B computes �P dam of each model Mi by comparing
+ASR[i] with W: It sums the indices in W of the small-
+est successful perturbations for the model Mi. It then
+divides the result by the square of the number of obser-
+vations and by the number of models.
+Finally, the algorithm returns �P dam.
+
+Selecting Models based on the Risk of Damage Caused by Adversarial Attacks
+Algorithm 2: Estimation of P dam without Detector
+Input
+:x1, . . . , xI, M1, . . . , MJ, A, d
+Output : �P dam for M1, M2, . . . , MJ
+1 W ← []
+2 ASR ← []
+3 �P dam ← []
+4 for j = 1 . . . J do
+▷ Part A
+5
+dmin ← ∞
+6
+ASR[j] ← []
+7
+for i = 1 . . . I do
+8
+for a ∈ A do
+9
+x′ ← a(xi, Mj)
+10
+if M(x′) ̸= Mj(xi) ∧ d(x′, xi) < dmin
+then
+11
+x′′ ← x′
+12
+dmin ← d(x′, x)
+13
+end
+14
+end
+15
+if x′′ ̸= xi then
+16
+push(W, d(x′′, xi))
+17
+push(ASR[j], d(x′′, xi))
+18
+else
+19
+push(W, ∞)
+20
+end
+21
+end
+22 end
+23 sort descending(W)
+24 for j = 1 . . . J do
+▷ Part B
+25
+�P dam[j] ← 0
+26
+for τ ∈ ASR[j] do
+27
+�P dam[j] ← �P dam[j] + index(W, τ)
+28
+end
+29
+�P dam[j] ←
+�
+P dam[j]
+I2·J
+30 end
+31 return �P dam
+E. Average Perturbation Size with Increasing
+Sample Size
+Figure 4 plots the average perturbation size calculated on a
+subset of the 200 observations. The non-robustness to out-
+liers causes the wide error bands, which leads to an overlap
+of the percentiles of Carmon-Semi and Rice-Overfit
+for estimates based on fewer than 50 observations. In con-
+trast, �P dam yields statistically significant results even for
+estimates on fewer than 50 observations, see Figure 2.
+

99FST4oBgHgl3EQfbzgH/content/tmp_files/2301.13800v1.pdf.txt

@@ -0,0 +1,1623 @@
+arXiv:2301.13800v1  [math.LO]  31 Jan 2023
+A monotone connection between model class size
+and description length
+Reijo Jaakkola
+Tampere University
+Finland
+Antti Kuusisto
+Tampere University
+University of Helsinki
+Finland
+Miikka Vilander
+Tampere University
+Finland
+Abstract
+This paper links sizes of model classes to the minimum lengths of their
+defining formulas, that is, to their description complexities. Limiting
+to models with a fixed domain of size n, we study description com-
+plexities with respect to the extension of propositional logic with the
+ability to count assignments. This logic, called GMLU, can alternati-
+vely be conceived as graded modal logic over Kripke models with the
+universal accessibility relation. While GMLU is expressively complete
+for defining multisets of assignments, we also investigate its fragments
+GMLU(d) that can count only up to the integer threshold d. We fo-
+cus in particular on description complexities of equivalence classes of
+GMLU(d).
+We show that, in restriction to a poset of type realiza-
+tions, the order of the equivalence classes based on size is identical to
+the order based on description complexities. This also demonstrates
+a monotone connection between Boltzmann entropies of model classes
+and description complexities. Furthermore, we characterize how the
+relation between domain size n and counting threshold d determines
+whether or not there exists a dominating class, which essentially means
+a model class with limit probability one. To obtain our results, we
+prove new estimates on r-associated Stirling numbers. As another cru-
+cial tool, we show that model classes split into two distinct cases in
+relation to their description complexity.
+1
+Introduction
+This paper investigates how sizes of model classes are linked to the minimum
+lengths of formulas needed to define the classes. In the scenarios we consider,
+we first fix a class M of models that share a domain of the same finite size n.
+The model classes M ⊆ M are then studied with respect to the extension
+of propositional logic with the ability to count propositional assignments.
+We call this logic GMLU, as it can alternatively be defined as graded modal
+1
+
+logic over Kripke models with the universal relation. In order to obtain more
+fine grained results, we parameterize GMLU and study also its fragments
+GMLUd that can count only up to the threshold d ∈ Z+. This also enables us
+to demonstrate how the relationship between minimum formula lengths and
+model class sizes develops when we gradually increase the expressive power
+of the logic used. For a model class M ⊆ M, the description complexity of
+M with respect to GMLUd is simply the minimum length of a formula of
+GMLUd needed to define M, if such a formula exists.
+In this paper we are particularly interested in the description complexi-
+ties Cd(M) of the logical equivalence classes M determined by GMLUd over
+M. Let us write M ≡d M′ if the models M, M′ ∈ M satisfy the same set of
+formulas of GMLUd. Note that Cd(M) of an equivalence class M of ≡d can
+also be regarded as the description complexity of each model M ∈ M, as the
+expressive power of GMLUd suffices precisely to describe M up to the equi-
+valence ≡d. From this perspective, description complexity is analogous to
+Kolmogorov complexity. There exist well known links between Kolmogorov
+complexity and Shannon entropy, see for example [11]. The recent work in
+[8],[7] demonstrates a way to conceive related results also in the scenario
+where relational structures are classified via logics. In particular, it is shown
+that the expected Boltzmann entropy of the equivalence classes of GMLU
+is asymptotically equivalent to the expected description complexity (with
+respect to GMLU) times the size of the vocabulary considered. It is also
+shown that for d = 1, the greatest equivalence class of GMLUd has maxi-
+mum description complexity among the classes. This paper builds on those
+results.
+Firstly, as a crucial tool for our proofs, we establish a classification of
+description complexities into two distinct classes.
+This division is based
+on the numbers ni of elements realizing different propositional types i ∈ I
+in models of the described model class; here I is just an index set for the
+types. The division is then determined by whether or not ni = d for at least
+two different types. Using this, we establish a strong connection between
+model class sizes and description complexities. For each model M ∈ M, let
+nM denote the tuple (ni)i∈I that gives the numbers ni of points realizing
+propositional types in M. Furthermore, instead of recording numbers ni
+greater than the counting threshold d, simply put d in nM. We define a
+poset (M, ⪯τ) over the models, where τ is the vocabulary and the order
+⪯τ is based on comparing the tuples nM coordinatewise. The order ⪯τ is
+directly inherited also by the classes of ≡d such that M ⪯τ M′ if and only
+if for some (or equivalently, all) models M and M′ in the respective classes,
+we have M ⪯τ M′. We will prove that for all classes M and M′ of ≡d such
+that M and M′ are ⪯τ-comparable, we have
+|M| < |M′| ⇔ Cd(M) < Cd(M′).
+2
+
+In other words, over ⪯τ, the ordering of model classes according to size is
+identical to the ordering based on description complexity. This is an intimate
+link between syntax and semantics. As a corollary, we obtain a correspon-
+ding relationship between Boltzmann entropies and description complexities
+of model classes.
+We then investigate how the classes of ≡d behave when we alter the
+domain size n and counting threshold d. Note that increasing d corresponds
+to moving to more and more expressive logics. First we observe that for
+thresholds d and d′ > d and the corresponding Shannon entropies HS(≡d)
+and HS(≡d′) of the model class distributions given by ≡d and ≡d′, we have
+HS(≡d) < HS(≡d′) when d′ is at most n/2, and
+HS(≡d) = H(≡d′) when d is at least n/2.
+A similar result also follows for expected Boltzmann entropies HB(≡d) and
+HB(≡d′), but with the orders reversed, that is HB(≡d) > HB(≡d′) for d′ at
+most n/2.
+To get a better sense of the relative sizes of the classes when n and d are
+altered, we prove an asymptotic characterization of the class distributions
+as n → ∞ and d is a function of n. Let us say that ≡d(n) has a dominating
+class if with limit probability one, a random model of size n belongs to a
+maximum size class in ≡d(n). Similarly, all classes in ≡d(n) are vanishing if
+with limit probability zero, a random model of size n belongs to a maximum
+size class. Then the following results hold as n → ∞.
+• If d(n) ≤ n/2|τ| − f(n) where f(n) = ω(√n), then ≡d(n) has a domi-
+nating class.
+• If d(n) ≥ n/2|τ| − f(n) where f(n) = o(√n), then ≡d(n) has no domi-
+nating class.
+• If d(n) ≥ n/2|τ| + f(n) where f(n) = ω(√n), then every class in ≡d(n)
+is vanishing.
+One corollary of these results is that for d(n) ≤ n/t−f(n), if f(n) = ω(√n),
+then with limit probability one, two random models of size n cannot be
+separated in GMLUd(n).
+Finally, we also give a non-asymptotic variant
+of the characterization of the class distributions for ≡d including explicit
+bounds on d for separating the cases where ≡d has a majority class or not.
+By a majority class, we mean a class containing more than half of all models
+in M.
+Concerning related work, as already mentioned, it is well known that
+entropy and Kolmogorov complexity are related. Indeed, for computable
+distributions, Shannon entropy links to Kolmogorov complexity to within
+a constant. This result is discussed, e.g., in [11], [4], [10]. However, it is
+shown in [15] that the general link fails for R´enyi and Tsallis entropies. See
+3
+
+for example [4], [10], [15] for R´enyi and Tsallis entropies. The first connec-
+tion between logical formula length and entropy has—to our knowledge—
+been obtained in [8], [7], where expected Boltzmann entropy is shown to be
+asymptotically equivalent to description complexity.
+Concerning further related work, we will next discuss the proof tech-
+niques used in the current paper. For proving bounds on formula sizes, we
+use formula size games for the logics GMLUd.
+Indeed, variants of stan-
+dard Ehrenfeucht-Fra¨ıss´e games and (graded) bisimulation games would not
+suffice, as we need to deal with formula length, and thereby with all logi-
+cal operators, including connectives. The formula size games for the logics
+GMLUd will be developed below based on a similar game used in [8], [7].
+That game builds on the game for standard modal logic ML used and devel-
+oped in [5] for proving a nonelementary succinctness gap between first-order
+logic and ML. The first formula size game, developed by Razborov in [13],
+dealt with propositional logic. A later variant of the game was defined by
+Adler and Immerman for CTL in [1]. Designing the games for GMLUd is
+relatively straightforward and based directly on similar earlier systems, but
+using them requires some nontrivial combinatorial arguments.
+In addition to games, we also make use of a range of techniques for esti-
+mating model class sizes and description complexity. These include Stirling’s
+approximations and Chernoff bounds. In particular, to obtain our results,
+we prove new estimates on r-associated Stirling numbers, which may be of
+independent interest.
+As a brief summary of our paper, the main objective is to elucidate the
+general picture of how description length relates to model class size. This
+also builds links between logic and notions of entropy. The logic GMLU is
+suitable for the current study, and it even allows simple access to a chain
+of increasingly expressive logics GMLUd via increasing d. The concluding
+section discusses possibilities for generalizing to further logics. While the
+current paper focuses on theory, the notion of description complexity is also
+relevant in a range of applications. For example, in some currently active
+research on explainability in AI, minimal length specifications can be used
+as explanations of longer formulas. For work on this topic see, e.g., [2], [6].
+The plan of the paper is as follows. After the preliminaries in Section
+2, we prove crucial lower bounds for description complexity in Section 3
+using games. In Section 4 we prove a monotone connection between model
+class size and description complexity, and in Section 5 we investigate phase
+transitions of class size distributions by varying n and d. Section 6 concludes
+the paper.
+4
+
+2
+Preliminaries
+We first define the logics studied in this work.
+Let τ be a finite set of
+proposition symbols. We consider τ to be fixed throughout the entire paper.
+The syntax of graded universal modal logic GMLU[τ] is generated as
+follows (the syntactic choices will be explained later on):
+ϕ :=♦≥kψ | ■<kψ | ♦=kψ | ■̸=kψ |
+ϕ ∧ ϕ | ϕ ∨ ϕ | ♦≥kϕ | ■<kϕ | ♦=kϕ | ■̸=kϕ
+ψ :=p | ¬p | ψ ∧ ψ | ψ ∨ ψ
+Here p ∈ τ and k ∈ N. Note that the formulas of GMLU[τ] have proposition
+symbols only in the scope of modal operators. Furthermore, all formulas are
+given in negation normal form. In the current paper, ¬ϕ will always mean
+a formula where ¬ has been pushed all the way to the level literals.
+Let M be a Kripke model with domain W. In this paper, modal logics
+will always have a unary vocabulary, so therefore Kripke models will not be
+associated with a binary accessibility relation. We define the semantics of
+the global graded modalities as follows: (M, w) ⊨ ♦≥kϕ ⇔ there exist at
+least d elements v ∈ W such that (M, v) ⊨ ϕ and (M, w) ⊨ ♦=kϕ ⇔ there
+exist exactly d elements v ∈ W such that (M, v) ⊨ ϕ. Additionally, (M, w) ⊨
+■<kϕ ⇔ (M, w) ⊨ ¬♦≥k¬ϕ and (M, w) ⊨ ■̸=kϕ ⇔ (M, w) ⊨ ¬♦=k¬ϕ. The
+semantics of the Boolean connectives ¬, ∧, ∨ is defined in the usual way.
+Notice that ♦≥k and ■<k as well as ♦=k and ■̸=k are dual to each other.
+Thus the modalities of GMLU are the diamonds ♦≥k, ♦=k and their duals
+■<k, ■̸=k. Intuitively ■<k (respectively, ■̸=k) means that all points satisfy
+ϕ, except for some number m < k (resp. m ̸= k) of exceptions.
+Let M be a Kripke model over τ and ϕ ∈ GMLU[τ]. The point-free truth
+relation is defined such that M ⊨ ϕ if and only if M, w ⊨ ϕ for all w ∈ W.
+As all proposition symbols occur in the scope of a global modality, M ⊨
+ϕ if and only if there exists some w ∈ W such that M, w ⊨ ϕ. Clearly
+truth of GMLU-formulas does not depend on the evaluation point w. This
+independence property is the reason behind the definition of the syntax of
+GMLU such that proposition symbols are guaranteed to be in the scope of
+modal operators.
+For a set M of pointed models, we denote M ⊨ ϕ ⇔
+(M, w) ⊨ ϕ for every (M, w) ∈ M.
+A propositional type π over τ is a maximally consistent set of literals
+(that is, proposition symbols and negated proposition symbols). Therefore
+π has exactly one of p, ¬p for each symbol p ∈ τ. We henceforth refer to
+propositional types as just types. The number of types over τ is denoted by
+t = 2|τ|.
+The counting depth of a formula ϕ ∈ GMLU[τ], denoted depth(ϕ), is
+defined as follows:
+• depth(α) = 0 for any literal α,
+5
+
+• depth(ϕ ∧ ψ) = depth(ϕ ∨ ψ)
+= max(depth(ϕ), depth(ψ)),
+• depth(♦≥kϕ) = depth(■<kϕ) = k,
+• depth(♦=kϕ) = depth(■̸=kϕ) = k + 1.
+We denote by GMLUd[τ] the counting depth d fragment of GMLU[τ],
+where the counting depth of formulas is restricted to at most d. The re-
+sults of this paper are formulated for the logics GMLUd[τ].
+Note that
+depth(♦=kϕ) = k + 1 while depth(♦≥kϕ) = k.
+We give some intuition
+to explain this choice. First of all, ♦=d−1ϕ ≡ ♦≥d−1ϕ ∧ ¬♦≥dϕ, so we see
+that when the counting depth of formulas is restricted to d, the allowed
+“exact counting” diamonds ♦=k always have k ≤ d − 1 and thus they add
+no expressive power over “threshold counting” diamonds ♦≥k with k ≤ d.
+Moreover, k + 1 also corresponds to the number of quantifiers required to
+express “exact counting” of k elements in monadic first-order logic.
+The size of a formula ϕ ∈ GMLU[τ], denoted size(ϕ), is defined as
+follows:
+• size(α) = 1 for any literal α,
+• size(ϕ ∧ ψ) = size(ϕ ∨ ψ) = size(ϕ) + size(ψ) + 1,
+• size(♦≥kϕ) = size(■<kϕ) = size(ϕ) + k,
+• size(♦=kϕ) = size(■̸=kϕ) = size(φ) + k + 1.
+Note that all literals have the same size. This is because we wish to consider
+negative (that is, negated) information and positive (that is, non-negated)
+information as equal in relation to formula size. This idea explains why we
+defined GMLU so that formulas are in negation normal form.
+Let M be the set of all τ-models with the fixed domain W = {1, . . . , n}.
+When M is clear from the context, a formula ϕ ∈ GMLUd is said to define
+a set M ⊆ M if for every M ∈ M, we have M ⊨ ϕ if and only if M ∈ M.
+The set M is then called GMLUd-definable.
+The GMLUd-description
+complexity C(M) of a GMLUd-definable set M is the minimum size of a
+formula ϕ ∈ GMLUd which defines M.
+We may write M ≡d N if the models M and N satisfy exactly the same
+GMLUd-formulas. The relation ≡d is clearly an equivalence relation and
+defines a natural related partition. For an example of description complexity,
+consider GMLU1[τ] for the case with the singleton alphabet τ = {p}. The
+model where p is true in every point constitutes a singleton class in the
+partition of models defined by ≡1. The description complexity of this class
+is 2 as a minimum size formula that defines the class is ■<1p.
+Let I := {1, . . . , t} and fix an enumeration (πi)i∈I of all the types over
+the propositional vocabulary τ. Now let M be an equivalence class of ≡d
+6
+
+over the set M of models of size n with domain W = {1, . . . , n}. For a type
+πi, all models in the class either have exactly ni points of type πi for some
+ni < d, or all the models in M have at least d points of type πi. In the latter
+case we define ni := d. We thus get a characterization of the classes of ≡d
+in terms of t-tuples n. A t-tuple n = (ni)i∈I is called (n, d)-admissible, if
+ni ≤ d for every i ∈ I, �
+i∈I ni ≤ n and either there is at least one i ∈ I
+such that ni = d or �
+i∈I ni = n. Note that (n, d)-admissible tuples n and
+equivalence classes of ≡d are in one-to-one correspondence. Thus we can
+write Mn for the class corresponding to n.
+Given an (n, d)-admissible tuple n and a type πi that has precisely the
+same number k of realizing points in every model M ∈ Mn, we denote this
+number k by |πi|n. (Note that k can be greater than d.) We will often omit
+the tuple n in the subscript of |πi|n when it is clear from the context.
+Following [7], we define the Boltzmann entropy of a class M as
+HB(M) := log(|M|). As discussed in [7], this terminology comes from sta-
+tistical mechanics, where Boltzmann entropy measures the randomness of a
+macrostate via the number of microstates that correspond to it. The idea
+is that a larger macrostate is “more random” (or “less specific”) since it
+is more likely to be hit by a random selection.
+In statistical mechanics,
+the formula for Boltzmann entropy is kB ln Ω, where Ω is the number of
+microstates and kB the Boltzmann constant. In our definition, we use the
+binary logarithm (and do not use kB). As a general intuition, it is natural
+to associate a formula ϕ (or the class it defines) with a macrostate, while
+the models of ϕ are then the corresponding microstates.
+Consider now the following natural probability distribution over the equi-
+valence classes of ≡d: p≡d(M) = |M|/|M| for each class M. We again refer
+to this distribution with the symbol ≡d (with slight abuse of notation). We
+define the Boltzmann entropy of the distribution ≡d as the expected
+value of HB over the distribution ≡d and we denote it by HB(≡d). In other
+words, we define HB(≡d) as �
+M∈M/≡d p≡d(M)HB(M). Roughly speaking,
+HB(≡d) is large when ≡d is far from the uniform distribution.
+The Boltzmann entropy of the distribution ≡d is closely related to the
+Shannon entropy HS(≡d) of ≡d, which we define as the expected value of
+− log(p≡d(M)) over the distribution ≡d. More explicitly, we define HS(≡d)
+as − �
+M∈M/≡d p≡d(M) log(p≡d(M)).
+In contrast to Boltzmann entropy,
+Shannon entropy measures randomness of ≡d by looking at how uniform the
+distribution is. Indeed, if ≡d contains a very large class, then its Shannon
+entropy is small, while its Boltzmann entropy is relatively large. The follow-
+ing result, which was shown in [7] in a more general setting (with slightly
+different notation), formally establishes that the two notions of entropy are
+complementary in nature.
+Proposition 2.1. HS(≡d) + HB(≡d) = |τ|n
+7
+
+3
+Description complexity
+In this section we investigate the GMLUd-description complexity of equi-
+valence classes in the partition ≡d. We will utilize a formula size game for
+GMLUd.
+Let I = {1, . . . , t} and let (πi)i∈I be an enumeration of the types of
+the set τ of proposition symbols. Let d ≤ n be the counting depth. We
+consider the partition induced by GMLUd for models of size n. For each
+admissible tuple n = (ni)i∈I there is an equivalence class Mn, where each
+type πi realized ni times, with ni = d meaning the type πi is realized at least
+d times. We denote the number of occurrences of d in n by kd.
+Such a class Mn can be defined via the following GMLUd formula:
+ϕ(n) :=
+�
+ni<d
+♦=niψ(πi) ∧
+�
+ni=d
+♦≥dψ(πi)
+The size of the formula ϕ(n) is �
+i∈I ni + t(2|τ| + 1) − kd − 1.
+If ni = d for at most one i ∈ I, then Mn can be defined by a smaller
+formula. Let πj be a type with maximal nj in n. Now Mn is also defined
+by the formula
+ϕ′(n) :=
+�
+i̸=j
+♦=niψ(πi).
+To see this, recall that we restrict to models of size n. As all other types
+have been exactly specified, the only option for the remaining points is the
+type πj. The size of ϕ′(n) is n−|πj|+(t−1)(2|τ|+1)−2, where |πj| denotes
+the number of points in models of Mn with type πj.
+We define the constant cτ := t(2|τ| + 1) − 1. By the formulas above we
+see that C(Mn) ≤ �
+i∈I ni+cτ if kd ≥ 2 and C(Mn) ≤ n−|πj|+cτ if kd ≤ 1.
+We will use the formula size game for GMLUd to show that these bounds
+are optimal up to the constant cτ.
+Let r0 ∈ N and let A0, B0 be sets of τ-models. The GMLUd-formula size
+game GAMEd(r0, A0, B0) has two players, S and D. Positions of the game
+are of the form P = (r, A, B) and the starting position is P0 = (r0, A0, B0).
+In a position P, if r = 0, then D wins. Otherwise S chooses between the
+following moves:
+p-move: S chooses a τ-literal α. The game ends. If A ⊨ α and B ⊨ ¬α, then
+S wins. Otherwise D wins. S cannot make this move if he has not made a
+modal move so far.
+∨-move: S chooses A1, A2 ⊆ A such that A1 ∪ A2 = A and r1, r2 ≥ 1 such
+that r1 + r2 + 1 = r. D chooses whether the next position is (r1, A1, B) or
+(r2, A2, B).
+∧-move: The same as a ∨-move with the roles of A and B switched.
+8
+
+♦≥d-move: S chooses a number k ∈ N with k ≤ d and k < r. For every
+(M, w) ∈ A, S chooses k different points v ∈ W. Let A′ be the set of models
+(M, v) chosen this way. For every (M, w) ∈ B, S chooses n − k + 1 different
+points v ∈ W. Let B′ again be the set of models chosen. The next position
+of the game is (r − k, A′, B′).
+■<d-move: The same as a ♦≥d-move with the roles of A and B switched.
+♦=d-move: S chooses a number k ∈ N with k < d and k < r. For every
+(M, w) ∈ A, S chooses a set PM,w of k different points.
+Let NM,w :=
+W \ PM,w. For every (M, w) ∈ B, S chooses either a set PM,w of k + 1
+different points or a set NM,w of |M| − k + 1 different points. Finally we let
+A′ := {(M, v) | (M, w) ∈ A ∪ B, v ∈ PM,w} and B′ := {(M, v) | (M, w) ∈
+A ∪ B, v ∈ NM,w}. The next position of the game is (r − k − 1, A′, B′).
+■̸=d-move: The same as a ♦=d-move with the roles of A and B switched.
+The formula size game characterizes the size of formulas that separate
+model classes. This is formalized in the following theorem:
+Theorem 3.1. The following statements are equivalent:
+1. S has a winning strategy in the game GAMEd(r, A, B).
+2. There is ϕ ∈ GMLUd[τ] with size at most r such that A ⊨ ϕ and
+B ⊨ ¬ϕ.
+Proof. Easy proof by induction. See [5] for a version of the proof for basic
+modal logic.
+It will be useful for the proofs below to note that if (essentially) the same
+model is on both sides of the game, then D has an easy winning strategy.
+Lemma 3.2. Let P = (r, A, B) be a position of a game GAMEd(r0, A0, B0).
+Let there be propositionally equivalent versions (M, w) ∈ A and (M, v) ∈ B
+of the same model M. Now D has a winning strategy from position P.
+Proof. It is easy to see that the pair of propositionally equivalent versions
+of the same model is maintained through any modal move of S. For ∨-moves
+and ∧-moves one of the two possible following positions will always have
+such a pair of models and the strategy of D is to choose this position.
+Let n be an admissible tuple and assume that n1 is one of the largest
+coordinates. We first define the sets An and Bn of models for the game. All
+models have the same universe W = {1, . . . , n}. We denote supp(n) := {i ∈
+I | ni > 0}. We first define a model M0 as follows. For each i ∈ I, i ̸= 1, the
+type πi is realized precisely ni times. The type π1 is realized n − �
+i̸=1 ni
+times. Additionally, the point 1 is of type π1. Intuitively the model M0 is a
+model of the class Mn, where all points not fixed by the tuple n are of type
+π1. We set An := {(M0, 1)}.
+9
+
+For the set Bn we define models Mi→j for some pairs (i, j) ∈ supp(n) ×
+supp(n) as follows. For i ̸= 1 and any j ∈ supp(n), the model Mi→j is
+obtained from the model M0 by changing one point of type πi to type πj.
+If i = 1 and nj = d, then the model Mi→j is obtained from M0 by changing
+|π1|M0 − d + 1 points w ̸= 1 of type π1 to type πj. If i = 1 and nj < d,
+no model is defined for the pair (i, j). We set Bn := {(Mi→j, 1) | i, j ∈
+supp(n)}.
+In terms of their tuples n′, the models Mi→j have n′
+i = ni − 1 and
+n′
+j = nj + 1, except if nj = d, in which case n′
+j = nj = d. For all other
+indices ℓ, n′
+ℓ = nℓ. All models (M0, 1) and (Mi→j, 1) are propositionally
+equivalent as they realize the type π1 in the point 1.
+Let P = (r, A, B) be a position of the game GAMEd(r, An, Bn).
+We
+define a directed graph G(A, B) := (V, E), where V = supp(n) and (i, j) ∈ E
+if there are propositionally equivalent (M0, w) ∈ A and (Mi→j, v) ∈ B, or
+vice versa with respect to A and B. We call a set C ⊆ supp(n) a cover of
+G(A, B) if for every (i, j) ∈ E, we have that either i ∈ C or both j ∈ C and
+nj < d. The cost r(C) of a cover C is
+r(C) :=
+�
+i∈C
+ni.
+Intuitively, the definition of a cover means that including an index i ∈ I
+generally covers all edges to and from i. The exception is that when ni = d,
+incoming edges are not covered.
+In the proof of the following lemma we use the notation Md(A) = {M |
+(M, w) ∈ A, w ∈ W} for the set of models that have at least one pointed
+version in the set A of pointed models. We also denote by tp(A) the set of
+propositional types realized in the set A of points in a model M that will
+below be clear from the set of points.
+Lemma 3.3. Let P = (r, A, B) be a position of the game GAMEd(r, An, Bn)
+and let
+R(P) := min{r(C) | C is a cover of G(A, B)}.
+If r < R(P), then D has a winning strategy from position P.
+Proof. We show for all possible moves of S that either the condition r <
+R(P) is maintained or D has a winning strategy for some other reason.
+Since the definition of the graph G(A, B) is symmetrical with respect to A
+and B, we only need to handle one of each pair of dual moves.
+p-move: Since 0 < r < R(P), we have propositionally equivalent models on
+both sides of the game and thus D wins if S makes any p-move.
+∨-move: Let A1, A2 ⊆ A and r1, r2 ≥ 1 be the choices of S and let P1 =
+(r1, A1, B) and P2 = (r2, A2, B).
+For each (i, j) ∈ E there is a pair of
+10
+
+propositionally equivalent models (M, w) ∈ A and (M′, v) ∈ B as witnesses.
+Since A1∪A2 = A, each model (M, w) ∈ A is in A1 or A2. The set B remains
+unchanged so for each (i, j) ∈ E we have (i, j) ∈ E1 or (i, j) ∈ E2. Now if
+r1 ≥ R(P1) and r2 ≥ R(P2), then there is a minimal cover C1 of position P1
+with r(C1) ≤ r1 and the same for P2. Now the set C := C1 ∪ C2 is a cover
+of position P with r(C) ≤ r1 + r2 < r, which is a contradiction. Therefore
+we have r1 < R(P1) or r2 < R(P2) and D can maintain the condition by
+choosing such a position.
+♦≥d-move: Let k ≤ d be the number chosen by S. The following position
+is P ′ = (r − k, A′, B′). We first note that if M0 ∈ Md(A) ∩ Md(B), then S
+must choose k points from the version in A and n − k + 1 points from the
+version in B. Thus the next position P ′ will have propositionally equivalent
+versions (M0, w) ∈ A′ and (M0, v) ∈ B′.
+By Lemma 3.2 this gives D a
+winning strategy so we assume M0 is only present on one side of the game.
+Case A: M0 ∈ Md(A). For each (M0, w) ∈ A, S chooses a set PM0,w of k
+points. Let PM0 := �
+w∈W PM0,w. We consider the following cases:
+1) We have �
+πi∈tp(PM0) ni > k. Let (i, j) ∈ E. Since the model Mi→j
+only differs from M0 by n′
+i = ni − 1 and n′
+j ≥ nj, the model Mi→j has
+at least k points with types from tp(PM0). Thus, when S chooses the set
+NMi→j of n − k + 1 points, it contains at least one point with a type from
+tp(PM0). Thus the pair of propositionally equivalent models is maintained
+and (i, j) ∈ E′.
+2) We have �
+πi∈tp(PM0) ni = k. Let (i, j) ∈ E. Assume πi /∈ tp(PM0) or
+πj ∈ tp(PM0). Now the model Mi→j has at least k points with types from
+tp(PM0) and (i, j) ∈ E′ as in case 1.
+Assume then that πi ∈ tp(PM0) and πj /∈ tp(PM0). All edges of this kind
+being eliminated is an acceptable worst case, so we assume that (i, j) /∈ E′.
+Summing up Case A, the worst case is that S chooses a set tp(PM0)
+of types with �
+πi∈tp(PM0) ni = k ≤ d and eliminates all edges with πi ∈
+tp(PM0) and πj /∈ tp(PM0).
+Case B: M0 ∈ Md(B). For each (M0, w) ∈ B, S chooses a set NM0,w of
+n − k + 1 points. Let NM0 := �
+w∈W NM0,w and let Π be the complement of
+tp(NM0). We note that ni < d for each πi ∈ Π since M0 only has at most
+k − 1 points with types from Π.
+Let (i, j) ∈ E and assume πi ∈ Π or πj /∈ Π.
+Now πi /∈ tp(NM0)
+or πj ∈ tp(NM0) so the model Mi→j has at least n − k + 1 points with
+types from tp(NM0). Thus any set PMi→j,v of k points contains at least one
+point with a type from tp(NM0). Thus the pair of propositionally equivalent
+models is maintained and (i, j) ∈ E′.
+Now assume πi /∈ Π and πj ∈ Π. If Mi→j has at least n − k + 1 points
+with types from tp(NM0), then (i, j) ∈ E′ as above. We thus assume that
+Mi→j has less than n − k + 1 points with types from tp(NM0), meaning it
+11
+
+has at least k points with types from Π. Since nj < d and Mi→j differs from
+M0 only by n′
+i = ni − 1 and n′
+j = nj + 1, the only remaining option is that
+Mi→j has exactly k points with types from Π. Thus by the same reasoning
+M0 has exactly k − 1 points with types from Π and since k − 1 < d, we have
+�
+i∈Π ni = k − 1. We again accept S eliminating these edges as a worst case
+and assume (i, j) /∈ E′.
+Summing up Case B, the worst case is that S chooses a set Π of types
+with �
+πi∈Π ni = k −1 < d and eliminates all edges with πi /∈ Π and πj ∈ Π.
+We now consider the condition r − k < R(P ′) in the following position.
+From the above arguments we see that the only way for S to eliminate
+edges moving from G(A, B) to G(A′, B′) is to choose in each model (M0, w)
+or (Mi→j, v) in A exactly all points that satisfy some set Π of types. If
+M0 ∈ Md(A), we have �
+πi∈Π ni = k ≤ d and S can eliminate all edges from
+types in Π to other types. If M0 ∈ Md(B), we have �
+πi∈Π ni = k − 1 and
+ni < d for all πi ∈ Π. In this case S can eliminate all edges from other types
+to types in Π. In both cases, the set C = {i ∈ supp(n) | πi ∈ Π} covers all
+eliminated edges. The cost of C is r(C) = �
+i∈C ni ≤ k. Let C′ be a cover
+of P ′ with minimal cost so r(C′) = R(P ′). Now C ∪ C′ is a cover of P so
+r < R(P) ≤ R(P ′) + r(C) ≤ R(P ′) + k. Thus r − k < R(P ′).
+♦=d-move: Let k < d be the number chosen by S. The following position
+is P ′ = (r − k − 1, A′, B′). As for the ♦≥d-move, we may assume that M0 is
+only present on one side of the game.
+Case A: M0 ∈ Md(A). For each (M0, w) ∈ A, S chooses a k point set
+PM0,w and NM0,w = W \ PM0,w.
+We denote PM0 := �
+w∈W PM0,w and
+NM0 := �
+w∈W NM0,w. We consider the following cases:
+1) We have �
+πi∈tp(PM0) ni > k. Thus there are propositionally equivalent
+w ∈ PM0 and v ∈ NM0. This means that in the following position P ′ there
+are propositionally equivalent versions of the model M0 on both sides of the
+game. D has a winning strategy from P ′ by Lemma 3.2.
+2) We have �
+πi∈tp(PM0) ni = k. Note that since k < d, also ni < d for all
+πi ∈ tp(PM0). Let (i, j) ∈ E.
+Assume πi, πj /∈ tp(PM0). Now the model Mi→j has exactly k points
+with types from tp(PM0). Thus if S chooses a k + 1 point set PMi→j, then
+one of those points v′ has a type πℓ /∈ tp(PM0). By the definition of the
+models, nℓ ̸= 0 so there is a point w′ in the model M0 of type πℓ. Now
+there are propositionally equivalent (Mi→j, v′) ∈ A′ and (M0, w′) ∈ B′ so
+(i, j) ∈ E′.
+Similarly, if S chooses for Mi→j a set NMi→j of n − k + 1 points, then
+this set contains at least one point v′ of a type πℓ ∈ tp(PM0). Now there
+is a point w′ ∈ PM0 of type πℓ. Thus there are propositionally equivalent
+(M0, w′) ∈ A′ and (Mi→j, v′) ∈ B′ so (i, j) ∈ E′.
+If πi ∈ tp(PM0) or πj ∈ tp(PM0), we assume (i, j) /∈ E′. As for the
+12
+
+♦≥d-move, this is an acceptable worst case for our lower bound.
+Summing up Case A, the worst case is that S chooses a set tp(PM0) of
+types with �
+πi∈tp(PM0) ni = k < d and eliminates all edges (i, j) ∈ E with
+πi ∈ tp(PM0) or πj ∈ tp(PM0).
+Case B: M0 ∈ Md(B). For each (M0, w) ∈ B, S chooses either a set PM0,w
+of k + 1 points or a set NM0,w of n − k + 1 points. There are three cases:
+1) Assume first that there are (M0, w), (M0, w′) ∈ B with sets PM0,w and
+NM0,w′ chosen. Since k + 1 + n − k + 1 > n, there is v ∈ PM0,w ∩ NM0,w′. In
+the following position P ′ the model (M0, v) is on both sides of the game so
+by Lemma 3.2 D has a winning strategy from P ′.
+2) Assume then that S chooses a set PM0,w of k+1 points for each (M0, w) ∈
+B and let PM0 = �
+w∈W PM0,w. Let (i, j) ∈ E and let (Mi→j, v) ∈ A and
+(M0, w) ∈ B be the corresponding models.
+Assume πi /∈ tp(PM0) or πj ∈ tp(PM0). Now the model Mi→j has at
+least k+1 points with types from tp(PM0). Thus the set NMi→j,v of size n−k
+has at least one point with a type from tp(PM0). Thus the propositionally
+equivalent pair of models is maintained and (i, j) ∈ E′.
+Assume πi ∈ tp(PM0) and πj /∈ tp(PM0). If Mi→j has at least k+1 points
+with types from tp(PM0), the above argument again works and (i, j) ∈ E′.
+Since k < d and Mi→j only differs from M0 by n′
+i = ni − 1 and n′
+j ≤ nj + 1,
+the only remaining option is that Mi→j has exactly k points with types from
+tp(PM0). We obtain �
+i∈tp(PM0) n′
+i = k. We again assume as a worst case
+that (i, j) /∈ E′.
+3) Finally assume that S chooses a set NM0,w of n − k + 1 points for each
+(M0, w) ∈ B and let NM0 = �
+w∈W NM0,w. Let Π be the complement of
+tp(NM0).
+Assume πi ∈ Π or πj /∈ Π. Now πi /∈ tp(NM0) or πj ∈ tp(NM0) so the
+model Mi→j has at least n−k+1 points with types from tp(NM0). Thus the
+set PMi→j,v of size k has at least one point with a type from tp(NM0). Thus
+the propositionally equivalent pair of models is maintained and (i, j) ∈ E′.
+Assume πi /∈ Π and πj ∈ Π. If Mi→j has at least n − k + 1 points with
+types from tp(NM0), the above argument again works and (i, j) ∈ E′.
+We assume Mi→j has less than n−k+1 points with types from tp(NM0),
+meaning it has at least k points with types from Π. Since k < d and Mi→j
+only differs from M0 by n′
+i = ni − 1 and n′
+j ≤ nj + 1, the only remaining
+option is that Mi→j has exactly k points with types from Π. We obtain
+�
+i∈Π n′
+i = k. We again assume as a worst case that (i, j) /∈ E′. Edges
+eliminated this way are ones from other types to types in Π. In particular,
+we note that for any edge (i, j) eliminated this way, nj < d since M0 has at
+most n − (n − k + 1) = k − 1 points of type πj.
+Summing up Case B, the worst case is that S chooses a set Π of types
+and eliminates either all edges (i, j) ∈ E with πi ∈ Π or πj /∈ Π or all edges
+13
+
+(i, j) ∈ E with πi /∈ Π or πj ∈ Π. In both cases �
+i∈Π n′
+i = k for the model
+Mi→j and for the latter case nj < d.
+We now consider the condition r−k−1 < R(P ′) in the following position.
+From the above arguments, we see that the only way for S to eliminate
+edges moving from G(A, B) to G(A′, B′) is to choose in each model (M0, w)
+or (Mi→j, v) in A exactly all points that satisfy some set Π of types. If
+M0 ∈ Md(A), we have �
+πi∈Π ni = k < d and S can eliminate all edges to
+and from the set Π of types. If M0 ∈ Md(B), we have �
+πi∈Π n′
+i = k < d and
+S can eliminate either all edges to or all edges from the set Π of types. In
+particular, if nj = d for πj ∈ Π, then only outgoing edges can be eliminated.
+In all cases, the set C = {i ∈ supp(n) | πi ∈ Π} covers all eliminated edges.
+The cost of C is r(C) = �
+i∈C ni ≤ k + 1 since �
+i∈C ni ≤ �
+i∈C n′
+i + 1.
+Let C′ be a cover of P ′ with minimal cost so r(C′) = R(P ′). Now C ∪ C′
+is a cover of P so r < R(P) ≤ R(P ′) + r(C) ≤ R(P ′) + k + 1.
+Thus
+r − k − 1 < R(P ′).
+A lower bound for the description complexity of an arbitrary class Mn
+can now be obtained by simply calculating the minimum cost of a cover of
+G(An, Bn).
+Lemma 3.4. Let Mn be a class with ni = d for at least two different i ∈ I.
+Then C(Mn) ≥ �
+i∈I
+ni.
+Proof. We assume that n1 = d. We consider the graph G(An, Bn) = {(i, j) ∈
+supp(n) | i ̸= 1 or nj = d}. This graph has edges from all types πi with
+ni ̸= 0 to each other, with the exception that there are no edges from π1 to
+πj with nj < d.
+We first note that C = supp(n) is a cover of G(An, Bn) with cost �
+i∈I ni.
+For C′ ̸= C if i /∈ C′ with i ̸= 1, then by the definition of a cover, the edge
+(i, 1) ∈ E is not covered, since i /∈ C and n1 = d. If 1 /∈ C′, then let j ∈ I
+be the other index with nj = d besides 1. Now the edge (1, d) ∈ E is not
+covered since 1 /∈ C and nj = d. We see that C is a minimal cost cover and
+by Lemma 3.3 and Theorem 3.1 the claim holds.
+Lemma 3.5. Let Mn be a class with ni = d for at most one i ∈ I. Let πj
+be one of the types with the most realizing points in Mn. Then C(Mn) ≥
+�
+i∈I\{j} ni = n − |πj|.
+Proof. We assume n1 = max{ni | i ∈ I}. Thus π1 is one of the types with
+the most realizing points and if n1 < d, then ni < d for all i ∈ I.
+We consider the graph G(An, Bn). From the definition we get G(An, Bn) =
+{(i, j) ∈ supp(n) | i ̸= 1}. This graph has edges from all types πi with ni ̸= 0
+to each other, with the exception that there are no edges originating from
+1.
+14
+
+We first note that C = supp(n)\{1} is a cover of G(An, Bn) with r(C) =
+�
+i∈I\{1} ni = n − |π1|. Clearly supp(n) is a cover with higher cost.
+Let C′ ⊂ supp(n). If there are i, j ∈ supp(n) with i, j /∈ C′, then the
+edge (i, j) or (j, i) is in E and is not covered. If there is only one i ∈ supp(n)
+with i /∈ C′ and i ̸= 1, then r(C′) = �
+i∈I\{j} ni ≥ r(C) since π1 is one of
+the types with the largest ni. We see that C is a minimal cost cover and by
+Lemma 3.3 and Theorem 3.1 the claim holds.
+We sum up the above lemmas into the Theorem below:
+Theorem 3.6. Let n be an (n, d)-admissible tuple and let Mn be the cor-
+responding equivalence class of ≡d. If ni = d for at least two i ∈ I, then
+C(Mn) ≥ �
+i∈I ni. Otherwise C(Mn) ≥ �
+i∈I\{j} ni = n − |πj|.
+4
+A monotone connection
+Consider the following strict partial orders on the set of all (n, d)-admissible
+tuples.
+1. n <s n′ iff |Mn| < |Mn′|
+2. n <c n′ iff C(Mn) < C(Mn′)
+Note that neither |Mn| = |Mn′| nor C(Mn) = C(Mn′) necessarily entails
+that n = n′, as demonstrated by the tuples (1, 2, d) and (2, 1, d). We let ≤s
+and ≤c denote the partial orders obtained by adding loops to <s and <c
+respectively. The main purpose of this section is to show that there exists
+a natural and non-trivial partial order which is contained both in ≤s and
+in ≤c (provided that n is sufficiently large w.r.t. d), which then gives us a
+monotone connection between sizes of model classes and their description
+complexities. To establish this result, we will use the heavy machinery devel-
+oped in the previous section together with further combinatorial arguments
+that involve r-associated Stirling numbers.
+4.1
+r-associated Stirling numbers
+We will use r-associated Stirling numbers to count the number of models in
+a given model class. Given positive integers n, m, r such that n ≥ mr, we
+define
+�n
+m
+�
+≥r
+to be the number of partitions of [n] which partition [n] into m, each set
+having size at least r. When r = 1 these numbers are also known as the
+Stirling numbers of the second kind and they simply count the number of
+partitions of [n] into m sets [3].
+15
+
+The following lemma will play the key role when we estimate the sizes
+of the model classes. Even though the proof is simple and elementary, we
+were not able to find these estimates in the existing literature.
+Lemma 4.1. Let n, m, r ∈ Z+. Suppose that n ≥ mr. Then
+mn
+mmr ≤
+�n
+m
+�
+≥r
+≤ mn
+m!
+Proof. For the upper bound note that
+�n
+m
+�
+≥r
+≤
+�n
+m
+�
+≥1
+≤ mn
+m! .
+The last inequality follows from the fact that mn counts the number of
+mappings f : [n] → [m], while m! ·
+� n
+m
+�
+≥1 only counts those f that are
+surjections. For the lower bound, we use the fact that m!·
+� n
+m
+�
+≥r counts the
+number of mappings f : [n] → [m] which have the property that for every
+1 ≤ k ≤ m we have |f −1({k})| ≥ r. To give a lower bound on the number
+of such functions, we count the number of mappings f : [n] → [m] which
+have the property that the sets {(k − 1) · r + 1, . . . , k · r}, where 1 ≤ k ≤ m,
+are mapped to distinct elements. Since the remaining n − mr elements can
+be mapped arbitrarily, the number of such mappings is simply m! · mn−mr.
+Thus
+m! · mn−mr ≤ m! ·
+�n
+m
+�
+≥r
+,
+giving the wanted lower bound after dividing by m!.
+We note that if m and r are much smaller than n — as they will be in
+our applications — the estimates of Lemma 4.1 are (perhaps surprisingly)
+quite sharp.
+One important consequence of Lemma 4.1 is the following lemma. Again,
+we emphasize that we were unable to find an estimate of this form in the
+existing literature.
+Lemma 4.2. Let n, m, r ∈ Z+. If n ≥ mr + 1, then
+�n
+m
+�
+≥r
+≤ mmr+1
+m!
+�n − 1
+m
+�
+≥r
+.
+Proof. Follows immediately from Lemma 4.1.
+If m and r are fixed, Lemma 4.2 bounds the growth rate of r-associated
+Stirling numbers as n increases.
+16
+
+4.2
+Combinatorics of model classes
+In this subsection we use the above results on r-associated Stirling numbers
+to investigate the sizes of model classes in terms of their (n, d)-admissible
+tuples. We begin with the following lemma gives a simple and closed formula
+for the size of a model class.
+Lemma 4.3. Let n be an (n, d)-admissible tuple. Let i1, . . . , ik ∈ I be the
+indices for which niℓ < d. We then have that
+|Mn| =
+�
+n
+ni1, . . . , nik, m
+�
+· kd! ·
+�m
+kd
+�
+≥d
+,
+where m := n − �
+i̸∈{i1,...,ik} ni and kd := t − k.
+Proof. Each model in Mn can be constructed as follows.
+(1) We first pick k subsets of {1, . . . , n} of sizes ni1, . . . , nik and define that
+each element in the iℓth set realizes the iℓth type. The number of ways this
+can be done is given by
+�
+n
+ni1,...,nik,m
+�
+.
+(2) We then partition the remaining subset of size m to kd pieces, each piece
+having size at least d. The number of ways this can be done is given by the
+associated Stirling number
+�m
+kd
+�
+≥d.
+(3) For each piece we select a unique type from the remaining types — the
+number of which is kd — and define that each element in a piece realizes
+the type associated with that piece. The number of ways this can be done
+is given by kd!.
+Multiplying the above factors gives us the result.
+The following lemmas establish how the size of a model class changes
+when we modify its admissible tuple.
+Lemma 4.4. Fix d ∈ Z+ and let n be an (n, d)-admissible tuple. Suppose
+that n is sufficiently large with respect to d and |τ|. Then for every i ∈ I
+such that ni < d − 1 we have that
+|Mn| < |Mn′|,
+where n′
+i = ni + 1 and n′
+j = nj, for every j ̸= i.
+Proof. For notational simplicity we assume that nℓ < d iff ℓ ≤ k. By Lemma
+4.3 the inequality that we need to establish is
+�
+n
+n1, . . . , nk, m
+�
+· kd! ·
+�m
+kd
+�
+≥d
+<
+�
+n
+n1, . . . , ni + 1, . . . , nk, m − 1
+�
+· kd! ·
+�m − 1
+kd
+�
+≥d
+,
+17
+
+where m = n − �k
+ℓ=1 nℓ and kd = t − k. Note that since n is large enough
+w.r.t. d, kd ≥ 1. Simplifying this gives us the equivalent inequality
+�m
+kd
+�
+≥d
+<
+m
+ni + 1
+�m − 1
+kd
+�
+≥d
+,
+which follows from Lemma 4.2, as long as
+m
+ni + 1 > kkdd+1
+d
+kd!
+⇔ n > (ni + 1)kkdd+1
+d
+kd!
++
+k
+�
+ℓ=1
+nℓ
+Using nℓ < d, which holds for every 1 ≤ ℓ ≤ k, and 1 ≤ kd ≤ t gives us the
+desired result.
+Lemma 4.5. Fix d ∈ Z+ and let n be an (n, d)-admissible tuple. Suppose
+that n is sufficiently large with respect to d and |τ|. Then for every i ∈ I
+such that ni < d we have that
+|Mn| < |Mn′|,
+where n′
+j = d, when j = i, and n′
+j = nj otherwise.
+Proof. For notational simplicity we assume that nℓ < d iff ℓ ≤ k. By Lemma
+4.3 the inequality that we need to establish is
+�
+n
+n1, . . . , ni, . . . , nk, m
+�
+· kd! ·
+�m
+kd
+�
+≥d
+<
+�
+n
+n1, . . . , ni−1, ni+1, . . . , nk, m + ni
+�
+· (kd + 1)! ·
+�m + ni
+kd + 1
+�
+≥d
+,
+where m = n − �k
+ℓ=1 nℓ and kd = t − k. Note that since n is large enough
+w.r.t. d, kd ≥ 1. Simplifying this gives us the equivalent inequality
+�m
+kd
+�
+≥d
+<
+m!ni!
+(m + ni)! · (kd + 1) ·
+�m + ni
+kd + 1
+�
+≥d
+It follows from Lemma 4.1 that
+�m + ni
+kd + 1
+�
+≥d
+≥ (kd + 1)m+ni
+(kd + 1)(kd+1)d
+and
+�m
+kd
+�
+≥d
+≤ km
+d
+kd!
+18
+
+Hence we only have to show that
+km
+d
+kd! <
+m!ni!
+(m + ni)! · (kd + 1) · (kd + 1)m+ni
+(kd + 1)(kd+1)d
+or equivalently that
+(kd + 1)(kd+1)d
+(kd + 1)ni+1kd!ni! < m!(kd + 1)m
+km
+d (m + ni)!
+=
+(kd + 1)m
+km
+d
+�ni−1
+j=0 (m + ni − j)
+=
+� kd + 1
+kd
+� �� �
+>1
+�m� ni−1
+�
+j=0
+(m + ni − j)
+Recall 0 ≤ ni < d and 1 ≤ kd ≤ t. Hence in the above inequality the left
+hand side is a constant, and thus it suffices to show that the right hand side
+formula tends to infinity as n grows (recall m = n−�k
+ℓ=1 nℓ, i.e., m is just n
+minus a constant). However, this is clear, since the right hand side is of the
+form f(n)/g(n), where f grows exponentially w.r.t. n while g grows only
+polynomially w.r.t. n.
+4.3
+Connecting size and description complexity
+Let Pn,d denote the set of all (n, d)-admissible tuples. We define a natural
+partial order ⪯ on Pn,d as follows: n ⪯ n′ if and only if ni ≤ n′
+i, for every
+i ∈ I. By writing n ≺ n′ we mean that n ⪯ n′ and n ̸= n′.
+Lemma 4.6. Suppose that n is sufficiently large with respect to d and |τ|.
+Let n and n′ be (n, d)-admissible tuples such that n ≺ n′. Then
+|Mn| < |Mn′|
+Proof. It suffices to show that if n′ is an immediate successor of n, then
+|Mn| < |Mn′|. Now, there must exist exactly one i ∈ I such that n′
+i = ni + 1
+and n′
+j = nj for every j ̸= i. If n′
+i < d, then the claim follows from Lemma
+4.4, while if n′
+i = d, then the claim follows from Lemma 4.5.
+Recall the constant cτ = 2|τ|(2|τ| + 1) − 1 from the previous section. We
+define yet another partial order ⪯τ on Pn,d as follows: n ⪯τ n′ if and only
+if the following two conditions hold:
+1. For every i ∈ I we have that ni ≤ n′
+i.
+2. Either n = n′ or �
+i∈I(n′
+i − ni) > cτ.
+19
+
+Roughly speaking n ⪯τ n′ means that if the tuples are distinct, then the
+distance between them w.r.t. to the order ⪯ is more than cτ. Again, by
+writing n ≺τ n′ we mean that n ⪯τ n′ and n ̸= n′. For example (1, d) ≺τ
+(2 + cτ, d), but (1, d) ̸⪯τ (1 + cτ, d).
+Lemma 4.7. Suppose that n is sufficiently large with respect to d and |τ|.
+Let n and n′ be (n, d)-admissible tuples such that n ≺τ n′. Then we have
+that
+C(Mn) < C(Mn′)
+Proof. Suppose first that d occurs at least twice in n′. Theorem 3.6 entails
+that C(Mn′) ≥ �
+i∈I n′
+i. We now have two cases based on whether or not d
+occurs at least twice in n. Suppose first that d occurs exactly once in n say,
+nj = d. Then Mn can be defined by a formula of size
+cτ +
+�
+i∈I−{j}
+ni <
+�
+i∈I−{j}
+n′
+i ≤ C(Mn′)
+and hence C(Mn) < C(Mn′). On the other hand, if d occurs at least twice
+in n, then Mn can be defined by a formula of size
+cτ +
+�
+i∈I
+ni <
+�
+i∈I
+n′
+i ≤ C(Mn′)
+and hence C(Mn) < C(Mn′).
+Suppose then that d occurs exactly once in n′, say n′
+j = d. Theorem 3.6
+entails that C(Mn′) ≥ �
+i∈I−{j} n′
+i. Since n ≺τ n′, we have that ni < d, for
+every i ̸= j. Since d must occur at least once in n — as n is (n, d)-admissible
+— we have that nj = d. Now Mn can be defined by a formula of size
+cτ +
+�
+i∈I−{j}
+ni <
+�
+i∈I−{j}
+n′
+i ≤ C(Mn′)
+and hence C(Mn) < C(Mn′).
+Recall the partial orderings ≤s and ≤c on Pn,d, which were introduced at
+the beginning of this section. The following theorem formalizes a connection
+between sizes of model classes and their description complexities.
+Theorem 4.8. If n is sufficiently large with respect to d, then
+⪯τ ⊆ ≤s ∩ ≤c .
+In particular, if n and n′ are two distinct ⪯τ-comparable tuples, then
+|Mn| < |Mn′| ⇔ C(Mn) < C(Mn′).
+20
+
+Proof. Suppose that n ≺τ n′. Lemmas 4.6 and 4.7 guarantee that if n is
+large enough w.r.t. d, then |Mn| < |Mn′| and C(Mn) < C(Mn′). Hence
+n ≤s n′ and n ≤c n′, which proves the first claim. The second claim follows
+directly from the first.
+The intuitive content of the above theorem is that the partial order ⪯τ
+approximates both ≤s and ≤c. Hence ⪯τ can be viewed as a highly non-
+trivial monotone connection between ≤s and ≤c.
+We note that the above theorem also works for Boltzmann entropy. This
+is because the order ≤s is the same as the order based on Boltzmann entropy,
+since logarithm is an increasing function. We formulate the latter of the two
+claims as a corollary.
+Corollary 4.9. Assume n is sufficiently large with respect to d. If n and n′
+are two distinct ⪯τ-comparable tuples, then
+HB(Mn) < HB(Mn′) ⇔ C(Mn) < C(Mn′).
+5
+The phase transitions of class size distributions
+In this section we move our attention from single classes to the entire pro-
+bability distribution given by ≡d. By allowing d to depend on n, we obtain
+qualitative results which link the growth rate of d with the emergence of a
+dominating class, i.e., a class which contains almost all the models. For fixed
+n, we also obtain quantitative results on how the relationship between d and
+n determines whether there exists a class in ≡d which contains majority of
+all the models of size n. We will also point out consequences of these results
+on the “average-case” expressive power of GMLUd.
+Throughout this section we continue to use our previous convention that
+t = 2|τ| denotes the number of types π of the alphabet τ. We start with the
+following observation, which gives a sense of what happens in the distribu-
+tion, when the counting depth is increased.
+Proposition 5.1. Suppose that d < d′ ≤ n/2. Then
+HS(≡d) < HS(≡d′) and HB(≡d) > HB(≡d′).
+Furthermore, for every n/2 ≤ d ≤ d′ we have that
+HS(≡d) = HS(≡d′) and HB(≡d) = HB(≡d′).
+Proof. By Proposition 2.1 HS(≡d) + HB(≡d) = |τ|n. Hence it suffices to
+establish the claims in the case of Boltzmann entropy. We first note that if
+n/2 ≤ d ≤ d′, then HB(≡d) = HB(≡d′), since the logic GMLUd can already
+specify each structure up to isomorphism.
+21
+
+Suppose then that d < d′ ≤ n/2. Consider the class Mn, where n =
+(0, . . . , 0, d, d). As the depth is increased to d′ > d, this class is divided to
+at least two smaller classes with tuples (0, . . . , 0, d, d′) and (0, . . . , 0, d′, d).
+Meanwhile, clearly no class increases in size so we see that HB(≡d) >
+HB(≡d′).
+We take this opportunity to point out an easy corollary of the previous
+result. In [7] it was established that HB(≡n) ∼ |τ|n, by which we mean
+that limn→∞ HB(≡n)/|τ|n = 1. When combined with Proposition 5.1, this
+result yields quite directly the following.
+Corollary 5.2. For any counting depth d(n), we have
+HB(≡d(n)) ∼ |τ|n
+Proof. W.l.o.g. we assume that d(n) ≤ n, for every n. Since log(|M|) ≤ |τ|n,
+for any class M, we have that HB(≡d(n)) ≤ |τ|n. Since HB(≡n) ∼ |τ|n, for
+every ε > 0 we have that if n is large enough, then HB(≡n) ≥ (1 − ε)|τ|n.
+Since HB(≡d(n)) ≥ HB(≡n), for every ε > 0 we have that HB(≡d(n)) ≥
+(1 − ε)|τ|n, provided that n is sufficiently large. Combining these bounds
+yields the desired result.
+Thus, from an asymptotic point of view, the dependence of the counting
+depth d on n has no effect on the Boltzmann entropy of ≡d. By virtue of
+Proposition 2.1, the same is true for the Shannon entropy of ≡d.
+We proceed now to further analyze the effect of counting depth on the
+distribution. To formulate some of our results, we will use standard asymp-
+totic notation, which we recall here. Let f, g : N → R>0. We use f = o(g)
+to denote that limn→∞ f(n)/g(n) = 0. Furthermore, we use f = ω(g) to
+denote that limn→∞ f(n)/g(n) = ∞.
+We first show that if the counting depth grows slowly enough with respect
+to n, then the distribution contains a class which contains almost all of the
+models of size n. More formally, fix a function d : Z+ → N and thereby
+a sequence ≡d(n) of equivalence relations. A class sequence M(n) with
+respect to the sequence ≡d(n) is a function that outputs a single class for each
+individual equivalence relation in the sequence. Each class sequence M(n)
+is naturally associated with the corresponding probability sequence
+pn := p≡d(n)(M(n)) = |M(n)|
+2n|τ| .
+We say that ≡d(n) has a dominating class if there exists a class sequence
+M(n) such that pn → 1 as n → ∞. Then the class sequence M(n) is said to
+dominate ≡d(n). Intuitively, a class (sequence) is dominating if a random
+model of size n belongs to it with limit probability one.
+Our proof uses the well-known Chernoff bounds. The following lemma
+will require the lower-tail estimate, while the upper-tail estimate will be
+used later in this section.
+22
+
+Proposition 5.3 (Chernoff bounds [12]). Let X := �n
+i=1 Xi be a sum of
+independent 0-1-valued random variables, where Xi = 1 with probability p
+and Xi = 0 with probability 1 − p. Then for every δ ≥ 0 we have that
+(Lower tail) Pr[X ≤ (1 − δ)np] ≤ e−δ2 np
+2
+(Upper tail) Pr[X ≥ (1 + δ)np] ≤ e−δ2 np
+2+δ
+Lemma 5.4. For any f(n) = ω(√n), if the counting depth is d(n) ≤ n/t −
+f(n), then the class sequence Mn, where n = (d(n), . . . , d(n)), dominates
+≡d(n).
+Proof. We will show that for each f(n), such that f(n) = ω(√n) and f(n) ≤
+n/t, we have that with limit probability one a random model realizes each
+type more than n/t − f(n) times.
+Given a type π, we let Xπ denote a
+random variable which counts the number of times π is realized. Now Xπ :=
+�
+1≤i≤n Xπ,i, where Xπ,i is an indicator random variable for the event that
+the element i realizes the type π. Since the success probability of Xπ,i is
+t−1, we have that E(Xπ) = n/t.
+Now, Chernoff bound give us that for every n and for every 0 �� δ we
+have
+Pr
+�
+Xπ ≤ (1 − δ)n
+t
+�
+≤ e−δ2 n
+2t .
+Note that δ can indeed depend on n. Setting δ(n) :=
+�
+2tg(n)/n, for any
+g(n) = ω(1), we obtain that
+e−δ2 n
+2t = e−g(n) → 0.
+Furthermore
+δ(n)n
+t =
+√
+2t · 1
+t
+� �� �
+=:C
+·
+�
+g(n) · √n = C
+�
+ng(n).
+Hence for any function g(n) = ω(1) we have that with high probability the
+type π is realized more than n/t − C
+�
+ng(n) many times.
+Since π was
+arbitrary, it follows from the union bound that with high probability every
+type π is realized more than n/t − C
+�
+ng(n) times. Now, if f(n) = ω(√n),
+then by setting g(n) = (f(n)/C√n)2 we obtain that every type is realized
+more than n/t − f(n) times.
+We showed that when the counting depth is low enough, the distribution
+has a dominating class. Next we will show that for a high enough counting
+depth, there is no dominating class. For this, we will utilize the following
+inequality version of Stirling’s approximation, due to Robbins [14].
+23
+
+Proposition 5.5 (Stirling’s approximation [14]). For all n ∈ N, n > 0, we
+have
+√
+2πn
+�n
+e
+�n
+e
+1
+12n+1 < n! <
+√
+2πn
+�n
+e
+�n
+e
+1
+12n
+Lemma 5.6. For any f(n) = o(√n), if the counting depth is d(n) ≥ n/t −
+f(n), then ≡d(n) has no dominating class as n → ∞.
+Proof. Let M(n) be an arbitrary sequence of classes. We will show that
+there is some n0 such that for every n ≥ n0 the probability of a model
+belonging to class M(n) is at most half.
+Let Mn be a class of the sequence M(n) with i, j ≤ t such that ni ̸= nj.
+Let n′ be obtained from n by switching the numbers ni and nj in the tuple.
+We clearly have |Mn′| = |Mn| by symmetry so the probability of a model
+belonging to Mn is at most half.
+It remains to show that classes of the sequence M(n) with tuples re-
+peating only one number have probability at most half, when n is large
+enough. Let d ≥ n/t − f(n), where f(n) = o(√n). Assume d < n/t and let
+M = Mn, where n = (d, . . . , d). Note that this is the only (n, d)-admissible
+tuple that repeats only one number. We show that this class M has proba-
+bility less than half if n is large enough. In fact, we establish the stronger
+claim that the sequence of such classes with tuples (d(n), . . . , d(n)) has limit
+probability 0.
+The size of the above class M is given by the sum
+|M| =
+�
+n1+···+nt=n
+ni≥d
+�
+n
+n1, . . . , nt
+�
+.
+First we note that a multinomial coefficient is largest, when the numbers
+n1, . . . , nt are equal. Using this and Stirling’s approximation, we get
+�
+n
+n1, . . . , nt
+�
+≤
+�
+n
+n
+t , . . . , n
+t
+�
+≤
+√
+2πn(n
+e )ne
+1
+12n
+(�2π n
+t ( n
+et)
+n
+t e
+1
+12 n
+t +1)t
+=
+√
+2πn(n
+e )ne
+1
+12n
+�2π n
+t
+t(n
+e )n 1
+tn e
+t
+12 n
+t +1
+=
+�
+tt
+(2π)t−1 ·
+1
+√
+nt−1 · eg(n) · tn
+≤
+�
+tt
+(2π)t−1 ·
+1
+√
+nt−1 · 2n|τ|
+24
+
+The exponent of e above is
+g(n) =
+1
+12n −
+t
+12n
+t + 1 =
+1
+12n −
+t2
+12n + t
+= 12n + t − 12t2n
+12n(12n + t)
+= (1 − t2)12n + t
+12n(12n + t) .
+Clearly g(n) < 0 for all positive n so eg(n) < 1 and the above estimate holds.
+By the stars and bars method, the original sum has
+�tf(n) + t − 1
+t − 1
+�
+≤ (2t)t−1 · f(n)t−1
+terms so the final estimate is
+|M| ≤
+�
+tt
+(2π)t−1 · (2t)t−1 · f(n)t−1
+√
+nt−1 · 2n|τ|
+Recall that there are 2n|τ| models of size n in total and f(n) = o(√n).
+We obtain the following limit:
+lim
+n→∞
+�
+tt
+(2π)t−1 · (2t)t−1 · f(n)t−1
+√
+nt−1 · 2n|τ|
+2n|τ|
+= 0.
+We see that the sequence of classes Mn with n = (d, . . . , d) has limit pro-
+bability 0. Thus for a single such class, the probability is certainly at most
+half if n is large enough.
+Now assume d ≥ n/t, and consider the sequence of classes Mn, where n =
+(n/t, . . . , n/t). This is again the only (n, d)-admissible tuple that repeats
+only one number. The size of these classes is given by
+�
+n
+n
+t ,..., n
+t
+�
+and it is easy
+to see from the above that this sequence also has limit probability 0. Thus
+a single such class has probability less than half for large enough n.
+Intuitively, a class (sequence) is vanishing, if a random model of size n
+belongs to it with limit probability zero. To define this notion formally, fix
+a sequence ≡d(n). We say that all classes in ≡d(n) are vanishing as n → ∞
+if for all class sequences M(n) for ≡d(n), we have pn → 0 as n → ∞. We
+now show that if the counting depth is high enough, then all classes in the
+distribution sequence are vanishing.
+Lemma 5.7. For any f(n) such that f(n) = ω(√n), if the counting depth
+is d(n) ≥ n/t + f(n), then all classes in ≡d(n) are vanishing as n → ∞.
+Proof. Fix some function f(n) such that f(n) = ω(√n).
+We first show
+that with limit probability zero a random model realizes each type less
+than n/t + f(n) times.
+Without loss of generality we can assume that
+25
+
+f(n) ≤ n−n/t. Given a type π, we let Xπ denote the same random variable
+as in the proof of Lemma 5.4. Chernoff bound give us again that for every
+n and for every 0 < δ ≤ 1 we have
+Pr
+�
+Xπ ≥ (1 + δ)n
+t
+�
+≤ e−δ2
+n
+t(2+δ) ≤ e−δ2 n
+3t
+Note that δ can depend on n. Setting δ(n) :=
+�
+3tg(n)/n, for any g(n) such
+that g(n) = ω(1) and g(n) ≤ n/(3t) (to guarantee that δ(n) ≤ 1), we get
+that
+e−δ2 n
+3t = e−g(n) → 0.
+Furthermore
+δ(n)n
+t =
+√
+3t · 1
+t
+� �� �
+=:C
+·
+�
+g(n) · √n = C
+�
+ng(n).
+Hence for any function g(n) such that g(n) = ω(1) and g(n) ≤ n/(3t) we
+have that with high probability the type π is realized less than n
+t −C
+�
+ng(n)
+many times. Since π was arbitrary, it follows from the union bound that
+with high probability every type π is realized less than n
+t − C
+�
+ng(n) many
+times. By setting g(n) = (f(n)/C√n)2 we obtain that every type is realized
+less than n/t − f(n) times.
+Now consider an arbitrary class sequence M(n). We want to show that
+for every ε > 0 we have that pn < ε, provided that n is sufficiently large.
+Consider a class M(n) and let n be the corresponding (n, d(n))-admissible
+tuple, i.e., M(n) is the class Mn. If there is i ∈ I such that ni = d(n) =
+n/t + f(n), then it follows from the previous result that p≡d(n)(Mn) < ε, as
+long as n is sufficiently large. Suppose then that ni < n/t + f(n), for every
+i ∈ I. In this case the class Mn is an isomorphism class, which means that
+the probability that a random model belongs to Mn is simply
+�
+n
+n1, . . . , nt
+�
+/2n|τ|,
+which, as calculated in the proof of Lemma 5.6, is at most a constant times
+1/
+√
+nt−1. This latter quantity is certainly less than ε, provided that n is
+sufficiently large.
+Hence p≡d(n)(Mn) < ε, provided that n is sufficiently
+large.
+We gather the above results in the following theorem:
+Theorem 5.8. The following statements hold for counting depth d(n) as
+n → ∞.
+• If d(n) ≤ n/t−f(n) where f(n) = ω(√n), then ≡d(n) has a dominating
+class.
+26
+
+• If d(n) ≥ n/t − f(n) where f(n) = o(√n), then ≡d(n) has no domina-
+ting class.
+• If d(n) ≥ n/t + f(n) where f(n) = ω(√n), then every class in ≡d(n)
+is vanishing.
+We point out a corollary of the above result. When the counting depth
+is too low, almost all models are in the same dominating class in terms
+of GMLUd definability. Conversely, if the counting depth is high enough,
+GMLUd can separate the models into classes that vanish. These observations
+directly give us the following result:
+Corollary 5.9. Let f(n) = ω(√n).
+If d(n) ≤ n/t − f(n), then with
+limit probability one, two random models of size n cannot be separated in
+GMLUd(n). If d(n) ≥ n/t + f(n), then with limit probability one, two ran-
+dom models of size n can be separated in GMLUd(n).
+We say that a class is a majority class, if it contains more than half of
+all the models. The following theorem is a quantitative version of Theorem
+5.8.
+Theorem 5.10. Let n ∈ Z+.
+• If d ≤ n/t − c1
+√n, where
+c1 := 1
+t
+�
+2t ln(2t),
+then the distribution ≡d for models of size n has a majority class.
+• If d ≥ n/t − c2
+√n, where
+c2 :=
+�
+π
+2t3(4t)1/(t−1) < c1
+then the distribution ≡d for models of size n does not have a majority
+class.
+Proof. Suppose first that d ≤ n/t − c1
+√n. Set δ = (tc1)/√n, in which case
+δ · n
+t = c1
+√n and δ2 · n/(2t) = (tc1)2/(2t). Applying Chernoff bound and
+the union bound we obtain, in a similar manner as in the proof of Lemma
+5.4, that with probability strictly greater than (1 − te−(tc1)2/(2t)) every type
+is realized at least n/2 − c1
+√n-times. A quick calculation shows that this
+latter probability is equal to 1/2 (hence the choice of c1).
+Consider then the case d ≥ n/t − c2
+√n.
+Let M = Mn, where n =
+(d, . . . , d). Using the estimate from the proof of Lemma 5.6 with f(n) =
+c2
+√n, we obtain
+|M|
+2n|τ| ≤
+�
+tt
+(2π)t−1 · (2t)t−1 · (c2
+√n)t−1
+√
+nt−1
+= 1/2
+27
+
+Thus M is not a majority class.
+The same reasoning as in the proof of
+Lemma 5.6 shows there is no other majority class.
+We of course obtain a corresponding quantitative version of Corollary
+5.9.
+Corollary 5.11. Let n ∈ Z+. If d ≤ n/t − c1
+√n, then the probability that
+two random models of size n can be separated in GMLUd is less than 1/4.
+If d ≥ n/t − c2
+√n, then the probability that two random models of size n
+can be separated in GMLUd is at least 1/2.
+6
+Conclusion
+We have established an interesting monotone connection between model
+class sizes and description complexities, also obtaining related results for
+entropy. Furthermore, we have characterized the phase transitions of model
+class size when the domain size n and expressive power (in the form of count-
+ing depth d) is altered. These results elucidate the interplay of class size and
+formula length. While focusing on GMLUd, the results have been intended
+to give a general overview of related phenomena. Thereby, an obvious future
+direction involves investigating how these results lift into the framework of
+first-order logic. There, natural parametrizations—analogous to varying the
+counting depth d—can possibly be obtained by using quantifier depth and
+the number of variables.
+Moving beyond first-order logic, it would be interesting to investigate
+the expressively Turing-complete logic CL, or computation logic, introduced
+in [9]. Studying description complexities within that framework would lead
+to an even closer link to Kolmogorov complexity.
+Acknowledgments. Antti Kuusisto and Miikka Vilander were supported
+by the Academy of Finland project Explaining AI via Logic (XAILOG),
+grant number 345612 (Kuusisto). Antti Kuusisto was also supported by the
+Academy of Finland project Theory of computational logics, grant numbers
+324435, 328987, 352419, 352420, 352419, 353027.
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+tion of Algorithms for Solving Problems with Combinatorial Explosion,
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+30
+

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+Balanced Datasets for IoT IDS 
+Alaa Alhowaide*  
+Computer Science Department 
+Memorial University of Newfoundland 
+St. John’s, NL, Canada 
+azalhowaide@mun.ca 
+Izzat Alsmadi 
+Department of Computing and 
+Cybersecurity 
+Texas A&M University-San Antonio 
+San Antonio, Texas, United States 
+ialsmadi@tamusa.edu 
+Jian Tang 
+Computer Science Department 
+Memorial University of 
+Newfoundland 
+St. John’s, NL, Canada 
+jian@mun.ca 
+ 
+Abstract 
+As the Internet of Things (IoT) continues to grow, cyberattacks are becoming increasingly common. The security 
+of IoT networks relies heavily on intrusion detection systems (IDSs). The development of an IDS that is accurate 
+and efficient is a challenging task. As a result, this challenge is made more challenging by the absence of balanced 
+datasets for training and testing the proposed IDS. In this study, four commonly used datasets are visualized and 
+analyzed visually. Moreover, it proposes a sampling algorithm that generates a sample that represents the original 
+dataset. In addition, it proposes an algorithm to generate a balanced dataset. Researchers can use this paper as a 
+starting point when investigating cybersecurity and machine learning. 
+The proposed sampling algorithms showed reliability in generating well-representing and balanced samples from 
+NSL-KDD, UNSW-NB15, BotNetIoT-01, and BoTIoT datasets. 
+Keywords: Sampling, Balanced Datasets, Cybersecurity, IDS, IoT 
+ 
+1 
+Introduction 
+A rapid growth of applications in our lives depending on Internet of Things (IoT). IoT applications connect 
+everything from thermostats, toaster, fridge to complete systems to the Internet [1]. Most of the common IoT 
+applications include personal healthcare [2], smart transportations [3] [2], smart grid [4], smart industrial 
+automation [5], and intelligent emergency response systems [2] ranging from monitoring to decision making. All 
+these systems aim to provide a more comfortable lifestyle and improve our capabilities to experience a 
+considerably better life [3], which looks “smart.”. The number of connected devices reached 9.5 billion devices 
+at the end of 2019, according to IoT Analytics [6]. IoT Analytics expects the previous figure to reach 22 billion 
+by 2025 [7]. This rapid growth aligned with lake of standers and low-cost manufacturing enormous increased the 
+number and types of security threats [8]. Consequently, IoT network security becomes mandatory.  
+IDS is the first defense line against cyber-attacks. It refers to a device or software strategically allocated at a 
+specific point on a network to monitor all the traffic [9]. IDS reports malicious behaviors to the network 
+administrator, stop the malicious behaviors, and encompass intruders’ identification [8]. IDS are categorized into 
+an anomaly, and signature detection, and hybrid. The anomaly-based detection depends on the behavioral methods 
+in which it defines two types of behaviors; the normal and abnormal behaviors [10]. Most of IDS types of highly 
+depends on Machine Learning (ML) [9], [11], and [12], . ML plays a significant role in developing IDSs to detect 
+
+malicious threats. The performance of ML models highly depends on the dataset(s) that are used to train, validate, 
+and test the MLs. With the absence of balanced datasets and lake of standards and details of how samples are 
+selected from the training/testing datasets, results become unreliable.  
+The contributions of this research are (i) visual analysis of commonly used datasets; (ii) sampling algorithm 
+that generates a representative sample of the original datasets (iii) sampling algorithm that generates representative 
+balanced samples from the imbalanced datasets; (iii) two methods to measure the statistical significance of the 
+produced results. 
+Section two provides a literature review. Section three explains the proposed sampling algorithms, while 
+sections four and five discuss experiments and results, respectively. Lastly, section six concludes the study. 
+2 
+Datasets in literature 
+This research considers four datasets: NSL-KDD [13], UNSW-NB15 [14], BotNetIoT-01 [15], and BoTIoT [16] 
+datasets. The NSL-KDD dataset is the most commonly used in intrusion detection, which contains regular network 
+traffic. Similarly, UNSW-NB15 contains regular network traffic. NSL-KDD and UNSW-NB15 represent a 
+benchmark for intrusion detection. In this regard, they allow the current research to be compared with previous 
+studies. BotNetIoT-01 dataset contains the traffic of nine IoT devices. Mirai and Gafgyt botnet attacks are the 
+only attacks in the BotNetIoT-01 dataset. BotNetIoT-01 dataset has 23 feature. This dataset is available at [17]. 
+BoTIoT is a recently published dataset of simulated IoT network traffic. BoTIoT has a variety of recent/new IoT 
+attacks. A detailed analysis of these datasets features and traffic is available at [15] and [9].  
+3 
+Proposed sampling methods 
+This section presents two sampling algorithms. Algorithm 1 select a random sample that is similar to the original 
+dataset. It uses the permutation to increase randomness of selecting the sample. Then it simply selects the first 
+num of records in the reindexed dataset. After selecting a random sample, it checks the similarity of the selected 
+labels’ distributions with the original dataset labels’ distribution. If they were similar, it returns the random sample. 
+Algorithm 1: The algorithm for sampling 
+Input: DS: dataset, num: sample size 
+Output: S: Sample of size num 
+Procedure: getSample(DS, num) 
+1 
+S ← Ø 
+2 
+Repeat as S is not similar to DS 
+3 
+    S = DS.reindex(np.random.permutation(dataframe.index)) 
+4 
+    S = S.head(num) 
+5 
+Return S 
+Algorithm 2 generates a balanced sample by first identifying the least represented class label in the original 
+dataset, referred as MinLabel. Then, it selects from the dominant class label a number or records equals to the 
+number of MinLabel using Algorithm 1. 
+ 
+Algorithm 2: The algorithm for generating a balanced sample 
+Input: DS: dataset 
+Output: B : Balanced sample 
+
+Procedure: getBalancedSample(DS) 
+1 
+B ← Ø 
+2 
+Define minimum class label, MinLabel 
+3 
+minClassSet=DS[‘label’== MinLabel] 
+4 
+S=getSample(DS[‘label’!= MinLabel], minimum class label count) 
+5 
+B=concat(minClassSet, S) 
+6 
+Return B 
+4 
+Experiments 
+All experiments were implemented using Python 3.8.8. A label feature was added or edited in every dataset to 
+represent the normal and attack traffics by 0 and 1, respectively. Furthermore, all the categorical features values 
+were transformed into numerical values to facilitate the visual representation using the PCA. The duplicate records 
+were removed from all the datasets as well. In NSL-KDD, the features num_outbound_cmds and is_hot_login 
+were dropped because they have zero value for all records. The generated balanced samples are available at [**]. 
+The Z-test experiments were executed with a freedom value equal to zero. The alternative hypothesis (H1) was: 
+The difference in means is not equal to zero. The experiments considered two methods to measure the significant 
+difference between the datasets and their balanced samples. The first method considered the original features of a 
+corresponding dataset. The method measured the similarity using Z-test for each feature. If all features were found 
+similar, then the balances sample is considered as a good representative sample of the original dataset. 
+The second method considered three dimensional PCA model to represent each of the original datasets and the 
+balanced sample. Similarly, it tests the similarity of each of the PCAs’ dimensions. The two sets are similar if all 
+the PCA’s dimensions are similar. 
+5 
+Results 
+Figures and Tables from 1 to 12 display the distributions and counts, respectively, for each traffic type in the 
+datasets and the generated samples. The blue color in the pie charts stands for normal traffic, while other colors 
+stand for different attacks. The normal traffic is the dominant class in NSL-KDD and UNSW-NB15 datasets, 
+while it is the opposite case for BoTIoT BotNetIoT-01 datasets. 
+Significantly, the 50% sample pie charts are identical to the original datasets for all the datasets. This similarity 
+of distribution leads to the success of Algorithm 1 proposed in section 3. Notably, the attacks distribution of 
+BotNetIoT-01 and BoTIoT balanced samples is quite similar to the corresponding original datasets. It is notable 
+that the highly represented attack type in the original dataset still reserves its representation in the 50% and 
+balanced datasets.  
+ 
+ 
+ 
+ 
+
+ 
+Fig. 1 NSL full dataset traffic 
+distribution. 
+ 
+Fig. 2 NSL 50% dataset traffic 
+distribution. 
+ 
+Fig. 3 NSL balanced dataset traffic 
+distribution. 
+ 
+Table 1 NSL full dataset traffic counts. 
+Traffic Type 
+count 
+% 
+Normal. 
+87832 
+0.60329977 
+neptune. 
+51820 
+0.35594082 
+back. 
+968 
+0.00664899 
+teardrop. 
+918 
+0.00630555 
+satan. 
+906 
+0.00622313 
+warezclient. 
+893 
+0.00613383 
+ipsweep. 
+651 
+0.00447158 
+smurf. 
+641 
+0.0044029 
+portsweep. 
+416 
+0.00285742 
+pod. 
+206 
+0.00141497 
+nmap. 
+158 
+0.00108527 
+guess_passwd. 
+53 
+0.00036405 
+buffer_overflow 
+30 
+0.00020606 
+warezmaster. 
+20 
+0.00013738 
+land. 
+19 
+0.00013051 
+imap. 
+12 
+8.2426E-05 
+rootkit. 
+10 
+6.8688E-05 
+loadmodule. 
+9 
+6.1819E-05 
+ftp_write. 
+8 
+5.495E-05 
+multihop. 
+7 
+4.8082E-05 
+phf. 
+4 
+2.7475E-05 
+perl. 
+3 
+2.0606E-05 
+spy. 
+2 
+1.3738E-05 
+Total 
+145586 
+  
+ 
+ 
+Table 2 NSL 50% dataset traffic counts. 
+Traffic Type 
+count 
+% 
+Normal. 
+43717 
+0.60056599 
+neptune. 
+26118 
+0.35879824 
+back. 
+506 
+0.00695122 
+teardrop. 
+464 
+0.00637424 
+satan. 
+461 
+0.00633303 
+warezclient. 
+451 
+0.00619565 
+smurf. 
+304 
+0.00417623 
+ipsweep. 
+302 
+0.00414875 
+portsweep. 
+213 
+0.00292611 
+pod. 
+104 
+0.00142871 
+nmap. 
+79 
+0.00108527 
+guess_passwd. 
+21 
+0.00028849 
+buffer_overflow 
+16 
+0.0002198 
+warezmaster. 
+11 
+0.00015111 
+rootkit. 
+6 
+8.2426E-05 
+imap. 
+5 
+6.8688E-05 
+ftp_write. 
+4 
+5.495E-05 
+land. 
+4 
+5.495E-05 
+multihop. 
+4 
+5.495E-05 
+phf. 
+1 
+1.3738E-05 
+loadmodule. 
+1 
+1.3738E-05 
+perl. 
+1 
+1.3738E-05 
+Total 
+72793 
+  
+ 
+Table 3 NSL balanced dataset traffic 
+counts. 
+Traffic Type 
+count 
+% 
+Normal. 
+57754 
+0.5 
+neptune. 
+51820 
+0.44862693 
+back. 
+968 
+0.00838037 
+teardrop. 
+918 
+0.0079475 
+satan. 
+906 
+0.00784361 
+warezclient. 
+893 
+0.00773107 
+ipsweep. 
+651 
+0.00563597 
+smurf. 
+641 
+0.0055494 
+portsweep. 
+416 
+0.00360148 
+pod. 
+206 
+0.00178343 
+nmap. 
+158 
+0.00136787 
+guess_passwd. 
+53 
+0.00045884 
+buffer_overflow 
+30 
+0.00025972 
+warezmaster. 
+20 
+0.00017315 
+land. 
+19 
+0.00016449 
+imap. 
+12 
+0.00010389 
+rootkit. 
+10 
+8.6574E-05 
+loadmodule. 
+9 
+7.7917E-05 
+ftp_write. 
+8 
+6.9259E-05 
+multihop. 
+7 
+6.0602E-05 
+phf. 
+4 
+3.463E-05 
+perl. 
+3 
+2.5972E-05 
+spy. 
+2 
+1.7315E-05 
+Total 
+115508 
+  
+ 
+ 
+ 
+Fig. 4 UNSW-NB15 full dataset traffic 
+distribution. 
+ 
+Fig. 5 UNSW-NB15 50% dataset traffic 
+distribution. 
+ 
+Fig. 6 UNSW-NB15 balanced dataset 
+traffic distribution. 
+Table 4 UNSW-NB15 full dataset traffic 
+counts. 
+Traffic Type 
+count 
+% 
+Normal 
+34205 
+0.61141499 
+Exploits 
+7609 
+0.13601101 
+Fuzzers 
+4838 
+0.08647934 
+Generic 
+3657 
+0.06536894 
+Reconnaissance 
+2703 
+0.04831617 
+DoS 
+1718 
+0.03070928 
+Analysis 
+446 
+0.00797226 
+Shellcode 
+378 
+0.00675676 
+Backdoor 
+346 
+0.00618476 
+Worms 
+44 
+0.0007865 
+Total 
+55944 
+  
+ 
+Table 5 UNSW-NB15 50% dataset 
+traffic counts. 
+Traffic Type 
+count 
+% 
+Normal 
+17185 
+0.61436436 
+Exploits 
+3797 
+0.13574289 
+Fuzzers 
+2380 
+0.08508509 
+Generic 
+1769 
+0.06324181 
+Reconnaissance 
+1321 
+0.0472258 
+DoS 
+892 
+0.03188903 
+Analysis 
+236 
+0.00843701 
+Shellcode 
+192 
+0.00686401 
+Backdoor 
+183 
+0.00654226 
+Worms 
+17 
+0.00060775 
+Total 
+27972 
+  
+ 
+Table 6 UNSW-NB15 balanced dataset 
+traffic counts. 
+Traffic Type 
+count 
+% 
+Normal 
+21739 
+0.5 
+Exploits 
+7609 
+0.17500805 
+Fuzzers 
+4838 
+0.11127467 
+Generic 
+3657 
+0.0841115 
+Reconnaissance 
+2703 
+0.06216937 
+DoS 
+1718 
+0.03951424 
+Analysis 
+446 
+0.01025806 
+Shellcode 
+378 
+0.00869405 
+Backdoor 
+346 
+0.00795805 
+Worms 
+44 
+0.00101201 
+Total 
+43478 
+ 
+ 
+ 
+
+Traffic Types:
+1% 1%
+0%
+1%,
+Inormal
+1%
+neptune.
+ back.
+teardrop.
+ satan.
+warezclient.
+■ipsweep.
+36%
+ smurf.
+■portsweep.
+ pod.
+ nmap.
+60%
+guess_passwd.
+■buffer_overflow.
+■warezmaster.
+ land.
+imap.
+rootkit.
+loadmodule.
+ftp_write.50
+1%_1%
+0%,
+Traffic,Types:
+1% ,
+1%.
+I normal
+I neptune.
+- bark.
+ tearcirop.
+I satan.
+36%
+I warezclient.
+jμnuis 
+Iipsveep.
+60%
+ portsweep.
+I pod.
+I nmap.
+ssed ssang 
+d.
+[uaro" ianq
+OU.1%
+1%
+1%.
+1%
+Traffic Types:
+(%
+■normal.
+1%
+neptune.
+ back.
+teardrop.
+ satan.
+warezclient.
+■ipsweep.
+ smurf.
+portsweep.
+50%
+ pod.
+45%
+ nmap.
+guess_passwd.
+■buffer_overflow
+■warezmaster.
+ land.
+ imap.
+rootkit.
+loadmodule
+■ftp_write.1%
+1%
+.1%
+0%
+Traffic Types:
+■Normal
+3%
+5%
+■Exploits
+6%
+ Fuzzers
+Generic
+9%
+Reconnaissance
+ DoS
+13%
+61%
+■Analysis
+Shellcode
+■Backdoor
+■Worms1%
+1% 
+.1%
+.0%
+Traffic Type:
+3%
+Normal
+5%
+■Exploits
+6%
+Fuzzers
+ Generic
+8%
+■Reconnaissance
+ Dos
+14%
+61%
+■Analysis
+■Shellcode
+■Backdoor
+Worms1%
+.1%
+1%
+0%
+Traffic Types:
+4%
+Normal
+6%
+■Exploits
+Fuzzers
+8%
+Generic
+Reconnaissance
+50%
+11%
+ Dos
+Analysis
+Shellcode
+18%
+■Backdoor
+■Worms 
+Fig. 7 BotIoT full dataset traffic 
+distribution. 
+ 
+Fig. 8 BotIoT 50% dataset traffic 
+distribution. 
+ 
+Fig. 9 BotIoT balanced dataset traffic 
+distribution. 
+Table 7 BotIoT full dataset traffic 
+counts. 
+Traffic Type 
+count 
+% 
+Normal 
+477 
+0.00013003 
+UDP 
+198123
+0 
+0.54006218 
+TCP 
+159318
+0 
+0.43428389 
+Service_Scan 
+73168 
+0.01994482 
+OS_Fingerprint 
+17914 
+0.00488317 
+HTTP 
+2474 
+0.00067439 
+Keylogging 
+73 
+1.9899E-05 
+Data_Exfiltration 
+6 
+1.6355E-06 
+ Total 
+366852
+2 
+  
+ 
+Table 8  BotIoT 50% dataset traffic 
+counts. 
+Traffic Type 
+count 
+% 
+Normal 
+231 
+0.000125936 
+UDP 
+990566 
+0.540035469 
+TCP 
+796673 
+0.434329139 
+Service_Scan 
+36604 
+0.019955721 
+OS_Fingerprint 
+8916 
+0.004860813 
+HTTP 
+1239 
+0.000675476 
+Keylogging 
+29 
+1.58102E-05 
+Data_Exfiltration 
+3 
+1.63554E-06 
+Total 
+183426
+1 
+  
+ 
+Table 9  BotIoT balanced dataset traffic 
+counts. 
+Traffic Type 
+count 
+% 
+Normal 
+477 
+0.5 
+TCP 
+235 
+0.24633124 
+UDP 
+232 
+0.24318658 
+Service_Scan 
+10 
+0.01048218 
+Total 
+954 
+  
+ 
+ 
+ 
+Fig. 10 BotNetIoT-01 full dataset traffic 
+distribution. 
+ 
+Fig. 11 BotNetIoT-01 50% dataset traffic 
+distribution. 
+ 
+Fig. 12 BotNetIoT-01 balanced dataset 
+traffic distribution. 
+Table 10 BotNetIoT-01 full dataset 
+traffic counts. 
+Traffic 
+Type 
+count 
+% 
+Normal 
+555932 
+0.07871485 
+udp 
+2176365 
+0.30815325 
+tcp 
+859850 
+0.12174685 
+scan 
+793090 
+0.11229424 
+syn 
+733299 
+0.10382839 
+ack 
+643821 
+0.09115913 
+udpplain 
+523304 
+0.07409503 
+combo 
+515156 
+0.07294135 
+junk 
+261789 
+0.03706691 
+Total 
+7062606 
+  
+ 
+Table 11 BotNetIoT-01 50% dataset 
+traffic counts. 
+Traffic 
+Type 
+count 
+% 
+Normal 
+277631 
+0.07861999 
+udp 
+1088615 
+0.30827573 
+tcp 
+429603 
+0.12165566 
+scan 
+397213 
+0.11248341 
+syn 
+366084 
+0.10366825 
+ack 
+322252 
+0.09125583 
+udpplain 
+261385 
+0.07401942 
+combo 
+257771 
+0.072996 
+junk 
+130749 
+0.03702571 
+Total 
+3531303 
+ 
+ 
+Table 12 BotNetIoT-01 balanced dataset 
+traffic counts. 
+Traffic 
+Type 
+count 
+% 
+Normal 
+555932 
+0.5 
+udp 
+186062 
+0.16734241 
+tcp 
+73405 
+0.06601977 
+scan 
+67626 
+0.06082219 
+syn 
+62917 
+0.05658696 
+ack 
+54864 
+0.04934416 
+udpplain 
+44624 
+0.04013441 
+combo 
+44091 
+0.03965503 
+junk 
+22343 
+0.02009508 
+Total 
+1111864 
+  
+ 
+To increase the reliability of the produced results by algorithms 1 and 2, we executed Z-test to measure the 
+statistical significance. Z-test was applied in two ways to measure if there is a significant difference between the 
+original dataset and the balanced dataset. Table 13 shows the two methods results of comparing the 50% and the 
+balanced samples compared with their corresponding original datasets. Only the NSL 50% sample was found 
+similar using the first method. On the other hand, all samples were found similar using the second method. 
+ 
+ 
+
+2%_
+1%_
+0% _0%
+0%_0%
+Traffic Type:
+ Normal
+ UDP
+ TCP
+43%
+ Service_Scan
+54%
+ OS_Fingerprint
+= HTTP
+ Keylogging
+Data_Exfiltration1%
+%0
+_0%
+.0%_0%
+2% ~
+Traffic Type:
+ Normal
+■ UDP
+= TCP
+43%
+ Service_Scan
+54%
+- OS_Fingerprint
+■ HTTP
+ Keylogging
+Data_Exfiltration1%
+24%
+Traffic Type:
+ Normal
+■ TCP
+50%
+■UDP
+Service_Scan
+25%0.04 
+Traffic Type:
+0.08
+0.07
+ Normal
+I udp
+= tcp
+0.09
+I scan
+0.31
+I syn
+ ack
+0.10
+I udpplain
+I combo
+0.11
+0.12
+ junk0.04 
+Traffic Type:
+0.08
+0.07
+ Normal
+I udp
+= tcp
+ scan
+0.09
+0.31
+ syn
+ ack
+0.10
+ udpplain
+ combo
+0.11
+0.12
+ junk2%
+Traffic Type:
+6t
+ Normal
+5%
+ udp
+6%
+ tcp
+ scan
+6%
+50%
+ syn
+■ ack
+6%
+ udpplain
+■combo
+17%
+ junkTable 13 Z-test results using All-feature and PCA comparisons. 
+Dataset 
+Sample 
+Features 
+PCA 
+NSL 
+50%  
+similar 
+similar 
+Balanced DS 
+13 similar, 27 different features 
+similar 
+NB15 
+50% DS 
+6 similar, 37 different 
+similar 
+Balanced DS 
+4 similar, 39 different features 
+similar 
+Botnetiot 
+50% DS 
+24 similar, 0 different 
+similar 
+Balanced DS 
+0 similar, 24 different features 
+similar 
+Botiot 
+50% DS 
+43 same, 0 different 
+similar 
+Balanced DS 
+4 similar, 39 different features 
+similar 
+To better understand these results, we plot a visual representation of the original datasets, 50%, and balanced 
+samples using the PCA. Table 14 presents all the 3D PCA plot for all datasets and samples. Also, the explained 
+variance of each dimension of the PCA and the accumulative explained variance are displayed above each plot. 
+Interestingly, the samples have similar plot as their corresponding datasets, without exception. From Table 14, the 
+second method that used PCA seems to be more accurate. 
+Table 14 3D visualization of all datasets and samples using PCA. 
+Dataset 
+Full 
+50% sample 
+Balanced sample 
+NSL 
+Dim Var=[0.99888489 0.99999954 0.99999999] 
+Acc Var=0.9999999897744162
+ 
+Dim Var=[0.99945449 0.99999977 0.99999999] 
+Acc Var=0.999999994890656
+ 
+Dim Var=[0.99894055 0.99999965 0.99999999] 
+Acc Var0.9999999918070609
+ 
+UNSW-NB15 
+Dim Var= [0.74721778 0.99154115 
+0.99999804] 
+Acc Var=0.9999980361116154
+ 
+Dim Var= [0.74798519 0.99129133 
+0.99999805] 
+Acc Var=0.9999980485289511 
+ 
+Dim Var= [0.74993195 0.9919089 0.99999836] 
+Acc Var=0.9999983606509643 
+ 
+BotIoT 
+Dim Var=[0.61936997 0.87203041 0.97291457] 
+Acc Var=0.9729145691941684 
+ 
+Dim Var= [0.57665246 0.83934756 
+0.96824873] 
+Acc Var=0.9682487261636522 
+ 
+Dim Var= [0.83554185 0.97786429 
+0.99720171] 
+Acc Var= 0.9972017146909109 
+ 
+BotNetIoT-01 
+Dim Var=[1. 1. 1.] 
+Acc Var=1 
+ 
+Dim Var=[1. 1. 1.] 
+Acc Var=0.9999999999996793 
+ 
+Dim Var=[1. 1. 1.] 
+Acc Var=0.9999999999998491 
+ 
+
+60000
+50000
+40000
+3000PC3
+20000
+10000
+0
+0
+3
+le8
+1
+2
+2PC2
+3
+4
+1
+PC1
+6
+040000
+30000
+20000
+PC3
+10000
+0
+0
+3
+les
+1
+2
+2 PC2
+3
+PC1
+5
+1
+6
+040000
+30000
+2000C3
+10000
+0
+0
+3
+le8
+1
+2
+2 PC2
+3
+PC1
+4
+5
+1
+6
+05
+4
+3 PC3
+2
+1
+0
+3
+2
+1
+le9
+-2
+-1
+0
+1
+PC2
+PC1
+N
+-2
+3
+-35
+3 PC3
+2
+1
+0
+m
+2
+1
+le9
+-2
+-1
+PC2
+PC1
+2
+-2
+3
+-35
+3 PC3
+2
+1
+0
+m
+2
+1
+le9
+2
+1
+0
+-1
+PC2
+PC1
+2
+-2
+3
+-3150
+1.0
+0.5 PC3
+0.0
+0.5
+1
+0.0
+le8
+0.5
+1.0
+0
+PC2
+1.5
+2.0
+-1
+PC1
+2.5
+30150
+1.0
+0.5 PC3
+0.0
+0.5
+1
+0.0
+le8
+0.5
+1.0
+1.5
+0
+PC2
+2.0
+-1
+PC1
+2.56
+4
+2 PC3
+0
+2
+-4
+1.5
+1.0
+0.0
+0.5
+le8
+0.5
+1.0
+0.0 PC2
+1.5
+PC1
+2.0
+2.5
+0.5600000
+400009c3
+500000
+B00000
+200000
+100000
+0
+2
+le17
+D
+2
+2
+PC2
+3
+4
+PC1600000
+400009c3
+500000
+B00000
+200000
+100000
+0
+2
+le17
+2
+2
+PC2
+3
+4
+PC1000008
+600000
+00000z
+1.2
+0.6
+0
+0.2 PC2
+0.4
+le17
+0.0
+PC1
+0.26 
+Conclusion 
+As IoT network evolves and expands, the need for reliable and efficient IDSs increases. Therefore, it is 
+mandatory to have high-quality datasets to train and test these security systems. It is significant to have balanced 
+datasets to generate reliable results. In this research, four commonly used datasets to build IDSs were used to 
+extract balanced samples. Two novel sampling algorithms were proposed. Furthermore, two methods that measure 
+the goodness of the generated samples were proposed. 
+Training the IDS on balanced datasets has a significant role in learning reliable models. It increases our 
+confidence in the detection model's results. The proposed algorithms showed the ability to generate good 
+representing samples out from the used datasets based on the Z-test and the 3D PCA plots. 
+7 References 
+[1] 
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+Proceedings of the Fifth Cybersecurity Symposium on - CyberSec ’18, Coeur d’ Alene, Idaho, Apr. 2018, pp. 1–4. doi: 
+10.1145/3212687.3212872. 
+[2] 
+L. Atzori, A. Lera, and G. Morabito, “The Internet of Things: A survey | Elsevier Enhanced Reader,” Computer 
+networks, vol. 54, no. 15, pp. 2787–2805, 2010, doi: 10.1016/j.comnet.2010.05.010. 
+[3] 
+A. Al-Fuqaha, M. Guizani, M. Mohammadi, M. Aledhari, and M. Ayyash, “Internet of Things: A Survey on Enabling 
+Technologies, Protocols, and Applications,” IEEE Communications Surveys Tutorials, vol. 17, no. 4, pp. 2347–2376, 
+Fourthquarter 2015, doi: 10.1109/COMST.2015.2444095. 
+[4] 
+K. J. Kaur and A. Hahn, “Exploring ensemble classifiers for detecting attacks in the smart grids,” in Proceedings of 
+the Fifth Cybersecurity Symposium, Coeur d’ Alene Idaho, Apr. 2018, pp. 1–4. doi: 10.1145/3212687.3212873. 
+[5] 
+S. Shen, L. Huang, H. Zhou, S. Yu, E. Fan, and Q. Cao, “Multistage Signaling Game-Based Optimal Detection 
+Strategies for Suppressing Malware Diffusion in Fog-Cloud-Based IoT Networks,” IEEE Internet of Things Journal, vol. 5, 
+no. 2, pp. 1043–1054, Apr. 2018, doi: 10.1109/JIOT.2018.2795549. 
+[6] 
+K. Lueth, “IoT 2019 in Review: The 10 Most Relevant IoT Developments of the Year.” https://iot-analytics.com/iot-
+2019-in-review/ (accessed May 27, 2020). 
+[7] 
+K. Lueth, “State of the IoT 2018: Number of IoT devices now at 7B – Market accelerating.” https://iot-
+analytics.com/state-of-the-iot-update-q1-q2-2018-number-of-iot-devices-now-7b/ (accessed May 27, 2020). 
+[8] 
+A. Alhowaide, I. Alsmadi, and J. Tang, “Ensemble Detection Model for IoT IDS,” Internet of Things, p. 100435, 
+Jul. 2021, doi: 10.1016/j.iot.2021.100435. 
+[9] 
+A. Alhowaide, I. Alsmadi, and J. Tang, “Towards the design of real-time autonomous IoT NIDS,” Cluster Comput, 
+Jan. 2021, doi: 10.1007/s10586-021-03231-5. 
+[10] 
+M. Aldwairi, W. Mardini, and A. Alhowaide, “Anomaly Payload Signature Generation System Based on Efficient 
+Tokenization Methodology,” International Journal on Communications Antenna and Propagation (IRECAP) (2018), Nov. 
+2018. 
+[11] 
+A. Alhowaide, I. Alsmadi, and J. Tang, “PCA, Random-Forest and Pearson Correlation for Dimensionality 
+Reduction in IoT IDS,” in 2020 IEEE International IOT, Electronics and Mechatronics Conference (IEMTRONICS), Sep. 
+2020, pp. 1–6. doi: 10.1109/IEMTRONICS51293.2020.9216388. 
+[12] 
+A. Alhowaide, I. Alsmadi, and J. Tang, “An Ensemble Feature Selection Method for IoT IDS,” in 2020 IEEE 6th 
+International Conference on Dependability in Sensor, Cloud and Big Data Systems and Application (DependSys), Dec. 2020, 
+pp. 41–48. doi: 10.1109/DependSys51298.2020.00015. 
+
+[13] 
+“NSL-KDD 
+| 
+Datasets 
+| 
+Research 
+| 
+Canadian 
+Institute 
+for 
+Cybersecurity 
+| 
+UNB.” 
+https://www.unb.ca/cic/datasets/nsl.html (accessed Nov. 20, 2019). 
+[14] 
+N. Moustafa and J. Slay, “UNSW-NB15: a comprehensive data set for network intrusion detection systems (UNSW-
+NB15 network data set),” in 2015 Military Communications and Information Systems Conference (MilCIS), Nov. 2015, pp. 
+1–6. doi: 10.1109/MilCIS.2015.7348942. 
+[15] 
+A. Alhowaide, I. Alsmadi, and J. Tang, “Features Quality Impact on Cyber Physical Security Systems,” in 2019 
+IEEE 10th Annual Information Technology, Electronics and Mobile Communication Conference (IEMCON), Vancouver, BC, 
+Canada, Oct. 2019, pp. 0332–0339. doi: 10.1109/IEMCON.2019.8936280. 
+[16] 
+“The 
+BoT-IoT 
+Dataset.” 
+https://www.unsw.adfa.edu.au/unsw-canberra-cyber/cybersecurity/ADFA-NB15-
+Datasets/bot_iot.php (accessed Dec. 12, 2019). 
+[17] 
+A. Alhowaide, “IoT dataset for Intrusion Detection Systems (IDS).” https://kaggle.com/azalhowaide/iot-dataset-for-
+intrusion-detection-systems-ids (accessed Nov. 09, 2021). 
+ 
+ 
+

D9AyT4oBgHgl3EQfevgJ/content/tmp_files/2301.00325v1.pdf.txt

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+Improved inference for MCP-Mod approach for
+time-to-event endpoints with small sample sizes
+Márcio A. Diniz∗
+Diego I. Gallardo†
+Tiago M. Magalhães‡
+Created: July 7, 2020, updated: July 2, 2022
+Abstract
+The Multiple Comparison Procedures with Modeling Techniques (MCP-Mod) frame-
+work has been recently approved by the U.S. Food and Administration and European
+Medicines Agency as fit-per-purpose for phase II studies. Nonetheless, this approach relies
+on the asymptotic properties of Maximum Likelihood (ML) estimators, which might not be
+reasonable for small sample sizes. In this paper, we derived improved ML estimators and
+correction for their covariance matrices in the censored Weibull regression model based on
+the corrective and preventive approaches. We performed two simulation studies to eval-
+uate ML and improved ML estimators with their covariance matrices in (i) a regression
+framework (ii) the Multiple Comparison Procedures with Modeling Techniques framework.
+We have shown that improved ML estimators are less biased than ML estimators yielding
+Wald-type statistics that controls type I error without loss of power in both frameworks.
+Therefore, we recommend the use of improved ML estimators in the MCP-Mod approach
+to control type I error at nominal value for sample sizes ranging from 5 to 25 subjects per
+dose.
+Keywords: MCP-Mod approach; small sample size; Weibull model; bias correction; co-
+variance refinement.
+MSC 2010: Primary 62Fxx; secondary 62F12
+∗Biostatistics and Bioinformatics Research Center, Samuel Oschin Comprehensive Cancer Center, Cedars
+Sinai Medical Center, Los Angeles, California, USA, e-mail to marcio.diniz@cshs.org.
+†Department of Mathematics, Engineering School, University of Atacama, Copiapó, Chile, e-mail to
+diego.gallardo@uda.cl.
+‡Department of Statistics, Institute of Exact Sciences, Federal University of Juiz de Fora, Juiz de Fora,
+Brazil, e-mail to tiago.magalhaes@ufjf.br.
+arXiv:2301.00325v1  [stat.ME]  1 Jan 2023
+
+2
+1
+Introduction
+Adequate designs for early-phase trials is essential to a successful clinical drug development.
+Traditionally, investigators evaluate safety in phase I trials, proof-of-concept (PoC) in phase IIa
+trials, and efficacy in phase IIb trials. When drugs are promising in these early stages, phase
+III trials are designed accruing a large number of patients to provide a definitive evidence of
+efficacy. Nonetheless, Jardim et al.1 showed that 20% of 80 cancer drug programs submitted
+between 2009 to 2015 to the Food and Drug Administration (FDA) did not have any data for
+phase II trials such that 31% obtained FDA approval; 46% had positive PoC with of 76% FDA-
+approval; and 34% had negative PoC with only 15% of FDA-approval. Therefore, attaining
+proof-of-concept is predictor to a successful phase III trial.
+Bretz et al.2 proposed a framework named Multiple Comparison Procedures with Modeling
+Techniques (MCP-Mod) unifying phase IIa and IIb trials into a seamless design while taking
+advantage of both traditional approaches for normally distributed endpoints. Later, Pinheiro
+et al.3 extended this methodology to general parametric models using generalized least squares
+estimation allowing statisticians to consider more complex designs and other types of endpoints
+such as binary and time to event. Regulatory agencies have stated their approval of the MCP-
+Mod framework as an adequate and efficient methodology for design and analysis of phase II
+dose-finding studies that will guide dose selection for phase III trials.
+The MCP-Mod framework is implemented in two steps: (i) MCP-step consists of a trend
+test to assess the presence of a dose response signal among a set of pre-specified candidate
+models while preserving FWER; (ii) Mod-step corresponds to estimate dose-response curves in
+order to identify the optimal dose that achieves a desired level of response in comparison to
+placebo among the models that were selected in the MCP-step. Therefore, the properties of
+the trend test defined as a Wald statistic used in the MCP-step and the maximum likelihood
+(ML) estimators with their covariance matrix used in the Mod-step are essential to the success-
+ful implementation of MCP-Mod approach. However, both steps rely on the the asymptotic
+properties of the ML estimators, which are only valid for large sample sizes.
+In cancer mouse studies, time-to-death is an endpoint that could be used to guide the identi-
+fication of the minimum effective dose (MED) with large expected effect sizes and small sample
+sizes. Nonetheless, the application of the MCP-Mod framework is limited due the underly-
+
+3
+ing asymptotic assumptions. In this context, parametric survival models such as the censored
+Weibull regression model (WRM) was showed to be useful given that provides clinical mean-
+ingful interpretation based on event time ratios or hazard ratios.4 Moreover, the asymptotic
+properties of the ML estimators can be studied and refined for small sample sizes. Recently,
+Magalhães et al.5 obtained the skewness coefficient of the distribution of the maximum likeli-
+hood estimators for WRM, and Magalhães et al.6 derived improved test statistics for LR, score
+and gradient tests but not for the Wald statistic.
+Our main goal is to derive improved ML estimators for the regression parameters and its
+second-order covariance matrix for WRM, then use them as input to the generalized least
+squares procedure proposed by Pinheiro et al.3 yielding a type I error probability closer to the
+nominal value when testing proof-of-activity and more accurate MED estimates. Moreover, we
+particularize the results from Cox and Snell7 and Magalhães et al.8 that are very general to
+WRM and they can be in much broader context than the MCP-Mod framework.
+The remaining paper is organized as follows: in Section 2, we revisit the Weibull distribution
+and its properties; in Section 3, we review the main concepts of the MCP-Mod framework; in
+Section 4, we introduce the improved estimators; in Section 5, we conducted a simulation study
+to evalute the use of improved estimators in the MCP-Mod framework; finally, in Section 6, we
+presented some concluding remarks.
+2
+Weibull distribution
+The Weibull distribution is commonly used to analysis of time-to-event or lifetime data and
+a continuous random variable T is called Weibull, if its probability density function (pdf) is
+f(t; λ, σ) =
+1
+σλ1/σ t1/σ−1 exp
+�
+− (t/λ)1/σ�
+, t > 0,
+(1)
+where σ > 0 is the shape parameter and λ > 0 is the scale parameter, it says T ∼ WE(λ, σ).
+Two particular models under this parametrization are obtained for σ = 1 and σ = 1/2, which
+represents the exponential and the Rayleigh models with means λ and λ
+�
+π/2, respectively. In
+this work, we focused on those models. However, if σ is unknown, we assume that it can be
+
+4
+replaced by consistent estimate. The ρ% survival time is given by
+tρ = λ[− log(ρ)]σ
+(2)
+The regression structure can be incorporated in (1) by making
+log(λi) = x⊤
+i β
+(3)
+where β is a p-vector of unknown parameters and xi is a vector of predictors related to the ith
+observation.
+In lifetime data, there is the censoring restriction, i.e, if T1, . . . , Tn are a random sample
+from (1), instead of Ti, we observe, under right censoring, ti = min(Ti, Li), where Li is the
+censoring time, independent of Ti, i = 1, . . . , n. Here, we consider an hybrid censoring scheme,
+where the study is finalized when a pre-fixed number r ≤ n out of n observations have failed, as
+well as when a prefixed time, say L1 = · · · = Ln = L, has been reached. The type I censoring
+is a particular case for r = n and the type II censoring appears when L1, . . . , Ln = +∞.
+Additionally, we add the non-informative censoring assumption, i.e., the random variables Li
+does not depend on λ. Usually, the regression modeling considers the distribution of Yi = log(Ti)
+instead of Ti, which is an accelerated lifetime model form, see Kalbfleisch and Prentice.9 The
+distribution of Yi is of the extreme value form with pdf given by
+f(yi; xi) = 1
+σ exp
+�yi − µi
+σ
+− exp
+�yi − µi
+σ
+��
+,
+−∞ < yi < ∞,
+(4)
+where µi = log λi. From this moment, we assume that σ is known, Then, the log-likelihood
+function derived from (4) is given by
+ℓ(β) =
+n
+�
+i=1
+�
+δi
+�
+−n log σ + yi − µi
+σ
+�
+− exp
+�yi − µi
+σ
+��
+.
+The total score function and the total Fisher information matrix for β are, respectively,
+U β = σ−1X⊤W 1/2v and K = Kββ = σ−2X⊤W X, where X = (x1, . . . , xn)⊤, the model ma-
+trix, assuming rank(X) = p, W = diag(w1, . . . , wn), wi = E
+�
+exp
+� yi−µi
+σ
+��
+and v = (v1, . . . , vn)⊤,
+vi =
+�
+−δi + exp
+� yi−µi
+σ
+��
+w−1/2
+i
+. It can observed that the value of wi depends on the mechanism
+
+5
+of censoring. That means wi = q ×
+�
+1 − exp
+�
+−L1/σ
+i
+exp(−µi/σ)
+��
++ (1 − q) × (r/n), where
+q = P
+�
+W(r) ≤ log Li
+�
+and W(r) denotes the rth order statistic from W1, . . . , Wn. Note that
+q = 1 and q = 0 for types I and II censoring, respectively, as showed in Magalhães et al.5
+The maximum likelihood estimator (MLE) of β, �β, is the solution of U β = 0. The �β can
+not be expressed in closed-form. It is typically obtained by numerically maximizing the log-
+likelihood function using a Newton or quasi-Newton nonlinear optimization algorithm. Under
+mild regularity conditions and in large samples,
+�β ∼ Np
+�
+β, K−1
+ββ
+�
+,
+(5)
+approximately. The classic Wald test10 statistic is
+WMLE =
+�
+C�β − Cβ(0)�⊤ �
+C �
+KC⊤�−1 �
+C�β − Cβ(0)�
+,
+(6)
+where C is a matrix of contrasts m × p. Under the null hypothesis H : Cβ = Cβ(0), WMLE
+has a χ2
+p distribution up to an error of order n−1. The null hypothesis is rejected for a given
+nominal level, α say, if the test statistic exceeds the upper 100(1 − α)% quantile of the χ2
+p
+distribution.
+3
+MCP-Mod General approach
+We briefly summarize the two-stage procedure discussed by Pinheiro et al.3 We consider
+that time-to-event responses tij for doses xi given to jth subject for i = 0, . . . , p and j = 1, . . . , ni
+can be described by the Weibull distribution with scale parameters λi and shape parameter σ
+defined in (1) and (3). Then, we consider the parameter µ as our response for the dose-response
+model such that it could be defined as the median survival time (2) for ρ = 0.5 or, alternatively,
+log(λ).
+Initially, a set of candidate dose-response models µi = f(xi, θ) is considered such that each
+model can be rewritten as function of standardized model as below
+
+6
+fm(x, θ) = θ0 + θ1f 0
+m(x, θ0)
+(7)
+for m = 1, . . . , M.
+In this work, we consider (M = 5) standardized models: (i) linear:
+f 0(x, θ0) = x; (ii) emax : f 0(x, θ0) = x/(x + ED50) where ED50 can be interpreted as the dose
+that produces the desired response on 50% of subjects; (iii) exponential: f 0(x, θ0) = exp{x/δ}−
+1, where δ is the exponential rate; (iv) logistic: f 0(x, θ0) = 1/(1 + exp{(ED50 − x)/δ}) (v)
+beta: f 0(x, θ0) = β(δ1, δ2)(x/scal)δ1(1 − x/scal)δ2.
+3.1
+MCP-step
+In this step, a set of contrasts corresponding to the candidate models will be tested. For
+each candidate model, an optimal contrast copt that maximizes the probability of rejecting the
+hypothesis of non-signal dose-response is derived assuming that the candidate model is correct
+and guess estimates for θ0:
+copt ∝ S−1
+�
+µ0
+m − µ0
+m
+⊤S−11
+1S−11⊤
+�
+,
+(8)
+where µ0
+m = (f 0
+m(x1, θ0), . . . , f 0
+m(xp, θ0)). Then, the test of hypotheses for proof-of-concept can
+be translated to H0 : copt
+m µ = 0 vs. H1 : copt
+m µ > 0 for candidate model m based on the Wald
+test statistic
+W (m) = (copt
+m ˆµ)⊤ �
+Copt ˆSCopt⊤�−1
+m,m copt
+m ˆµ
+(9)
+where Copt is the matrix m × p of optimal contrast with [A]m,m denoting the mth diagonal
+element of matrix A for m = 1, . . . , M. Critical values for tests are derived based on the joint
+distribution for W = (W (1), . . . , W (M)) allowing one to calculate multiplicity adjusted p-values
+controlling the FWER at a prespecificed nominal type I error α.
+
+7
+3.2
+Mod-step
+In this step, the estimation of non-linear dose responses models is performed in two stages. In
+the first stage, the parameters µ = [µ1, . . . , µp] are estimated using standard software packages
+with analysis of variance (ANOVA) parametrization for the design matrix resulting into a
+separate parameter µi for each dose level xi for i = 1, . . . , D. Let ˆµ denote the estimated
+dose-response parameter vector under ANOVA parametrization, such that we assume that ˆµ
+follows approximately Np(µ, S), where S is the covariance matrix of ˆµ that is estimated by ˆS.
+In particular for µ = log(λ), we have µ = β and S = K1
+ββ.
+In the second stage, the non-linear dose-response model f(x, θ) is fitted by minimizing the
+generalized linear squares (GLS) criterion:
+Ψ(θ) = (ˆµ − f(x, θ))′ ˆS
+−1(ˆµ − f(x, θ))
+(10)
+with respect to θ.
+Then, the minimum effective dose can be estimated as �
+MED = {x|f(x, ˆθ) > f(0, ˆθ) + ∆},
+where ∆ is a clinical meaningful threshold and ˆθ minimizes (10).
+4
+Improved inference
+4.1
+Bias correction
+Inferences based on maximum likelihood method depend strongly on asymptotic properties.
+Among these properties, the MLE is approximately non-biased, in other words, E( �β − β) =
+O(n−1), which is essential to define the mean of the normal distribution of �β.
+Therefore,
+likelihood inferences based on asymptotic approximation may not be reliable when sample sizes
+are small or moderate, and two approaches are available to correct the MLE.
+The corrective approach.
+The bias of the MLE can be written as E( �β − β) = B(β) +
+O(n−2), where B(β) is a term of order O(n−1), a function of the derivatives of the log-likelihood
+function. Cox and Snell7 proposed a bias-corrected maximum likelihood estimator (BCE), that
+can be expressed as �β = �β − B( �β), where B(β) is the term of order n−1, evaluated in �β and
+E(�λ − λ) = O(n−2), i.e., less biased then MLE of β. The Cox and Snell’s method is known as
+
+8
+a corrective approach because the MLE is calculated and then, the bias correction is applied.
+For the censored Weibull regression model, the expression of B( �β) has the form
+B( �β) = − 1
+2σ3P Zd (W + 2σW ′) 1,
+(11)
+where P = K−1X⊤, Z = XK−1X⊤, Zd is a diagonal matrix with diagonal given by the
+diagonal of Z, W ′ = diag(w′
+1, . . . , w′
+n), w′
+i = −σ−1L1/σ
+i
+exp{−L1/σ
+i
+exp(−µi/σ) − µi/σ} and 1
+is a n-dimensional vector of ones.
+The preventive approach.
+As alternative to the corrective approach, Firth11 proposed the
+following modification in the score vector:
+U ⋆
+β = Uβ − KββB(β),
+(12)
+where B(β) is given by (11). The estimator ˇβ, solution of U ⋆
+β = 0, has a bias of order O(n−2).
+This is a preventive approach because the procedure already computes a less biased estimator
+than the regular MLE.
+4.2
+Covariance correction
+From the general result of Magalhães et al.,8 we derived the specific matrix expression for
+the MLE and BCE second-order covariance matrices for the censored Weibull regression model
+and it is given by
+Covτ
+2(β⋆) = K−1 + K−1 �
+∆ + ∆⊤�
+K−1 + O(n−3),
+(13)
+where ∆ = −0.5∆(1) + 0.25∆(2) + 0.5τ2∆(3) with
+∆(1) = 1
+σ4X⊤W ⋆ZdX,
+∆(2) = − 1
+σ6X⊤ �
+W Z(2)W − 2σW Z(2)W ′ − 6σ2W ′Z(2)W ′�
+X,
+∆(3) = 1
+σ5X⊤W ′W ⋆⋆X,
+W ⋆ = diag(w⋆
+1, . . . , w⋆
+n), w⋆
+i = wi(wi − 2) − 2σw′
+i + στ1(w′
+i + 2σw′′
+i ), Z(2) = Z ⊙ Z, with ⊙
+
+9
+representing a direct product of matrices (Hadamard product), W ⋆⋆ is a diagonal matrix, with
+Z(W +2σW ′)Zd1 as its diagonal, W ′′ = diag(w′′
+1, . . . , w′′
+n), w′′
+i = −σ−1w′
+i
+�
+L1/σ
+i
+exp(−µi/σ) − 1
+�
+,
+τ = (τ1, τ2) = (1, 1) indicating the second-order covariance matrix of the MLE β⋆ = �β denoted
+by Cov2( �β) and τ = (0, −1) indicating the second-order covariance matrix of the BCE β⋆ = �β
+denoted by Cov2( �β).
+4.3
+Wald-type test
+Let Cov−1
+2 ( �β) and Cov−1
+2 ( �β) the inverse of Cov2( �β) and Cov2( �β), respectively and con-
+sidering also the partitions and the notation for the Fisher information matrix discussed in the
+introductory section, we can propose two modification to the Wald test in (6):
+WMLE2 =
+�
+C �β − Cβ(0)�⊤ �
+C �Cov2( �β)C⊤�−1 �
+C �β − Cβ(0)�
+,
+(14)
+WBCE =
+�
+C �β − Cβ(0)�⊤ �
+C �
+KC⊤�−1 �
+C �β − Cβ(0)�
+,
+(15)
+WBCE2 =
+�
+C �β − Cβ(0)�⊤ �
+C �Cov2( �β)C⊤�−1 �
+C �β − Cβ(0)�
+,
+(16)
+WFirth =
+�
+C ˇβ − Cβ(0)�⊤ �
+C ˇ
+KC⊤�−1 �
+C ˇβ − Cβ(0)�
+,
+(17)
+where �Cov2( �β) is the matrix Cov2(β) evaluated at �β, �
+K is the Fisher information evaluated
+at �β, �Cov2( �β) is the matrix Cov2( �β) evaluated at �β, ˇ
+K is the Fisher information evaluated
+at ˇβ,. Under H, WMLE, WBCE, WBCE2, WFirth follow a χ2
+p distribution.
+4.4
+Improved Estimator strategies
+In the supplemental material and the next section, we studied the statistical properties of
+improved estimators for β and its covariance matrix with the following strategies: the classi-
+cal ML estimator with the Fisher information as covariance matrix (MLE); the classical ML
+estimator with the corrected covariance matrix defined in (13) (MLE2); the bias corrected esti-
+mator (BCE) given in (11) with the Fisher information as covariance matrix; the bias corrected
+estimator with the corrected covariance matrix (BCE2); and the Firth estimator defined in (12)
+with its Fisher information as covariance matrix.
+
+10
+5
+Simulation study
+In our collaborative work, investigators wanted to establish a dose-response relationship
+between a new inhibitor agent for pancreatic cancer in combination with a given dose of gem-
+citabine in mouse models. Based on preliminary data, a survival median time of 4 months was
+estimated in control-treated KPC mice model such that previous studies showed no survival
+benefit with only gemcitabine.12 Assuming a Weibull distribution with σ = 0.5, we calculated
+the placebo effect equal to 1.57 and maximum effect of 2.96 considering a hazard ratio of 4.
+Investigators were interested in the minimum effective dose yielding a minimum hazard ratio
+of 2 yielding ∆ = 0.693.
+Model
+Constraints
+Guess estimates/True parameters
+True MED
+θ0
+θ1
+θ2
+Linear
+-
+E0 = 1.569
+δ = 0.0139
+-
+-
+Emax
+50% at x4
+E0 = 1.569
+EMax = 2.079
+ED50 = 50.000
+25.00
+Exponential
+10% at x1
+E0 = 1.569
+E1 = 0.017
+δ = 22.756
+84.51
+Logistic
+10% at x3
+80% at x4
+E0 = 1.569
+EMax = 1.391
+ED50 = 40.329
+δ = 6.976
+40.37
+Beta
+30% at x2
+E0 = 1.569
+EMax = 1.386
+δ1 = 0.749
+δ2 = 1.049
+10.61
+Table 1: Scenarios (Emax, Exponential, Logistic and Beta) and candidate models (Linear,
+Emax, Exponential, Logistic and Beta) defined based on, respectively, true parameters and
+guess estimates. True parameters/guess estimates were calculated based on placebo effect of
+1.57, maximum effect of 2.96 and constraints with scale parameter of 120 for Beta model. True
+MED was calculated based on ∆ = 0.693.
+Five scenarios (constant, emax, exponential, logistic and beta model) presented in Table 1
+were studied. True parameters were defined based on the aforementioned placebo and maxi-
+mum effects, and the percent of maximum effect that is achieved at givens dose as discussed in
+Bornkamp et al.13 For each scenario, the doses 0, 5, 25, 50 and 100 (mg/kg) were considered
+such that true MED was calculated as continuous dose given in Table 1. Five candidate mod-
+els (linear, emax, exponential, logistic and beta model) were considered with guess estimates
+defined as the true parameters to calculate the optimal contrasts in (8). For each model, the
+null hypothesis of non-signal of the new inhibitor agent was tested at 5% significance level, and
+we used Akaike Information Criteria (AIC) to choose the model to estimate MED when more
+than one model rejected the non-signal hypothesis.
+
+11
+For both steps of the MCP-Mod framework, the five strategies (MLE, MLE2, BCE, BCE2
+and Firth) were evaluated based on a Monte Carlo simulation study with 100,000 replicates for
+sample sizes of 5, 10, 15, 20 and 25 mice per dose and censoring rate of 10%, 25% and 50%. The
+following operating characteristics were studied: (i) convergence rate of GLS algorithm in the
+MCP-Mod General framework; (ii) Probability of incorrectly detecting a dose-response signal,
+i.e., type I error; (iii) Probability of correctly detecting a dose-response signal, i.e., power; (iv)
+Probability of selecting the true model given that there is a signal; (v) Bias of MED estimate
+and (vi) RMSE of MED estimate.
+5.1
+Results
+In Figure 1, convergence rates when calculating estimators and applying them as input for
+the MCP-Mod framework is presented for different censoring rates and true models. When
+censoring is 10%, there is no difference among estimators; when censoring is 25%, similar con-
+clusion can be drawn but the estimators Firth, MLE2 and BCE2 have lower convergence rates
+than MLE and BCE for the true model Exponential; when censoring is 50%, the lower conver-
+gence rates of the estimators Firth, MLE2 and BCE2 are also observed in other scenarios with
+dose-relationship signal. These lower convergence rates are attained because small sample sizes
+in combination with high censoring rates result into doses with no events (in our case, deaths),
+therefore, very large estimates for the regression coefficients are obtained and, consequently,
+singular covariance matrix estimates.
+For the MCP-step, type I error probability is displayed for different censoring rates and
+true models in Figure 2. When censoring is 10%, corrected estimators (MLE2, BCE, BCE2,
+Firth) show observed type I error probability closer to the nominal value (0.05) than MLE for
+all sample sizes. In particular, MLE shows type I error probability more than twice than the
+nominal value for sample size of 5 and almost twice the nominal value for 10 mice per dose;
+when censoring is 25%, corrected estimators are slightly conservative for small sample sizes
+of 5 and 10, while type I error probability of MLE is still inflated for all sample sizes; when
+censoring is 50%, corrected estimators are overly conservative for all sample sizes and MLE
+produces the nominal type I probability. As sample size increases, it is expected that type I
+error probability converges to the nominal value (0.05) for all estimators.
+
+12
+In Figure 3A, the probability of correctly detecting dose-response signal in the MCP-step
+is showed as function of censoring rates and true models. It is expected that MLE show higher
+probability than corrected estimators given that the power function is inflated as seen in Figure
+2. A fair comparison would require us to re-adjust critical values to reject the null hypothesis
+such that the observed type I probability was set at 0.05 for all estimators.
+Nonetheless,
+the probability of correctly detecting dose-response signal is similar among estimators when
+censoring rate is 10% because of the large effect sizes. When censoring rate is 25%, all estimators
+are still comparable except when the true model is Exponential with BCE showing a similar or
+superior performance than MLE; BCE2 having a higher probability than MLE for sample sizes
+from 10 to 25; Firth estimator displaying comparable performance with MLE only when sample
+size is at least 15 mice per dose; and MLE2 having the poorest performance for all sample sizes.
+For 50% of censoring, there is no clear pattern, but MLE presents the highest probability when
+the sample size is 5 mice per dose in three out of four scenarios, while corrected estimators
+have poorer performance that becomes comparable with MLE as sample size increases, except
+MLE2.
+In Figure 3B, the probability of correctly selecting dose-response model using AIC is cal-
+culated given that we selected at least one model in the MCP-step, i.e., it is a conditional
+probability. Conclusions are somewhat similar to 3A: when censoring is 10%, MLE shows a
+slight higher probability for Exponential, Logistic and Beta as true models such that this pat-
+tern becomes more prominent for censoring rates of 25% and 50%. For Emax as true model,
+BCE and BCE2 show consistently higher performance when censoring rates are 25% and 50%
+and no differences can be seen for censoring rate of 10%.
+For Mod-step, we calculated the relative bias and RMSE of �
+MED estimator in Figure 4A
+and B, respectively, for different censoring rates and true models. When censoring is 10%, MLE
+and corrected estimators have negligible differences for bias and RMSE; when censoring is 25%,
+discrepancies can be see for bias and RMSE when true models are Emax and Exponential such
+that MLE consistently shows a better performance than corrected estimators; when censoring
+is 50%, conclusions are hard to be drawn for sample sizes of 5 and 10 such that estimators
+perform similar - but MLE2 - for sample sizes greater than 15 mice per dose.
+
+13
+6
+Concluding Remarks
+We have derived improved inferences based on the Wald statistic for WRM particularizing
+general results from Cox and Snell7 and Magalhães et al.,8 which complements previous results
+for improved inference based on likelihood ratio, Rao score and gradient statistics discussed in
+Magalhães and Gallardo.6 Few authors have presented improved inference for survival models
+with small sample sizes under the classical approach: Cordeiro and Colosimo14,15 and Medeiros16
+derived those statistics, respectively, for censored exponential regression models (ERM), a par-
+ticular case of censored WRM. Also for ERM, Lemonte17 presented the second-order covariance
+matrix of the MLE.
+We have also proposed to use bias-corrected (BCE, BCE2 and Firth) and second-order
+covariance matrices (MLE2, BCE2) as input for the general MCP-Mod framework introduced
+by Pinheiros et al.,3 which addresses the issue of relying on the asymptotic properties of MLE,
+which might not be valid for small sample sizes.
+To the best of our knowledge, this work
+is the first attempt to apply refined estimators for small sample sizes in the general MCP-
+Mod framework. Two simulations studies were performed to study the properties of refined
+estimators in an usual context of regression models and relevant operating characteristics in
+the MCP-Mod framework.
+In the simulation study presented in the Supplementary material, we showed numerical
+evidences that BCE and Firth estimator have lower bias and RMSE than MLE; second-order
+covariance matrices evaluated at MLE and BCE are closer to their respective empirical covari-
+ance matrices in comparison to the first-order covariance matrices; and Wald statistics derived
+from the combination between bias-corrected estimators and second-order covariance matrices
+yielded type I error probability closer to the nominal value than the standard Wald statistic
+with no loss of power.
+In the simulation study for the MCP-Mod framework, we have found that refined estimators
+and second-order covariance matrices approximate type I error probability of its nominal value
+in the MCP-step, while there are negligible differences between MLE and refined estimators in
+the probability of correctly detecting the dose-response signal, probability correctly selecting the
+dose-response model, bias and RMSE when censoring rates are up to 25%. For censoring rate
+of 50%, we found convergence issues for the corrected estimators and second-order covariance
+
+14
+matrices and poorer performance in the assessed operating characteristics.
+In conclusion, we recommend the use of refined estimators for small sample sizes when
+the expected censoring rate is at most 25% in the MCP-Mod framework. In the context of
+basic science with limited sample sizes and large effect sizes, we do not expect large censoring
+rates in mouse experiments. In human trials, smaller effect sizes are pursued such that larger
+sample sizes are required. In this case, refined estimators are not needed because all estimators
+present comparable performance for large sample sizes. Nonetheless, we showed that type I
+error probability is still inflated even for sample sizes of 25 subjects per dose, therefore, the use
+of refined estimators would avoid to dedicate further efforts on non-promising drugs.
+We hope that refined estimators allow statisticians to implement the MCP-Mod framework
+with small sample sizes accelerating the pre-clinical and clinical drug development process.
+Similar ideas can be applied to other distributions such as Binomial, Negative Binomial and
+Poisson, and they are currently under investigation.
+
+15
+10%
+25%
+50%
+Constant
+Emax
+Exponential
+Logistic
+Beta
+5
+10
+15
+20
+25
+5
+10
+15
+20
+25
+5
+10
+15
+20
+25
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Sample size per dose
+MCP Convergence Rate
+Method
+MLE
+MLE2
+BCE
+BCE2
+Firth
+Figure 1: Convergence rate when calculating MLE, MLE2, BCE, BCE2 and Firth estimators
+and applying them as input for the MCP-Mod framework as function of censoring rate and
+true model.
+10%
+25%
+50%
+5
+10
+15
+20
+25
+5
+10
+15
+20
+25
+5
+10
+15
+20
+25
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+Sample size per dose
+Type I error probability
+Method
+MLE
+MLE2
+BCE
+BCE2
+Firth
+Figure 2: Type I error probability when in the MCP-step using MLE, MLE2, BCE, BCE2 and
+Firth estimators as function of censoring rate.
+
+16
+10%
+25%
+50%
+Emax
+Exponential
+Logistic
+Beta
+5
+10
+15
+20
+25
+5
+10
+15
+20
+25
+5
+10
+15
+20
+25
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+Sample size per dose
+Probability of Correctly Detecting Dose Response Signal
+A
+10%
+25%
+50%
+Emax
+Exponential
+Logistic
+Beta
+5
+10
+15
+20
+25
+5
+10
+15
+20
+25
+5
+10
+15
+20
+25
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+Sample size per dose
+Probability of Correctly Selecting Dose Response Model
+B
+Method
+MLE
+MLE2
+BCE
+BCE2
+Firth
+Figure 3: A. Probability of correctly detecting dose-response signal when in the MCP-step
+using MLE, MLE2, BCE, BCE2 and Firth estimators as function of censoring rate and true
+model. B. Probability of correctly selecting dose-response model using AIC in the Mod-step
+using MLE, MLE2, BCE, BCE2 and Firth estimators as function of censoring rate and true
+model.
+
+17
+10%
+25%
+50%
+Emax
+Exponential
+Logistic
+Beta
+5
+10
+15
+20
+25
+5
+10
+15
+20
+25
+5
+10
+15
+20
+25
+−50%
+−25%
+0%
+25%
+−75%
+−50%
+−25%
+0%
+−80%
+−60%
+−40%
+−20%
+0%
+−50%
+−25%
+0%
+Sample size per dose
+Relative Bias (%)
+A
+10%
+25%
+50%
+Emax
+Exponential
+Logistic
+Beta
+5
+10
+15
+20
+25
+5
+10
+15
+20
+25
+5
+10
+15
+20
+25
+10
+15
+20
+25
+30
+0
+50
+100
+150
+200
+10
+20
+30
+5
+10
+15
+Sample size per dose
+RMSE
+B
+Method
+MLE
+MLE2
+BCE
+BCE2
+Firth
+Figure 4: A. Bias of �
+MED derived from the MCP-Mod framework when using MLE, MLE2,
+BCE, BCE2 and Firth estimators as function of censoring rate and true model. B. Root-Mean-
+Square Error of �
+MED derived from the MCP-Mod framework when using MLE, MLE2, BCE,
+BCE2 and Firth estimators as function of censoring rate and true model.
+
+18
+References
+1 Jardim Denis L, Groves Eric S, Breitfeld Philip P, Kurzrock Razelle. Factors associated
+with failure of oncology drugs in late-stage clinical development: a systematic review Cancer
+treatment reviews. 2017;52:12–21.
+2 Bretz Frank, Pinheiro José C, Branson Michael. Combining multiple comparisons and mod-
+eling techniques in dose-response studies Biometrics. 2005;61:738–748.
+3 Pinheiro José, Bornkamp Björn, Glimm Ekkehard, Bretz Frank. Model-based dose finding un-
+der model uncertainty using general parametric models Statistics in medicine. 2014;33:1646–
+1661.
+4 Carroll Kevin J. On the use and utility of the Weibull model in the analysis of survival data
+Controlled clinical trials. 2003;24:682–701.
+5 Magalhães Tiago M., Gallardo Diego I., Gómez H. W.. Skewness of maximum likelihood
+estimators in the Weibull censored data Symmetry. 2019;11:1351.
+6 Magalhães Tiago M., Gallardo Diego I.. Bartlett and Bartlett-type corrections for censored
+data from a Weibull distribution SORT - Statistics and Operations Research Transactions.
+2020;44:127–140.
+7 Cox David R., Snell E. J.. A general definition of residuals Journal of the Royal Statistical
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+order covariance matrices - an application to dispersion models Brazilian Journal of Proba-
+bility and Statistics. 2020. To appear.
+9 Kalbfleisch John D., Prentice Ross L.. The statistical analysis of failure time data.
+New
+Jersey: John Wiley & Sons2 ed. 2002.
+10 Wald Abraham. Test of statistical hypotheses concerning several parameter when the number
+of observations is large Transactions of the American Mathematical Society. 1943;54:426–482.
+11 Firth David. Bias reduction of maximum likelihood estimates Biometrika. 1993;80:27–38.
+
+19
+12 Olive Kenneth P, Jacobetz Michael A, Davidson Christian J, et al. Inhibition of Hedgehog
+signaling enhances delivery of chemotherapy in a mouse model of pancreatic cancer Science.
+2009;324:1457–1461.
+13 Bornkamp Björn, Pinheiro José, Bretz Frank, others . MCPMod: An R package for the design
+and analysis of dose-finding studies Journal of Statistical Software. 2009;29:1–23.
+14 Cordeiro Gaus M., Colosimo Enrico A.. Improved likelihood ratio tests for exponential cen-
+sored data Journal of Statistical Computation and Simulation. 1997;56:303–315.
+15 Cordeiro Gaus M., Colosimo Enrico A.. Corrected score tests for exponential censored data
+Statistics & Probability Letters. 1999;44:365–373.
+16 Medeiros Francisco M. C., Lemonte Artur J.. Likelihood-based inference in censored exponen-
+tial regression models Communications in Statistics - Theory and Methods. 2021;50:3214–
+3233.
+17 Lemonte Artur J.. Covariance matrix of maximum likelihood estimators in censored exponen-
+tial regression models Communications in Statistics - Theory and Methods. 2022;51:1765–
+1777.
+
+Improved inference for MCP-Mod approach for
+time-to-event endpoints with small sample sizes -
+Supplementary material
+Márcio A. Diniz∗
+Diego I. Gallardo†
+Tiago M. Magalhães‡
+July 2, 2022
+We performed a simulation study with 10,000 Monte Carlo replicates where censored Weibull
+data was generated with the regression structure established in (3). We considered scenarios
+with p = 3, 5, 7 covariates following the standard normal distribution associated with the first p
+components of the regression coefficient vector β = (−2, 1.5, −1, 2.5, −1.3, 1.8, −0.5). Censoring
+was assumed as 10%, 25% and 50%. Bias and RMSE for MLE, Firth and BCE estimators are
+presented in Supplemental Figures S1-S6.
+Furthermore, we calculated matrix distances between the sampling covariance matrix -
+Cov(β∗) - and the Fisher Information - K(β∗) - and the corrected covariance matrix Cov2(β∗)
+from 13 evaluated at MLE (�β) and BCE (�β) in Table S1. The following distances between
+matrices A and B were considered:
+• d1 = max diag|A − B|;
+• d2 =
+�
+tr((A − B)⊤(A − B));
+• d3 = �
+i
+�
+j |aij − bij|;
+Finally, we evaluated type I error testing the composite null hypothesis H : β1 = β(0)
+1
+against a composite alternative hypothesis A : H is false, where β(0)
+1
+is a specified vector,
+β1 is a q-dimensional vector and β2 contains the remaining p − q parameters. This partition
+∗Biostatistics and Bioinformatics Research Center, Samuel Oschin Cancer Center, Cedars Sinai Medical
+Center, Los Angeles, California, USA, e-mail to marcio.diniz@cshs.org.
+†Department of Mathematics, Engineering School, University of Atacama, Copiapó, Chile, e-mail to
+diego.gallardo@uda.cl.
+‡Department of Statistics, Institute of Exact Sciences, Federal University of Juiz de Fora, Juiz de Fora,
+Brazil, e-mail to tiago.magalhaes@ufjf.br.
+arXiv:2301.00325v1  [stat.ME]  1 Jan 2023
+
+2
+β = (β⊤
+1 , β⊤
+2 )⊤ induces the corresponding partitions
+K =
+�
+�
+�
+K11
+K12
+K21
+K22
+�
+�
+� = σ−2
+�
+�
+�
+X⊤
+1 W X1
+X⊤
+1 W X2
+X⊤
+2 W X1
+X⊤
+2 W X2
+�
+�
+� and K−1 =
+�
+�
+�
+K11
+K12
+K21
+K22
+�
+�
+� ,
+where X = [X1 X2], X1, X2 being n × q and n × (p − q), respectively. We consider the
+following Wald test[1] statistics:
+WMLE =
+�
+�β1 − β(0)
+1
+�⊤ �
+�
+K11�−1 �
+�β1 − β(0)
+1
+�
+,
+(1)
+WMLE2 =
+�
+�β1 − β(0)
+1
+�⊤ �
+�Cov11
+2 ( �β)
+�−1 �
+�β1 − β(0)
+1
+�
+,
+(2)
+WBCE =
+�
+�β1 − β(0)
+1
+�⊤ �
+�
+K11�−1 �
+�β1 − β(0)
+1
+�
+,
+(3)
+WBCE2 =
+�
+�β1 − β(0)
+1
+�⊤ �
+�Cov11
+2 ( �β)
+�−1 �
+�β1 − β(0)
+1
+�
+,
+(4)
+WFirth =
+�
+ˇβ1 − β(0)
+1
+�⊤ � ˇ
+K11�−1 �
+ˇβ1 − β(0)
+1
+�
+.
+(5)
+In the censored Weibull regression model the statistic (1)-(5) can be rewritten as
+W =
+�
+β1 − β(0)
+1
+�⊤ �
+R⊤W R
+� �
+β1 − β(0)
+1
+�
+,
+(6)
+with R = X1 − X2C, C =
+�
+X⊤
+2 W X2
+�−1 X⊤
+2 W X1 represents a (p − q) × q matrix whose
+columns are the vectors of regression coefficients obtained in the weighted normal linear regres-
+sion of the columns of X1 on the model matrix X2 with W as a weight matrix.
+Under the null hypothesis H, W has a χ2
+q distribution up to an error of order n−1. The null
+hypothesis is rejected for a given nominal level, α say, if the test statistic exceeds the upper
+100(1 − α)% quantile of the χ2
+q distribution.
+Similarly, we evaluated power with the composite alternative hypothesis A : β⊤ = (ψ1q, 0p−q)⊤
+for ψ = 0.05, 0.10, 0.25, 0.50, 1.00, 2.00. Results are presented in Table S3.
+1
+Results
+1.1
+Assessing the bias for different estimation procedures
+
+3
+p = 3
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.06
+0.08
+0.10
+0.12
+0.14
+Sample size
+Bias β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.06
+0.08
+0.10
+0.12
+Sample size
+Bias β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.06
+0.08
+0.10
+0.12
+Sample size
+Bias β2
+Method
+MLE
+BCE
+Firth
+p = 5
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.06
+0.09
+0.12
+0.15
+Sample size
+Bias β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.06
+0.09
+0.12
+0.15
+Sample size
+Bias β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.06
+0.08
+0.10
+0.12
+0.14
+0.16
+Sample size
+Bias β3
+Method
+MLE
+BCE
+Firth
+p = 7
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.08
+0.10
+0.12
+0.14
+Sample size
+Bias β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.08
+0.10
+0.12
+0.14
+Sample size
+Bias β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.08
+0.10
+0.12
+Sample size
+Bias β6
+Method
+MLE
+BCE
+Firth
+Figure S1: Empirical bias for different estimators. Case σ = 0.5.
+
+4
+p = 3
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.075
+0.100
+0.125
+0.150
+Sample size
+RMSE β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.075
+0.100
+0.125
+0.150
+Sample size
+RMSE β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.075
+0.100
+0.125
+0.150
+Sample size
+RMSE β2
+Method
+MLE
+BCE
+Firth
+p = 5
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.08
+0.12
+0.16
+0.20
+Sample size
+RMSE β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.08
+0.12
+0.16
+0.20
+Sample size
+RMSE β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+Sample size
+RMSE β3
+Method
+MLE
+BCE
+Firth
+p = 7
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.100
+0.125
+0.150
+0.175
+Sample size
+RMSE β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.100
+0.125
+0.150
+0.175
+Sample size
+RMSE β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.100
+0.125
+0.150
+Sample size
+RMSE β6
+Method
+MLE
+BCE
+Firth
+Figure S2: Empirical RMSE for different estimators. Case σ = 0.5.
+
+5
+p = 3
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.12
+0.16
+0.20
+0.24
+0.28
+Sample size
+Bias β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.12
+0.16
+0.20
+0.24
+Sample size
+Bias β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.12
+0.16
+0.20
+0.24
+Sample size
+Bias β2
+Method
+MLE
+BCE
+Firth
+p = 5
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.15
+0.20
+0.25
+0.30
+Sample size
+Bias β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.15
+0.20
+0.25
+0.30
+Sample size
+Bias β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.15
+0.20
+0.25
+0.30
+Sample size
+Bias β3
+Method
+MLE
+BCE
+Firth
+p = 7
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.2
+0.3
+0.4
+Sample size
+Bias β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.2
+0.3
+0.4
+Sample size
+Bias β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.2
+0.3
+Sample size
+Bias β6
+Method
+MLE
+BCE
+Firth
+Figure S3: Empirical bias for different estimators. Case σ = 1.
+
+6
+p = 3
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.15
+0.20
+0.25
+0.30
+Sample size
+RMSE β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.15
+0.20
+0.25
+0.30
+Sample size
+RMSE β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.15
+0.20
+0.25
+0.30
+Sample size
+RMSE β2
+Method
+MLE
+BCE
+Firth
+p = 5
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+Sample size
+RMSE β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+Sample size
+RMSE β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.15
+0.20
+0.25
+0.30
+0.35
+Sample size
+RMSE β3
+Method
+MLE
+BCE
+Firth
+p = 7
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.2
+0.3
+0.4
+Sample size
+RMSE β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.2
+0.3
+0.4
+0.5
+Sample size
+RMSE β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+Sample size
+RMSE β6
+Method
+MLE
+BCE
+Firth
+Figure S4: Empirical RMSE for different estimators. Case σ = 1.
+
+7
+p = 3
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.4
+0.5
+0.6
+0.7
+0.8
+Sample size
+Bias β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.4
+0.5
+0.6
+0.7
+Sample size
+Bias β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.4
+0.5
+0.6
+0.7
+Sample size
+Bias β2
+Method
+MLE
+BCE
+Firth
+p = 5
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.4
+0.6
+0.8
+Sample size
+Bias β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.4
+0.6
+0.8
+Sample size
+Bias β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.4
+0.5
+0.6
+0.7
+0.8
+Sample size
+Bias β3
+Method
+MLE
+BCE
+Firth
+p = 7
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.4
+0.6
+0.8
+1.0
+Sample size
+Bias β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.4
+0.6
+0.8
+1.0
+Sample size
+Bias β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Sample size
+Bias β6
+Method
+MLE
+BCE
+Firth
+Figure S5: Empirical bias for different estimators. Case σ = 3.
+
+8
+p = 3
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.5
+0.6
+0.7
+0.8
+0.9
+Sample size
+RMSE β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Sample size
+RMSE β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.4
+0.5
+0.6
+0.7
+0.8
+Sample size
+RMSE β2
+Method
+MLE
+BCE
+Firth
+p = 5
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.6
+0.8
+1.0
+Sample size
+RMSE β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.6
+0.8
+1.0
+Sample size
+RMSE β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.6
+0.8
+1.0
+Sample size
+RMSE β3
+Method
+MLE
+BCE
+Firth
+p = 7
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.6
+0.8
+1.0
+1.2
+Sample size
+RMSE β0
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.6
+0.8
+1.0
+1.2
+1.4
+Sample size
+RMSE β1
+Method
+MLE
+BCE
+Firth
+10%
+25%
+50%
+20
+30
+40 20
+30
+40 20
+30
+40
+0.6
+0.8
+1.0
+Sample size
+RMSE β6
+Method
+MLE
+BCE
+Firth
+Figure S6: Empirical RMSE for different estimators. Case σ = 3.
+
+9
+1.2
+Assessing the variance estimation
+Table S1: Distance between different estimated covariance matrices.
+n = 20
+n = 30
+n = 40
+Cov(�β) and
+Cov(�β) and
+Cov(�β) and
+Cov(�β) and
+Cov(�β) and
+Cov(�β) and
+c
+σ
+p d
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+10% 0.5
+3 d1 0.0634 0.0262 0.0593 0.0175 0.0431 0.0275 0.0403 0.0243 0.0320 0.0151 0.0297 0.0116
+d2 0.0070 0.0010 0.0059 0.0005 0.0026 0.0012 0.0022 0.0009 0.0017 0.0003 0.0015 0.0003
+d3 0.0129 0.0025 0.0115 0.0014 0.0053 0.0029 0.0044 0.0022 0.0033 0.0009 0.0031 0.0006
+5 d1 0.1046 0.0657 0.0988 0.0640 0.0616 0.0338 0.0588 0.0323 0.0485 0.0278 0.0468 0.0280
+d2 0.0232 0.0095 0.0207 0.0088 0.0085 0.0031 0.0078 0.0030 0.0051 0.0023 0.0048 0.0023
+d3 0.0642 0.0343 0.0590 0.0328 0.0257 0.0132 0.0241 0.0127 0.0168 0.0100 0.0161 0.0097
+7 d1 0.2691 0.2520 0.2694 0.2545 0.1790 0.1684 0.1783 0.1689 0.1470 0.1397 0.1463 0.1398
+d2 0.1799 0.1562 0.1786 0.1581 0.0867 0.0777 0.0863 0.0783 0.0583 0.0537 0.0579 0.0538
+d3 0.6681 0.6066 0.6620 0.6084 0.3999 0.3755 0.3982 0.3763 0.3020 0.2888 0.3004 0.2884
+1.0
+3 d1 0.1319 0.0680 0.1138 0.0419 0.0781 0.0555 0.0677 0.0488 0.0694 0.0342 0.0625 0.0265
+d2 0.0280 0.0070 0.0208 0.0028 0.0099 0.0059 0.0073 0.0037 0.0071 0.0016 0.0060 0.0012
+d3 0.0517 0.0194 0.0396 0.0061 0.0213 0.0160 0.0151 0.0094 0.0131 0.0038 0.0120 0.0025
+5 d1 0.1639 0.0630 0.1442 0.0494 0.1065 0.0512 0.0952 0.0441 0.0781 0.0452 0.0710 0.0402
+d2 0.0572 0.0103 0.0434 0.0064 0.0237 0.0059 0.0189 0.0044 0.0120 0.0031 0.0099 0.0024
+d3 0.1472 0.0461 0.1138 0.0258 0.0656 0.0233 0.0542 0.0173 0.0323 0.0114 0.0280 0.0091
+7 d1 0.2433 0.1433 0.2192 0.1368 0.1609 0.1026 0.1514 0.1032 0.1268 0.0893 0.1201 0.0882
+d2 0.1487 0.0496 0.1214 0.0459 0.0651 0.0274 0.0571 0.0267 0.0399 0.0205 0.0359 0.0200
+d3 0.4508 0.2061 0.3819 0.1819 0.2397 0.1385 0.2151 0.1325 0.1642 0.1096 0.1521 0.1068
+3.0
+3 d1 0.4335 0.2503 0.3222 0.2257 0.2526 0.1844 0.1633 0.1480 0.2072 0.0949 0.1570 0.0826
+d2 0.2772 0.0751 0.1308 0.0639 0.0996 0.0495 0.0446 0.0364 0.0695 0.0159 0.0388 0.0079
+d3 0.4908 0.1454 0.2683 0.1309 0.1865 0.0984 0.1088 0.0866 0.1349 0.0411 0.0744 0.0167
+5 d1 0.5378 0.1597 0.4069 0.1557 0.3359 0.1497 0.2678 0.1372 0.2472 0.1339 0.1933 0.1250
+d2 0.6043 0.1040 0.3160 0.0591 0.2407 0.0490 0.1397 0.0287 0.1232 0.0372 0.0726 0.0235
+d3 1.6158 0.4748 0.8530 0.2428 0.6192 0.2058 0.3797 0.0970 0.3509 0.1551 0.2081 0.0857
+7 d1 0.6234 0.2563 0.4373 0.2420 0.3905 0.1814 0.2912 0.1542 0.2880 0.1344 0.2240 0.1162
+d2 0.9863 0.2202 0.4766 0.1692 0.3766 0.0933 0.2042 0.0617 0.1996 0.0546 0.1183 0.0401
+
+10
+n = 20
+n = 30
+n = 40
+Cov(�β) and
+Cov(�β) and
+Cov(�β) and
+Cov(�β) and
+Cov(�β) and
+Cov(�β) and
+c
+σ
+p d
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+d3 3.3965 1.1968 1.9409 0.9714 1.2864 0.5180 0.7472 0.3326 0.7210 0.3036 0.4882 0.2401
+25% 0.5
+3 d1 0.0793 0.0369 0.0715 0.0276 0.0525 0.0256 0.0492 0.0226 0.0402 0.0098 0.0380 0.0134
+d2 0.0099 0.0022 0.0081 0.0014 0.0043 0.0011 0.0037 0.0008 0.0027 0.0003 0.0024 0.0004
+d3 0.0188 0.0054 0.0163 0.0037 0.0084 0.0027 0.0073 0.0020 0.0056 0.0009 0.0053 0.0010
+5 d1 0.1023 0.0246 0.0916 0.0227 0.0686 0.0288 0.0634 0.0277 0.0466 0.0159 0.0437 0.0141
+d2 0.0225 0.0023 0.0182 0.0021 0.0099 0.0022 0.0085 0.0019 0.0048 0.0008 0.0042 0.0007
+d3 0.0570 0.0108 0.0483 0.0093 0.0259 0.0097 0.0219 0.0084 0.0138 0.0029 0.0127 0.0028
+7 d1 0.2113 0.1726 0.2025 0.1731 0.1410 0.1185 0.1371 0.1180 0.1114 0.0967 0.1093 0.0965
+d2 0.1134 0.0757 0.1049 0.0757 0.0483 0.0345 0.0456 0.0341 0.0287 0.0217 0.0274 0.0214
+d3 0.4040 0.3062 0.3807 0.3053 0.1978 0.1613 0.1886 0.1585 0.1283 0.1086 0.1237 0.1067
+1.0
+3 d1 0.1674 0.0780 0.1431 0.0586 0.1119 0.0528 0.0982 0.0428 0.0757 0.0426 0.0680 0.0366
+d2 0.0434 0.0077 0.0312 0.0051 0.0197 0.0041 0.0151 0.0034 0.0089 0.0030 0.0069 0.0023
+d3 0.0842 0.0176 0.0673 0.0135 0.0378 0.0108 0.0312 0.0092 0.0189 0.0076 0.0160 0.0059
+5 d1 0.2079 0.0774 0.1749 0.0671 0.1493 0.0739 0.1369 0.0766 0.0958 0.0508 0.0851 0.0456
+d2 0.0885 0.0114 0.0631 0.0082 0.0457 0.0107 0.0366 0.0112 0.0183 0.0038 0.0145 0.0033
+d3 0.2280 0.0446 0.1818 0.0321 0.1186 0.0418 0.0972 0.0396 0.0517 0.0143 0.0451 0.0130
+7 d1 0.2791 0.1399 0.2393 0.1451 0.1852 0.1011 0.1676 0.1047 0.1350 0.0755 0.1247 0.0788
+d2 0.1993 0.0447 0.1473 0.0494 0.0853 0.0259 0.0689 0.0273 0.0446 0.0148 0.0370 0.0152
+d3 0.6023 0.2093 0.4748 0.2045 0.2781 0.1079 0.2396 0.1138 0.1589 0.0749 0.1406 0.0743
+3.0
+3 d1 0.4998 0.2301 0.3203 0.1692 0.3732 0.1706 0.2704 0.1684 0.2375 0.1330 0.1685 0.1164
+d2 0.4127 0.0684 0.1597 0.0329 0.2179 0.0321 0.1133 0.0328 0.0918 0.0263 0.0434 0.0219
+d3 0.7448 0.1563 0.3189 0.0655 0.3926 0.0630 0.2255 0.0641 0.1965 0.0726 0.1000 0.0556
+5 d1 0.6530 0.1756 0.4516 0.1994 0.4249 0.1260 0.3083 0.1536 0.2991 0.1105 0.2281 0.0834
+d2 0.9047 0.0906 0.4160 0.0909 0.3938 0.0506 0.2079 0.0514 0.1868 0.0277 0.0967 0.0190
+d3 2.3326 0.3686 1.3273 0.3487 1.0084 0.2307 0.5947 0.2009 0.4849 0.1211 0.2822 0.0791
+7 d1 0.8010 0.3253 0.5387 0.2408 0.5254 0.2072 0.3900 0.2308 0.3497 0.0975 0.2608 0.1008
+d2 1.5787 0.2777 0.6635 0.1951 0.6367 0.1091 0.3239 0.1048 0.3118 0.0512 0.1607 0.0418
+d3 5.2917 1.5483 2.7006 1.1317 2.1360 0.6376 1.2488 0.5553 1.0500 0.2858 0.6471 0.2486
+50% 0.5
+3 d1 0.1158 0.0483 0.1039 0.0485 0.0776 0.0370 0.0702 0.0343 0.0540 0.0204 0.0496 0.0183
+
+11
+n = 20
+n = 30
+n = 40
+Cov(�β) and
+Cov(�β) and
+Cov(�β) and
+Cov(�β) and
+Cov(�β) and
+Cov(�β) and
+c
+σ
+p d
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+�K
+Cov2(�β)
+d2 0.0228 0.0036 0.0182 0.0033 0.0096 0.0018 0.0081 0.0015 0.0047 0.0006 0.0039 0.0005
+d3 0.0422 0.0083 0.0348 0.0078 0.0174 0.0035 0.0148 0.0030 0.0091 0.0018 0.0075 0.0015
+5 d1 0.1607 0.0742 0.1458 0.0761 0.0990 0.0480 0.0913 0.0451 0.0686 0.0262 0.0637 0.0244
+d2 0.0562 0.0110 0.0462 0.0118 0.0211 0.0046 0.0178 0.0040 0.0099 0.0012 0.0085 0.0011
+d3 0.1446 0.0376 0.1217 0.0388 0.0548 0.0170 0.0475 0.0152 0.0257 0.0051 0.0218 0.0046
+7 d1 0.2964 0.2236 0.2852 0.2309 0.1937 0.1616 0.1862 0.1586 0.1552 0.1358 0.1508 0.1334
+d2 0.2162 0.1162 0.2001 0.1267 0.0896 0.0606 0.0825 0.0584 0.0571 0.0430 0.0536 0.0415
+d3 0.7277 0.4459 0.6687 0.4468 0.3187 0.2389 0.2954 0.2289 0.2305 0.1906 0.2185 0.1842
+1.0
+3 d1 0.2294 0.0855 0.1953 0.0953 0.1541 0.0762 0.1304 0.0728 0.1015 0.0218 0.0860 0.0189
+d2 0.0880 0.0133 0.0607 0.0150 0.0372 0.0085 0.0269 0.0076 0.0174 0.0020 0.0124 0.0018
+d3 0.1646 0.0348 0.1166 0.0382 0.0721 0.0187 0.0541 0.0169 0.0338 0.0044 0.0246 0.0040
+5 d1 0.3220 0.1557 0.2783 0.1709 0.1877 0.0797 0.1645 0.0756 0.1337 0.0480 0.1157 0.0444
+d2 0.2137 0.0411 0.1573 0.0531 0.0739 0.0113 0.0549 0.0107 0.0375 0.0047 0.0283 0.0042
+d3 0.5431 0.1338 0.4213 0.1656 0.1826 0.0381 0.1429 0.0364 0.0977 0.0205 0.0748 0.0181
+7 d1 0.4181 0.1732 0.3656 0.2374 0.2503 0.1409 0.2212 0.1384 0.1831 0.1091 0.1651 0.1053
+d2 0.4305 0.0704 0.3286 0.1362 0.1573 0.0470 0.1218 0.0469 0.0859 0.0297 0.0685 0.0276
+d3 1.4277 0.4023 1.0993 0.5507 0.4759 0.1787 0.3801 0.1720 0.2791 0.1271 0.2325 0.1182
+3.0
+3 d1 0.6912 0.3287 0.4826 0.3279 0.4278 0.1813 0.2997 0.1798 0.3416 0.1817 0.2595 0.1766
+d2 0.7694 0.1459 0.3561 0.1466 0.2957 0.0417 0.1362 0.0399 0.1864 0.0504 0.1004 0.0463
+d3 1.3934 0.3504 0.6465 0.3234 0.5378 0.1021 0.2642 0.0922 0.3608 0.1235 0.2068 0.1108
+5 d1 0.9271 0.4391 0.6588 0.4635 0.5571 0.2677 0.4015 0.2622 0.3910 0.1621 0.2830 0.1558
+d2 1.8814 0.3978 0.9394 0.4618 0.6351 0.1231 0.3188 0.1175 0.3187 0.0517 0.1633 0.0478
+d3 4.6010 1.4400 2.3371 1.4081 1.6206 0.4609 0.8959 0.4286 0.8370 0.2154 0.4862 0.1886
+7 d1 1.1551 0.2748 0.7808 0.5265 0.6514 0.2569 0.4563 0.2763 0.4654 0.2025 0.3365 0.2006
+d2 3.4455 0.5081 1.5612 0.7768 1.0745 0.2111 0.5275 0.2168 0.5367 0.1118 0.2744 0.1042
+d3 12.2895 3.0940 6.5278 4.0423 3.7371 1.1778 2.1761 1.1681 1.8224 0.6151 1.0733 0.5555
+1.3
+Assessing the type I error
+
+12
+Table S2: Simulated rejection rates for H0 : βq = 0q, with different combinations of σ, censoring,
+n, p and q.
+σ
+censoring
+n
+p
+q
+MLE
+MLE2
+BCE
+BCE2
+Firth
+0.5
+10%
+20
+3
+1
+0.0723
+0.0463
+0.0723
+0.0590
+0.0722
+2
+0.0874
+0.0474
+0.0877
+0.0627
+0.0869
+5
+1
+0.0794
+0.0415
+0.0796
+0.0580
+0.0779
+3
+0.1131
+0.0518
+0.1132
+0.0729
+0.1121
+7
+2
+0.1223
+0.0531
+0.1226
+0.0791
+0.1190
+4
+0.1495
+0.0580
+0.1492
+0.0853
+0.1460
+30
+3
+1
+0.0636
+0.0422
+0.0638
+0.0522
+0.0639
+2
+0.0770
+0.0473
+0.0769
+0.0605
+0.0767
+5
+1
+0.0806
+0.0462
+0.0806
+0.0626
+0.0793
+3
+0.0917
+0.0461
+0.0919
+0.0636
+0.0919
+7
+2
+0.0952
+0.0442
+0.0956
+0.0656
+0.0939
+4
+0.1179
+0.0503
+0.1182
+0.0733
+0.1153
+40
+3
+1
+0.0620
+0.0460
+0.0621
+0.0539
+0.0621
+2
+0.0684
+0.0471
+0.0684
+0.0570
+0.0687
+5
+1
+0.0703
+0.0430
+0.0708
+0.0561
+0.0704
+3
+0.0836
+0.0465
+0.0836
+0.0625
+0.0830
+7
+2
+0.0888
+0.0439
+0.0888
+0.0638
+0.0875
+4
+0.0993
+0.0471
+0.0997
+0.0684
+0.0992
+25%
+20
+3
+1
+0.0791
+0.0473
+0.0795
+0.0642
+0.0794
+2
+0.0922
+0.0495
+0.0925
+0.0674
+0.0925
+5
+1
+0.0955
+0.0515
+0.0972
+0.0758
+0.0956
+3
+0.1286
+0.0590
+0.1291
+0.0844
+0.1301
+7
+2
+0.1374
+0.0555
+0.1397
+0.0957
+0.1371
+4
+0.1666
+0.0675
+0.1680
+0.1024
+0.1652
+30
+3
+1
+0.0767
+0.0511
+0.0770
+0.0657
+0.0770
+2
+0.0888
+0.0544
+0.0886
+0.0700
+0.0878
+5
+1
+0.0884
+0.0536
+0.0892
+0.0738
+0.0887
+
+13
+σ
+censoring
+n
+p
+q
+MLE
+MLE2
+BCE
+BCE2
+Firth
+3
+0.1229
+0.0625
+0.1229
+0.0895
+0.1229
+7
+2
+0.1190
+0.0559
+0.1205
+0.0900
+0.1195
+4
+0.1490
+0.0599
+0.1501
+0.0978
+0.1480
+40
+3
+1
+0.0691
+0.0485
+0.0691
+0.0607
+0.0689
+2
+0.0786
+0.0518
+0.0787
+0.0666
+0.0787
+5
+1
+0.0767
+0.0503
+0.0768
+0.0666
+0.0773
+3
+0.0919
+0.0540
+0.0923
+0.0726
+0.0919
+7
+2
+0.0991
+0.0546
+0.0993
+0.0796
+0.0991
+4
+0.1195
+0.0545
+0.1198
+0.0849
+0.1199
+50%
+20
+3
+1
+0.0924
+0.0570
+0.0927
+0.0780
+0.0922
+2
+0.1121
+0.0657
+0.1131
+0.0898
+0.1123
+5
+1
+0.1059
+0.0571
+0.1108
+0.0872
+0.1096
+3
+0.1531
+0.0722
+0.1557
+0.1088
+0.1561
+7
+2
+0.1485
+0.0641
+0.1572
+0.1133
+0.1542
+4
+0.1992
+0.0845
+0.2042
+0.1373
+0.2038
+30
+3
+1
+0.0723
+0.0507
+0.0731
+0.0637
+0.0727
+2
+0.0938
+0.0588
+0.0940
+0.0794
+0.0939
+5
+1
+0.0863
+0.0543
+0.0874
+0.0743
+0.0868
+3
+0.1205
+0.0630
+0.1215
+0.0925
+0.1231
+7
+2
+0.1156
+0.0583
+0.1190
+0.0926
+0.1178
+4
+0.1572
+0.0714
+0.1584
+0.1144
+0.1589
+40
+3
+1
+0.0682
+0.0521
+0.0683
+0.0626
+0.0687
+2
+0.0824
+0.0555
+0.0825
+0.0722
+0.0828
+5
+1
+0.0844
+0.0570
+0.0849
+0.0764
+0.0855
+3
+0.1005
+0.0567
+0.1011
+0.0812
+0.1014
+7
+2
+0.1013
+0.0555
+0.1024
+0.0834
+0.1026
+4
+0.1305
+0.0652
+0.1323
+0.0978
+0.1326
+1.0
+10%
+20
+3
+1
+0.0732
+0.0441
+0.0733
+0.0570
+0.0727
+2
+0.0844
+0.0464
+0.0843
+0.0614
+0.0836
+
+14
+σ
+censoring
+n
+p
+q
+MLE
+MLE2
+BCE
+BCE2
+Firth
+5
+1
+0.0820
+0.0445
+0.0823
+0.0612
+0.0810
+3
+0.1159
+0.0555
+0.1161
+0.0744
+0.1141
+7
+2
+0.1146
+0.0476
+0.1158
+0.0734
+0.1131
+4
+0.1401
+0.0566
+0.1407
+0.0822
+0.1365
+30
+3
+1
+0.0657
+0.0444
+0.0657
+0.0547
+0.0657
+2
+0.0782
+0.0495
+0.0781
+0.0621
+0.0781
+5
+1
+0.0738
+0.0453
+0.0739
+0.0590
+0.0736
+3
+0.0960
+0.0483
+0.0960
+0.0676
+0.0946
+7
+2
+0.0964
+0.0447
+0.0968
+0.0661
+0.0966
+4
+0.1215
+0.0511
+0.1219
+0.0760
+0.1205
+40
+3
+1
+0.0660
+0.0474
+0.0660
+0.0547
+0.0659
+2
+0.0716
+0.0485
+0.0715
+0.0593
+0.0718
+5
+1
+0.0733
+0.0482
+0.0735
+0.0607
+0.0734
+3
+0.0880
+0.0478
+0.0881
+0.0636
+0.0877
+7
+2
+0.0876
+0.0432
+0.0870
+0.0637
+0.0855
+4
+0.1008
+0.0461
+0.1006
+0.0672
+0.1005
+25%
+20
+3
+1
+0.0863
+0.0490
+0.0865
+0.0677
+0.0852
+2
+0.0962
+0.0546
+0.0965
+0.0721
+0.0961
+5
+1
+0.0984
+0.0486
+0.0999
+0.0743
+0.0991
+3
+0.1306
+0.0608
+0.1307
+0.0866
+0.1292
+7
+2
+0.1296
+0.0535
+0.1326
+0.0934
+0.1293
+4
+0.1713
+0.0671
+0.1727
+0.1061
+0.1693
+30
+3
+1
+0.0769
+0.0482
+0.0772
+0.0633
+0.0765
+2
+0.0866
+0.0526
+0.0869
+0.0690
+0.0872
+5
+1
+0.0866
+0.0514
+0.0874
+0.0727
+0.0871
+3
+0.1125
+0.0548
+0.1125
+0.0806
+0.1116
+7
+2
+0.1143
+0.0541
+0.1154
+0.0869
+0.1142
+4
+0.1455
+0.0612
+0.1462
+0.1001
+0.1460
+40
+3
+1
+0.0672
+0.0480
+0.0671
+0.0591
+0.0671
+
+15
+σ
+censoring
+n
+p
+q
+MLE
+MLE2
+BCE
+BCE2
+Firth
+2
+0.0738
+0.0487
+0.0740
+0.0613
+0.0741
+5
+1
+0.0775
+0.0516
+0.0777
+0.0671
+0.0781
+3
+0.0990
+0.0539
+0.0990
+0.0768
+0.0989
+7
+2
+0.0960
+0.0490
+0.0966
+0.0782
+0.0963
+4
+0.1204
+0.0573
+0.1207
+0.0859
+0.1203
+50%
+20
+3
+1
+0.0885
+0.0542
+0.0894
+0.0743
+0.0886
+2
+0.1151
+0.0625
+0.1152
+0.0868
+0.1141
+5
+1
+0.1060
+0.0525
+0.1093
+0.0877
+0.1081
+3
+0.1511
+0.0719
+0.1540
+0.1080
+0.1531
+7
+2
+0.1477
+0.0683
+0.1578
+0.1175
+0.1550
+4
+0.2017
+0.0833
+0.2070
+0.1346
+0.2080
+30
+3
+1
+0.0720
+0.0502
+0.0723
+0.0649
+0.0721
+2
+0.0973
+0.0631
+0.0974
+0.0828
+0.0981
+5
+1
+0.0862
+0.0565
+0.0873
+0.0761
+0.0888
+3
+0.1239
+0.0637
+0.1246
+0.0974
+0.1251
+7
+2
+0.1184
+0.0608
+0.1224
+0.0919
+0.1209
+4
+0.1545
+0.0713
+0.1574
+0.1130
+0.1601
+40
+3
+1
+0.0668
+0.0505
+0.0670
+0.0602
+0.0671
+2
+0.0843
+0.0567
+0.0844
+0.0743
+0.0842
+5
+1
+0.0758
+0.0525
+0.0767
+0.0669
+0.0761
+3
+0.1082
+0.0612
+0.1088
+0.0881
+0.1091
+7
+2
+0.1068
+0.0586
+0.1084
+0.0878
+0.1080
+4
+0.1386
+0.0661
+0.1396
+0.1041
+0.1410
+3.0
+10%
+20
+3
+1
+0.0741
+0.0453
+0.0742
+0.0567
+0.0742
+2
+0.0873
+0.0484
+0.0873
+0.0626
+0.0872
+5
+1
+0.0900
+0.0475
+0.0903
+0.0657
+0.0885
+3
+0.1186
+0.0523
+0.1180
+0.0742
+0.1159
+7
+2
+0.1208
+0.0500
+0.1211
+0.0789
+0.1163
+4
+0.1492
+0.0581
+0.1497
+0.0846
+0.1475
+
+16
+σ
+censoring
+n
+p
+q
+MLE
+MLE2
+BCE
+BCE2
+Firth
+30
+3
+1
+0.0694
+0.0440
+0.0693
+0.0563
+0.0690
+2
+0.0808
+0.0497
+0.0806
+0.0634
+0.0805
+5
+1
+0.0782
+0.0489
+0.0787
+0.0638
+0.0781
+3
+0.0971
+0.0470
+0.0971
+0.0658
+0.0963
+7
+2
+0.0942
+0.0436
+0.0938
+0.0670
+0.0932
+4
+0.1125
+0.0475
+0.1126
+0.0694
+0.1109
+40
+3
+1
+0.0623
+0.0451
+0.0624
+0.0534
+0.0623
+2
+0.0706
+0.0486
+0.0706
+0.0574
+0.0706
+5
+1
+0.0718
+0.0473
+0.0718
+0.0607
+0.0716
+3
+0.0857
+0.0444
+0.0858
+0.0609
+0.0846
+7
+2
+0.0867
+0.0450
+0.0867
+0.0644
+0.0861
+4
+0.1072
+0.0487
+0.1071
+0.0727
+0.1066
+25%
+20
+3
+1
+0.0850
+0.0516
+0.0856
+0.0686
+0.0842
+2
+0.0982
+0.0522
+0.0984
+0.0711
+0.0994
+5
+1
+0.0978
+0.0527
+0.0987
+0.0775
+0.0982
+3
+0.1329
+0.0577
+0.1336
+0.0871
+0.1328
+7
+2
+0.1336
+0.0533
+0.1354
+0.0922
+0.1316
+4
+0.1709
+0.0640
+0.1731
+0.1029
+0.1682
+30
+3
+1
+0.0801
+0.0550
+0.0802
+0.0690
+0.0802
+2
+0.0889
+0.0533
+0.0888
+0.0713
+0.0883
+5
+1
+0.0904
+0.0547
+0.0909
+0.0757
+0.0906
+3
+0.1168
+0.0577
+0.1171
+0.0862
+0.1165
+7
+2
+0.1192
+0.0606
+0.1198
+0.0898
+0.1178
+4
+0.1496
+0.0642
+0.1499
+0.1006
+0.1486
+40
+3
+1
+0.0683
+0.0480
+0.0684
+0.0602
+0.0681
+2
+0.0717
+0.0457
+0.0718
+0.0601
+0.0717
+5
+1
+0.0773
+0.0511
+0.0776
+0.0675
+0.0776
+3
+0.0972
+0.0543
+0.0972
+0.0748
+0.0970
+7
+2
+0.0971
+0.0497
+0.0974
+0.0745
+0.0969
+
+17
+σ
+censoring
+n
+p
+q
+MLE
+MLE2
+BCE
+BCE2
+Firth
+4
+0.1166
+0.0523
+0.1169
+0.0836
+0.1159
+50%
+20
+3
+1
+0.0883
+0.0540
+0.0896
+0.0732
+0.0895
+2
+0.1125
+0.0643
+0.1132
+0.0881
+0.1135
+5
+1
+0.1022
+0.0567
+0.1054
+0.0852
+0.1062
+3
+0.1570
+0.0786
+0.1584
+0.1136
+0.1593
+7
+2
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+0.0613
+0.1501
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+4
+0.1962
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+3
+1
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+2
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+1
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+3
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+0.0978
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+7
+2
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+4
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+0.1096
+0.1572
+40
+3
+1
+0.0648
+0.0489
+0.0647
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+0.0650
+2
+0.0814
+0.0571
+0.0813
+0.0723
+0.0814
+5
+1
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+0.0783
+3
+0.1063
+0.0629
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+0.0878
+0.1066
+7
+2
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+0.0597
+0.1065
+0.0871
+0.1056
+4
+0.1296
+0.0629
+0.1306
+0.0961
+0.1320
+1.4
+Assessing the power of the tests
+Table S3: Simulated rejection rates for H0 : βq = 0q, with different combinations of σ, censoring,
+n and q when the true parameter vector is β⊤ = (ψ1⊤
+q , 0⊤
+p−q)
+σ
+censoring
+n
+q
+ψ
+MLE
+MLE2
+BCE
+BCE2
+Firth
+0.5
+10%
+20
+1
+0.05
+0.1047
+0.0583
+0.1047
+0.0799
+0.1030
+0.10
+0.1599
+0.0956
+0.1605
+0.1256
+0.1575
+0.25
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+0.3537
+0.4798
+0.4231
+0.4726
+0.50
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+0.8577
+0.9143
+0.8909
+0.9092
+1.00
+0.9993
+0.9982
+0.9993
+0.9989
+0.9993
+
+18
+σ
+censoring
+n
+q
+ψ
+MLE
+MLE2
+BCE
+BCE2
+Firth
+2.00
+1.0000
+1.0000
+1.0000
+1.0000
+1.0000
+3
+0.05
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+0.0673
+0.1446
+0.0938
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+0.10
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+0.1763
+0.2442
+0.25
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+0.6427
+0.7924
+0.7191
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+0.9891
+0.9964
+0.9941
+0.9958
+1.00
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+1
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+0.1478
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+0.10
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+0.50
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+0.1435
+0.2079
+
+19
+σ
+censoring
+n
+q
+ψ
+MLE
+MLE2
+BCE
+BCE2
+Firth
+1.00
+0.5019
+0.3272
+0.5016
+0.4028
+0.4946
+2.00
+0.9433
+0.8812
+0.9436
+0.9173
+0.9404
+25%
+20
+1
+0.05
+0.1030
+0.0649
+0.1032
+0.0816
+0.1015
+0.10
+0.1984
+0.1397
+0.1987
+0.1698
+0.1971
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+0.5724
+0.6663
+0.6237
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+0.05
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+0.1461
+0.1067
+0.1447
+0.10
+0.3205
+0.2030
+0.3208
+0.2578
+0.3192
+0.25
+0.9351
+0.8836
+0.9349
+0.9129
+0.9333
+0.50
+1.0000
+0.9998
+1.0000
+1.0000
+1.0000
+1.00
+1.0000
+1.0000
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+1.0000
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+1.0000
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+1
+0.05
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+0.0658
+0.0820
+0.10
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+0.0645
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+0.5685
+0.6671
+0.6228
+0.6630
+1.00
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+0.9777
+0.9872
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+0.9871
+2.00
+1.0000
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+1.0000
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+3
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+0.0539
+0.1096
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+0.1081
+0.10
+0.1457
+0.0780
+0.1457
+0.1085
+0.1451
+0.25
+0.4489
+0.3164
+0.4488
+0.3759
+0.4452
+0.50
+0.9321
+0.8766
+0.9319
+0.9073
+0.9300
+1.00
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+0.9999
+1.0000
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+1
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+0.0473
+0.0819
+0.0624
+0.0818
+0.25
+0.1026
+0.0650
+0.1024
+0.0816
+0.1013
+
+20
+σ
+censoring
+n
+q
+ψ
+MLE
+MLE2
+BCE
+BCE2
+Firth
+0.50
+0.1539
+0.1019
+0.1544
+0.1279
+0.1528
+1.00
+0.3941
+0.3050
+0.3942
+0.3538
+0.3914
+2.00
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+0.8058
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+3
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+0.0994
+0.1312
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+0.1493
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+0.6006
+0.6665
+2.00
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+0.9917
+0.9871
+0.9910
+50%
+20
+1
+0.05
+0.1095
+0.0766
+0.1097
+0.0937
+0.1089
+0.10
+0.2310
+0.1738
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+0.7904
+0.7657
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+0.3924
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+0.3907
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+0.9658
+0.9814
+0.9751
+0.9809
+0.50
+1.0000
+1.0000
+1.0000
+1.0000
+1.0000
+1.00
+1.0000
+1.0000
+1.0000
+1.0000
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+1.0000
+1.0000
+1.0000
+1.0000
+1.0000
+30
+1
+0.05
+0.0813
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+0.0815
+0.0681
+0.0810
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+0.0708
+0.1070
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+0.1064
+0.25
+0.3089
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+0.3091
+0.2799
+0.3080
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+0.7294
+0.7897
+0.7621
+0.7875
+1.00
+0.9977
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+0.9977
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+1.0000
+1.0000
+1.0000
+3
+0.05
+0.1060
+0.0597
+0.1060
+0.0794
+0.1056
+0.10
+0.1603
+0.0953
+0.1602
+0.1245
+0.1591
+
+21
+σ
+censoring
+n
+q
+ψ
+MLE
+MLE2
+BCE
+BCE2
+Firth
+0.25
+0.5550
+0.4358
+0.5550
+0.4939
+0.5534
+0.50
+0.9789
+0.9641
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+0.9782
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+1
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+10%
+20
+1
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+1
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+0.0862
+
+22
+σ
+censoring
+n
+q
+ψ
+MLE
+MLE2
+BCE
+BCE2
+Firth
+0.10
+0.0926
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+
+29
+References
+[1] Wald Abraham. Test of statistical hypotheses concerning several parameter when the
+number of observations is large Transactions of the American Mathematical Society.
+1943;54:426–482.
+

DNAyT4oBgHgl3EQf4frj/content/tmp_files/2301.00789v1.pdf.txt

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+Inertial migration in pressure-driven channel flow: beyond the Segre-Silberberg pinch
+Prateek Anand1 and Ganesh Subramanian1, ∗
+1Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India
+We examine theoretically the inertial migration of a neutrally buoyant rigid sphere in pressure-
+driven channel flow, accounting for its finite size relative to the channel width (the confinement
+ratio). For sufficiently large channel Reynolds numbers (Rec), a small but finite confinement ratio
+qualitatively alters the inertial lift velocity profiles obtained using a point-particle formulation. Fi-
+nite size effects are shown to lead to new equilibria, in addition to the well known Segre-Silberberg
+pinch locations. Consequently, a sphere can migrate to either the near-wall Segre-Silberberg equi-
+libria, or the new stable equilibria located closer to the channel centerline, depending on Rec and
+its initial position. Our findings are in accord with recent experiments and simulations, and have
+implications for passive sorting of particles based on size, shape and other physical characteristics,
+in microfluidic applications.
+Inertia-driven
+cross-stream
+migration
+of
+neutrally
+buoyant spheres in pipe flow, to an annular location
+between the centerline and walls, was first observed by
+Segre and Silberberg[1–3], the location termed the Segre-
+Silberberg annulus (henceforth, SS-annulus or equilib-
+ria). Equilibria arising from inertial lift forces have since
+been exploited in a range of microfluidic applications[4–
+8]. The first theoretical explanations of the phenomenon
+were for pressure-driven channel flow (the plane Poiseuille
+profile)[9, 10], and involved determining the inertial lift
+on a sphere for Rep, Rec ≪ 1, Rep and Rec being the
+particle and channel Reynolds numbers, respectively [11].
+The pair of zero-crossings of the O(Rep) lift profile, sym-
+metrically located about the centerline, corresponded to
+the SS-equilibria. The calculations were later extended
+to Rec ∼ O(1) and larger[12, 13], with the SS equilibria
+starting at a location intermediate between the walls and
+centerline for Rec ≪ 1, and moving wallward with in-
+creasing Rec. An analogous dependence on the Reynolds
+number was predicted for the SS-annulus in pipe flow[14],
+pointing to the similar physics governing inertial migra-
+tion in the two configurations.
+Later experiments[15], while confirming the original
+observations[1–3], revealed an additional inner annulus,
+this being the only equilibrium location beyond a cer-
+tain Rec[40]. The calculations above[9, 10, 12, 13] use a
+point-particle approximation, and predict only the pair of
+SS-equilibria in plane Poiseuille flow, and the SS-annulus
+alone in pipe flow[14], regardless of Rec. The inner an-
+nulus was therefore speculated to arise from finite-size
+effects[15].
+Although initially regarded as a transient
+feature[17], recent experiments[18] have confirmed the in-
+ner annulus to be a stable equilibrium, leading to the
+following migration scenario: all spherical particles fo-
+cus onto the SS-annulus at low Rec (Regime A); for Rec
+greater than a threshold, particles focus onto either the
+SS-annulus or the aforementioned inner annulus depend-
+ing on their radial distance from the centerline (Regime
+B); particles focus solely onto the inner annulus beyond
+a second threshold (Regime C). The threshold Rec’s de-
+marcating different regimes are observed to decrease with
+increasing λ, λ being the confinement ratio defined as the
+ratio of the sphere radius a to channel width H (or pipe
+radius). This scenario has been qualitatively confirmed
+by simulations[19, 20], although the parameter ranges ex-
+plored in the above studies are necessarily restricted.
+In this letter, for the first time, we move beyond earlier
+point-particle formulations, and theoretically examine in-
+ertial migration in plane Poiseuille flow for small but fi-
+nite λ, with Rec = VmaxH/ν being arbitrary; Vmax here
+is the centerline speed of plane Poiseuille flow, and ν the
+kinematic viscosity of the suspending fluid. Rep = λ2Rec
+is assumed small, allowing analytical progress based on
+a leading order Stokesian approximation. The O(λRep)
+finite-size contribution is shown to qualitatively alter the
+inertial lift profiles obtained from a point-particle for-
+mulation (λ = 0), for large Rec, in a manner consistent
+with the recent studies above. Our calculations show that
+a new pair of stable equilibria, closer to the centerline,
+emerges beyond a threshold Rec, even for Rep ≪ 1. We
+provide a complete characterization of migration scenar-
+ios in the λ−Rec plane for plane Poiseuille flow.
+FIG. 1: Neutrally buoyant sphere of radius a in
+pressure-driven flow through a channel of width H.
+For a neutrally buoyant rigid sphere in plane Poiseuille
+flow, at a non-dimensional distance of λ−1s from the
+lower wall (see Fig.1), use of the generalized reciprocal
+theorem leads to the following expression for the inertial
+arXiv:2301.00789v1  [physics.flu-dyn]  2 Jan 2023
+
+12
+d
+H
+s =
+H
+a
+r
+d
+r32
+lift velocity[21]:
+Vp(s) = −Rep
+�
+V F +V P
+�
+uSt ·
+�
+ustr · ∇U ∞ + U ∞ · ∇ustr
+�
+dV + λRep
+�
+−
+�
+V ∞ ui ·
+�
+us,i · ∇us,i
+�
+dV −
+�
+V ∞
+�
+ui ·
+�
+us,i · ∇u∞
++ u∞ · ∇us,i
+�
+− uSt,i ·
+�
+ustr,i · ∇U ∞ + U ∞ · ∇ustr,i
+��
+dV +
+�
+V P uSt,i ·
+�
+ustr,i · ∇U ∞ + U ∞ · ∇ustr,i
+�
+dV
+�
+,
+(1)
+for Rep small but finite, Vp in being scaled with Vmaxλ.
+The first integral in (1) is the point-particle contribu-
+tion examined earlier[9, 10, 12, 13], the domain of in-
+tegration (V F + V P ) being the total volume contained
+between the channel walls.
+The dominant contribu-
+tion to this integral comes from scales of O(H), whence
+the finite sphere size may be neglected.
+Thus, uSt
+and ustr in the integrand are, respectively, the veloc-
+ity fields due to a Stokeslet and a stresslet confined
+between the channel walls.
+The Stokeslet is oriented
+perpendicular to the walls, while the stresslet is pro-
+portional to the rate of strain tensor, associated with
+the plane Poiseuille flow, evaluated at the sphere loca-
+tion. This tensor is 1
+2β(1112 + 1211), with the Poiseuille
+flow given by U ∞ = (βr2 + γλr2
+2)11 in a reference
+frame moving with the fluid velocity at the sphere cen-
+ter; here, 11 is a unit vector along the flow direction, r2
+the gradient coordinate relative to the sphere center, and
+β = 4(1 − 2s) and γ = −4 denote the shear rate and cur-
+vature of the plane Poiseuille profile. The dependence of
+the point-particle contribution on H amounts to an Rec-
+dependence in non-dimensional terms, so the first term
+in (1) is of the form RepF1(s, Rec), with F1 determined
+semi-analytically for Rec ≪ 1[9, 10, 16], and numerically
+for Rec ≳ O(1)[12, 13, 16].
+The integrals within square brackets, in (1), are the
+O(λRep) contributions.
+The dominant contributions
+to the first two integrals arise from scales of O(a), so
+the channel walls may be neglected, the integration be-
+ing over an unbounded fluid domain (V ∞) outside the
+sphere. Thus, ui is the Stokesian velocity disturbance
+due to a sphere translating under a constant force nor-
+mal to the walls, and us,i is the Stokesian disturbance
+due to a force-free torque-free sphere in an ambient
+plane Poiseuille flow, both in an unbounded fluid do-
+main; u∞ = (βr2 + λγr2
+2 − λγ/3)11 is the Poiseuille
+flow in a reference frame translating with the sphere,
+and differs from U ∞ above since the sphere transla-
+tion speed includes a contribution (λγ/3) from the profile
+curvature[21, 22]. uSt,i and ustr,i denote the unbounded-
+domain Stokeslet and the stresslet, respectively, the for-
+mer given by the Oseen-Burgers tensor[22]; they differ
+from uSt and ustr above in not including the wall-image
+contributions [22, 23]. The third integral within brackets
+corrects for the inclusion of the sphere volume (V P ) in
+the domain of integration of the point-particle integral.
+The irrelevance of H for the finite-size integrals im-
+plies that the expression within square brackets, in (1),
+is only a function of s. Further, the simple domain of
+integration (V ∞ or V P ) leads to this s-dependence being
+evaluable in closed form[21], and (1) reduces to:
+Vp(s) = Rep
+�
+F1(s, Rec) + λ1141(1 − 2s)
+216
+�
+,
+(2)
+F1(s, Rec) in (2) can be computed numerically for any
+Rec using a shooting method[12, 13, 16].
+In Fig. 2a,
+the resulting (scaled) lift profiles are shown, for different
+Rec’s, over the lower half-channel with s ∈ [0, 0.5] (due
+to anti-symmetry about the centerline). In addition to
+the wallward movement of the lone zero-crossing (the SS-
+equilibrium), the magnitude of the lift at any fixed loca-
+tion, not in the neighbourhood of the wall[24], decreases
+sharply with increasing Rec[13]. This reflects the weak-
+ened particle-wall interaction when the walls recede be-
+yond the inertial screening length of O(HRe
+− 1
+2
+c
+), owing
+to a more rapid decay of the disturbance velocity field
+at these distances. Apart from the overall decrease in
+magnitude, the shape of the profile also changes, with
+an intermediate concave-downward portion emerging for
+Rec ≳ 296. An analogous scenario prevails for pipe flow,
+although the lift is smaller than that for channel flow at
+the same Rec[14, 18].
+The changes in the point-particle contribution above
+imply that the finite-size term in (2), although O(λ)
+smaller for Rec ≪ 1, becomes comparable for sufficiently
+large Rec. This is seen in Fig. 2b which shows the lift
+profiles, for λ = 0.01, for the same Rec’s as in Fig. 2a. For
+Rec = 50, the lift profile and the SS-equilibrium are only
+marginally affected. In contrast, for Rec = Rethreshold
+c1
+(≈ 665 for λ = 0.01), while the SS-equilibrium (expect-
+edly) has moved closer to the walls, a pair of stable and
+unstable equilibria appear between it and the centerline
+via a saddle-node bifurcation; the unstable equilibrium
+demarcating the basins of attraction of the SS equilib-
+rium and the inner stable equilibrium. The bifurcation
+arises due to finite-size effects causing the region of neg-
+ative curvature, in the point-particle profile, to cross the
+
+3
+(a) λ = 0 (point-particle)
+(b) λ = 0.01
+FIG. 2: Inertial lift profiles in the lower half-channel for Rec ∈(50, 2000): (a) λ = 0 (point-particle); (b) λ = 0.01; the
+two insets for λ = 0.01 provide a magnified view of the saddle-node bifurcations at Rethreshold
+c1
+≈ 665 and
+Rethreshold
+c2
+≈ 1500. The black, red and blue symbols denote the SS, unstable and stable equilibria, respectively. The
+arrow in (a) shows the movement of the SS equilibria with increasing Rec.
+zero-lift line (upper inset in Fig. 2b). As Rec increases to
+800, the unstable equilibrium moves towards the SS equi-
+librium even as both move wallward, while the inner sta-
+ble equilibrium moves towards the centerline. A second
+saddle-node bifurcation at Rec = Rethreshold
+c2
+(≈ 1500 for
+λ = 0.01) leads to the near-center stable equilibrium be-
+ing the only one in the half-channel for larger Rec (lower
+inset in Fig. 2b). Note that for Rec ∈ (50, 2000) as in
+Fig. 2b, and λ = 0.01, Rep ∈ (0.005,0.2), consistent with
+the theoretical assumption of weak fluid inertial effects
+on scales of O(a).
+Fig. 3a plots the equilibrium loci identified above, for
+λ = 0.01, as a function of Rec. The SS-branch is seen to
+start at s ≈ 0.182 for Rec ≪ 1, moving to smaller s with
+increasing Rec.
+The inner stable equilibrium emerges
+discontinuously at s ≈ 0.18 for Rec ≈ 665, moving to
+larger s thereafter (the SS-equilibrium is at s ≈ 0.09 for
+this Rec). The loci of the SS and the inner equilibria are
+shown as sequences of black and blue dots, respectively,
+with the unstable equilibrium locus connecting the two
+shown as a sequence of red dots. The fold that devel-
+ops in the interval Rec ∈ (665, 1500), bracketed by the
+two saddle-node bifurcations, implies a hysteretic behav-
+ior in an experiment [29]. A quasi-static protocol of in-
+creasing flow rate will lead to spheres remaining at the
+SS-equilibrium until Rec ≈ 1500, at which point they
+will jump onto the new stable branch closer to the cen-
+terline. In contrast, along a path of decreasing flow rate,
+spheres will remain at the inner stable equilibrium down
+to Rec ≈ 665, before jumping back to the SS-branch.
+A behavior analogous to that in Fig. 3a occurs for
+λ less than 0.01, with the pair of Rec-thresholds in-
+creasing with decreasing λ.
+However, the equilibrium
+loci undergo a qualitative change as λ increases.
+To
+see this, note that the SS-equilibrium, in the point-
+particle framework, emerges from a balance between an
+O(βγ) curvature-induced contribution driving migration
+towards higher shear rates (that is, away from the cen-
+terline), and an O(β2) wall-induced repulsion. Both con-
+tributions arise due to inertial forces acting on scales
+of O(H) for Rec ≪ 1[16], and decrease with increasing
+Rec. The O(β2) contribution decreases faster, leading to
+the wallward movement of the SS-equilibrium. At O(λ),
+there arises an additional curvature-induced contribution
+on scales of O(a), and that drives migration towards the
+centerline [32]. The opposing signs of the point-particle
+and finite-size curvature-induced contributions weakens
+the wallward movement (with increasing Rec) of the SS-
+equilibrium for larger λ. The profound effect of this weak-
+ening may be seen from Fig. 3b which shows the equilib-
+rium locus for λ = 0.025. The region of multiple equi-
+libria is now absent, with the SS-equilibrium smoothly
+transitioning from an initial wallward movement, to a
+movement towards the centerline, across Rec ≈ 200.
+By identifying the equilibrium loci as a function of
+Rec, for different λ, a ‘phase diagram’ of migration sce-
+narios in the λ − Rec plane may be constructed as in
+Fig. 4. The figure highlights the existence of three dis-
+tinct regions.
+Region
+1
+○, corresponding to the area
+below the red curve and outside the (gray) shaded re-
+gion, contains lift profiles with a single stable off-center
+equilibrium in the half-channel. Region 2
+○, correspond-
+ing to the shaded region, contains lift profiles with two
+stable equilibria, separated by an intervening unstable
+one, in the half-channel. The upper and lower bound-
+aries of this region are determined by the pair of turning
+points on the equilibrium locus, corresponding to saddle-
+node bifurcations - these were identified in Fig 3a for
+λ = 0.01. The two boundaries end in a cusp for the fold
+bifurcation under consideration [33, 34], corresponding to
+
+Rec=50
+0.2
+Rec=100
+Rec=300
+- Rec=665
+0.1
+Rec=800
+— Rec=1500
+ Rec=1700
+0.0
+-0.1
+-0.2
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+S0.0005
+Rec=50
+Rec=660
+0.2
+Rec=670
+- Rec=100
+Vp/Rep
+0.0000
+Rec=300
+ Rec=665
+0.0005
+0.1
+- Rec=800
+- Rec=1500
+0.0010
+0.16
+0.17
+0.18
+0.19
+0.20
+0.21
+— Rec=2000
+S
+0.0
+0.010
+-0.1
+Vp/Rep
+0.005
+Rec=1400
+Rec=1700
+0.000
+-0.2
+0.005
+0.06
+80'0
+0.10
+0.12
+0.14
+s
+0.0
+0.1
+0.2 
+0.3
+0.4
+0.5
+s4
+(a) λ = 0.01
+(b) λ = 0.025
+FIG. 3: Inertial equilibrium loci for (a)λ = 0.01 and (b)λ = 0.025; black, blue and red dots denote the SS, and the
+inner stable and unstable equilibria, respectively. The region of multiple equilibria in (a), for Rec ∈ (665, 1500),
+leads to hysteretic jumps in the equilibrium location marked by the vertical arrows A1 (s = 0.179 → 0.09) and
+A2(s = 0.08 → 0.31). Vertical dashed lines in (a) and (b) denote the laminar-turbulent transition.
+(λcritical, Recritical
+c
+) ≡ (0.0216, 296) in Fig. 4; see top right
+inset. Along either a vertical or a horizontal line, the lat-
+ter corresponding to an experimental path of changing
+flow rate, Region 2
+○ mediates a discontinuous transition
+from the SS-equilibrium to the inner stable equilibrium
+closer to the centerline. Region 3
+○, above the red curve,
+includes lift profiles with the centerline as the only sta-
+ble equilibrium.
+Note that the centerline is always an
+equilibrium by symmetry, albeit an unstable one in Re-
+gions 1
+○ and 2
+○. Insets in Fig. 4 show lift profiles for
+the following (λ, Rec) pairs: (0.3, 5); (b) (0.05, 10); (c)
+(0.01, 1000); (d) (0.015, 1500), which are consistent with
+the aforementioned features of Regions 1
+○- 3
+○.
+The black dot-dashed line in Fig. 4 corresponds to
+Rep = 1, and may be regarded as a rough threshold above
+which the present theoretical results may no longer be
+valid. This implies, for instance, that our results may not
+be quantitatively accurate beyond Rec = 100 at λ = 0.1.
+Importantly, the region of multiple equilibria and the
+associated hysteretic transitions, predicted here for the
+first time, lie well within this threshold. A second factor
+that limits the observability of Region 2
+○ is the laminar-
+turbulence transition. Although plane Poiseuille flow is
+predicted to become linearly unstable at Rec = 11544
+[35], experiments show a nonlinear subcritical transition
+to turbulence at a much lower Rec ∼ O(2000)[36, 37].
+This subcritical transition is shown as vertical dashed
+lines in both Figs. 3 and 4. The emergence of the region
+of multiple equilibria in the latter figure clearly occurs
+well before the transition threshold.
+While the migration pattern for a fixed λ, implied by
+Fig. 4, is in qualitative agreement with studies quoted at
+the beginning[15, 17–20], the inner annulus in these stud-
+ies is observed for higher λ (≳ 0.05) and for Rep ≳ O(1)
+- see hatched region in Fig. 4. The absence of multiple
+equilibria for smaller λ is very likely due to the develop-
+ment length, needed for a steady particle distribution,
+being larger than the pipe length used in the experi-
+ments. For Rec fixed, this length scales as O(λ−3)[15],
+increasing rapidly with decreasing particle size. Notwith-
+standing differences between the pipe and channel ge-
+ometries, experiments with longer pipes should lead to
+the hatched region in Fig. 4 extending down to smaller
+λ. There remains the provocative question of how the
+secondary finite-Rep region of multiple equilibria, iden-
+tified in the said studies, connects to the theoretically
+identified small-Rep region (Region 2
+○ in Fig. 4).
+FIG. 4: Migration scenarios in the λ − Rec plane.
+Inertial lift profiles for canonical (λ, Rec) pairs
+(highlighted by black dots) are shown (see text for
+details). The hatched region shows parameter ranges
+covered in earlier studies [15, 17–20].
+Apart from the fundamental significance of our find-
+ings, in terms of enriching the inertial migration land-
+scape, and providing an explanation for recent exper-
+iments and computations, Fig. 4 may be leveraged to-
+wards passive sorting in microfluidic applications. The
+simplest scenario pertains to separating spheres of two
+
+0.4
+Critical Re. for transition
+0.3
+to turbulence ～ 2000
+location
+A2
+0.1
+Lower wall
+(s = 0)
+～ 665
+～ 1500
+5
+10
+50
+100
+500
+1000
+Rec0.45F
+0.40
+0.35
+cation
+0.30
+O1
+Critical Rec for transition
+to turbulence ～ 2000
+0.25
+0.20
+Lower wall
+(s = 0)
+5
+10
+50
+100
+500
+1000
+Rec(a)
+(c)
+3
+0.500
+d
+(b)
+0.100
+0.050
+0.0220
+0.0215
+0.010
+0.0210
+2
+(0.02156,296)
+0.005
+0.0205
+296
+298
+300
+302
+304
+Rec
+1
+10
+100
+1000
+Critical Re.fortransition
+Rec
+toturbulence20005
+different sizes, corresponding to confinement ratios λ1
+and λ2 (λ2 > λ1). An experimental protocol of changing
+flow rate (Rec) for a bi-disperse suspension, with parti-
+cles of the aforementioned sizes, would appear as a pair
+of horizontal lines in Fig. 4, the upper one correspond-
+ing to λ2. With increasing flow rate, separation would
+be achieved at an Rec when the point on the λ2-line is
+above Region 2
+○ (after crossing it to the right), with the
+one on the λ1-line directly below.
+At this Rec, larger
+spheres would focus onto the pair of near-centerline sta-
+ble equilibria, with the smaller ones focusing onto the
+near-wall SS-equilibria. If the point on the λ1-line lies
+within Region 2
+○, rather than below it, as would be the
+case when λ2/λ1 is not far from unity, partial separation
+will be achieved owing to smaller spheres focusing onto
+both the SS and inner equilibria (the relative fractions
+determined by the pair of unstable equilibria).
+The implications of the near-centerline stable equilib-
+ria found here go well beyond the size-sorting scenario
+above.
+The dependence of the interval of existence of
+these equilibria, on inertial forces in a region of order the
+sphere size, implies a generic sensitivity of the finite-size
+contributions to the detailed characteristics of the sus-
+pended microstructure.
+Thus, for anisotropic particles
+such as spheroids or ellipsoids, the threshold Reynolds
+numbers (Rethreshold
+c1
+, Rethreshold
+c2
+) that characterize the
+region of multiple equilibria are expected to be functions
+of the particle aspect ratio(s).
+In contrast, the time-
+averaged inertial lift for spheroids with order unity aspect
+ratios, calculated within a point-particle framework for
+plane Poiseuille flow, may be shown to yield SS-equilibria
+identical to those for spheres[16]; the spheroid aspect
+ratio only affecting the magnitude of the point-particle
+lift profile, not the equilibrium locations.
+The aspect-
+ratio-dependence expected for the near-centerline equi-
+libria will therefore be crucial for shape-sorting proto-
+cols in microfluidic applications[38]. Along similar lines,
+there will be a dependence on the viscosity ratio for
+drops, allowing, in principle, for separation of weakly
+deformed drops based on both size and viscosity ratio
+differences[39]. Analogous remarks apply to other elastic
+microstructures such as vesicles, capsules or red blood
+cells. Migration phase diagrams for these cases will have
+at least one additional axis - this axis could be the appro-
+priate shape parameter for anisotropic particles (aspect
+ratio for spheroids) or the viscosity ratio for drops. In
+the latter case, the degree of deformability, as character-
+ized by the Capillary number, offers an additional degree
+of freedom. It would be of interest, in future, to quan-
+titatively determine these phase diagrams, allowing for
+rational design of passive sorting protocols.
+∗ sganesh@jncasr.ac.in
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+mechanics and convective transport processes (Vol. 7).
+Cambridge University Press.
+[24] Within a point-particle framework, the inertial lift veloc-
+ity increases to approximately 1.52, independent of Rec,
+on approach towards either wall[16].
+[25] Leal, L. G., & Hinch, E. J. (1971). The effect of weak
+Brownian rotations on particles in shear flow. Journal of
+Fluid Mechanics, 46(4), 685-703.
+[26] Dabade, V., Marath, N. K., & Subramanian, G. (2016).
+The effect of inertia on the orientation dynamics of
+anisotropic particles in simple shear flow. Journal of Fluid
+Mechanics, 791, 631-703.
+[27] Marath, N. K., & Subramanian, G. (2017). The effect
+of inertia on the time period of rotation of an anisotropic
+particle in simple shear flow. Journal of Fluid Mechanics,
+830, 165-210.
+[28] Okagawa, A., Cox, R. G., & Mason, S. G. (1973). The
+kinetics of flowing dispersions. VI. Transient orientation
+
+6
+and rheological phenomena of rods and discs in shear flow.
+Journal of Colloid and Interface Science, 45(2), 303-329.
+[29] Such a hysteresis will arise only in the absence of stochas-
+tic fluctuations, or for short channels. Positional fluctua-
+tions arising either from weak Brownian motion [25–27],
+or from pair-hydrodynamic interactions[28], will eliminate
+hysteretic behavior in sufficiently long channels.
+[30] Ho, B. P., & Leal, L. G. (1976). Migration of rigid spheres
+in a two-dimensional unidirectional shear flow of a second-
+order fluid. Journal of Fluid Mechanics, 76(4), 783-799.
+[31] Chan, P. H., & Leal, L. (1979). The motion of a de-
+formable drop in a second-order fluid. Journal of fluid me-
+chanics, 92(1), 131-170.
+[32] For cross-stream migration in plane Poiseuille flow
+arising from mechanisms other than inertia such as
+viscoelasticity[30] or drop deformation[31], the lift at lead-
+ing order arises from this curvature-induced finite size con-
+tribution, and drives migration towards the centerline.
+[33] Zeeman, E. C. (1976). Catastrophe theory. Scientific
+American, 234(4), 65-83.
+[34] Dubey, P., Roy, A., & Subramanian, G. (2022). Linear
+stability of a rotating liquid column revisited. Journal of
+Fluid Mechanics, 933.
+[35] Orszag, S. A. (1971). Journal of Fluid Mechanics, 50(4),
+689-703.
+[36] Carlson, D. R., Widnall, S. E., & Peeters, M. F. (1982).
+Journal of Fluid Mechanics, 121, 487-505.
+[37] Nishioka, M., & Asai, M. (1985). Journal of Fluid Me-
+chanics, 150, 441-450.
+[38] Masaeli, M., Sollier, E., Amini, H., Mao, W., Camacho,
+K., Doshi, N., ... & Di Carlo, D. (2012). Continuous iner-
+tial focusing and separation of particles by shape. Physical
+Review X, 2(3), 031017.
+[39] The SS-equilibria for drops, within the framework of a
+point-particle formulation, may be shown to be identical
+to those for rigid spheres. The point-particle disturbance
+velocity field depends only on the strength of the induced
+stresslet, and as a result, the inertial lift for a spherical
+drop may be obtained from that for a rigid sphere by mul-
+tiplication by the ratio of the corresponding stresslet co-
+efficients.
+[40] Wherever appropriate,
+Rec
+is taken to denote the
+Reynolds number based on the pipe diameter and the
+mean velocity of the flow through the pipe.
+
+1
+Supplemental material
+Following [S1] and [S2], one may use a generalized reciprocal theorem formulation to derive a formal expression for
+the inertial lift velocity of a neutrally buoyant sphere, in an ambient plane Poiseuille flow, for arbitrary Rep. In the
+limit Rep ≪ 1, the non-dimensional inertial lift is O(Rep), being given by:
+Vp = −Rep
+�
+V F u · (us · ∇us + us · ∇u∞ + u∞ · ∇us) dV.
+(S1)
+where Vp is scaled by Vmaxa/H or Vmaxλ.
+The actual problem in the reciprocal theorem framework corresponds to the one of interest, that is, a neutrally
+buoyant sphere freely moving in wall-bounded plane Poiseuille flow for Rep and λ small but finite, with Rec = λ−2Rep
+being arbitrary. For purposes of calculating the inertial lift to O(Rep), the disturbance velocity field in the actual
+problem may be replaced by its Stokesian approximation. Thus, us, in the inertial acceleration terms in (S1), is the
+Stokesian disturbance field due to a force-free and torque-free sphere translating with Up in an ambient plane Poiseuille
+flow. In a reference frame moving with the sphere center, the latter flow is given by u∞ = (α + βr2 + λγr2
+2)11 − Up
+where, with the sphere at a distance λ−1s (in units of a) from the lower wall, one has α = 4λ−1s(1 − s), β = 4(1 − 2s)
+and γ = −4. Here, α11 −Up and β are the ambient slip and shear rate at the sphere center, the latter varying linearly
+across the channel and equalling zero at the centerline (s = 0.5); γ denotes the constant curvature of the Poiseuille
+profile. Up is determined using the force-free constraint in Faxen’s law for translation. This leads to Up = (α+λγ/3)11
+and, as a result, the ambient flow in the said reference frame takes the form u∞ = (βr2 + λγr2
+2 − λγ/3)11. The test
+problem in the reciprocal theorem framework corresponds to the Stokesian translation of a sphere in an otherwise
+quiescent fluid confined between parallel walls (ones that bound the Poiseuille flow in the actual problem), under a
+constant force acting along the gradient direction. The test disturbance field u multiplies the inertial acceleration
+terms involving us in (S1).
+We now examine the length scales that contribute dominantly to the integral in (S1), beginning with the limit Rec ≪
+1, when the inertial screening length (HRe−1/2
+c
+) is much larger than the channel width, and therefore, irrelevant. The
+dominant contributions to the volume integral in this limit may arise from either scales of O(a) (the inner region) or
+those of O(H) (the outer region). In order to assess their relative magnitudes, we consider the intermediate asymptotic
+interval 1 ≪ r ≪ λ−1 (r is measured in units of a) where both the finite size of the sphere and wall-induced image
+contributions may be neglected at leading order. For r in this interval, u ∼ 1/r corresponding to the farfield Stokeslet,
+and us ∼ β/r2 + γλ/r3 corresponding to the farfield stresslet and force-quadrupole contributions associated with the
+linear and quadratic ambient flow components, respectively. Using these forms along with u∞ ∼ βr + γλr2, one
+obtains the following estimates for parts of the integrand involving the linear and nonlinear components of the inertial
+acceleration:
+• u · (us · ∇u∞ + u∞ · ∇us) ∼ β2
+r3 + λβγ
+r2
++ λγβ
+r4
++ γ2
+r3
+(linear),
+• u · (us · ∇us) ∼ β2
+r6 + λβγ
+r7
++ λ2γ2
+r8
+(nonlinear).
+Using dV ∼ O(r2dr), one obtains the following estimates for contributions to the lift velocity integral:
+V linear
+p
+∼ Rep
+� r
+dr′
+� β2
+r′ + λβγ + λγβ
+r′2 + λ2γ2
+r′
+�
+,
+∼ Rep
+�
+β2 ln r + λβγr + λγβ
+r
++ λ2γ2 ln r
+�
+,
+(S2a)
+V non-linear
+p
+∼ Rep
+� r
+dr′
+� β2
+r′4 + λβγ
+r′5 + λ2γ2
+r′6
+�
+,
+∼ Rep
+� β2
+r3 + λβγ
+r4
++ λ2γ2
+r5
+�
+.
+(S2b)
+The algebraically growing terms in (S2a) will be dominated by scales of O(H), and the resulting contributions to the
+lift velocity are obtained by cutting off the divergence (for r → ∞) at r ∼ O(λ−1); this cut-off recognizes the wall-
+induced screening of the unbounded-domain behavior that eventually leads to a more rapid decay for r ≫ O(λ−1),
+and thence, convergence. The algebraically decaying terms in both (S2a) and (S2b) will be dominated by scales of
+
+2
+O(a), with the lift velocity contributions now obtained by cutting off the divergence (for r → 0) at r ∼ O(1). The ln r
+terms in (S2a) imply the dominance of the intermediate (matching) interval 1 ≪ r ≪ λ−1, leading, in principle, to
+contributions of O(β2 ln λ−1) and O(λ2γ2 ln λ−1) to the lift velocity; logarithmically smaller contributions of O(β2)
+and O(λ2γ2) must arise from both scales of O(a) (r ∼ O(1)) and O(H) (r ∼ O(λ−1)). However, contributions from
+the inner and matching regions turn out to be zero by symmetry. Owing to the absence of walls at leading order, the
+O(β2) and O(β2 ln λ−1) inner and matching-region contributions must correspond to the lift on a neutrally buoyant
+sphere (or the equivalent point-particle singularity) in an unbounded simple shear flow; likewise, the O(λ2γ2) and
+O(λ2γ2 ln λ−1) inner and matching-region contributions must correspond to the lift on a sphere at the origin of an
+unbounded quadratic flow. In both these scenarios, however, the two lateral directions are equivalent, and there can
+be no lift. That these contributions are zero may also be seen from the fact that one cannot construct a true vector,
+the inertial lift velocity, from any quadratic combination of the velocity gradient tensor associated with an ambient
+linear flow (simple shear for the present case of an ambient Poiseuille flow), or from any quadratic combination of the
+third order tensor that would characterize a generic quadratic flow. Crucially, the O(β2) and O(λ2γ2) outer-region
+contributions are not subject to the above symmetry-argument-based limitation. This is due to the importance of
+walls at leading order, and the implied availability of an additional vector (the unit normal characterizing the wall
+orientations) to construct the lift velocity vector.
+Based on the above arguments, one is led to the following lift velocity contributions from the linear and nonlinear
+inertial terms in the integrand:
+V linear
+p
+∼ Rep
+�
+β2(outer) + βγ(outer) + λγβ(inner) + λ2γ2(outer)
+�
+,
+(S3a)
+V non-linear
+p
+∼ Rep λβγ(inner).
+(S3b)
+From (S3a) and (S3b), the leading order contribution to the lift velocity is seen to come from the linearized inertial
+terms, and from scales of O(H), with there being two such contributions: one proportional to β2 that characterizes
+wall-induced repulsion in an ambient linear flow, and the other proportional to βγ that denotes the contribution due
+to the ambient profile curvature. The dominance of scales of O(H) implies that, for purposes of evaluating the above
+contributions, the sphere in both the actual and test problems can be replaced by the corresponding point singularity.
+Thus, at leading order, one only need consider the terms in (S1) that are linear in us, and further, us and u may be
+approximated as the disturbance fields due to a stresslet (ustr) and Stokeslet (uSt), respectively. Note that uSt and
+ustr are a combination of the unbounded domain components and additional wall-image contributions, both of which
+are of comparable importance on scales of O(H); detailed expressions are given in [S2]. Thus, the inertial lift velocity
+at leading order in Rep and λ reduces to:
+Vp = −Rep
+�
+V F +V P uSt · (ustr · ∇U ∞ + U ∞ · ∇ustr) dV,
+(S4)
+where the O(λγ) term in u∞ has been neglected, so U ∞ = (βr2 + λγr2
+2)11 in (S4). Further, on account of the
+subdominance of scales of O(a), the domain of integration has been extended to include the particle volume (V P ),
+with an accompanying change in the integration variable that is now the position vector scaled by H (rather than a as
+in (S1)). (S4) is the point-particle approximation for the lift velocity for Rec ≪ 1, and corresponds to a dimensional
+lift velocity of O(V 2
+maxλ2a/ν). The use of a rescaled (with λ) integration variable, as mentioned above, leads to the
+integral in (S4) being only a function of s; the detailed evaluation of this integral, via a partial Fourier transform, is
+discussed in [S2].
+For Rec ≳ O(1), the scaling arguments used above to establish the dominance of the outer region contributions
+still hold. The outer region now corresponds to scales of order the inertial screening length or larger, so that the
+farfield Stokesian estimate for us, used above to establish outer-region dominance, remains valid only for 1 ≪ r ≪
+λ−1Re
+− 1
+2
+c
+, with the disturbance velocity field in the actual problem decaying more rapidly for larger r. The outer-
+region dominance for Rec ≳ O(1) implies that this disturbance field may still be approximated as being driven by
+a stresslet forcing, although the forcing appears in the linearized Navier-Stokes equations. While there still exist
+physically distinct contributions arising from profile curvature and wall-induced repulsion, the lift velocity can no
+longer be written as an additive superposition of the two. Furthermore, despite the relevance of the inertial screening
+length, asymptotically larger scales of O(H) continue to be relevant. The ratio of these two outer scales involves Rec
+which appears in the governing linearized equations of motion. Thus, the version of the reciprocal theorem integral
+in (S4) for Rec ≳ O(1), with us replaced by its finite-Rec analog, will be a function of both s and Rec. Although
+the actual calculation is more easily accomplished via a direct solution of the partially Fourier transformed ODE’s
+using a shooting method [S2, S3], one may nevertheless write the point-particle lift velocity contribution formally in
+the form (V 2
+maxλ2a/ν)F1(s; Rec).
+
+3
+The modification of the leading order lift velocity, for λ small but finite, arises from contributions in (S3) that
+are of a smaller order in λ than the outer-region contributions included in (S4). The largest such contributions are
+proportional to λ(βγ), and pertain to the inner region, arising from both the linear and nonlinear inertial terms
+in (S3a) and (S3b).
+The βγ-dependence implies that these contributions arise solely due to the coupling of the
+shear and curvature of the ambient profile, consistent with earlier symmetry arguments. Further, scales of O(H) are
+irrelevant, implying that the λ(βγ) terms will lead to contributions independent of Rec, with the dimensional lift
+velocity being of the form (V 2
+maxλ3a/ν)F2(s), as in equation (2) of the main manuscript. A second implication of the
+O(λβγ) contributions being from the inner region is that they arise independently of the outer-region point-particle
+contribution.
+Said differently, the (V 2
+maxλ2a/ν)F1(s; Rec) and (V 2
+maxλ3a/ν)F2(s) contributions to the inertial lift
+correspond to the leading order terms of the underlying asymptotic expansions of the integrand in the outer and inner
+regions, respectively. This implies that the O(V 2
+maxλ3a/ν)F2(s) contribution is not a correction to the leading order
+point-particle result, and thereby, not constrained to be small in comparison. This feature is especially significant
+since the emergence of multiple equilibria in the lift profiles (Region 2
+○ in Fig 4 of the main manuscript) is only made
+possible by allowing the finite-size contribution to be comparable to the leading point-particle one.
+Note that there are other corrections for finite λ: for instance, the O(λ2γ2) outer-region contribution in (S3), the
+correction to us arising from λ-dependent corrections to the stresslet coefficient, etc. While both of these may be
+shown to be of a smaller order in λ for Rec ≪ 1, they will be far smaller for large Rec owing to an overall weakening
+of wall-induced corrections. This weakening arises from the more rapid decay of the disturbance velocity field for
+distances larger than O(HRe
+− 1
+2
+c
+). This Rec-dependence is important since, as explained in the main manuscript, the
+finite-size contributions become significant only at large Rec owing to the reduction in the magnitude of the point-
+particle contribution with increasing Rec. Clearly, the only finite-size contributions of relevance correspond to the
+O(λβγ) terms in (S3a) and (S3b). To calculate these, we return to the original reciprocal theorem result, viz. (S1),
+and subtract and add back the leading point-particle contribution given by (S4):
+Vp = − Rep
+�
+V F +V P uSt ·
+�
+ustr · ∇U ∞ + U ∞ · ∇ustr
+�
+dV
+− Rep
+��
+V F u ·
+�
+us · ∇us + us · ∇u∞ + u∞ · ∇us
+�
+dV
+−
+�
+V F +V P uSt ·
+�
+ustr · ∇U ∞ + U ∞ · ∇ustr
+�
+dV
+�
+.
+(S5)
+Splitting the point-particle contribution within brackets into separate integrals over the fluid (V F ) and particle (V P )
+domains, separating the nonlinear and linear inertial terms in the original reciprocal theorem integral, and then
+combining the integrals involving the linear inertial terms, one obtains:
+Vp = − Rep
+�
+V F +V P uSt ·
+�
+ustr · ∇U ∞ + U ∞ · ∇ustr
+�
+dV
+− Rep
+��
+V F u ·
+�
+us · ∇us
+�
+dV +
+�
+V F
+�
+u ·
+�
+us · ∇u∞ + u∞ · ∇us
+�
+− uSt ·
+�
+ustr · ∇U ∞ + U ∞ · ∇ustr
+��
+dV −
+�
+V P uSt ·
+�
+ustr · ∇U ∞
++ U ∞ · ∇ustr
+�
+dV
+�
+.
+(S6)
+The last integral within brackets, over V P , evidently involves scales of O(a), and we therefore focus on the remaining
+two integrals. The first integral within brackets contains the nonlinear inertial term, and as already seen in (S2b), the
+integrand exhibits an algebraic decay for r ≫ 1. The second integral involves the difference between the linearized
+inertial terms, and their approximations based on point-particle representations of the corresponding velocity fields,
+as a result of which growing terms cancel out, and only the algebraically decaying one in (S3a) survives. The algebraic
+decay implies the dominance of scales of O(a), and therefore, that wall-induced image contributions do not contribute
+at leading order. Thus, the leading approximations of all of the integrals within brackets, in (S6), may be obtained by
+
+4
+replacing the original fluid domain (V F ) by a completely unbounded one (V ∞), and (S6) may be written in the form:
+Vp = − Rep
+�
+V F +V P uSt ·
+�
+ustr · ∇U ∞ + U ∞ · ∇ustr
+�
+dV.
++ λRep
+�
+−
+�
+V ∞ ui ·
+�
+us,i · ∇us,i
+�
+dV −
+�
+V ∞
+�
+ui ·
+�
+us,i · ∇u∞ + u∞ · ∇us,i
+�
+− uSt,i ·
+�
+ustr,i · ∇U ∞ + U ∞ · ∇u(1)
+str,i
+��
+dV +
+�
+V P uSt,i ·
+�
+ustr,i · ∇U ∞
++ U ∞ · ∇ustr,i
+�
+dV
+�
+,
+(S7)
+where both the actual velocity fields and their point-particle approximations, that appear in the three bracketed
+integrals, are now approximated by simpler expressions pertaining to an unbounded fluid domain; these are indicated
+by the additional subscript ‘i’, and are given below::
+ui = 12
+8π ·
+�I
+r + rr
+r3
+�
++ 12
+8π ·
+� I
+3r3 − rr
+r5
+�
+,
+(S8a)
+us,i = −5
+2
+(E : rr)r
+r5
+−
+�E · r
+r5
+− 5(E : rr)r
+2r7
+�
++ λγ
+�
+−
+1
+24r9
+�
+3r6 + r4(−23r2
+1 + r2
+2 − 8r2
+3 − 3) + 15r2�
+r2
+1(7r2
+2 + 1) + r2
+2
+�
+− 105r2
+1r2
+2
+�
+11 +
+5
+8r9 (r2 − 1)r1r2(3r2 − 7r2
+2)12 +
+5
+8r9 (r2 − 1)r1r3(r2 − 7r2
+2)13
+�
+,
+(S8b)
+uSt,i = 12
+8π ·
+�I
+r + rr
+r3
+�
+,
+(S8c)
+ustr,i = −5
+2
+(E : rr)r
+r5
+.
+(S8d)
+(S8a) is the disturbance velocity field induced by a translating sphere. (S8b) is the disturbance field due to a force-free
+sphere in an ambient flow with both linear and quadratic components: the part involving E (= β
+2 (1112 + 1211), the
+rate of strain tensor of the ambient Poiseuille flow), constitutes the disturbance in an ambient linear flow, while that
+proportional to γ constitutes the response to the quadratic component. As originally shown (but not used) by [S1],
+the latter may be obtained using an expansion in spherical harmonics [S4]. (S8c) and (S8d) are the usual velocity
+fields for a Stokeslet and the stresslet in an unbounded domain.
+Finally, note that the integrals over V P and V ∞ extend down to the origin (the sphere center) with both integrands
+including an O(β2) contribution arising from the coupling of the linear ambient flow component with the stresslet field,
+that is O(1/r3) for r → 0 (see initial integrand estimates); while this leads to a conditionally convergent behavior,
+doing the angular integration first leads to a trivial answer, as must be the case based on symmetry arguments
+above. For the integral over V ∞, a similar conditionally convergent behavior arises at infinity from the leading O(γ2)
+contribution, that is again resolved by appropriate choice of the order of integration. Even the leading point-particle
+integral is conditionally convergent at the origin, an issue (implicitly) addressed by a particular order of integration
+in the calculation procedure[S2].
+The volume integrals, within brackets in (S8), are readily calculated analytically:
+�
+V ∞ ui ·
+�
+us,i · ∇us,i
+�
+dV =6143 βγ
+120960 ,
+(S9)
+�
+V ∞
+�
+ui·
+�
+us,i·∇u∞+u∞·∇us,i
+�
+−uSt,i·
+�
+ustr,i·∇U ∞+U ∞·∇ustr,i
+��
+dV =37 βγ
+1260 ,
+(S10)
+�
+V P uSt,i ·
+�
+ustr,i · ∇U ∞ + U ∞ · ∇ustr,i
+�
+dV =− βγ
+4 .
+(S11)
+Substituting (S9-S11) in (S6), with β = 4(1 − 2s) and γ = −4, gives:
+Vp = Rep
+�
+F1(s, Rec) + λF2(s)
+�
+,
+(S12)
+with F2(s) = 1141(1−2s)
+216
+.
+
+5
+∗ sganesh@jncasr.ac.in
+[S1] Ho, B. P., & Leal, L. G. (1974). Journal of fluid mechanics, 65(2), 365-400.
+[S2] Anand, P., & Subramanian, G. (2022). To be submitted to JFM.
+[S3] Schonberg, J. A., & Hinch, E. J. (1989). Journal of Fluid Mechanics, 203, 517-524.
+[S4] Kim, S., & Karrila, S. J. (2013). Microhydrodynamics: principles and selected applications. Courier Corporation.
+

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+Pressure Reconstruction from the Measured Pressure Gradient
+Using Gaussian Process Regression
+Zejian You∗ and Qi Wang†
+San Diego State University, San Diego, California, 92182, USA
+Xiaofeng Liu‡
+San Diego State University, San Diego, California, 92182, USA
+Many numerical algorithms have been established to reconstruct pressure fields from mea-
+sured kinematic data with noise by Particle Image Velocimetry (PIV), such as the Pressure Pois-
+son solver and the Omni-Directional Integration (ODI) method. This study adopts Gaussian
+Process Regression (GPR), a probabilistic framework with an intrinsic de-noising mechanism
+to tackle drawbacks of traditional Pressure Poisson solver and compares the performance with
+ODI. To evaluate the accuracy of the algorithm, GPR and ODI are tested in detail in a canon-
+ical setup of forced homogeneous isotropic turbulence from the Johns Hopkins Turbulence
+Database. According to the result, GPR has the same level of accuracy as ODI with optimized
+hyper-parameters for the isotropic turbulence flow. However, GPR has the tendency to flatten
+impulsive signals. Therefore, without further modifications, it is not suitable to detect flow
+structures with impulsive true signals. The error propagation of the proposed framework is
+also analyzed and discussed in both physical and spectral spaces.
+Nomenclature
+𝑝(𝒙)
+=
+pressure field
+�𝑝(𝒙)
+=
+true pressure field
+𝒙
+=
+spatial coordinates
+𝒖
+=
+velocity field
+𝜌
+=
+density
+𝑡
+=
+time
+𝜇
+=
+dynamic viscosity
+𝑘
+=
+realization number
+¯𝑝(𝒙)
+=
+mean of the pressure field
+GP
+=
+Gaussian Process
+N
+=
+Gaussian distribution
+C
+=
+covariance matrix of the Gaussian process
+𝑙
+=
+correlation length in the radial basis function kernel
+𝑙𝑝
+=
+correlation length scale from the true pressure field
+𝜎(𝒙)
+=
+standard deviation of Gaussian Process at a spatial location 𝒙
+𝜎𝑝
+=
+standard deviation of the pressure field
+𝜎𝜖
+=
+standard deviation of assumed noise level in pressure gradient
+𝜎∇𝑝
+=
+standard deviation of embedded noise in pressure gradient
+𝑶
+=
+vector formed by observations of material derivatives
+𝑿∗, 𝑿
+=
+vector formed by observation locations and general spatial locations
+Σ𝑖 𝑗
+=
+Covariance matrices
+𝑅𝜆
+=
+Reynolds number based on Taylor micro scale
+𝑝𝐺𝑃𝑅, 𝑝𝑂𝐷𝐼
+=
+pressure field reconstructed by GPR or ODI
+∗Doctoral student, Department of Aerospace Engineering, San Diego State University, Student Member AIAA.
+†Assistant Professor, Department of Aerospace Engineering, San Diego State University, Member AIAA. E-mail: qwang4@sdsu.edu
+‡Associate Professor, Department of Aerospace Engineering, San Diego State University, Associate Fellow AIAA.
+1
+arXiv:2301.13282v1  [physics.flu-dyn]  30 Jan 2023
+
+I. Introduction
+A. Pressure field estimation from PIV
+U
+nderstanding the pressure field is essential in various turbulence and hydrodynamic researches. The generation
+of acoustic noise [1, 2] and the onset of boundary separation [3, 4], for example, are instances where an
+accurate estimation of the pressure is of paramount importance. However, non-intrusive measurements of the detailed
+instantaneous pressure field is a leading challenge in experimental studies. By applying Particle Image Velocimetry
+(PIV), the gradient information of pressure can be obtained from the balance of the Navier-Stokes equation, in which the
+material acceleration is the dominant term, while the viscous term is negligible at regions away from the wall for high
+Reynolds number flow [5],
+∇𝑝 = −𝜌 𝐷𝒖
+𝐷𝑡 + 𝜇∇2𝒖 ≈ −𝜌 𝐷𝒖
+𝐷𝑡 .
+(1)
+Numerical tools have been established to further reconstruct the instantaneous pressure field based on the measured
+pressure gradient. While the traditional approach solves the pressure-Poisson equation, which often utilizes an inaccurate
+boundary condition, the state-of-art tool, named Omni-directional Integration (ODI), integrates the pressure gradients
+in a collection of directions and averages the resulting field [5–8]. This has led to a robust de-noising framework
+that greatly mitigates the effect of measurement error [9]. A recent example shows that the time-averaged pressure
+reconstructed from Reynolds Averaged Navier-Stokes (RANS) equation based on stereo PIV measurements even reaches
+spatial resolution beyond that of PIV [10]. The goal of the current study is to extend the idea of denoising and establish
+a probabilistic framework using Gaussian Process Regression to reconstruct pressure from its gradient information with
+uncertainties.
+B. Gaussian process regression
+Gaussian process regression has been used to solve regression problem when the underlying function is unknown
+and hard to evaluate analytically. For example, apply Gaussian Process upper confidence bound sampling (GP-UCB)
+to provide movie recommendation[11]; reconstruct missing temporal and spatial sensor data of a dynamic nonlinear
+response [12, 13]; analyze motion trajectory of moving targets from sparse observations [14]. This non-parametric
+Bayesian approach has been proven to be very powerful in exploration and exploitation scenarios. Furthermore, GPR
+can capture a wide variety of relations between inputs and outputs by the kernel-encoded prior assumption. In this study,
+we reconstruct the pressure from pressure gradient by deliberately encoding the prior assumption with different kernel
+functions while previous works mainly focus on obtaining function from observations of the function itself.
+The rest of the paper is structured as follows: in §II we introduce the mathematical formulation for Gaussian
+Process Regression when applied to pressure field reconstruction. In §III we explain how we choose the optimal
+hyper-parameters for GPR and how we evaluate the performance of GPR and ODI through the forced homogeneous
+isotropic turbulence database from Johns Hopkins Turbulence Database (JHTDB). In §IV we show some comparison
+results and analyses of reconstructed pressure field by GPR and ODI. In §V, we conclude current results and propose
+some prospective future works.
+II. Mathematical formulation
+In the current study, we propose a new approach, adopting the idea of Gaussian Process Regression (GPR) [see 11,
+for a brief tutorial] This probability framework takes into account measurement noise and could help to perform field
+inversion from gradient information and mitigate the effect of measurement noise.
+In Gaussian process regression, we assume the observation of pressure gradient at any spatial location 𝒙 can be
+expressed by the true pressure gradient with an additional noise:
+∇𝑝(𝒙) = ∇�𝑝(𝒙) + 𝜖
+(2)
+where the noise term 𝜖 follows normal distribution
+𝜖 ∼ N (0, 𝜎2
+𝜖 ).
+(3)
+Furthermore, GPR regards the pressure field, 𝑝(𝒙) as a Gaussian process in infinite dimensional space, e.g.
+𝑝(𝒙) ∼ GP( ¯𝑝(𝒙), C(𝒙, 𝒙′)),
+(4)
+2
+
+Fig. 1
+(a): An example of one-dimensional GPR when observations of the values of a smooth function are
+available. Red dashed lines mark the location of observations while the black dashed lines are samples drawn
+from the posterior distribution. Two red solid lines mark the region within one standard deviation in the
+posterior distribution. (b): When observations of function derivatives are available, a similar approach can also
+be adopted to infer the values of the function.
+in which ¯𝑝(𝒙) represents the mean, or the expected value of the prior distribution for the pressure, ¯𝑝(𝒙) = E[𝑝(𝒙)],
+and 𝒙, 𝒙′ represent two different spatial locations. The kernel of the Gaussian process, C(𝒙, 𝒙′) represents the
+covariance of the uncertain pressure fields at two different spatial locations, 𝑝(𝒙) and 𝑝(𝒙′).
+Or equivalently,
+C(𝒙, 𝒙′) = E [(𝑝(𝒙) − ¯𝑝(𝒙))(𝑝(𝒙′) − ¯𝑝(𝒙′))].
+For a continuous and stationary random process, radial basis function kernel is popularly introduced. The Gaussian
+kernel, for example, reads
+C(𝒙, 𝒙′) = 𝜎(𝒙)𝜎(𝒙′) exp(−1
+2
+||𝒙 − 𝒙′||2
+𝑙2
+),
+(5)
+in which 𝜎(𝒙) is the standard deviation and 𝑙 is the correlation length. The standard deviation 𝜎(𝒙) will be updated
+during the inference and can naturally lead to uncertainty quantification (UQ) of the reconstruction. The correlation
+length 𝑙 is a hyper-parameter representing the smoothness of the pressure field, which can be tuned to achieve the best
+performance for different scenarios.
+The corresponding correlation function of the pressure field for the kernel function C(𝒙, 𝒙′) is defined as
+K(𝒙, 𝒙′) =
+C(𝒙, 𝒙′)
+𝜎(𝒙)𝜎(𝒙′)
+(6)
+where 𝜎𝑝 is the root mean square of entire pressure field. For radial basis function kernel, the correlation function
+K(𝒙, 𝒙′) is only function of distance 𝑟 between two point,
+K(𝒙, 𝒙′) = K(𝑟)
+(7)
+in which 𝑟 = ||𝒙 −𝒙′||. In order to better comprehend the physical meaning of pressure field, we introduce the correlation
+length scale 𝑙𝑝, which could be obtained by fitting the correlation function K(𝑟) of true pressure field into the correlation
+function of Gaussian kernel.
+While most applications of GPR deal with observing the values of the function directly [13], as shown in Figure
+1(𝑎), the formulation of GPR is general and can be applied to observations of the gradient information. An example of
+such reconstruction for a one-dimensional case is shown in Figure 1(𝑏), where the dashed lines marks the observation
+location for the gradient of the function 𝑝(𝑥). Given 𝑿∗ a vector collection of 𝑛 spatial locations 𝒙𝒏 where data are
+observed, the samples of the pressure gradient observations would follow a multivariate Gaussian distribution,
+𝑶 = ∇𝑝 (𝑿∗) ∼ N
+�
+∇ ¯𝑝
+��
+𝑿∗, ∇𝒙∇𝒙′C (𝑿∗, 𝑿∗) + 𝜎2
+𝜖 I∗
+�
+.
+(8)
+Here 𝜎𝜖 is the assumed noise level of synthetic noise introduced as a mimic of experimental data. I∗ is the identity
+matrix. Moreover, once the observations are drawn from the above distribution, the observations of pressure gradient
+at 𝑿∗ and unknown values of pressure field at 𝑿, a vector collection of spatial locations where reconstruction are
+3
+
+α)
+2
+1
+1
+p
+I
+1
+-1
+1
+1
+-
+-2
+2
+0.2
+0.2
+0.4
+0
+0.4
+0.6
+0.8
+0
+0.6
+0.8
+1
+1Fig. 2
+(a): True pressure field from an isotropic turbulence DNS database. (b)-(c): Pressure gradient obtained
+from the DNS pressure field by central finite difference method. (d)-(e): Sample realization of 1000 error
+embedded pressure gradient.
+conducted, would follow the joint Gaussian distribution,
+�������
+𝑶
+𝑝(𝑿)
+�������
+∼ N
+������
+�
+���������
+∇𝑝 (𝑿∗)
+¯𝑝(𝑿)
+���������
+,
+����������
+∇𝒙∇𝒙′C (𝑿∗, 𝑿∗) + 𝜎2
+𝜖 I∗
+��������������������������������������������������������
+Σ11
+, ∇𝒙′C (𝑿, 𝑿∗)
+��������������������������
+Σ12
+∇𝒙C (𝑿∗, 𝑿)
+������������������������
+Σ21
+, C (𝑿, 𝑿)
+��������������
+Σ22
+����������
+������
+�
+(9)
+The last piece of this joint Gaussian distribution is the reference pressure. Since the value of reference pressure does not
+influence the dynamic structure of reconstructed pressure fields, we include one additional observation of pressure
+𝑝(𝒙0) = 0 at location 𝒙0 in the formulation. Once noisy measurements for the pressure gradient become available, the
+pressure field can be recovered using Bayes’ theorem. And the posterior conditional distribution is given by,
+𝑝 (𝑿) ∼ N
+�
+¯𝑝 − Σ21Σ−1
+11
+�
+𝑶 − ∇ ¯𝑝
+��
+𝑿∗
+�
+, Σ22 − Σ21Σ−1
+11Σ12
+�
+.
+(10)
+The updated mean of the posterior distribution is regarded as the reconstructed pressure field from GPR,
+𝑝𝐺𝑃𝑅(𝑿) = ¯𝑝 − Σ21Σ−1
+11
+�
+𝑶 − ∇ ¯𝑝
+��
+𝑿∗
+�
+(11)
+Notice that the matrix inversion in the above expression requires O(𝑁3) operations, with 𝑁 being the number of
+observations. The inversion would therefore be computationally intractable for large 𝑁. Nevertheless, for large-scale
+computations, we could transform the matrix inversion into an iterative algorithm using the Conjugate Gradient method
+[15], which is part of future efforts.
+Moreover, the covariance matrix of the posterior probability distribution can be computed from,
+𝐶𝐺𝑃𝑅(𝑿, 𝑿) = Σ22 − Σ21Σ−1
+11Σ12.
+(12)
+Square roots of the diagonal elements of the above matrix are the standard deviation 𝜎𝑝(𝑿), representing the uncertainty
+of the reconstructed pressure fields at different spatial locations.
+III. Problem Setup
+A. Data acquisition
+As the first step, we test our algorithm in a homogeneous isotropic turbulence flow field with Reynolds number around
+𝑅𝜆 ∼ 433 based on Taylor microscale. Instead of using data from experiments, we extract data in the Johns Hopkins
+4
+
+a
+1.0
+0.8
+0.5
+0.6
+0.0
+0.4
+-0.5
+0.2
+-1.0
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+ab)
+c)
+0.8
+30
+0.6
+20
+0.4
+0.2
+10
+0.0
+0
+0.8
+-10
+0.6
+0.4
+-20
+0.2
+30
+0.0
+0.00
+0.25
+0.50
+0.75
+0.00
+0.25
+0.50
+0.75Turbulent database [16–18] from a direct numerical simulation (DNS) as a surrogate. The simulated data provides
+access to the fully resolved velocity and pressure fields, enabling us to test our algorithm thoroughly. Although the
+database is three-dimensional, the algorithm of both ODI and GPR is able to reconstruct pressure on a two-dimensional
+plane given observations of the in-plane components of the gradient vectors, as shown in Figure 2.
+The observations are the projection of material derivatives onto the 𝑥 − 𝑦 plane at certain observation locations
+extracted from the database, at four times the DNS grid spacing in 𝑥 and 𝑦 directions. A total number of 150 × 150
+observations of pressure gradient are obtained on a two-dimensional plane in the turbulent field on the 𝑥 − 𝑦 plane with
+𝐿𝑥 = 𝐿𝑦 = 0.9.
+In order to compare the performance of GPR and ODI, we adopted 1000 realizations of error-embedded pressure
+gradient generated by Liu and Moreto [9] and reconstruct the pressure field by ODI as well as GPR. The error-
+embedded pressure gradient is generated by adding random noise of uniform distribution with the magnitude as 40%
+of (|∇𝑝|DNS)max, the maximum magnitude of the true pressure gradient, at each point. The true pressure field, true
+pressure gradient as well as one sample realization of error embedded pressure gradient are shown in Figure 2.
+B. Evaluation of pressure reconstruction
+To quantify the accuracy of pressure reconstruction when subject to noise in the measurements, as often observed in
+PIV, we evaluate the cumulative error over 1000 realizations of noisy measurements. First, we calculate the error of the
+reconstructed pressure field by GPR and ODI by subtracting the true pressure field �𝑝(𝒙) at each point, i.e.,
+𝜖𝑖 𝑗 = 𝑝𝑖 𝑗 − �𝑝𝑖 𝑗,
+(13)
+where 𝑖, 𝑗 refer to Cartesian indices. Then, the standard deviation of error 𝜖std is defined as
+𝜖std =
+�
+�
+�
+�
+1
+𝑁𝑥 × 𝑁𝑦 − 1
+𝑁𝑦
+∑︁
+𝑗=1
+𝑁𝑥
+∑︁
+𝑖=1
+(𝜖𝑖 𝑗 − 𝜖𝑖 𝑗)2.
+(14)
+Finally, we calculate the averaged standard deviation over 1000 realizations, the cumulative error 𝜀std, as
+𝜀std = 1
+𝑘
+𝑘
+∑︁
+𝑛=1
+� 𝜖std
+𝑝std
+�
+𝑛
+.
+(15)
+C. Hyper-parameter Optimization
+The performance of both ODI and GPR methods depends on setting up proper parameters, or hyper-parameters.
+For Omni-directional Integration, we adopt the state-of-art formulation with Rotating Parallel Rays to create different
+integration paths [see 7, for details]. The parallel rays are rotated with an increment of 0.2 degrees, which creates a
+total number of 1800 different orientations of parallel integration paths over the entire domain of calculation. The
+optimal separation between adjacent parallel rays at 0.4 times of the grid size as suggested by [9] is adopted in the ODI
+calculation. For GPR, the prior distribution is set with mean ¯𝑝(𝒙) = 0 and variance 𝜎(𝒙) = 1. However, correlation
+length 𝑙 and assumed noise level 𝜎𝜖 still need to be optimized, which is going to be elaborated in the following sections.
+In order to optimize the performance of GPR, we calculate the averaged error of 10 realizations with different
+hyper-parameters sets and find the optimal hyper-parameters with the lowest error as the correlation length 𝑙 = 0.0625
+and the assumed noise level 𝜎𝜖 = 6. The error is slightly smaller with assumed noise level 𝜎𝜖 less than 6. However,
+because the influence of equivalent noise level is not very significant once it is smaller than 6, we choose 𝜎𝜖 = 6 as
+our optimal hyper-parameter. The correlation length scale 𝑙𝑝 could be obtained by fitting the Gaussian kernel into the
+correlation function of the true pressure field, which is equal to 0.0574 for the isotropic turbulence. So the correlation
+length scale 𝑙𝑝 = 0.0574. Embedded noise is a uniform distribution from -12 to 12, the standard deviation of which
+is 6.928. Thus the magnitude of embedded noise is 𝜎∇𝑝 = 6.918. We could find out that optimal hyper-parameters
+correlation length 𝑙 approximates to correlation length scale 𝑙𝑝 while assumed noise level 𝜎𝜖 approximates to the
+magnitude of embedded noise 𝜎∇𝑝. This phenomenon proves that the kernel of GPR depends on the correlation function
+of the true pressure field as well as the standard deviation of noise in the observations.
+5
+
+Fig. 3
+(a): Correlation function 𝐾(𝑟) of the true pressure field and the optimal Gaussian kernel function from
+curve-fitting. (b): The cumulative error of 10 realizations with different correlation length 𝑙 and assumed noise
+level 𝜎𝜖 as well as the location of optimal hyper-parameters.
+IV. Results
+A. Error Analysis in Physical Space
+The cumulative error of GPR and ODI on 150 by 150 grids as well as the cumulative error of ODI on 254 by
+254 grids are shown in Figure 4. From the result, we could observe that error of GPR converges to 0.153 over 1000
+realizations. Furthermore, GPR has a similar level of accuracy with ODI method on 150 by 150 grids domain. However,
+ODI on 254 by 254 grids can reach to lower cumulative error 0.148, the same as the result of previous work by Liu and
+Moreto [9]. But GPR requires more memory to conduct the matrix inversion in the regression. Therefore, for large-scale
+problems, it is necessary to convert matrix inversion into an iterative algorithm, such as the conjugate gradient iteration
+for better efficiency, which is part of future work.
+To further investigate the performance of GPR and ODI, Figure 5 shows the exact pressure field �𝑝, the pressure
+field reconstructed by GPR for the instance of the worst case scenario among the 1000 realizations, pressure field
+reconstructed by ODI for the instance of the worst case scenario among the 1000 realizations, the standard deviation of
+reconstructed pressure field by GPR, error of GPR result as well as error of ODI result of the worst performance case
+among 1000 realizations. In Figure 5, we could observe that the reconstructed pressure field by GPR is significantly
+smoother than ODI result and both reconstructed pressure fields are fairly accurate compared to the exact pressure
+field. Although the reconstructed pressure field by GPR has a smaller global error, it also has a larger local error
+compared to the reconstructed pressure field by ODI, indicating the tendency of GPR in flattening impulsive local
+pressure changes. Furthermore, the reconstructed pressure field by ODI preserves more fine structures (combined with
+noise and high-frequency pressure signal) while GPR seems to have a stronger denoising effect according to the smooth
+distribution.
+While the mean of the posterior distribution can be used to represent the reconstructed pressure field, we could also
+evaluate the accuracy of reconstruction based on the standard deviation of the posterior distribution. Notice that adding
+or subtracting a constant field on the pressure does not alter the agreement with the observation of the pressure gradient.
+This property, sometimes coined as “gauge invariance", has to be eliminated in order to obtain meaningful results for
+uncertainty analysis. Therefore, we assume that the reconstructed pressure at a given reference point, e.g. 𝒙0 at the
+center of the domain is solely due to such freedom of adding an arbitrary constant field. The variances of the pressure
+field at other locations further take into account the effect of 𝜎𝑝(𝒙0), which should be subtracted to obtain a reasonable
+estimation of the reconstruction uncertainty without the influence of such gauge invariance. For this reason, we here
+plot 𝜎𝑝(𝑿) − 𝜎𝑝(𝒙0) in Figure 5(d).
+In order to visualize how the error is propagated throughout the entire pressure field by different methods, we add a
+6
+
+a)
+b)
+×10-1
+1.0
+correlation function of p
+★
+optimal parameters
+4
+1.6
+fitted correlation function
+0.8
+1.4
+0.6
+1.2 
+2
+0.4
+三 1.0 +
+p$3
+6
+0.2 
+0.8
+1
+0.0
+0.6
+-0.2
+0.4 -
+0.0
+0.1
+0.2
+0.3
+0.4
+100
+5 × 100
+1/lpFig. 4
+Cumulative error 𝜀std and standard deviation (𝜖std)𝑘 of 1000 realizations with correlation length 𝑙 =
+0.0625 and assumed noise level 𝜎𝜖 = 6. The bottom histogram shows the standard deviation of error (𝜖std)𝑘 in
+the 𝑘-th realization. The blue bar represents the standard deviation of error by GPR and red line represents
+the standard deviation of error by ODI in 150 × 150 grids. Three converged lines on the top show the cumulative
+error of GPR and ODI in 150 × 150 and 254 × 254 grids.
+unit impulse in 𝜕𝑝
+𝜕𝑥 at the middle of the computational domain and use GPR and ODI to reconstruct the pressure field,
+respectively. Differences between the reconstructed pressure field with and without the unit impulse are then visualized
+in Figure 6 to quantify the domain of influence for such point-wise perturbation. Such approaches have been adopted in
+previous research to study the domain of influence or the domain of dependence to fully understand the forward and
+backward propagation of perturbations in a dynamical system or data assimilation algorithm [19, 20].
+The results of this impulse response have profound implications. First of all, the ODI method indicates a clear
+singularity point at the location of perturbation, manifested by very large positive and negative values. The implication
+here is that although ODI averages the error across the whole computational domain, the reconstructed pressure still
+relies heavily on the local pressure gradient information near the point of interest. In other words, most of the local error
+remains local, and correspondingly, the error diffusion is relatively not strong in comparison with that of GPR. On the
+other hand, the GPR method exhibits nearly zero influence at the point of perturbation and has a larger influence slightly
+farther away. This difference could explain the stronger de-noising effect in GPR than in ODI.
+B. Error Analysis in Wave Number Space
+The different behavior of GPR and ODI in Figure 5 leads to a comparison of power spectrum density of the
+reconstructed pressure field by GPR and ODI, as shown in Figure 7. From Figure 7(a), we could see that both methods
+perform well on low wave number space: the power spectrum density of GPR and ODI coincides with the power
+spectrum density of true pressure field perfectly. However, in high wave number space, the power spectrum density
+of different pressure field seem to diverge: GPR has a lower power spectrum density while ODI has a larger power
+spectrum density compared to the power spectrum density of true pressure field. This phenomenon validates previous
+observations in the pressure field shown in Figure 5, as well as the impulse response in Figure 6. GPR has a stronger
+denoising effect but also smooths out some pressure information in high wave number space while ODI preserves
+more fine structures. Some of them are dynamic behavior of the pressure field, others are noise in the observation.
+Furthermore, Figure 7(b) clearly indicates that if the pressure gradient information is accurate, ODI can faithfully
+replicate the dynamic behavior of the pressure field over the entire spectral range. However, in contrast, because
+7
+
+0.50
+0.20
+GPR (150×150)
+0.45
+ODI (254×254)
+ODI (150×150)
+0.18
+0.40
+0.35
+0.16
+Estd
+p4s
+0.30
+0.14
+0.25
+0.20
+0.12
+0.15
+0.10
+0.10
+0
+200
+400
+600
+800
+1000
+Realization kFig. 5
+(a): True pressure field from isotropic turbulence DNS database. (b): Instant of realization of recon-
+structed pressure field by GPR with the largest standard deviation of error 𝜖std. (c): Instant of realization of
+reconstructed pressure field by ODI with the largest standard deviation of error 𝜖std. (d): Standard deviation
+of reconstructed pressure field by GPR, 𝜎(𝑿) subtracted by the standard deviation of reference point 𝜎(𝒙0).
+(e): Error distribution of reconstructed pressure field by GPR, obtained by subtracting (a) from (b). (f): Error
+distribution of reconstructed pressure field by ODI, obtained by subtracting (a) from (c).
+Fig. 6
+Error Propagation of GPR and ODI, represented by the change of reconstructed pressure field when
+perturbation in the form of a unit impulse of 𝜕𝑝𝜕𝑥 is added at the center of the computational domain.
+8
+
+a)
+b)
+c)
+1.0
+0.8
+0.5
+0.6
+Pressure
+0.0
+0.4-
+-0.5
+0.2
+-1.0
+0.0 -
+a
+d)
+e)
+X10-4
+2
+0.8
+4
+1
+2
+0.6
+E
+Error
+0
+0
+0.4
+-2
+0.2
+-1
+-4
+0.0 -
+-2
+0.25
+0.50
+0.75
+0.25
+0.50
+0.75
+0.00
+0.00
+0.00
+0.25
+0.50
+0.75ODI
+GPR
+×10-4
+0.8
+2
+1
+0.6
+0
+0.4
+-1
+0.2
+-2
+0.0 -
+0.00
+0.25
+0.50
+0.75
+0.00
+0.25
+0.50
+0.75
+a
+aFig. 7
+(a): Power spectrum density of true pressure field as well as reconstructed pressure field by GPR and
+ODI from error embedded pressure gradients of 150 by 150 grids. (b): Power spectrum density of true pressure
+field as well as reconstructed pressure field by GPR and ODI from true pressure gradients of 150 by 150 grids
+the hyperparameters used in this GPR computation were optimized with the error-embedded data (therefore are flow
+dependent), GPR produces a reconstructed pressure spectrum with an overall lower fluctuation amplitude over the
+entire spectral domain. This indicates that the optimization of GPR needs to be adjusted according to the actual flow
+properties. This problem might be solved by switching a proper kernel function rather than a Gaussian kernel in this
+case since its denoising effect is too strong to preserve necessary information, which requires future work.
+V. Conclusion and future work
+We adopt the framework of Gaussian Process Regression (GPR) to the problem of determining the pressure fields
+from measured pressure gradients, with the potential of reconstructing pressure from sparsely measured data. The
+formulation naturally avoids the burden of solving Poisson equation with inaccurate boundary conditions and takes into
+account the effect of measurement noise. Furthermore, this framework provides more possibilities to improvement.
+From the comparison between the reconstructed pressure field by GPR and ODI, we are able to conclude that the
+reconstructed pressure field by GPR achieves accuracy comparable to the state-of-art Omni-directional integration
+method (ODI) and has a stronger denoising effect compared to ODI. However, pressure reconstruction by GPR might
+also smooth out some pressure information in high wave number space by mistake, especially dealing with accurate
+pressure gradient data. This problem might be able to be solved by switching the Gaussian kernel, which requires
+further investigation.
+In future directions, the following will be studied in detail: (a) improvement of computational efficiency for
+large-scale problems. (b) different kernel functions for the Gaussian Process. (c) effect of the sparseness of the
+observation in terms of reconstruction quality.
+VI. Acknowledgments
+The support from San Diego State University is gratefully acknowledged. Thanks to Jose Moreto for his assistance in
+the usage of the parallel-ray Omni-directional integration code and for providing the error-embedded pressure gradients
+data in previous study by Liu and Moreto [9].
+References
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+9
+
+a)
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+real pressure
+TI
+GPR
+GPR
+T
+ODI
+ODI
+10-1
+10-1,
+88
+@
+更
+x
+10-3,
+10-3,
+10-4
+10-4 ↓
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+10-1
+100
+10-2
+10-1
+100
+rn
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+10
+

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+SHUNIT: Style Harmonization for Unpaired Image-to-Image Translation
+Seokbeom Song1, Suhyeon Lee1, Hongje Seong1, Kyoungwon Min2, and Euntai Kim1*
+1Yonsei University, Seoul, Korea
+2Korea Electronics Technology Institute, Seongnam, Korea
+{lgs5751, hyeon93, hjseong, etkim}@yonsei.ac.kr, minkw@keti.re.kr
+light
+red light
+red light
+red light
+light
+red light
+red light
+red light
+light
+light
+light
+light
+input image
+translated 
+image
+I2I translation
+annotation
+: source domain
+: target domain
+: source-to-target
+style mapping
+Red, blue, and purple:
+styles from different
+categories
+zoom-in
+zoom-in
+zoom-in
+target domain images
+source domain images
+(a) Global I2I
+(b) Class-level I2I
+(c) Style harmonization I2I
+Figure 1: Illustration of the concepts in unpaired I2I translation. The results are obtained on Cityscapes → ACDC (night) setting.
+In the image, many head and tail lamps should be bright at night. (a) Global I2I (Huang et al. 2018b) converts all classes to
+bright because it translates the image with a single source-to-target style mapping function. (b) Class-level I2I (Jeong et al.
+2021) leverages additional annotations to address the problem of (a) and performs per-class source-to-target style mapping. It
+can effectively deal with multiple classes in an image, but loses the original style: All white lights and red lights become white
+lights. (c) Style harmonization I2I also performs class-wise style mapping, while adaptively preserving the original styles.
+Abstract
+We propose a novel solution for unpaired image-to-image
+(I2I) translation. To translate complex images with a wide
+range of objects to a different domain, recent approaches of-
+ten use the object annotations to perform per-class source-to-
+target style mapping. However, there remains a point for us to
+exploit in the I2I. An object in each class consists of multiple
+components, and all the sub-object components have differ-
+ent characteristics. For example, a car in CAR class consists
+of a car body, tires, windows and head and tail lamps, etc.,
+and they should be handled separately for realistic I2I trans-
+lation. The simplest solution to the problem will be to use
+more detailed annotations with sub-object component annota-
+tions than the simple object annotations, but it is not possible.
+The key idea of this paper is to bypass the sub-object com-
+*Corresponding authors.
+ponent annotations by leveraging the original style of the in-
+put image because the original style will include the informa-
+tion about the characteristics of the sub-object components.
+Specifically, for each pixel, we use not only the per-class
+style gap between the source and target domains but also the
+pixel’s original style to determine the target style of a pixel.
+To this end, we present Style Harmonization for unpaired I2I
+translation (SHUNIT). Our SHUNIT generates a new style
+by harmonizing the target domain style retrieved from a class
+memory and an original source image style. Instead of direct
+source-to-target style mapping, we aim for source and target
+styles harmonization. We validate our method with extensive
+experiments and achieve state-of-the-art performance on the
+latest benchmark sets. The source code is available online:
+https://github.com/bluejangbaljang/SHUNIT.
+arXiv:2301.04685v1  [cs.CV]  11 Jan 2023
+
+Introduction
+Unpaired image-to-image (I2I) translation aims to learn
+source-to-target style mapping, where source and target im-
+ages are unpaired. It can be applied to data augmenta-
+tion (Antoniou, Storkey, and Edwards 2017; Mariani et al.
+2018; Huang et al. 2018a; Xie et al. 2020), domain adapta-
+tion (Hoffman et al. 2018; Murez et al. 2018) and various im-
+age editing applications, such as style transfer (Gatys, Ecker,
+and Bethge 2016; Huang and Belongie 2017; Ulyanov,
+Vedaldi, and Lempitsky 2017), colorization (Zhang, Isola,
+and Efros 2016; Zhang et al. 2017), and image inpaint-
+ing (Iizuka, Simo-Serra, and Ishikawa 2017; Pathak et al.
+2016).
+In the I2I translation, the biggest problem is how to deal
+with the style variations among objects or classes. In other
+words, when a global style gap is applied to an entire im-
+age as in Fig. 1a, the I2I translation often results in unreal-
+istic images because each class has different the style gaps
+between the source domain and target domain. Recent ad-
+vanced methods (Shen et al. 2019; Bhattacharjee et al. 2020;
+Jeong et al. 2021; Kim et al. 2022) addressed the problem
+by leveraging additional object annotations. They simplify
+the task into class-level I2I translation and then perform per-
+class source-to-target style mapping. This enables the net-
+works to explicitly estimate class-wise target styles, but it
+has a critical limitation: An object in each class consists
+of multiple components, and all the sub-object components
+might also have different characteristics. Let us consider the
+example given in Fig. 1b.
+Understandably, when a road image taken on a sunny day
+is translated into a night image, head and tail lamps in a car
+should be brighter while the rest of the components, such as
+car body, tires, and windows, should be darker than before.
+Therefore, each component in a car should be handled sep-
+arately for realistic I2I translation. However, if the previous
+approaches are applied to perform per-class source-to-target
+style mapping, they will translate all head and tail lamps into
+white lights, making unrealistic images, as shown in Fig. 1b.
+Here, one might think that this issue can be addressed by an-
+notating more detailed sub-object components than the sim-
+ple object categories, but it is actually impossible. A brief
+example is as follows. A car consists of body, window, and
+tires. A tire consists of wheel and gum. In this way, sub-
+object components can be divided endlessly.
+To solve the above limitation of the previous class-level
+I2I methods, we present Style Harmonization for unpaired
+I2I translation (SHUNIT). The key idea of SHUNIT is to by-
+pass the sub-object component annotations by leveraging the
+original style of the input image because the original style
+will include the information about the characteristics of the
+sub-object components. Thus, instead of mapping source-
+to-target style directly, SHUNIT harmonizes the source and
+target styles to realize realistic and practical I2I translation.
+As illustrated in Fig. 1c, SHUNIT uses not only the per-
+class style gap between the source and target domains but
+also the pixel’s original style to determine the target style
+of a pixel. To achieve this, we disentangle the target style
+into class-aware memory style and image-specific compo-
+nent style. The class-aware memory style is stored in a style
+memory, and image-specific component style is taken from
+the original input image.
+The goal of the style memory is to obtain class-wise
+source-to-target style gaps and it is motivated by (Jeong et al.
+2021). Compared to the memory in (Jeong et al. 2021), our
+style memory differs in two aspects. First, the output from
+the style memory was used alone as a target style in (Jeong
+et al. 2021), but the output from the memory is adaptively
+aggregated (=harmonized) in this paper with the style of
+the original input image to make a target style. Second, the
+memory was simply updated in (Jeong et al. 2021), whereas
+our style memory is jointly trained and optimized with the
+other parts of SHUNIT. Specifically, the class-aware mem-
+ory in (Jeong et al. 2021) was not trained but simply was
+updated using the input features during the training, memo-
+rizing the style features from the target domain. The gradi-
+ent was not propagated to the style memory. Thus, the style
+memory in (Jeong et al. 2021) cannot update their param-
+eters based on the error of memory. In SHUNIT, however,
+we overcome this problem by enabling the memory to learn
+through backpropagation. To this end, we train the style
+memory from randomly initialized parameters and introduce
+style contrastive loss to constrain the memory to learn class-
+wise style representations. The backpropagation forces the
+style memory to reduce the final loss jointly and effectively
+along with the other parts of SHUNIT. To demonstrate the
+superiority of our SHUNIT, we conduct extensive experi-
+ments on the latest benchmark sets and achieve state-of-the-
+art performance.
+Overall, the contributions of our work are summarized as
+follows:
+• We present a novel challenge in I2I translation: an object
+might have various styles.
+• We propose a new I2I method, style harmonization, that
+leverages two distinct styles: class-aware memory style
+and image-specific component style. To the best of our
+knowledge, the style harmonization is the first method
+to estimate the target style in multiple perspectives for
+unpaired I2I translation.
+• We achieve new state-of-the-art performance on latest
+benchmarks and provide extensive experimental results
+with analysis.
+Related Work
+Image-to-image translation.
+The goal of I2I is to
+learn source-to-target style mapping. For I2I translation,
+pix2pix (Isola et al. 2017) proposes a general solution us-
+ing conditional generative adversarial networks (Mirza and
+Osindero 2014). However, it has a significant limitation:
+paired training data should be used for training networks.
+CycleGAN (Zhu et al. 2017) successfully addresses this
+problem with a cycle consistency loss. The loss allows us to
+train the networks with unpaired training data by supervis-
+ing the reconstructed original image only. Based on Cycle-
+GAN, many approaches (Kim et al. 2017; Choi et al. 2018)
+have been proposed to tackle I2I translation take in an un-
+paired manner. UNIT (Liu, Breuel, and Kautz 2017) pro-
+poses another unpaired I2I translation solution by mapping
+
+𝑳𝒙
+𝐸��
+𝐸��
+𝑰𝒙
+𝒄𝒙
+𝒔𝒙
+Read
+𝐺�
+𝑰�𝒚
+Learnable Style Memory 
+𝑘
+𝑣
+class 1
+class 2
+class N
+𝑴𝒚
+𝒔�𝒚
+SH ResBlk
+SH ResBlk
+SH ResBlk
+ReLU
+SHL
+Conv
+𝒔�𝒚
+𝒔𝒙
+ReLU
+SHL
+Conv
+𝒔�𝒚
+𝒔���
+(a) Overall architecture
+(b) SH ResBlk
+Figure 2: An overview of SHUNIT. (a) From a pair of image Ix and label Lx in source domain X, the two encoders (i.e.,
+Ex
+c and Ex
+s ) extract the content cx and component style sx, respectively. The content retrieves the memory style ˆsy. The three
+features cx, sx, and ˆsy are fed to the target generator Gy, which consists of several Style Harmonization Residual Blocks (SH
+ResBlk). The generator Gy outputs the translated image ˆIy. (b) The input of the SH ResBlk is the content cx for the first layer,
+and the output of the previous block is used as input for the remainders. Each SH ResBlk includes two Style Harmonization
+Layers (SHL) that transfer target styles, i.e., sx and ˆsy.
+two images in different domains into the same latent code
+in a shared-latent space. MUNIT (Huang et al. 2018b) and
+DRIT (Lee et al. 2018) introduce a disentangled representa-
+tion to achieve diverse and multi-modal I2I translation from
+unpaired data. Basically, they perform global I2I translation
+which focuses on mapping a global style on all pixels in an
+image. Although they work well on object-centric images,
+they bring severe artifacts for complex images, such as mul-
+tiple objects being presented or large domain gap scenarios,
+as illustrated in Fig. 1a. To complement the problem, recent
+approaches leverage additional object annotations and per-
+form class-level I2I translation.
+Class-level image-to-image translation.
+Recent several
+approaches (Mo, Cho, and Shin 2019; Shen et al. 2019;
+Bhattacharjee et al. 2020; Jeong et al. 2021; Kim et al.
+2022) propose class-level image-to-image translation so-
+lutions with object annotations. Specifically, INIT (Shen
+et al. 2019) generates instance-wise target domain images.
+DUNIT (Bhattacharjee et al. 2020) additionally employs
+an object detection network and jointly trains it with the
+I2I translation network. MGUIT (Jeong et al. 2021) pro-
+poses an approach to store and read class-wise style repre-
+sentations with key-value memory networks (Miller et al.
+2016). This approach, however, cannot directly supervise
+the memory with objective functions for I2I translation. In-
+staformer (Kim et al. 2022) proposes a transformer-based
+(Vaswani et al. 2017; Dosovitskiy et al. 2021) architec-
+ture that mixes instance-aware content and style represen-
+tations. The existing methods that leverage object annota-
+tions learns the direct class-wise source-to-target style map-
+ping, as shown in Fig. 1b. This effectively simplifies the I2I
+translation problem into per-class I2I translation, but they
+overlook an important point that not all pixels in the same
+class should be translated with the same style. Our approach,
+style harmonization, addresses this problem by introducing
+the component style that facilitates preserving the original
+style of the source image, as illustrated in Fig. 1c.
+Proposed Method
+Definition and Overview
+Let X and Y be the visual source and target domains, respec-
+tively. Given an image and the corresponding label (=bound-
+ing box or segmentation mask) in X domain, our frame-
+work generates a new image in Y domain while remain-
+ing the semantic information in the given image. We assume
+that each domain consists of images and labels denoted by
+(Ix, Lx) ∈ X and (Iy, Ly) ∈ Y, and both domains have the
+same set of N classes. Our framework contains the source
+encoder Ex = {Ex
+c , Ex
+s }, target generator Gy, and target
+style memory M y for source-to-target mapping, and the tar-
+get encoder Ey = {Ey
+c , Ey
+s }, source generator Gx, source
+style memory M x for target-to-source mapping. For conve-
+nience, we will only describe the source-to-target direction,
+and the overview of our framework is depicted in Fig. 2.
+Following the previous studies (Huang et al. 2018b; Lee
+et al. 2018), we assume that an image can be disentangled
+into domain-invariant content and domain-specific style. For
+this, we basically follow the MUNIT (Huang et al. 2018b)
+architecture. The content encoder Ex
+c consists of several
+strided convolutional layers and residual blocks (He et al.
+2016), and all the convolutional layers are followed by
+Instance Normalization (Ulyanov, Vedaldi, and Lempitsky
+2016). The content encoder extracts the domain-invariant
+content feature cx from the image Ix and label Lx. The
+style encoder Ex
+s also consists of several strided convolu-
+tional layers and residual blocks, and it extracts the compo-
+
+𝒄𝒙
+𝒔�𝒚
+class-wise separation
+𝑣�
+𝑘�
+𝑑
+softmax
+class 1
+𝑣�
+𝑘�
+class 2
+𝑣�
+𝑘�
+class N
+•••
+𝑐�
+�
+𝑐�
+�
+𝑐�
+�
+𝑠̂�
+�
+𝑠̂�
+�
+𝑠̂�
+�
+Figure 3: Read operation in style memory. The content
+feature cx is separated by the label Lx. For each class, the
+memory read is independently performed. After class-wise
+memory read, the memory style ˆsy is obtained by gathering
+the retrieved class-wise values into the original locations. �
+denotes matrix multiplication.
+nent style feature sx from the image Ix. The memory style
+ˆsy is read by retrieving from the learnable style memory M y
+to the content feature cx. The generator Gy consists of sev-
+eral style harmonization layers and residual blocks, and it
+produces the translated image ˆIy from cx, sx, and ˆsy.
+Style Harmonization for Unpaired Image-to-Image
+Translation (SHUNIT)
+The important point of SHUNIT is that two styles are em-
+ployed to determine the target style: One is image-specific
+component style, and the other one is class-aware memory
+style. We focus on extracting two distinct styles accurately
+and then harmonizing them. In what follows, we describe
+the detail of each step.
+Component style.
+The style encoder Ex
+s takes the im-
+age Ix as input and extracts the style feature sx of size
+H × W × C, where H, W, and C are the height, width, and
+number of channels of the feature, respectively. Here, sx ex-
+plicitly represents the style of the input image, thus we use
+it as image-specific component style. The component style
+is used together with the memory style in the style harmo-
+nization layer to reduce the artifacts of the generated target
+image.
+Memory style.
+The component style is not sufficient to
+handle complex scenes with multiple objects. Therefore, we
+exploit the class-specific style that leverages an object anno-
+tation. To this end, we construct the class-wise style mem-
+ory retrieved by the content feature. The target style memory
+M y consists of N class memories to store class-wise style
+representations of the target domain Y. Each class memory
+M y
+n has U key-value pairs (ky, vy) (Jeong et al. 2021), con-
+sidering that various styles exist in one class (e.g., different
+styles of headlamp and tires in CAR class). The key ky is
+used for matching with the content feature and the value vy
+has the class-aware style representations. The key and value
+are learnable vectors, each one of size 1 × 1 × C.
+Fig. 3 shows a detailed implementation of the process of
+reading the corresponding memory style ˆsy from the mem-
+𝒔𝒙
+𝜷𝒙
+𝜸𝒙
+𝒔�𝒚
+𝜷𝒚
+𝜸𝒚 
+𝛼� 
+1 − 𝛼� 
+Conv
+Conv
+Conv
+Conv
+𝜷
+𝜸
+input
+output
+•••
+class-wise 𝛼 
+broadcasting
+norm
+Figure 4: Style harmonization layer. From the component
+style sx and memory style ˆsy, scale (γx and γy) and shift (βx
+and βy) factors are extracted via four convolutional layers.
+The alpha mask ˆα is obtained by broadcasting the class-wise
+alpha α with the semantic label Lx. We weighted sum the
+image and memory styles with the alpha mask, and transfer
+it to the input feature. ⊙ denotes Hadamard product.
+ory M y. With the semantic label, we separate the content
+feature cx into {cx
+1, cx
+2, · · · , cx
+N}, where cx
+n denotes source
+content feature for the n-th class. Let cx
+n,i be the i-th pixel
+of the n-th class source content feature and (ky
+n,j, vy
+n,j) be
+the j-th key-value pair of the n-th class target style memory
+M y
+n. In this work, we aim to read the target memory style
+ˆsy
+n,i corresponding to the source content cx
+n,i using the simi-
+larity between the source content and key of target memory.
+To this end, we calculate the similarity wx
+n,i,j between cx
+n,i
+and ky
+n,j as:
+wx
+n,i,j =
+exp
+�
+d(cx
+n,i, ky
+n,j)
+�
+�U
+u=1 exp
+�
+d(cx
+n,i, ky
+n,u)
+�
+(1)
+where d(·, ·) is the cosine similarity. We then read the
+memory style ˆsy
+n,i corresponding to cx
+n,i by calculating the
+weighted sum of values in n-th class style memory:
+ˆsy
+n,i =
+U
+�
+j=1
+wx
+n,i,jvy
+n,j.
+(2)
+The same process is applied for content features of other
+classes, finally extracting spatially varying target memory
+style features ˆsy of size H × W × C.
+Different from the previous key-value memory net-
+works (Jeong et al. 2021) that learn the memory via updating
+mechanism, we learn the memory through backpropagation.
+The updating mechanism is used to directly store the ex-
+ternal input features. However, it has a critical drawback:
+The memory cannot be trained with the network jointly with
+the same objective function because the gradient should be
+stopped at the updated memory. To solve the problem, we
+discard the update mechanism and learn the memory with
+the loss functions presented in Eq. (7). The effectiveness of
+our memory learning strategy is validated in the experiments
+section.
+Style harmonization layer.
+Our goal is harmonizing the
+source and target styles instead of mapping source-to-target
+
+style directly. To this end, we propose the style harmoniza-
+tion layer to adaptively aggregate the component style and
+memory style. The style harmonization layer consists of sev-
+eral convolution layers and class-wise alpha parameters, and
+it is illustrated in Fig. 4. Here we use three conditional in-
+puts: memory style, component style, and label. Convolu-
+tional layers are used to compute pixel-wise scale γ and shift
+β factors from the two styles. Following (Jiang et al. 2020;
+Park et al. 2019; Zhu et al. 2020; Ling et al. 2021), we trans-
+fer the harmonized target style by scaling and shifting the
+normalized input feature with the computed factors (i.e., γ
+and β). In the layer, we additionally set class-wise alpha pa-
+rameters (α1, α2, · · · , αN). It is used to decide which style
+has more influence for each class in the generated images.
+If the alpha value is large, the component style sx has more
+influence, and vice versa.
+Let f, of size H × W × C, be the input feature of the
+current style harmonization layer in the generator Gy. With
+the style harmonizing scale γ and shift β factors, the feature
+is denormalized by
+γc,h,w
+(fc,h,w − µc)
+σc
++ βc,h,w
+(3)
+where µc and σc are the mean and standard deviation of the
+input feature f at the channel c, respectively. The modula-
+tion parameters γc,h,w and βc,h,w are obtained from γx
+c,h,w,
+γy
+c,h,w, βx
+c,h,w, and βy
+c,h,w, which are the scale and shift fac-
+tors of the component style sx and memory style, sy respec-
+tively, and they are computed by
+γc,h,w = ˆαh,wγx
+c,h,w + (1 − ˆαh,w)γy
+c,h,w,
+βc,h,w = ˆαh,wβx
+c,h,w + (1 − ˆαh,w)βy
+c,h,w,
+(4)
+where ˆα denotes the alpha mask. It is obtained by broadcast-
+ing the class-wise alpha parameters to their corresponding
+semantic regions of the label Lx. We experimentally demon-
+strate that our style harmonization layer adaptively controls
+the style of each object well, and the results are given in the
+experiments section.
+Loss Functions
+We leverage standard loss functions used in MUNIT (Huang
+et al. 2018b) to generate proper target domain images. It in-
+cludes self-reconstruction Lself (Zhu et al. 2017), cycle con-
+sistency Lcycle (Zhu et al. 2017), perceptual Lperc (Johnson,
+Alahi, and Fei-Fei 2016) and adversarial loss Ladv (Good-
+fellow et al. 2014). The detailed explanations of those loss
+functions are given in the supplementary material.
+In this paper, we propose two advanced loss functions to
+facilitate style harmonization: content contrastive loss and
+style contrastive loss. It is used with the aforementioned
+standard loss functions jointly. In what follows, we intro-
+duce the proposed two loss functions.
+Content contrastive loss.
+To extract domain invariant
+content features from the content encoder, MUNIT (Huang
+et al. 2018b) simply reduces the L1 distances between cx and
+ˆcy. We replace this with contrastive representation learning
+to improve discrimination within a class. For a content fea-
+ture ˆcy
+i at pixel i, which is the content feature extracted from
+translated target image, we set the positive sample to cx
+i and
+we set the remaining features at the other pixels as negative
+samples. The content contrastive loss is defined with the a
+form of InfoNCE (Oord, Li, and Vinyals 2018) as:
+Lcontent = −
+HW
+�
+i=1
+log
+�
+exp((cx
+i · ˆcy
+i )/τ)
+�HW
+j=1 exp((cx
+j · ˆcy
+i )/τ)
+�
+(5)
+where τ is a temperature parameter. In this equation, the
+features in the same class at the pixel i can be considered
+as negative samples. This encourages the content encoder
+to extract more diverse style representations from the style
+memory within the same class. It is also applied to the target-
+to-source pipeline with ˆcx and cy
+Style constrative loss.
+We propose the style contrastive
+loss to allow the style memory to learn class-wise style rep-
+resentations. Similar to the content contrastive loss, for a
+style feature ˆsx
+i at pixel i, which is the memory style of
+target-to-source mapping, we set the positive sample to the
+source component style sx
+i and we set the remaining features
+at the other pixels as negative samples. The style constrative
+loss is defined as follows:
+Lstyle = −
+HW
+�
+i=1
+log
+�
+exp((sx
+i · ˆsx
+i )/τ)
+�HW
+j=1 exp((sx
+j · ˆsx
+i )/τ)
+�
+(6)
+This loss directly supervises the memory style ˆsx for the
+translated target image. This improves the stability of cycle
+consistency learning for reconstructing Ix from ˆIy. It is also
+applied to the target styles, i.e., ˆsy and sy
+Finally, all loss functions are summarized as follows:
+min
+(Ex,Ey,Gx,Gy) max
+(Dx,Dy)L(Ex, Ey, Gx, Gy, Dx, Dy) =
+λselfLself + λcycleLcycle + λpercLperc
++λadvLadv + λcontentLcontent + λstyleLstyle
+(7)
+where Dx and Dy denote the multi-scale discrimina-
+tors (Wang et al. 2018) for each visual domain, X and Y.
+The details of Lself, Lcycle, Lperc, and Ladv are described
+in the supplementary material.
+Experiments
+In this section, we present extensive experimental results and
+analysis. To demonstrate the superiority of our method, we
+compare our SHUNIT with state-of-the-art I2I translation
+methods. The implementation details of our method are pro-
+vided in the supplementary material.
+Datasets
+We evaluate our SHUNIT on three I2I translation scenar-
+ios: Cityscapes (Cordts et al. 2016) → ACDC (Sakaridis,
+Dai, and Van Gool 2021) and INIT (Shen et al. 2019),
+and KITTI (Geiger et al. 2013) → Cityscapes (Cordts
+et al. 2016). In all scenarios, INIT (Shen et al. 2019),
+DUNIT (Bhattacharjee et al. 2020), MGUIT (Jeong et al.
+2021), InstaFormer (Kim et al. 2022), and our method use
+semantic labels provided in each dataset.
+
+Input
+CycleGAN
+MUNIT
+MGUIT
+SHUNIT (ours)
+Target domain
+Figure 5: Qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night). From the given clear image (first
+column), we generate four adverse condition images using (Zhu et al. 2017; Huang et al. 2018b; Jeong et al. 2021) and SHUNIT.
+In the last column, we show a sample of the real image for each adverse condition.
+clear → snow
+clear → rain
+clear → fog
+clear → night
+cFID ↓
+mIoU ↑
+cFID ↓
+mIoU ↑
+cFID ↓
+mIoU ↑
+cFID ↓
+mIoU ↑
+CycleGAN (Zhu et al. 2017)
+21.88
+23.68
+20.16
+35.96
+31.72
+13.73
+15.26
+31.33
+UNIT (Liu, Breuel, and Kautz 2017)
+13.89
+31.24
+16.25
+39.39
+29.36
+28.70
+12.28
+35.29
+MUNIT (Huang et al. 2018b)
+13.79
+33.83
+12.62
+44.20
+29.34
+27.44
+12.56
+37.43
+TSIT (Jiang et al. 2020)
+10.47
+38.08
+14.16
+46.40
+25.16
+36.68
+11.62
+35.92
+MGUIT (Jeong et al. 2021)
+8.75
+33.33
+10.76
+42.60
+24.36
+10.22
+15.83
+31.36
+SHUNIT (ours)
+6.62
+45.15
+8.47
+48.84
+6.53
+38.96
+14.08
+33.66
+Table 1: Quantitative comparison on Cityscapes → ACDC. We measure class-wise FID (lower is better) and mIoU (higher
+is better). For brevity, class-wise FID is written as cFID.
+clear → snow
+clear → rain
+clear → fog
+clear → night
+AdaptSegNet (Tsai et al. 2018)
+35.3
+49.0
+31.8
+29.7
+ADVENT (Vu et al. 2019)
+32.1
+44.3
+32.9
+31.7
+BDL (Li, Yuan, and Vasconcelos 2019)
+36.4
+49.7
+37.7
+33.8
+CLAN (Luo et al. 2019)
+37.7
+44.0
+39.0
+31.6
+FDA (Yang and Soatto 2020)
+46.9
+53.3
+39.5
+37.1
+SIM (Wang et al. 2020)
+33.3
+44.5
+36.6
+28.0
+MRNet (Zheng and Yang 2021)
+38.7
+45.4
+38.8
+27.9
+SHUNIT (ours)
+45.2
+48.8
+39.0
+33.7
+Table 2: Quantitative Comparison on domain adaptation for semantic segmentation. We report mIoU for Cityscapes →
+ACDC.
+Cityscapes → ACDC
+Cityscapes (Cordts et al. 2016)
+is one of the most popular urban scene dataset. ACDC
+(Sakaridis, Dai, and Van Gool 2021) is the latest dataset
+with multiple adverse condition images and consists of
+four conditions of street scenes: snow, rain, fog, and night.
+ACDC dataset provides images with corresponding dense
+pixel-level semantic annotations, and it has 19 classes the
+same as Cityscapes dataset for all adverse conditions. Fol-
+lowing (Sakaridis, Dai, and Van Gool 2021), we leverage
+Cityscapes dataset as a clear condition and translate it to the
+adverse conditions (i.e., snow, rain, fog, and night) in ACDC
+dataset. Therefore, this scenario is challenging because not
+only the weather conditions, but also layouts, such as cam-
+era model, view, and angle, are different. To train the net-
+works, 2975, 400, 400, 400, and 400 images are used for
+clear, snow, rain, fog, and night conditions, respectively. For
+a fair comparison on this benchmark, we reproduce existing
+state-of-the-art methods (Zhu et al. 2017; Liu, Breuel, and
+Kautz 2017; Huang et al. 2018b; Jiang et al. 2020; Jeong
+et al. 2021) in our system. For fair comparison, we set the
+number of key-value pairs for style memory to be the same
+as our setting and use segmentation mask for reproducing
+
+国Input
+CycleGAN
+UNIT
+MUNIT
+DRIT
+MGUIT
+InstaFormer
+SHUNIT (ours)
+Figure 6: Qualitative comparison on INIT dataset. (Top to bottom) sunny→night, night→sunny, cloudy→sunny results. Our
+method preserves object details and looks more realistic.
+sunny → night
+night → sunny
+sunny → rainy
+sunny → cloudy
+cloudy → sunny
+Average
+CIS ↑
+IS ↑
+CIS ↑
+IS ↑
+CIS ↑
+IS ↑
+CIS ↑
+IS ↑
+CIS ↑
+IS ↑
+CIS ↑
+IS ↑
+CycleGAN (Zhu et al. 2017)
+0.014
+1.026
+0.012
+1.023
+0.011
+1.073
+0.014
+1.097
+0.090
+1.033
+0.025
+1.057
+UNIT (Liu, Breuel, and Kautz 2017)
+0.082
+1.030
+0.027
+1.024
+0.097
+1.075
+0.081
+1.134
+0.219
+1.046
+0.087
+1.055
+MUNIT (Huang et al. 2018b)
+1.159
+1.278
+1.036
+1.051
+1.012
+1.146
+1.008
+1.095
+1.026
+1.321
+1.032
+1.166
+DRIT (Lee et al. 2018)
+1.058
+1.224
+1.024
+1.099
+1.007
+1.207
+1.025
+1.104
+1.046
+1.321
+1.031
+1.164
+INIT (Shen et al. 2019)
+1.060
+1.118
+1.045
+1.080
+1.036
+1.152
+1.040
+1.142
+1.016
+1.460
+1.043
+1.179
+DUNIT (Bhattacharjee et al. 2020)
+1.166
+1.259
+1.083
+1.108
+1.029
+1.225
+1.033
+1.149
+1.077
+1.472
+1.079
+1.223
+MGUIT (Jeong et al. 2021)
+1.176
+1.271
+1.115
+1.130
+1.092
+1.213
+1.052
+1.218
+1.136
+1.489
+1.112
+1.254
+Instaformer (Kim et al. 2022)
+1.200
+1.404
+1.115
+1.127
+1.158
+1.394
+1.130
+1.257
+1.141
+1.585
+1.149
+1.353
+SHUNIT (ours)
+1.205
+1.503
+1.308
+1.585
+1.136
+1.609
+1.111
+1.405
+1.085
+1.315
+1.169
+1.483
+Table 3: Quantitative Comparison on INIT dataset. We measure CIS and IS (higher is better).
+(Jeong et al. 2021).
+INIT
+INIT (Shen et al. 2019) is a public benchmark set for
+I2I translation. It contains street scenes images including 4
+weather categories (i.e., sunny, night, rainy, and cloudy) with
+the corresponding bounding box labels. Following (Shen
+et al. 2019), we split the 155K images into 85% for train-
+ing and 15% for testing. We conduct five translation exper-
+iments: sunny ↔ night, sunny ↔ cloudy, sunny → rainy.
+In this dataset, we directly copied the results of the existing
+methods from (Shen et al. 2019; Bhattacharjee et al. 2020;
+Jeong et al. 2021; Kim et al. 2022). Similarly, for fair com-
+parison with MGUIT, the number of key-value pairs in style
+memory is set equally.
+KITTI → Cityscapes
+KITTI is a public benchmark set
+for object detection. It contains 7481 images with bounding
+boxes annotations for training and 7518 images for testing.
+Following the previous I2I translation methods (Bhattachar-
+jee et al. 2020; Jeong et al. 2021; Kim et al. 2022), we select
+the common 4 object classes (person, car, truck, bicycle) for
+evaluatation.
+Qualitative Comparison
+Fig. 5 shows qualitative results on Cityscapes → ACDC.
+Since our I2I translation setting, Cityscapes → ACDC, is
+very challenging as discussed in the datasets section, ex-
+isting methods cannot generate realistic images in several
+scenarios. Specifically, CycleGAN (Zhu et al. 2017) often
+destroys the semantic layout. MUNIT (Huang et al. 2018b)
+translates images with a global style, thus it also often gen-
+erates artifacts, as shown in the snow, rain, and fog images.
+MGUIT (Jeong et al. 2021) also includes artifacts in the car
+even though leveraging memory style. It shows the limita-
+tion of the updating mechanism for training memory style,
+and the limitation is clearly depicted in the challenging sce-
+nario. In contrast to them, our SHUNIT accurately generates
+images in the target domains without losing the original style
+in the input image. In the supplementary material, we further
+provide the results of UNIT (Liu, Breuel, and Kautz 2017)
+and TSIT (Jiang et al. 2020).
+As shown in Fig. 6, which depicts qualitative results on
+INIT dataset, our method generates high-quality images in
+various scenarios. In the night → sunny scenario (second
+row), InstaFormer (Kim et al. 2022) translates the color of
+the road lane to yellow. On the other hand, our method keeps
+the color of the lane as white and generates a sunny scene by
+harmonizing the target domain style retrieved from a style
+memory and an image style.
+Quantitative Comparison
+The quantitative results on Cityscapes → ACDC are pre-
+sented in Table 1. To quantify the per-class image-to-image
+translation quality, we measure class-wise FID (Shim et al.
+2022). We further measure mIoU on ACDC test set. The
+
+Pers.
+Car
+Truc.
+Bic.
+mAP
+DT (Inoue et al. 2018)
+28.5
+40.7
+25.9
+29.7
+31.2
+DAF (Huang et al. 2018b)
+39.2
+40.2
+25.7
+48.9
+38.5
+DARL (Kim et al. 2019)
+46.4
+58.7
+27.0
+49.1
+45.3
+DAOD (Rodriguez and Mikolajczyk 2019)
+47.3
+59.1
+28.3
+49.6
+46.1
+DUNIT (Bhattacharjee et al. 2020)
+60.7
+65.1
+32.7
+57.7
+54.1
+MGUIT (Jeong et al. 2021)
+58.3
+68.2
+33.4
+58.4
+54.6
+InstaFormer (Kim et al. 2022)
+61.8
+69.5
+35.3
+55.3
+55.5
+SHUNIT (ours)
+56.3
+74.4
+51.9
+53.2
+59.0
+Table 4: Quantitative Comparison on domain adaptation for object detection. We report per-class AP for KITTI →
+Cityscapes.
+clear → snow
+clear → rain
+Mem. Comp.
+α
+cFID ↓ mIoU ↑ cFID ↓ mIoU ↑
+✓
+12.93
+26.33
+7.83
+39.06
+✓
+✓
+8.72
+37.20
+8.47
+38.97
+✓
+✓
+✓
+6.62
+44.35
+8.47
+42.77
+(a) Style ablation study on SHL.
+clear → snow
+clear → rain
+Lcontent
+Lstyle
+cFID ↓ mIoU ↑ cFID ↓ mIoU ↑
+✓
+11.06
+32.43
+13.23
+38.39
+✓
+12.38
+30.89
+9.33
+39.47
+✓
+✓
+6.62
+44.35
+8.47
+42.77
+(b) Ablation study on loss functions.
+clear → snow
+clear → rain
+cFID ↓
+mIoU ↑
+cFID ↓
+mIoU ↑
+Updating
+10.55
+36.70
+12.05
+38.21
+Backprop.
+6.62
+44.35
+8.47
+42.77
+(c) Style memory training strategies.
+clear → snow
+clear → rain
+cFID ↓
+mIoU ↑
+cFID ↓
+mIoU ↑
+L1
+12.21
+38.98
+7.72
+39.78
+Contrastive
+6.62
+44.35
+8.47
+42.77
+(d) L1 vs. Contrastive loss for style and content
+losses.
+clear → snow
+clear → rain
+cFID ↓ mIoU ↑
+cFID ↓ mIoU ↑
+w/o
+6.73
+44.11
+9.03
+38.72
+w/
+6.62
+44.35
+8.47
+42.77
+(e) Experimental results of label input for
+content encoder.
+Table 5: Ablation Study. We report class-wise FID and mIoU in two scenarios: clear → {snow, rain}.
+mIoU metric is used to validate the results on the practical
+problem, semantic segmentation. We generate a training set
+by Cityscapes → ACDC and then train DeepLabV2 (Chen
+et al. 2017) on it. The mIoU score is obtained with the
+trained DeepLabV2 by evaluating on ACDC test set. As
+shown in Table 1, we surpass the state-of-the-art I2I trans-
+lation methods by a significant margin in most scenarios,
+demonstrating the superiority of our style harmonization for
+unpaired I2I translation. Table 2 shows the quantitative re-
+sults of domain adaptation for semantic segmentation with
+the state-of-the-art methods. Despite we trained DeepLabV2
+with only a simple cross-entropy loss, our SHUNIT achieves
+comparable performance with the domain adaptation meth-
+ods that were trained DeepLabV2 with additional loss func-
+tions and several techniques for boosting the performance of
+mIoU score on the target domain.
+Table 3 shows another quantitative results on the testing
+split of INIT (Shen et al. 2019). To directly compare our
+method with the public results, we evaluate our method with
+Inception Score (IS) (Salimans et al. 2016) and Conditional
+Inception Score (CIS) (Huang et al. 2018b). As shown in
+Table 3, we achieve the best performance in most scenarios.
+We further evaluate our method on domain adaptation
+benchrmark following DUNIT (Bhattacharjee et al. 2020).
+We use Faster-RCNN (Ren et al. 2015) trained on the source
+domain as a detector. As shown in Table 4, we achieve the
+state-of-the-art performance.
+Ablation Study
+In this section, we study the effectiveness of each component
+in our method. We validate on Cityscapes (clear) → ACDC
+(snow/rain) scenarios and use ACDC validation set for both
+class-wise FID and mIoU1.
+Ablation study on style harmonization layer.
+We ablate
+the memory style, component style, and class-wise α in the
+style harmonization layer, and they are denoted as “Mem.”,
+“Comp.”, and “α” in Table 5a, respectively. As shown in the
+table, the memory style-only is far behind the full model.
+With component style, we can achieve performance im-
+provement on clear → snow while decreasing on clear →
+rain. We obtain significant improvement on most scenarios
+with class-wise α. The results demonstrate that the exist-
+ing approach, which only leverages the memory style, is not
+sufficient for I2I translation, and we successfully address the
+problem by adaptively harmonizing two styles.
+Ablation study on content and style losses.
+We study the
+effectiveness of the proposed two loss functions, Lstyle and
+Lcontent, by ablating them step-by-step, and the results are
+1mIoU on test set should be evaluated on the online server
+(Sakaridis, Dai, and Van Gool 2021) and it has a limit on the num-
+ber of submissions. Therefore, we use validation set for ablation
+study.
+
+given in Table 5b. As shown in the table, our model is effec-
+tive when two losses are used jointly.
+Style memory training strategy.
+As described in the pro-
+posed method section, we opt for backpropagation to train
+the style memory rather than updating the mechanism used
+in (Jeong et al. 2021). The results are shown in Table 5c. We
+surpass the existing updating method by a large margin.
+L1 vs. Contrastive loss.
+Table 5d shows the efficacy of our
+contrastive-based approach by replacing Lcontent and Lstyle
+with L1 losses as used in MUNIT (Huang et al. 2018b). The
+L1 losses are designed to reduce L1 distances within posi-
+tive pairs without consideration of negative pairs. As we dis-
+cussed in loss functions section, our method effectively en-
+courages extracting more diverse style representations, lead-
+ing to performance improvement.
+Label input for content encoder.
+Table 5e shows that la-
+bel input is not significant but always leads to performance
+improvements. Therefore, we have no reason to omit the la-
+bel input.
+Limitations
+Since our framework leverages the component style of the
+source image, the generated image’s quality relies on the
+source image’s quality. If the source image has a very bright
+colors, SHUNIT often generates a relatively bright night im-
+age. In Fig. 5, SHUNIT struggles to generate geometrically
+distinct lights from the source image. Due to the above prob-
+lems, SHUNIT cannot achieve the best performance on clear
+→ night scenario in Table. 1. We believe that these problems
+can be alleviated by leveraging geometric information such
+as depth or camera pose. Additionally, our approach can give
+limited benefit to some I2I translation scenarios, such as dog
+→ cat, because these tasks need to change the content; how-
+ever, we tackle the unpaired I2I translation task under the
+condition that the content will not be changed, and only the
+style will be changed.
+Conclusion
+We present a new perspective of the target style: It can
+be disentangled into class-aware and image-specific styles.
+Furthermore, our SHUNIT effectively harmonizes the two
+styles, and its superiority is demonstrated through extensive
+experiments. We believe that our proposal has the potential
+to break new ground in style-based image editing applica-
+tions such as style transfer, colorization, and image inpaint-
+ing.
+Acknowledgement
+This work was supported by Institute of Information &
+communications Technology Planning & Evaluation (IITP)
+grant funded by the Korea government (MSIT) (No.2021-
+0-00800, Development of Driving Environment Data Trans-
+formation and Data Verification Technology for the Mutual
+Utilization of Self-driving Learning Data for Different Vehi-
+cles).
+References
+Antoniou, A.; Storkey, A.; and Edwards, H. 2017. Data aug-
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+arXiv:1711.04340.
+Bhattacharjee, D.; Kim, S.; Vizier, G.; and Salzmann, M.
+2020. Dunit: Detection-based unsupervised image-to-image
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+
+Implementation Details
+Experiments Settings
+We implement our model with 1.7.1 version of Py-
+Torch (Paszke et al. 2017) framework. To train SHUNIT,
+we use a fixed learning rate of 10−4 and use the Adam op-
+timizer (Kingma and Ba 2015) with β1 and β2 of 0.5 and
+0.999, respectively. The weight decay is set to 10−4. Our
+models are trained with a single NVIDIA RTX A6000 GPU.
+For the experiments on Cityscapes (Cordts et al. 2016) →
+ACDC (Sakaridis, Dai, and Van Gool 2021), we use RGB
+images with a size of 512 × 256 during both training and
+inference. We train the model for 50K iterations. Since the
+ACDC dataset has been published recently, no official re-
+sults are available on this dataset for the existing meth-
+ods. Therefore, we reproduced CycleGAN (Zhu et al. 2017),
+UNIT (Liu, Breuel, and Kautz 2017), MUNIT (Huang et al.
+2018b), TSIT (Jiang et al. 2020), and MGUIT (Jeong et al.
+2021) with their official implementation code1, 2, 3, 4.
+For the experiments on INIT (Shen et al. 2019), we use
+RGB images with a size of 360 × 360 during training while
+we use RGB images with a size of 572 × 360 during infer-
+ence. We train the model for 250K iterations. For the ex-
+periment on KITTI (Geiger et al. 2013) → Cityscapes, we
+use RGB images with a size of 540 × 360 during training
+while we use RGB images with a size of 1242 × 375 during
+inference. We train the model for 200K iterations.
+For fair comparisons with INIT (Shen et al. 2019),
+DUNIT (Bhattacharjee et al. 2020), MGUIT (Jeong et al.
+2021), and InstaFormer (Kim et al. 2022), which used
+bounding-box labels as an additional object annotation,
+we did not use segmentation labels but used bounding-
+box labels for all experiments on both INIT and KITTI
+→ Cityscapes scenarios. In addition, we reproduced
+MGUIT (Jeong et al. 2021) on Cityscapes → ACDC exper-
+iments with segmentation labels.
+Details of Network Architecture
+In the content encoder Ex
+c , we employ two convolutional
+networks, one network takes RGB input, and the other takes
+one-hot encoded semantic label input. Each network con-
+sists of Conv(input channel, 64, 7, 1, 3, IN, relu) - Conv(64,
+128, 4, 2, 1, IN, relu) - Conv(128, 256, 4, 2, 1, IN, relu) -
+Resblk ×4 - Conv(256, 128, 1, 1, IN, relu), where IN de-
+notes the instance normalization layer (Ulyanov, Vedaldi,
+and Lempitsky 2016); Resblk denotes the residual block (He
+et al. 2016); and the components in Conv(·) denotes (input
+channel, output channel, kernel size, stride, padding, nor-
+malization, activation). The encoded RGB and semantic la-
+bel features are then concatenated along the channel dimen-
+sion, and it is the content feature cx. The style encoder Ex
+s
+1The code of CycleGAN was taken from
+https://github.com/junyanz/pytorch-CycleGAN-and-pix2pix
+2The code of UNIT and MUNIT were taken from
+https://github.com/NVlabs/MUNIT
+3The code of TSIT was taken from
+https://github.com/EndlessSora/TSIT
+4The code of MGUIT was taken from
+https://github.com/somijeong/MGUIT
+has the same network architecture as a convolutional net-
+work in the content encoder, except for the normalization
+layer. The style encoder does not use normalization layers
+to keep the original style of the input image. The generator
+Gy consists of SH Resblk ×4 - Conv(256, 128, 5, 1, 2, IN,
+relu) - Conv(128, 64, 5, 1, 2, IN, relu) - Conv(64, 3, 7, 1, 3, -,
+tanh). For the experiments on Cityscapes → ACDC, we use
+20 key-value pairs for each class in the style memory M y
+n.
+In the rest of the experiments, we follow key-value pairs of
+MGUIT (Jeong et al. 2021).
+Details of Loss Functions
+Following MUNIT (Huang et al. 2018b), we leverage self-
+reconstruction Lself (Zhu et al. 2017), cycle consistency
+Lcycle (Zhu et al. 2017), perceptual Lperc (Johnson, Alahi,
+and Fei-Fei 2016), and adversarial Ladv (Goodfellow et al.
+2014) losses as follows:
+Lself = ∥Ix − Gx(cx, sx)∥1 + ∥Iy − Gy(cy, sy)∥1 , (8)
+Lcycle = ∥Ix − Gx(ˆcy, ˆsx)∥1 + ∥Iy − Gy(ˆcx, ˆsy)∥1 , (9)
+Lperc =
+���F(Ix) − F( ˆIy)
+���
+1 +
+���F(Iy) − F( ˆIx)
+���
+1 , (10)
+Ladv =
+�
+log (1 − Dx(ˆIx)) + log Dx(Ix)
+�
++
+�
+log (1 − Dy(ˆIy)) + log Dy(Iy)
+�
+,
+(11)
+where F(·) is a feature map extracted from relu5 3
+layer of the ImageNet pretrained VGG-16 network (Si-
+monyan and Zisserman 2014); Dx(·) and Dy(·) are
+features
+extracted
+from
+the
+multi-scale
+discrimina-
+tors (Wang et al. 2018) for each visual domain, X
+and Y, respectively. We empirically set the hyperpa-
+rameters
+{λself, λcycle, λperc, λadv, λcontent, λstyle}
+to
+{10, 10, 1, 1, 10, 10}. The temperature τ in Lcontent and
+Lstyle is set to 0.7.
+Analysis of class-wise FID
+Recent work (Shim et al. 2022) introduced class-wise FID
+(cFID) to quantify the per-class image quality in the image
+generation task. We realized that the I2I translation task also
+has advantages if we evaluate the results with cFID. In this
+section, we provide a detailed analysis of the necessity for
+cFID on our benchmark set and the implementation detail of
+cFID for I2I translation.
+Necessity of CFID.
+By the nature of unsupervised I2I, as
+shown in Fig. 7, the class statistics between the source do-
+main (Cityscapes (Cordts et al. 2016)) and the target do-
+main (ACDC (Sakaridis, Dai, and Van Gool 2021)) are un-
+matched. However, FID ignores the unmatched class statis-
+tics because FID only uses the global embeddings to com-
+pute the distance. Therefore, the different class statistics pre-
+vent FID improvement, regardless of the generated image
+quality.
+
+Figure 7: The class statistics in Cityscapes (Cordts et al. 2016) validation set vs. ACDC (Sakaridis, Dai, and Van Gool
+2021) validation set. Even though the two datasets share the class categories, the class statistics are naturally different.
+To demonstrate the problem of FID intuitively, we il-
+lustrate an example in Fig. 8. In the figure, we translate
+a clear image sampled from Cityscapes dataset to snow
+with three methods: (a) CycleGAN (Zhu et al. 2017), (b)
+MGUIT (Jeong et al. 2021), and (c) SHUNIT. As can be
+compared qualitatively, SHUNIT generates a more realis-
+tic snow image than CycleGAN. For a quantitatively com-
+parison, we take a similar strategy used in FID: We extract
+global embeddings from a real snow image sampled from
+ACDC dataset and the generated image using Inception-
+V3 (Szegedy et al. 2016). Then, we compute the squared Eu-
+clidean distance between the two global embeddings. How-
+ever, as depicted in Fig. 8, CycleGAN (267.47) achieves bet-
+ter performance than SHUNIT (295.45) in global distance.
+The reason is that CycleGAN destroys the layout in the input
+image and follows the layout of the real snow image, result-
+ing in severe artifacts in the generated image while better
+performance in the global distance. If we extract class-wise
+embeddings and compute class-wise distance then average
+them, we can obtain a reasonable rank: SHUNIT (6.64)
+achieves the best, MGUIT (10.61) achieves the second-best,
+and CycleGAN (31.07) achieves the third. As can be ob-
+served in this example, comparing class-wise embeddings is
+a more reasonable method to evaluate the generated qual-
+ity than comparing global embeddings in unsupervised I2I
+translation.
+To further demonstrate the necessity of cFID, we plot the
+pixel-wise features of Inception-V3 (Szegedy et al. 2016) in
+Fig. 9. In Figs. 9(a), 9(b), and 9(c), Cityscapes validation
+set is used as clear images for I2I translation. In Fig. 9(d),
+ACDC validation set is used as real snow images. In the
+figure, we use the same color for the same category of the
+pixel-wise feature. As shown in the zoom-in regions, pixel-
+wise features extracted from (d) real snow images are well
+separated by class categories because the pretrained network
+is used. However, (a) CycleGAN includes various categories
+in the zoom-in region, which clearly indicates that the qual-
+ity of the generated images is low. In contrast, (c) features in
+SHUNIT are well separated by class categories as in (d) real
+snow images. This class information should be considered
+to evaluate the quality of the generated images.
+However, as shown in Table 6, CycleGAN achieves the
+best performance of FID in clear → snow scenario. There-
+fore, FID score is not reliable in unsupervised I2I transla-
+tion with multiple classes. In this paper, we simply and ef-
+fectively address the problem by computing class-wise dis-
+tances separately.
+Implementation detail of cFID for I2I translation.
+We
+measure cFID based on the official implementation of
+FID (Seitzer 2020). cFID is calculated with the Inception-
+V3 (Szegedy et al. 2016) features. We extract the features
+from the first block of the network to obtain fine-scale fea-
+tures, which effectively include features of small objects. To
+extract class-wise embeddings, we upsample the features to
+their original input size by bilinear interpolation, and then
+semantic region pooling is applied to the features. Here, we
+use the ground truth of the semantic segmentation for the
+semantic region pooling. With the class-wise embeddings,
+we compute FID for each class separately, and then cFID is
+obtained by averaging them.
+Stability of SHUNIT
+To show the stability of our model, we train SHUNIT five
+times with different random seeds on Cityscapes (clear) →
+ACDC (snow, rain), and the results are given in Table 7.
+As shown in the table, the performance gap between the
+best and worst results is reasonably small. In addition, the
+worst result also achieves the state-of-the-art performance
+over previous works. This result demonstrates that our re-
+sults for quantitative comparison are not cherry-picked and
+actually lead to performance improvements. Furthermore,
+the result promises that our strong performance can be easily
+reproduced.
+
+Semantics statistics
+Cityscapes val
+ACDC val
+0.30
+0.25
+semantic ratio
+0.20
+0.15
+0.10
+0.05
+0.00
+0
+15
+16
+17
+1819I2I
+Global distance: 267.47
+Class-wise distance: 31.07
+Global distance: 357.54
+Class-wise distance: 10.61
+Global distance: 295.45
+Class-wise distance: 6.64
+(a) CycleGAN (Zhu et al. 2017)
+Input
+(b) MGUIT (Jeong et al. 2021)
+(d) Target domain
+(c) SHUNIT (ours)
+Figure 8: Illustration of comparison of global distance and class-wise distance. From a clear image, we generate snow
+images using (a) CycleGAN (Zhu et al. 2017), (b) MGUIT (Jeong et al. 2021), and (c) SHUNIT. To quantitatively evaluate the
+results, we compute distances between (a,b,c) the generated images and (d) real snow image in two methods: One is computed
+with global embeddings of Inception-V3 and the other one is computed with class-wise embeddings.
+clear → snow
+clear → rain
+clear → fog
+clear → night
+FID ↓ CFID ↓ mIoU ↑ FID ↓ CFID ↓ mIoU ↑ FID ↓ CFID ↓ mIoU ↑ FID ↓ CFID ↓ mIoU ↑
+CycleGAN (Zhu et al. 2017)
+134.21 21.88
+23.68 160.91 20.16
+35.96 154.23 31.72
+13.73 105.40 15.26
+31.33
+UNIT (Liu, Breuel, and Kautz 2017) 136.63 13.89
+31.24 145.79 16.25
+39.39 190.47 29.36
+28.70 116.39 12.28
+35.29
+MUNIT (Huang et al. 2018b)
+136.91 13.79
+33.83 151.48 12.62
+44.20 195.28 29.34
+27.44 114.94 12.56
+37.43
+TSIT (Jiang et al. 2020)
+162.19 10.47
+38.08 158.12 14.16
+46.40 197.62 25.16
+36.68 127.09 11.62
+35.92
+MGUIT (Jeong et al. 2021)
+166.76
+8.75
+33.33 178.59 10.76
+42.60 190.86 24.36
+10.22 119.11 15.83
+31.36
+SHUNIT (ours)
+143.21
+6.62
+45.15 158.96
+8.47
+48.84 153.19
+6.53
+38.96 125.80 14.08
+33.66
+Table 6: Quantitative comparison on Cityscapes → ACDC. We measure FID, CFID (lower is better) and mIoU (higher is
+better).
+More Qualitative Results
+Cityscapes → ACDC
+We show more qualitative results on Cityscapes (clear) →
+ACDC (snow/rain/fog/night) in Figs 10 to 13. In the results,
+we additionally show the results of UNIT (Liu, Breuel, and
+Kautz 2017) and TSIT (Jiang et al. 2020), which are omitted
+in the main paper. The qualitative results demonstrate that
+our method generate more realistic images than the previ-
+ous works (Zhu et al. 2017; Liu, Breuel, and Kautz 2017;
+Huang et al. 2018b; Jiang et al. 2020; Jeong et al. 2021) in
+most cases. We further provide a comparison video in the
+supplementary material.
+KITTI → Cityscapes
+Fig. 14 shows the visual comparison of InstaFormer (Kim
+et al. 2022) and our method for domain adaptive detection on
+KITTI → Cityscapes. As shown in the figure, InstaFormer
+generates artifacts in the sky and struggles to maintain the
+sub-object components of a small object, such a car located
+in the center. In contrast, our method translates the sky to
+the Cityscapes style without artifacts, and preserves the sub-
+objects components of a small object. The more realistic
+image translation is demonstrated by a detection result: the
+Faster-RCNN (Ren et al. 2015) detects the small car located
+in the center in our image (Fig. 14c) while cannot detect it
+in InstaFormer image (Fig. 14b)
+More Quantitative Results
+Tables 8 to 11 provide the per-class IoU performance on
+ACDC test set. As shown in the tables, we achieve the best
+performance not only in mIoU but also in per-class IoU of
+the most classes on snow, rain, and fog.
+
+zoom-in
+zoom-in
+zoom-in
+zoom-in
+zoom-in
+zoom-in
+zoom-in
+zoom-in
+(a) CycleGAN (Zhu et al. 2017)
+(b) MGUIT (Jeong et al. 2021)
+(c) SHUNIT (ours)
+(d) Target domain
+Figure 9: t-SNE (Van der Maaten and Hinton 2008) visualization. We plot pixel-wise features extracted from Inception-
+V3 (Szegedy et al. 2016). In (a), (b), and (c), the generated snow images, which are translated from clear images, are used for
+extracting features, while in (d), the real snow images in the ACDC validation set are used. In the figure, the same color of dots
+are in the same class category.
+clear → snow
+clear → rain
+cFID ↓
+mIoU ↑
+cFID ↓
+mIoU ↑
+CycleGAN (Zhu et al. 2017)
+21.88
+23.68
+20.16
+35.96
+UNIT (Liu, Breuel, and Kautz 2017)
+13.89
+31.24
+16.25
+39.39
+MUNIT (Huang et al. 2018b)
+13.79
+33.83
+12.62
+44.20
+TSIT (Jiang et al. 2020)
+10.47
+38.08
+14.16
+46.40
+MGUIT (Jeong et al. 2021)
+8.75
+33.33
+10.76
+42.60
+SHUNIT (trial 1, reported in the main paper)
+6.62
+45.15
+8.47
+48.84
+SHUNIT (trial 2)
+7.18
+44.71
+9.53
+49.14
+SHUNIT (trial 3)
+6.41
+45.94
+8.44
+48.01
+SHUNIT (trial 4)
+7.56
+38.41
+8.34
+48.67
+SHUNIT (trial 5)
+6.86
+45.41
+8.56
+48.70
+Table 7: Results of five trials on Cityscapes → ACDC. We
+measure class-wise FID (lower is better) and mIoU (higher
+is better). In our result, the best results are bold-faced while
+the worst results are red-colored. For brevity, class-wise FID
+is written as cFID.
+
+oo
+1
+.
+:Methods
+road
+sidewalk
+building
+wall
+fence
+pole
+traffic light
+traffic sign
+vegetation
+terrain
+sky
+person
+rider
+car
+truck
+bus
+train
+motorcycle
+bicycle
+mIoU
+CycleGAN (Zhu et al. 2017)
+54.55 12.74 41.24 8.42 9.71 11.78 23.77 28.34 38.86 1.68 2.07 17.85 17.20 62.00 6.05 34.29 50.93 8.94 19.57 23.68
+UNIT (Liu, Breuel, and Kautz 2017) 58.88 16.27 43.10 9.74 14.35 17.47 38.47 39.71 47.97 6.98 4.17 41.25 30.11 73.35 20.49 36.20 60.74 6.75 27.48 31.24
+MUNIT (Huang et al. 2018b)
+65.09 21.06 44.42 13.53 18.20 19.04 41.24 40.68 51.58 7.94 5.85 41.21 27.10 77.44 24.08 31.01 60.19 18.93 34.14 33.83
+TSIT (Jiang et al. 2020)
+79.32 42.91 47.03 13.01 18.48 21.82 45.05 42.71 55.98 10.05 7.55 47.79 39.11 78.97 19.21 49.75 58.25 19.55 27.07 38.08
+MGUIT (Jeong et al. 2021)
+69.14 31.58 49.37 7.13 6.84 18.31 28.06 32.98 64.92 9.20 51.08 32.05 37.06 74.26 4.57 30.31 60.42 6.81 19.15 33.33
+SHUNIT(ours)
+81.62 46.12 52.33 16.03 27.34 25.07 49.22 43.19 80.08 9.41 54.61 52.95 36.23 77.66 20.22 48.18 69.04 26.47 43.12 45.15
+Table 8: Per-class IoU results on Cityscapes (clear) → ACDC (snow).
+Methods
+road
+sidewalk
+building
+wall
+fence
+pole
+traffic light
+traffic sign
+vegetation
+terrain
+sky
+person
+rider
+car
+truck
+bus
+train
+motorcycle
+bicycle
+mIoU
+CycleGAN (Zhu et al. 2017)
+73.51 31.73 51.75 20.75 18.66 20.99 24.80 26.39 71.58 24.46 28.91 34.06 11.75 74.66 37.32 23.02 52.42 16.18 40.19 35.96
+UNIT (Liu, Breuel, and Kautz 2017) 74.20 37.16 58.67 24.76 17.98 23.51 38.65 44.07 58.10 20.11 18.63 40.51 15.17 78.89 41.53 40.16 57.62 18.57 40.20 39.39
+MUNIT (Huang et al. 2018b)
+77.68 40.79 71.64 20.81 27.55 26.40 44.48 43.77 67.15 24.84 55.84 42.86 15.69 79.60 41.18 38.71 57.21 20.00 43.53 44.20
+TSIT (Jiang et al. 2020)
+78.08 46.43 71.28 29.44 25.12 29.23 44.95 46.64 78.14 24.68 67.75 44.24 16.37 80.94 46.28 31.09 57.90 25.87 41.74 46.40
+MGUIT (Jeong et al. 2021)
+73.69 31.94 80.06 19.54 14.44 24.74 37.17 34.91 79.99 21.61 93.74 34.46 13.60 76.69 27.54 38.64 60.43 16.32 31.04 42.60
+SHUNIT(ours)
+76.41 28.42 82.03 23.44 28.12 28.81 46.15 46.78 86.54 36.26 94.36 41.39 12.79 80.74 46.90 25.71 58.57 29.93 45.09 48.84
+Table 9: Per-class IoU results on Cityscapes (clear) → ACDC (rain).
+Methods
+road
+sidewalk
+building
+wall
+fence
+pole
+traffic light
+traffic sign
+vegetation
+terrain
+sky
+person
+rider
+car
+truck
+bus
+train
+motorcycle
+bicycle
+mIoU
+CycleGAN (Zhu et al. 2017)
+46.89 35.49 33.79 27.63 27.07 17.65 14.74 10.70 7.82 7.14 6.96 5.39 5.28 3.81 3.08 2.57 2.28 2.25 0.32 13.73
+UNIT (Liu, Breuel, and Kautz 2017) 64.21 26.65 30.18 16.54 11.36 10.77 23.89 30.93 31.44 25.16 1.68 14.22 4.55 58.45 34.28 45.42 41.64 27.20 5.79 28.70
+MUNIT (Huang et al. 2018b)
+54.73 26.48 28.43 14.60 8.54 7.36 30.30 29.62 33.86 24.07 3.95 8.33 32.37 56.23 30.05 58.80 51.89 12.87 8.93 27.44
+TSIT (Jiang et al. 2020)
+80.10 46.44 19.39 14.77 15.83 16.11 27.27 39.82 72.30 44.35 2.44 25.78 41.78 65.67 41.98 49.48 42.54 31.12 19.71 36.68
+MGUIT (Jeong et al. 2021)
+54.26 11.66 21.85 5.81 0.89 8.73 3.86 11.28 22.51 7.30 1.37 1.97 3.08 34.31 1.08 0.05 3.86 0.00 0.34 10.22
+SHUNIT(ours)
+72.35 43.36 41.68 21.42 18.13 16.08 39.11 35.90 71.56 35.10 67.93 27.53 50.45 52.55 37.31 37.06 28.32 27.64 16.57 38.96
+Table 10: Per-class IoU results on Cityscapes (clear) → ACDC (fog).
+Methods
+road
+sidewalk
+building
+wall
+fence
+pole
+traffic light
+traffic sign
+vegetation
+terrain
+sky
+person
+rider
+car
+truck
+bus
+train
+motorcycle
+bicycle
+mIoU
+CycleGAN (Zhu et al. 2017)
+86.03 46.64 60.57 24.69 15.16 26.56 11.75 31.92 42.44 27.05 4.47 36.84 10.44 61.32 0.04 15.97 51.54 16.27 25.51 31.33
+UNIT (Liu, Breuel, and Kautz 2017) 86.03 46.74 63.33 26.69 14.80 28.41 19.25 30.92 43.94 35.14 15.34 40.08 19.55 64.43 2.51 30.81 44.15 29.44 29.00 35.29
+MUNIT (Huang et al. 2018b)
+87.73 68.58 66.92 54.84 51.74 44.49 42.45 35.94 33.97 32.46 30.67 29.12 27.61 26.24 20.50 19.91 18.52 16.51 3.01 37.43
+TSIT (Jiang et al. 2020)
+88.69 56.81 63.49 21.67 16.59 27.28 21.60 33.81 43.53 38.21 3.50 42.42 19.94 67.13 4.08 30.27 45.36 25.55 32.62 35.92
+MGUIT (Jeong et al. 2021)
+86.67 44.75 66.36 24.50 16.13 28.26 8.87 32.15 42.24 8.32 8.66 37.88 10.85 61.65 2.38 19.17 46.22 20.52 30.16 31.36
+SHUNIT(ours)
+85.65 44.84 67.98 24.53 18.98 26.66 20.57 30.32 45.02 26.21 16.61 39.13 14.40 62.68 0.56 21.13 42.92 23.59 27.76 33.66
+Table 11: Per-class IoU results on Cityscapes (clear) → ACDC (night).
+
+Input
+CycleGAN
+UNIT
+MUNIT
+TSIT
+MGUIT
+SHUNIT (ours)
+Target domain
+Figure 10: More qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night). From the given clear image
+(first row), we generate four adverse condition images using (Zhu et al. 2017; Liu, Breuel, and Kautz 2017; Huang et al. 2018b;
+Jiang et al. 2020; Jeong et al. 2021) and SHUNIT (top row to bottom row order). In the last row, we show a sample of the real
+image for each adverse condition.
+
+Input
+CycleGAN
+UNIT
+MUNIT
+TSIT
+MGUIT
+SHUNIT (ours)
+Target domain
+Figure 11: More qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night). From the given clear image
+(first row), we generate four adverse condition images using (Zhu et al. 2017; Liu, Breuel, and Kautz 2017; Huang et al. 2018b;
+Jiang et al. 2020; Jeong et al. 2021) and SHUNIT (top row to bottom row order). In the last row, we show a sample of the real
+image for each adverse condition.
+
+Input
+CycleGAN
+UNIT
+MUNIT
+TSIT
+MGUIT
+SHUNIT (ours)
+Target domain
+Figure 12: More qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night). From the given clear image
+(first row), we generate four adverse condition images using (Zhu et al. 2017; Liu, Breuel, and Kautz 2017; Huang et al. 2018b;
+Jiang et al. 2020; Jeong et al. 2021) and SHUNIT (top row to bottom row order). In the last row, we show a sample of the real
+image for each adverse condition.
+
+HOTEL-RESTAURANT
+HAGNAUERHOFInput
+CycleGAN
+UNIT
+MUNIT
+TSIT
+MGUIT
+SHUNIT (ours)
+Target domain
+Figure 13: More qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night). From the given clear image
+(first row), we generate four adverse condition images using (Zhu et al. 2017; Liu, Breuel, and Kautz 2017; Huang et al. 2018b;
+Jiang et al. 2020; Jeong et al. 2021) and SHUNIT (top row to bottom row order). In the last row, we show a sample of the real
+image for each adverse condition.
+
+(b) InstaFormer
+(a) Input image (KITTI)
+(c) SHUNIT (ours)
+Figure 14: Qualitative comparison on domain adaptive detection for KITTI → Cityscapes. Given (a) input image (KITTI),
+(b) and (c) show the translated image and the object detection result generated by InstaFormer and our method, respectively.
+Note that the results are taken from InstaFormer paper.
+

JdAyT4oBgHgl3EQf5_qu/content/tmp_files/2301.00815v1.pdf.txt

@@ -0,0 +1,898 @@
+NeuroExplainer: Fine-Grained Attention
+Decoding to Uncover Cortical Development
+Patterns of Preterm Infants
+Chenyu Xue1, Fan Wang2, Yuanzhuo Zhu2, Hui Li3, Deyu Meng1, Dinggang
+Shen4, and Chunfeng Lian1
+1 School of Mathematics and Statistics,Xi’an Jiaotong University, Xi’an, China
+chunfeng.lian@xjtu.edu.cn
+2 Key Laboratory of Biomedical Information Engineering of Ministry of Education,
+School of Life Science and Technology, Xi’an Jiaotong University, Xi’an, China
+fan.wang@xjtu.edu.cn
+3 Department of Neonatology, The First Affiliated Hospital of Xi’an Jiaotong
+University, Xi’an, China
+4 School of Biomedical Engineering, ShanghaiTech University, Shanghai, China
+Abstract. Deploying reliable deep learning techniques in interdisciplinary
+applications needs learned models to output accurate and (even more im-
+portantly) explainable predictions. Existing approaches typically expli-
+cate network outputs in a post-hoc fashion, under an implicit assumption
+that faithful explanations come from accurate predictions/classifications.
+We have an opposite claim that explanations boost (or even determine)
+classification. That is, end-to-end learning of explanation factors to aug-
+ment discriminative representation extraction could be a more intu-
+itive strategy to inversely assure fine-grained explainability, e.g., in those
+neuroimaging and neuroscience studies with high-dimensional data con-
+taining noisy, redundant, and task-irrelevant information. In this pa-
+per, we propose such an explainable geometric deep network dubbed
+as NeuroExplainer, with applications to uncover altered infant cortical
+development patterns associated with preterm birth. Given fundamen-
+tal cortical attributes as network input, our NeuroExplainer adopts a
+hierarchical attention-decoding framework to learn fine-grained atten-
+tions and respective discriminative representations to accurately rec-
+ognize preterm infants from term-born infants at term-equivalent age.
+NeuroExplainer learns the hierarchical attention-decoding modules un-
+der subject-level weak supervision coupled with targeted regularizers de-
+duced from domain knowledge regarding brain development. These prior-
+guided constraints implicitly maximizes the explainability metrics (i.e.,
+fidelity, sparsity, and stability) in network training, driving the learned
+network to output detailed explanations and accurate classifications. Ex-
+perimental results on the public dHCP benchmark suggest that NeuroExplainer
+led to quantitatively reliable explanation results that are qualitatively
+consistent with representative neuroimaging studies.
+Keywords: Geometric Deep Learning · Explainability · Infant Brain
+Cortical Development · AI for Neuroscience
+arXiv:2301.00815v1  [cs.LG]  1 Jan 2023
+
+2
+Authors Suppressed Due to Excessive Length
+1
+Introduction
+Due to the capacity of learning highly nonlinear representations in task-oriented
+fashions, deep neural networks are showing promising applications in many in-
+terdisciplinary communities, including biomedical image computing, neuroimag-
+ing, and neuroscience studies [33]. To quantitatively analyze brain develop-
+ment/degeneration or timely diagnose associated disorders, various deep learn-
+ing methods have been proposed [45,28]. Most of these studies focused on the
+designs of network architectures and learning strategies to produce accurate pre-
+dictions/classifications. A critical challenge is that the learned models typically
+lack computational interpretability and results’ explainability. From the aspect
+of practical usage, neuroimaging and neuroscience studies desire AI tools that
+can make accurate and (even more importantly) explainable predictions. For
+example, a key value of a brain disease diagnosis network is to identify from
+high-dimensional neuroimage data individualized subtle changes leading to ac-
+curate classification, which can be clues for human experts to analyze disease
+heterogeneity [45]. Also, the classification task to differentiate between preterm
+and term-born (or male and female) infants may not be practically meaningful;
+in contrast, fine-grained differences on brain cortical surfaces, identified by the
+learned classification network, could be valuable factors for better understanding
+featured cortical development patterns of infants from different groups.
+Recently, explainable and interpretable deep learning is being actively stud-
+ied in the machine learning community, with obviously more works on gridded
+data (e.g., images) [11] than on non-Euclidean data (e.g., 3D meshes) [46]. Accu-
+rate classification and faithful explanation are highly correlated and inseparable.
+Existing methods typically adopt post-hoc techniques to explain a deep network
+[46], which is first trained for a specific classification task, and then the underly-
+ing (sparse) correlations between its input and output are analyzed offline, e.g.,
+by backpropagating prediction gradients to the shallow layers [34]. Notably, such
+post-hoc approaches are established upon a common assumption that reliable ex-
+planations are the results caused by accurate predictions. This assumption could
+work in general applications that have large-scale training data, while cannot al-
+ways hold for neuroimaging and neuroscience research, where available data are
+typically small-sized and much more complex (e.g., high-resolution cortical sur-
+faces containing noisy, highly redundant, and task-irrelevant information).
+Distinct to existing post-hoc methods, we have an opposite claim that ex-
+plainability boosts or even determines classification, especially in those chal-
+lenging tasks related to brain cortical development analyses based on high-
+dimensional neuroimaging data. The key is how to construct an end-to-end
+framework, where fine-grained explanation factors can be identified in a fully
+learnable fashion to enhance discriminative representation extraction and finally
+output accurate classification. This paper presents such an explainable geometric
+deep network, called NeuroExplainer, with applications to uncover altered infant
+cortical development patterns associated with preterm birth. NeuroExplainer
+adopts high-resolution cortical attributes as the input to develop a hierarchi-
+cal attention-decoding architecture. In the framework of weakly supervised dis-
+
+NeuroExplainer
+3
+criminative localization, our NeuroExplainer is trained by minimizing general
+classification losses coupled with a set of constraints designed according to prior
+knowledge regarding brain development. These targeted regularizers drive the
+network to implicitly optimize the explainability metrics from multiple aspects
+(i.e., fidelity, sparsity, and stability), thus capturing fine-grained explanation fac-
+tors to explicitly improve classification accuracies. Experimental results on the
+public dHCP benchmark suggest that our NeuroExplainer led to quantitatively
+reliable explanation results that are qualitatively consistent with representative
+neuroimaging studies, implying that it could be a practically useful AI tool for
+other related cortical surface-based neuroimaging studies.
+2
+Related Work
+Deep Learning for Cortical Surface Analyses. The human cerebral cortex
+is a highly folded and thin sheet of gray matter [15]. By leveraging structural
+magnetic resonance imaging (MRI), such complex topology can be rendered as a
+3D mesh, with each vertex/cell presenting fundamental cortical attributes, e.g.,
+cortical thickness, mean curvature, and average convexity. For brain develop-
+ment/degeneration analyses, advanced geometric deep learning methods can be
+potentially applied to learning from cortical surface data a powerful and efficient
+classification/prediction model, either over the original 3D mesh or after mapping
+it onto a spherical surface to ease the computation [47]. Although these existing
+studies suggested promising accuracies of geometric deep learning in multiple
+tasks (e.g., parcellation [47], registration [36], and longitudinal prediction [22]),
+the learned models typically lack explainability and interpretability.
+Explainability in Deep Neural Networks. The instance-level explanation
+approaches aim to study why a deep model makes a specific prediction for
+a given input, e.g., which part of the input contributes more to its output
+classification score. Such input-dependent explanation methods can be roughly
+categorized into four branches, including gradient/feature-based, perturbation-
+based, decomposition-based, and surrogate-based methods [46]. As the most
+straightforward strategy, gradient/feature-based approaches have been actively
+studied in deep learning over gridded data [11], with some extensions to non-
+Euclidean cases (e.g., graphs) [46,24]. Currently, research regarding explainable
+geometric deep learning on 3D meshes (like brain cortical surfaces) is still lim-
+ited [30]. More importantly, existing explanation methods (in both gridded and
+non-Euclidean spaces) typically investigate network outputs in a post-hoc fash-
+ion [11,46,23], assuming that faithful explanations are the results of accurate pre-
+dictions/classifications. In this paper, we have an opposite claim that leveraging
+domain knowledge to perform end-to-end learning of feature-based explanation
+factors for discriminative representation extraction (and classification perfor-
+mance enhancement) could be a more intuitive strategy to produce fine-grained
+explainability in challenging tasks.
+
+4
+Authors Suppressed Due to Excessive Length
+EB-1
+DB-1
+A
+EB-4
+T A
+�𝑠�, 𝑠�}
+�𝑠�, 𝑠�}
+DB-2
+A
+�𝑠�, 𝑠�}
+DB-3
+A
+�𝑠�, 𝑠�}
+C
+EB-2
+EB-3
+Cortical attributes
+Left
+Right
+Convexity
+Mean curvature
+Cortical thickness
+Spherical 
+mapping
+1D
+Conv
+1D
+Conv
+⋯
+⋯
+Input features
+Share 
+weights
+GAP
+�𝑠�, 𝑠�}
+⊙
+Attention-gated
+features
+Spherical attention mechanism
+Left
+Right
+Left
+Right
+Left
+Right
+Classification 
+score
+Attention maps
+EB
+: Hexagonal Conv + Pooling (except EB-4)
+: Cross-hemisphere Self-Attention
+T
+A : Spherical Attention Mechanism
+DB
+: Hexagonal Transposed Conv
++ Skip connection
++ Hexagonal Conv
+C : GAP + 1D Conv
+: Concatenation
+: Multiplication
+⊙: Dot-product
+Fig. 1: The schematic diagram of our NeuroExplainer that learns to capture fine-
+grained explanation factors in an end-to-end attention-decoding architecture to
+boost discriminative representation extraction from cortical-surface data.
+3
+Method
+As the schematic diagram shown in Fig. 1, our NeuroExplainer works on the
+high-resolution spherical surfaces of both brain hemispheres (each with 10, 242
+vertices). The inputs are fundamental vertex-wise cortical attributes, i.e., thick-
+ness, mean curvature, and convexity. The architecture has two main parts, in-
+cluding an encoding branch to produce initial task-related attentions on down-
+sampled hemispheric surfaces, and a set of attention decoding blocks to hier-
+archically propagate such vertex-wise attentions onto higher-resolution spheres,
+finally capturing fine-grained explanation factors on the input high-resolution
+surfaces to boost the prediction task (i.e., the differentiation between preterm
+and fullterm infants in our study). The whole network is trained end-to-end by
+minimizing multi-resolution classification losses, under the constraints provided
+by prior-induced regularizations to enhance explanation metrics.
+3.1
+Spherical Attention Encoding
+The starting components of the encoding branch are four spherical convolution
+blocks (i.e., EB-1 to EB-4 in Fig. 1), with the learnable parameters shared across
+two hemispheric surfaces. Each EB adopts 1-ring hexagonal convolution [47]
+followed by batch normalization (BN) and ReLU activation to extract vertex-
+wise representations, which are then downsampled by hexagonal max pooling
+[47] (except in EB-4) to serve as the input of the subsequent layer. Based on
+the outputs from EB, we propose a learnable spherical attention mechanism to
+conduct weakly-supervised discriminative localization.
+Specifically, let Fl and Fr ∈ R162×M0 be the vertex-wise representations
+(produced by EB-4) for the left and right hemispheres, respectively. We first
+concatenate them as a 324 × M0 matrix, on which a self-attention operation
+[38] is applied to capturing cross-hemisphere long-range dependencies to refine
+the vertex-wise representations from both hemispheric surfaces, resulting in a
+unified feature matrix denoted as F0 = [ˆFl; ˆFr] ∈ R324×M0. As shown in Fig.
+
+10242lrvatureNeuroExplainer
+5
+1, F0 is further global average pooled (GAP) across all vertices to be a holistic
+feature vector f0 ∈ R1×M0 representing the whole cerebral cortex. Both F0 and
+f0 are then mapped by a same vertex-wise 1D convolution (i.e., W0 ∈ RM0×2,
+without bias) into the categorical space, denoted as A0 = [Al
+0; Ar
+0] ∈ R324×2
+and s0, respectively. Notably, so is supervised by the one-hot code of subject’s
+categorical label, by which Al
+0 and Ar
+0 highlight discriminative vertices on the
+(down-sampled) left and right surfaces, respectively, considering that
+s0[i] ∝
+�
+1T F0
+�
+W0[:, i] = 1T �
+[ˆFl; ˆFr]W0[:, i]
+�
+= 1T Al
+0[:, i] + 1T Ar
+0[:, i],
+(1)
+where so[i] (i = 0 or 1) in our study denote the prediction scores of preterm
+and fullterm, respectively, and 1 is an unit vector having the same row size
+with the subsequent matrix. Finally, we define the hemispheric attentions as
+¯Al
+0 = �1
+i=0 s0[i]Al
+0[:, i] and ¯Ar
+0 = �1
+i=0 s0[i]Ar
+0[:, i] ∈ R324×1, respectively, with
+values spatially varying and depending on the relevance to subject’s category.
+3.2
+Hierarchically Spherical Attention Decoding
+The explanation factors captured by the encoding branch are relatively coarse,
+as the receptive field of a cell on the downsampled surfaces (with 162 vertices
+after three pooling operations) is no smaller than a hexagonal region of 343
+cells on the input surfaces (with 10, 242 vertices). To tackle this challenge, we
+design a spherical attention decoding strategy to hierarchically propagate coarse
+attentions (from lower-resolution spheres) onto higher-resolution spheres, based
+on which fine-grained attentions are finally produced to improve classification.
+Specifically, NeuroExplainer contains three consecutive decoding blocks (i.e.,
+DB-1 to DB-3 in Fig. 1). Each DB adopts both the attention-gated discriminative
+representations from the preceding DB (except DB-1 that uses EB-4 outputs)
+and the local-detailed representations from the symmetric EB (at the same reso-
+lution) as the input. Let the attention-gated representations from the preceding
+DB be Fl
+G =
+� ¯Al
+in11×Min
+�
+⊙ ˆFl
+in and Fr
+G =
+� ¯Ar
+in11×Min
+�
+⊙ ˆFr
+in, respectively,
+where each row of ˆFin has Min channels, and ⊙ denotes element-wise dot prod-
+uct. We first upsample Fl
+G and Fr
+G to the spatial resolution of the current DB,
+by using hexagonal transposed convolutions [47] with learnable weights shared
+across hemispheres. Then, the upsampled discriminative representations from
+each hemisphere (say ˜Fl
+G and ˜Fr
+G) are channel-wisely concatenated with the lo-
+cal representations from the corresponding EB (say Fl
+E and Fr
+E), followed by an
+1-ring convolution to produce a unified feature matrix, such as
+FD = [Cθ(˜Fl
+G ⊕ Fl
+E); Cθ(˜Fr
+G ⊕ Fr
+E)],
+(2)
+where Cθ(·) denotes 1-ring conv parameterized by θ, and ⊕ stands for channel
+concatenation. In terms of FD, the attention mechanism described in (1) is
+further applied to producing refined spherical attentions and classification scores.
+Finally, as shown in Fig. 1, based on the fine-grained attentions over the input
+surfaces (each with 10, 242 vertices), we use GAP to aggregate the attention-
+gated representations and apply an 1D conv to output the classification score.
+
+6
+Authors Suppressed Due to Excessive Length
+Original convexity
+…
+Random coarsening
+-15
+10
+Atypical subject
+⋯
+Feature matrix 𝐅�
+�
+Attention 𝐀�
+�
+⋯
+Feature matrix 𝐅�
+�
+Attention 𝐀�
+�
+Typical subject
+Pull
+Discriminative 
+representation space
+(a)
+(b)
+Fig. 2: Brief illustrations of (a) the explanation fidelity-aware contrastive learning
+strategy, and (b) explanation stability-aware data augmentation strategy.
+3.3
+Domain Knowledge-Guided Explanation Enhancement
+In this study, we design NeuroExplainer under a central idea that task-oriented
+learning of explanation factors to boost discriminative representation extraction
+can inversely assure fine-grained explainability in challenging cases like learn-
+ing on complex cortical-surface data. To effectively train our network for such a
+purpose, we design a set of targeted regularization strategies by considering fun-
+damental domain knowledge regarding infant brain development. Specifically, it
+is reasonable to assume that human brains in infancy have generally consistent
+developments, while the structural/functional discrepancies between different
+groups (e.g., preterm and term-born) are typically localized [37,10]. Accordingly,
+we require the preterm-altered cortical development patterns captured by our
+NeuroExplainer to be discriminative, spatially sparse, and robust, which sug-
+gests the design of the following constraints that concurrently optimize fidelity,
+sparsity, and stability metrics [46] in deploying an explainable deep network.
+Explanation Fidelity-Aware Contrastive Learning. Given the spherical
+attention block at a specific resolution, we have A+
+i
+and A−
+j ∈ RV ×1 as the
+output attentions for a positive and negative subjects (i.e., preterm and fullterm
+infants in our study), respectively, and F+
+i and F−
+j ∈ RV ×M are the correspond-
+ing representation matrices. Based on the prior knowledge regarding infant brain
+development, it is reasonable to assume that A+
+i highlights atypically-developed
+cortical regions caused by preterm birth. In contrast, the remaining part of the
+cerebral cortex of a preterm infant (corresponding to 1 − A+
+i ) still growths nor-
+mally, i.e., looking globally similar to the cortex of a term-born infant.
+Accordingly, as the illustration shown in Fig. 2(a), we design a fidelity-
+aware contrastive penalty to regularize the learning of the attention maps and
+associated representations to improve their discriminative power. Let f +
+i
+=
+1T �
+A+
+i 11×M ⊙ F+
+i
+�
+and ¯f +
+i
+= 1T �
+{1 − A+
+i }11×M ⊙ F+
+i
+�
+be the holistic fea-
+ture vector and its inverse for the ith (positive) sample, respectively. Similarly,
+f −
+j = 1T �
+A−
+j 11×M ⊙ F−
+j
+�
+denotes the holistic feature vector for the compared
+jth (negative) sample. By pushing f +
+i
+away from both ¯f +
+i
+and f −
+j , while pulling
+
+Original convexitNeuroExplainer
+7
+¯f +
+i
+close to f −
+j , we define the respective loss as
+Lcontrast =
+N
+�
+i̸=j
+|| ¯f +
+i − f −
+j || + max(m − || ¯f +
+i − f +
+i ||, 0) + max(m − ||f −
+j − f +
+i ||, 0),
+(3)
+where i and j indicate any a pair of positive and negative cases from totally N
+training samples, and m is a margin setting as 1 in our implementation.
+Explanation Sparsity-Aware Regularization. According to the specified
+prior knowledge regarding infant brain development, the attention maps pro-
+duced by our NeuroExplainer should have two featured properties in terms of
+sparsity. That is, the attention map for a preterm infant (e.g., A+
+i ) should be
+sparse, considering that altered cortical developments are assumed to be local-
+ized. In contrast, the attention map for a healthy term-born infant (e.g., A−
+j )
+should not be spatially informative, as all brain regions growth typically without
+abnormality. To this end, we design a straightforward entropy-based regulariza-
+tion to enhance results’ explainability, such as
+Lentropy =
+N
+�
+i̸=j
+1T �
+A+
+i ⊙ log(A+
+i ) − A−
+j ⊙ log(A−
+j )
+�
+,
+(4)
+where i and j indicate a positive and a negative cases from totally N training
+samples, respectively, and 1 is an unit vector to sum up the values of all vertices.
+Explanation Stability-Aware Regularization. We enhance the explanation
+stability of our NeuroExplainer from two aspects. First, we require the spherical
+attention mechanisms to robustly decode from complex cortical-surface data (po-
+tentially containing perturbed and noisy information) fine-grained explanation
+factors to produce accurate predictions. To this end, a customized data augmen-
+tation strategy is designed to explicitly increase perturbations and variances in
+preprocessing the data for training such a network. Specifically, the inputs of our
+network are cortical surfaces with 10, 242 vertices that are downsampled from
+the original data with 163, 842 vertices. We randomize this surface coarsening
+step by quantifying a vertex’s cortical attributes (on the downsampled surface)
+as the average of a random subset of the vertices from the respective hexagonal
+region of the highest-resolution surface. As the examples summarized in Fig.
+2(b), such a data augmentation strategy can generate partially different training
+samples from one single subject. Considering that the network is trained to pro-
+duce consistently accurate predictions for all these variants with perturbations,
+it inversely enhances the stability of learned explanation factors.
+Second, as described in Sec. 3.2, the fine-grained explanation factors over
+high-resolution cortical surface are captured by the proposed hierarchical atten-
+tion decoding strategy, where coarse results from the encoder part serve as the
+foundation. To further enhance the explanation stability, we design a cross-scale
+consistency regularization to refine the decoding branch. Specifically, let Al
+i and
+
+8
+Authors Suppressed Due to Excessive Length
+Ah
+i be the spherical attentions from two different DB blocks. We simply minimize
+Lconsistent =
+N
+�
+i=1
+�
+Al
+i − Ah
+i
+�2,
+(5)
+which encourages spherical attention maps at different spatial resolutions to be
+consistent in network training.
+Implementation Details. In our implementation, the feature representations
+produced by EB-1 to EB-4 in Fig. 1 have 32, 64, 128, and 256 channels, respec-
+tively. Correspondingly, DB-1 to DB-3, and the final classification layer have 256,
+128, 64, and 32 channels, respectively. The network was trained end-to-end by
+minimizing the cross-entropy classification losses defined at three different spa-
+tial resolutions (overall denoted as LCE), coupled with the regularization terms
+introduced in Sec. 3.3, such as
+L = LCE + λ1Lcontrast + λ2Lentropy + λ3Lconsistent,
+(6)
+where the tuning parameters were empirically set as λ1 = 0.2, λ3 = 0.5, and
+λ3 = 0.1. The network parameters were updated by using Adam optimizer for
+500 epochs, with the initial learning rate setting as 0.001 and bath size as 20.
+4
+Experiments
+Dataset & Experimental Setup. We conducted experiments on the dHCP
+benchmark [26]. The structural MRIs of 700 infants scanned at term-equivalent
+ages (35-44 weeks postmenstrual age) were studied, including 143 preterm and
+557 term-born infants. These subjects were randomly split as a training set of 500
+infants (89 preterm and 411 fullterm), and a test set of the remaining 200 infants
+(54 preterm and 146 fullterm). Using the data-augmentation strategy described
+in Sec. 3.3, the training set was augmented to have roughly 1, 250 subjects from
+each category for balanced network training. The input spherical surfaces contain
+10, 242 vertices, and each of them has three morphological attributes, i.e., cortical
+thickness, mean curvature, and convexity.
+For classification, our NeuroExplainer was compared with three representa-
+tive geometric deep networks, including a spherical network based on 1-ring con-
+volution (SphericalCNN) [47], a MoNet reimplementation working on spherical
+surfaces (SphericalMoNet) [36], and SubdivNet [16] working on original cor-
+tical meshes. In addition, we conducted detailed ablation studies to verify the
+efficacy of each prior-guided regularization introduced in Sec. 3.3. The classifi-
+cation performance was quantified in terms of accuracy (ACC), area under the
+ROC curve (AUC), sensitivity (SEN), and specificity (SPE).
+On the other hand, the explanation performance of our NeuroExplainer was
+compared with two representative feature-based explanation approaches, i.e.,
+CAM [49] and Grad-CAM [31], which were coupled with the geometric net-
+works described above for post-hoc analysis. The explanation performance was
+
+NeuroExplainer
+9
+Table 1: Classification results obtained by the competing geometric deep net-
+works and different variants of our NeuroExplainer.
+Competing Mehtods
+ACC
+AUC
+SEN
+SPE
+SphericalCNN [47]
+0.93
+0.92
+0.76
+0.98
+SphericalMoNet [36]
+0.85
+0.93
+0.65
+0.92
+SubdivNet [16]
+0.79
+0.67
+0.74
+0.80
+NeuroExplainer (ours)
+0.95
+0.97
+0.94
+0.95
+w/o Lcontrast (3)
+0.88
+0.89
+0.80
+0.91
+w/o Lentropy (4)
+0.91
+0.96
+0.74
+0.97
+w/o Lconsistent (5)
+0.89
+0.95
+0.89
+0.88
+quantitatively evaluated in terms of three metrics [46], i.e., Fidelity that mea-
+sures the classification differences between the network captured explanation
+factors and the remaining part, Sparsity of the captured explanation factors
+compared to the whole surface, and Stability that measures the average clas-
+sification accuracy on different perturbations of a single subject. Please refer
+to [46] for more details regarding these explainability evaluation metrics.
+Classification Results. The classification results obtained by different com-
+peting methods are summarized in Table 1, from which we can have at least
+three observations. 1) Compared with SubdivNet working on original meshes,
+and the other two networks working on spherical surfaces (i.e., SphericalCNN
+and SphericalMoNet), our NeuroExplainer consistently led to better classifica-
+tion accuracies in terms of all metrics. 2) The improvements brought by our
+method are especially significant in terms of SEN and AUC, suggesting that
+it can reliably identify featured development patterns associated with preterm
+birth to make accurate predictions in such an imbalanced learning task. These
+results imply that our idea to capture fine-grained explanation factors in an end-
+to-end fashion to boost discriminative representation extraction is beneficial for
+deploying an accurate classification model in the task of learning on complex
+surface data containing noisy, redundant, and task-irrelevant information.
+3) To check the efficacy of the prior-induced regularization strategies, we
+orderly removed them from the loss function (6) to quantify the respective in-
+fluence on classification results. From Table 1, we can see that all the three reg-
+ularizations (i.e., Lcontrast, Lentropy, and Lconsistent) demonstrated significant
+but different improvements on classification. Specifically, according to the com-
+parison between NeuroExplainer and its variant w/o Lentropy, we can see that
+such an explanation sparsity-aware regularization boosted SEN by 20%, imply-
+ing its efficacy in capturing localized patterns related to preterm-altered cortical
+development. By comparing NeuroExplainer with its variant w/o Lconsistent,
+we can see that the explanation stability-ware regularization helped stabilize
+the learning of the hierarchical attention-decoding branch, leading to relatively
+large improvements of overall classification ACC (by 6%). Finally, we can see
+that the explanation fidelity-aware contrastive learning strategy brought overall
+the largest improvements of both ACC and AUC (by 7% and 8%, respectively),
+
+10
+Authors Suppressed Due to Excessive Length
+Table 2: Quantitative explanation results obtained by the competing post-hoc
+approaches and our end-to-end NeuroExplainer.
+Competing Methods
+Fidelity
+Sparsity Stability
+CAM [49] +
+SphericalCNN
+0.24
+0.91
+0.77
+SphericalMoNet
+0.55
+0.93
+0.58
+SubdivNet
+0.06
+0.97
+0.53
+Grad-CAM [31] +
+SphericalCNN
+0.22
+0.99
+0.77
+SphericalMoNet
+0.42
+0.98
+0.58
+SubdivNet
+0.16
+0.96
+0.53
+NeuroExplainer (ours)
+0.56
+0.73
+0.96
+0
+1
+Left
+Right
+Subject 1
+Subject 2
+Grad-CAM
+Spherical
+CNN
+Spherical
+MoNet
+CAM
+Spherical
+MoNet
+Spherical
+CNN
+Ours
+Fine-
+grained
+Coarse-
+scaled
+Ours
+Fine-
+grained
+Coarse-
+scaled
+CAM
+Spherical
+MoNet
+Spherical
+CNN
+Grad-CAM
+Spherical
+CNN
+Spherical
+MoNet
+Fig. 3: Typical examples of the explanation factors captured by the com-
+peting post-hoc approaches (CAM and Grad-CAM) and our end-to-end
+NeuroExplainer. Higher values indicate larger links to preterm birth.
+implying its efficacy in transforming domain knowledge regarding atypical brain
+development to capture fine-grained explanation factors that boost discrimina-
+tive representation learning.
+Explanation Results. The quantitative explanation results obtained by our
+NeuroExplainer and other feature-based explanation methods (i.e., CAM and
+Grad-CAM coupled with the trained geometric deep networks, respectively)
+are summarized in Table 2. From Table 2, we can observe that our end-to-end
+NeuroExplainer outperformed other post-hoc explanation approaches by a large
+margin in terms of the three explainability metrics. Notably, the three metrics
+should be analyzed concurrently in evaluating a network’s explainability [46], as
+the isolated quantification of a single metric could be biased. For example, al-
+though Grad-CAM+SphericalCNN led to the largest Sparsity value (= 0.99) in
+our experiment, the corresponding Fidelity and Stability values are significantly
+low (= 0.22 and 0.77, respectively), indicating that the very sparse explanation
+factors captured by Grad-CAM+SphericalCNN are relatively uninformative and
+random. In contrast, our NeuroExplainer led to significantly better Fidelity and
+Stability, under reasonable Sparsity, suggesting that it can robustly identify lo-
+calized preterm-altered cortical patterns from high-dimensional inputs to boost
+discriminative representation learning for preterm infant recognition.
+
+NeuroExplainer
+11
+-10
+10
+0
+1
+Our 
+NeuroExplainer
+Existing Multi-Modal (dMRI + sMRI) Studies
+Mean Diffusivity
+Neurite Density
+Cortical Thickness
+Fig. 4: Comparison of the individualized preterm-altered cortical development
+patterns uncovered by NeuroExplainer with the group-wise patterns identified
+by existing multi-modal studies [10].
+In addition to the above quantitative evaluations, we also visually compared
+the attention maps produced by different competing methods, with two typical
+examples presented in Fig. 3. From Fig. 3, we can have two main observations. 1)
+Compared with post-hoc explanation methods (i.e., CAM and Grad-CAM), our
+end-to-end NeuroExplainer stably produced more reasonable attentions. Specif-
+ically, given a subject (e.g., Subject 2 in Fig. 3), CAM and Grad-CAM could
+produce very different explanation results for a trained classification model (e.g.,
+SphericalCNN). Similarly, a post-hoc approach (e.g., CAM) could produce dis-
+tinct results for two different classification models on the same subject. Due
+to the nature of end-to-end learning of explanation factors to establish classi-
+fication models, our NeuroExplainer effectively avoided such a problem. More
+importantly, across different subjects, our NeuroExplainer led to group-wisely
+more consistent explanations than these post-hoc approaches. Also, it produced
+more consistent results across hemispheres, without any related constraints dur-
+ing network training. 2) We can see that the coarse attentions produced by
+NeuroExplainer’s encoder are consistent with the final fine-grained outputs of
+the decoder, which implies the positive effect of the cross-scale consistency reg-
+ularization in enhancing explainability.
+Finally, we compared the individualized preterm-altered cortical development
+patterns uncovered by our NeuroExplainer with representative group-wise neu-
+roimaging studies in the literature, e.g., the multi-modal (dMRI and sMRI)
+quantitative analyses presented in [10]. According to the comparisons shown in
+Fig. 3, we can see that our observations in this paper are consistent with [10].
+The discriminative cortical regions captured by our NeuroExplainer (using solely
+morphological features) are largely overlapped with the group-wise significantly
+different regions identified by [10] in terms of the mean diffusivity, neurite den-
+sity, and cortical thickness, respectively. For example, they both highlighted some
+specific regions in the inferior parietal, medial occipital, and superior temporal
+lobe, and posterior insula, which is worth deeper evaluations in the future.
+
+12
+Authors Suppressed Due to Excessive Length
+5
+Conclusion
+In the paper, we have proposed an geometric deep network, i.e., NeuroExplainer,
+to learn fine-grained explanation factors from complex cortical-surface data to
+boost discriminative representation extraction and accurate classification model
+construction. On the benchmark dHCP database, the applications of our NeuroExplainer
+to uncover preterm-altered infant cortical development patterns achieved bet-
+ter performance in terms of both explainability and prediction accuracy, when
+compared with representative post-hoc approaches coupled with state-of-the-
+art geometric deep networks. The proposed method could be a promising AI
+tool applied to other similar cortical surface-based neuroimage and neuroscience
+studies.
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+49. Zhou, B., et al.: Learning deep features for discriminative localization. In: ICCV.
+pp. 2921–2929 (2016)
+50. Zhou, Z.H.: A brief introduction to weakly supervised learning. National science
+review 5(1), 44–53 (2018)
+

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+Factors that affect Camera based Self-Monitoring of Vitals in the Wild
+Nikhil S. Narayan
+Shashanka B. R. *
+Rohit Damodaran∗
+Dr. Chandrashekhar Jayaram∗
+Dr. M. A. Kareem
+Dr. Mamta P.∗
+Dr. Saravanan K. R.∗
+Dr. Monu Krishnan∗
+Dr. Raja Indana
+MFine
+Abstract
+The reliability of the results of self monitoring of the
+vitals in the wild using medical devices or wearables or
+camera based smart phone solutions is subject to variabil-
+ities such as position of placement, hardware of the device
+and environmental factors. In this first of its kind study,
+we demonstrate that this variability in self monitoring of
+Blood Pressure (BP), Blood oxygen saturation level (SpO2)
+and Heart rate (HR) is statistically significant (p < 0.05)
+on 203 healthy subjects by quantifying positional and hard-
+ware variability. We also establish the existance of this vari-
+ability in camera based solutions for self-monitoring of vi-
+tals in smart phones and thus prove that the use of camera
+based smart phone solutions is similar to the use of medical
+devices or wearables for self-monitoring in the wild.
+1. Introduction
+Monitoring of vitals is important for identifying and
+managing of diseases. Glasziou et. al. [11] identify five
+phases of monitoring a disease from a chronic condition
+perspective along with the interval for monitoring. These
+include continuous monitoring of vitals to (a) detect abnor-
+mality; (b) confirm abnormality; (c) establish plan of action
+to treat the abnormality; (d) make adjustments to the treat-
+ment; and (e) to confirm the success of treatment. These
+phases can very well be extended to acute conditions with-
+out loss of generality except for the fact that the monitor-
+ing intervals may be much shorter than that for chronic
+conditions and may totally be based on self-monitoring.
+Self-monitoring refers to monitoring of conditions by pa-
+tients/users by using medical devices that are available off-
+*The authors were at MFine when the research was carried out. Prior
+consent has been obtained from the authors to publish this work
+the-shelf at a place of the patient’s convenience without the
+intervention of a qualified medical professional to monitor
+the condition. Self monitoring in the wild refers to monitor-
+ing the vitals in uncontrolled environments such as outdoors
+or house where the conditions with respect to lightihg, phys-
+ical activity etc., dynamically change.
+At certain points in time, when access to care is made
+difficult either due to the unavailability of care providers
+or due to environmental factors such as the most recent
+COVID-19 pandemic where patients were forced to mon-
+itor their conditions at home before approaching a health-
+care provider, self-monitoring plays a crucial role in saving
+human lives. Given the importance of self-monitoring of
+diseases in recent times, there has been a flurry of activity in
+research circles to come up with novel technologies that en-
+able self-monitoring of vitals such as Blood Pressure (BP),
+Blood oxygen saturation level (SpO2), Heart rate (HR),
+Blood Glucose etc. While some of these have made it to
+commercial devices such as wearables and mobile phones, a
+vast majority of the technology is still restricted to academia
+and research circles.
+1.1. Prior art
+Of the technologies that are being researched for self-
+monitoring of vitals, computer vision has received consid-
+erable amount of attention the the last decade. One of the
+early attempts in this direction was by Jonathan et. al., [16]
+where the authors used a Nokia device to record videos at
+15 FPS of a user before and after performing a physical ac-
+tivity and the change in HR calculated by employing us-
+ing Photoplethysmography(PPG) Imaging. Thereafter there
+have been quite a few attempts to demonstrate the capability
+of using PPG signals from videos/image stream to estimate
+HR in an individual [4,13,14,19,24,32].
+The PPG imaging methodology employed for HR is also
+a popular method to estimate SpO2 by both signal process-
+1
+arXiv:2301.12943v1  [cs.CV]  30 Jan 2023
+
+ing/mathematical models [6, 21, 31] and machine learning
+[2, 3, 5, 17, 18]. A majority of smart phone PPG solutions
+for SpO2 consider the video of a finger tip placed against
+the back camera of the phone, hereafter referred to as Fin-
+ger tip (FT) PPG, as the input to extract the PPG signals
+and estimate the vital. A non-contact way to monitor SpO2
+is explored in [15, 35]where a video stream of user’s face,
+hereafter referred to as Face (FC) PPG, is used as input for
+PPG signal extraction and SpO2 Estimation.
+There has been growing interest in recent times to esti-
+mate BP using PPG signals extracted either from FT PPG or
+FC PPG. While, the exact relationship between PPG the sig-
+nal and Blood Pressure has not been clinically established,
+fitting an ML model on top of either the raw signal, or some
+features extracted from the PPG signal seems to work and
+is the general direction of work so far. Neural Networks
+[1,9,20,33], Ensemble methods [10] and LSTM’s [27] are
+some popular machine learning algorithms that have been
+employed to estimate BP using either the raw PPG signal
+or features extracted from the raw PPG signal as inputs to
+the algorithms. Efforts towards developing contact-less so-
+lutions include using a camera to capture facial videos of
+the user and use deep Learning models on the extracted
+rPPG(remote PPG) signals to estimate BP [29]. Transder-
+mal Optical Imaging technology is another way to capture
+facial blood flow changes and estimate BP in a contact-less
+manner [23].
+1.2. Motivation
+The algorithms discussed in Section 1.1 either use pub-
+licly available datasets or data available from a limited set of
+devices (usually one) to develop camera based vitals moni-
+toring solutions. PhysioNet [12] is a popular database com-
+monly used to develop algorithms based on Finger tip PPG.
+Algorithms based on Face PPG use either [7] or [36] or
+both to train the models. It is interesting to note that these
+datasets are constructed under strictly controlled environ-
+ment such as fixed lighting settings, fixed background, fixed
+distance from camera etc., and do not account for interde-
+vice variability. Thus, these datasets do not account for vari-
+abilities that arise when the devices are used in the wild or
+when multiple devices of different kinds are used for self-
+monitoring. Some examples of variabilities that may arise
+while acquiring data for self monitoring in the wild are:
+• Positional Variance: (a) Wrong positioning of the BP
+cuff of a digital monitor during data acquisition for
+BP algorithms. Previous studies have established that
+there is a significant Inter Arm BP difference when a
+digital monitor is used for BP measurement [22, 26].
+Several clinical studies in the past have also estab-
+lished the existence of variabilities in the technique
+used to measure BP by experts and the negative im-
+pact that it has while monitoring in a clinical setting
+Table 1. Mobile applications and wearables that support vitals
+monitoring
+Technology
+Name
+Vitals support
+Solution
+BP
+SpO2
+HR
+Wearable
+Apple Watch
+Y
+Y
+IoT
+GOQii
+Y
+Y
+Y
+IoT
+boAT Xtend
+Y
+Y
+IoT
+One Plus
+Y
+Y
+IoT
+Fitbit
+Y
+Y
+IoT
+Oura ring
+Y
+IoT
+Omron
+Y
+IoT
+Mobile App
+Careplix
+Y
+Y
+Camera (FT)
+MFine
+Y
+Y
+Y
+Camera (FT)
+ICICI
+Y
+Y
+Y
+Camera (FC)
+Anura
+Y
+Y
+Camera (FC)
+[28,30,34]. When such variabilities exist in measure-
+ments taken by experts themselves, it is not uncom-
+mon to expect the variability to exist while measuring
+BP in a self monitoring setting. (b) Similar positional
+variances exist while measuring SpO2 and HR using
+pulseoximeters.
+• Hardware Variance: Variabilities in the hardware used
+to acquire Image / Video signals for PPG based vitals
+analysis. It is a well known fact that manufacturers
+of different brands of smart phones employ sensors of
+different Original Equipment Manufacturers (OEM’s)
+to have a competitive edge. This will directly result
+in the variability of the quality of PPG signal obtained
+from different sensors.
+• Environmental Variance: the mobility aspect of smart
+phones by default will introduce variances related to
+lighting, motion etc.
+In this paper we address the following questions quan-
+titatively/statistically by undertaking a systematic clinical
+study: (a) What variabilities exist in self monitoring of the
+following vitals: BP, SpO2 and HR? (b) How do these vari-
+abilities compare to the measurements obtained by an ex-
+pert who is a qualified medical professional? (c) If the vari-
+abilities between self-monitoring and the measurements by
+an expert are similar, then how do these variabilities affect
+the training of computer vision based solutions? for vitals
+monitoring? (d) What do we need to do to minimise the
+effect of this variability in AI based solutions?
+The focus of this paper is on the variabilities associated
+with Position and Hardware while Environmental Variance
+will be picked up as an extension to this study to do justice
+to the number of environmental factors that may influence
+the outcome of self-monitoring in the wild. It should also
+be noted that the primary goal of this paper is to validate
+2
+
+Figure 1. Study setup
+commercially available solutions at this point in time as we
+wish to evaluate that solution which is easily accessible to
+a user in the current situation given the pandemic and the
+global burden on healthcare infrastructure.
+1.3. Contributions
+To the best of our knowledge, there is no prior work that
+has formally quantified the variabilities that exist in self-
+monitoring of vitals. Since a majority of solutions in lit-
+erature are targeted towards self-monitoring of vitals, it is
+crucial to identify this variability and determine it’s signif-
+icance in order to establish error bounds for measurements
+of vitals on smartphones and wearables in the future. In this
+first-of-a-kind study, we establish statistically that there is
+a significant variability in the Vitals when measured by self
+using medical devices that are available off the shelf. Ad-
+ditionally, we also establish the existance of this variability
+in camera based solutions for self-monitoring of vitals and
+thus prove that the use of camera based smart phone solu-
+tions is similar to the use of medical devices or wearables
+for self-monitoring. Finally, discuss some methods that can
+potentially be used to reduce this variability in camera based
+solutions.
+2. Materials and Method
+2.1. Data
+The study was conducted on 203 healthy subjects com-
+prising of 112 (55%) men and 91 (45%) women in the age
+range [20,55] years with an average age of 28.77 ± 5.877
+years. The sample size was estimated for a 5% error at 95%
+confidence interval based on the computations in [8,22,25].
+Prior written consent was obtained from the subjects who
+volunteered for the study as per the IRB guidelines and ap-
+provals obtained for the study.
+In order to monitor the vitals on medical devices, mobile
+phones and wearables, we will require at least:
+1. one pair of mobile phones/smartphones with one of
+the phones pre-installed with a mobile application that
+measures vitals by extracting PPG signals from finger
+tip image stream/video and the other from face videos
+from front camera of the phone. These mobile appli-
+cations are assumed to employ any one of the methods
+described in Section 1.1. The exact details on the na-
+ture of the algorithm are unavailable at this point in
+time as the developers of the applications have not dis-
+closed it publicly as either patents or publications.
+The phones of the pair should be of the same
+brand to eliminate the variabilities that may arise due
+to differences in the hardware and software used to ac-
+quire the signals (Image/Video). Since the study also
+involves observing the variabilities associated with
+changes in hardware and software of the smartphones,
+we employ 4 pairs of phones in this study. The cri-
+teria for selection of the phone brand/model is based
+on price of the phone and global availability of the
+phone. Lower the price, higher is the reach to the peo-
+ple who would need access to affordable healthcare.
+The price range under consideration for this study is
+US$100-US$250. Accordingly, the following 4 mod-
+els of phones are used: 1.(a) Xiaomi Note 9 Pro (Xi
+N9); 1.(b) Xiaomi Note 8 Pro (Xi N8); 1.(c) Oppo A15
+(Oppo); and 1.(d) Samsung M31 (SM31);
+The mobile applications should be capable of
+measuring all the vitals that are considered for this
+study. Table 1 shows different commercially available
+mobile applications and wearables along with the vi-
+tals supported by each. We select one mobile appli-
+cation each for Finger tip PPG based monitoring and
+Face PPG based monitoring from the Mobile App cat-
+egory in Table 1. Accordingly, the following mobile
+applications are considered for this study: MFine for
+Finger tip PPG based monitoring and ICICI mobile ap-
+plication for Face PPG based monitoring.
+2. one wearable. As with the mobile phones, the wear-
+able selected for this study should cover all the vitals
+considered for the study. Froom Table 1 it can be seen
+that the GOQii smartwatch supports all vitals and is
+thus used in the study. Since we also want to study
+the variabilities between wearables, we have also in-
+cluded Apple Watch Series 7 in our study even though
+BP monitoring capability is absent in it;
+3. a digital BP monitor. Omron HEM 7121J fully auto-
+matic Digital Blood Pressure monitor is used in this
+3
+
+1
+Observation Station 1
+Observer 1
+Digital BP
+Mercury
+★ Pulse Oximeter
+monitor
+Sphygmomanometer
+1
+Observation Station 2
+Observer 2
+Finger Tip PPG
+FacePPG
+Wearables
+1
+Observation Station 3
+Observer 3
+Cardiac monitorFigure 2. Quantifying the (a,c) transverse variability; and (b,d)
+angular variability in the placement of the sensor of the BP cuff
+for self-monitoring
+Figure 3. Variations in measuring SpO2 and HR by placing a
+pulseoximeter on the index finger of the hand (a) resting on the
+table; (b) resting on the table with index figer at maximum angle;
+and (c) in the air.
+study;
+4. a mercury based sphygmomanometer. Diamond Mer-
+curial Blood Pressure apparatus was used for this
+study;
+5. a pulseoximeter. BPL Smary Oxy pulseoximeter was
+used for this study to measure SpO2 and Heart Rate;
+and
+6. a cardiac monitor that will be used as gold standard to
+eliminate the interobserver variability that may arise
+with manual measurements with a sphygmomanome-
+ter. Yonker YK 8000C Multi-parameter patient moni-
+tor was used in this study.
+Figure 4. Commercially available vitals monitoring solutions that
+use (a) face videos (selfie); and (b) Finger tip videos to measure
+vitals such as BP, SpO2 and HR.
+2.2. Methodology
+The study setup comprises of three observation stations
+in a well lit room manned by three independent observers
+who are qualified medical practitioners for : (a) self mon-
+itoring; (b) monitoring with camera based mobile applica-
+tion and wearables; and (c) monitoring with a cardiac mon-
+itor. Figure 1 illustrates the setup used for this study.
+The first station is where the variability in self monitor-
+ing is studied. The self monitoring exercise starts with the
+subjects being asked to measure BP using a digital BP mon-
+itor that is placed in front of him/her without any instruc-
+tions given on how to operate the BP monitor (which also
+includes the placement of the cuff on the arm). The observer
+of the station then notes down the position of the sensor in
+the cuff on the arm with respect to: (a) the displacement
+along the arm which is quantified as per the grading in Fig
+2 (c); and (b) the approximate angle it makes with the cen-
+treline (0o) of Fig 2 (d). Once the BP measurement and the
+corresponding variations are recorded by the observer, the
+subjects are then asked to measure the SpO2 and HR by
+placing the pulseoximeter on the index finger and starting
+the measurement. The observer notes down the SpO2 and
+HR readings after 15s of the start of the measurement at
+each of the positions indicated in Fig 3. The last step in sta-
+tion 1 involves the observer measuring the Blood pressure
+of the subject using a mercury based sphygmomanometer.
+The subject is then asked to proceed to the second station
+where the vitals are monitored using mobile based applica-
+tions and wearables. As described in Section 2.1, we use 4
+pairs. of mobile phones of and two wearables per subject to
+monitor the vitals. One phone of each pair is used for vitals
+monitoring using Face PPG and the other is used for vitals
+monitoring using Finger Tip PPG. Phones in each pair be-
+long to the same brand and measurements on both phones
+4
+
+Posterior (P)
+Median nerve
+Brachial artery
+Mediannerve
+Brachial artery
+Anterior (A)
+(a)
+(b)
+p
+180°
+202.5°
+157.5°
+cm
+....
+2
+cm
+225°
+135°
+1 cm
+0 cm
+.-1 cm
+:-2 cm
+247.5°
+112.5°
+-3 cm
+270°
+90°
+292.5°
+67.5°
+.....
+45°
+315°
+...
+....
+337.5°
+22.5°
+0°
+A
+(c)
+(d)1. Hand resting on
+1. Hand resting on
+1. Hand 5 cm
+table
+table
+abovetable
+2. Index Finger with
+2. Index Finger with
+Pulseoximeter
+Pulseoximeter at
+resting on table
+maximum lift22:41网02
+72%
+21:24国·
+20%
+22:52国网·
+l 70%
+Knowyourhealth
+<
+Blood Pressure
+01:48
+124/81
+mmHg
+Whatdoes itmean
+Your BP appears to be within normal. However, this is merely a
+ConstructingthePPG
+high-level analysis. We do not claim 100% accuracy.
+SYS
+60
+80
+100
+120
+140
+160
+180
+200
+220
+DIA
+30
+50
+70
+90
+110
+130
+150
+170
+190
+Legends
+oVery Low
+●Normal
+·HighBP
+A
+MeasurementusingMFine's BPMonitorisonlyadvisedfor
+general wellbeing and fitness.
+sec
+Itisnotintendedformedicaluse
+HeartRate
+OxygenSaturation
+92 beats pm
+99%
+RespiratoryRate
+Heart Rate Variability
+21breaths pm
+114 ms
+Stress Level
+Blood Pressure
+Normal
+Systolic -123 mm Hg
+Diastolic -78 mm Hg
+Hold yourfinger overthecamera inthe same
+position.
+STOP
+ Tell a friend
+MeasureAgain
+川I
+0
+(a)
+(q)Figure 5. Vitals monitoring using mobile phones and wearables in
+the 2nd station
+are taken simultaneously. The phone used for Face PPG is
+placed on a mobile phone holder with the angle adjusted so
+that the front camera points to the face of the subject (with
+the full face in view). Figure 4 illustrates the face based
+and finger tip based methods to measure Vitals on mobile
+phones. Figure 5 shows the procedure followed to measure
+the vitals in the second observation station. Subjects who
+have completed measuring their vitals in the second station
+are asked to proceed to the final observation station where
+their vitals are monitored by a cardiac monitor. The mea-
+surements of the final station are used as the gold standard
+for evaluating the measurements obtained at each observa-
+tion station.
+3. Experimental results
+In this section we demonstrate the variabilities that ex-
+ist in self monitoring using medical devices, mobile phones
+and wearables, respectively. We use oneway ANOVA and
+one tailed Student’s t-test to establish the statistical signifi-
+cance of the variabilities using p-values. Where variabilities
+are concerned, we use the mean and variance of differences
+between the measurements of the device under considera-
+tion and the gold standard which is the cardiac monitor.
+3.1. Self monitoring of vitals using medical devices
+BP
+The plots in Figure 6 show the variability in both
+Systolic and Diastolic BP measurements when the sensor
+in the cuff of the digital BP monitor is placed at different
+angles around the arm and at different positions along the
+arm (Left/Right). It is interesting to note that the Systole
+BP measurements obtained on the Left hand showed a sta-
+tistically significant difference (p < 0.05) with the read-
+ings obtained from the cardiac monitor. Difference in mea-
+surements between cardiac monitor and digital BP moni-
+tor for Systolic BP measurement was statistically significant
+(p < 0.05) while Diastolic measurements were within sta-
+Table 2. Error in camera based vitals measurement on mobile
+phone when readings of cardiac monitor are used as gold standard
+Brand
+Method
+BP (S)
+BP (D)
+SpO2
+HR
+(mm/hg)
+(mm/hg)
+(%)
+(/min)
+Xi N9
+FT
+2.432
+−0.680
+−2.277
+−3.30
+±13.66
+±10.25
+±3.28
+±8.69
+FC
+−2.929
+−2.596
+−1.368
+5.758
+±20.70
+±13.60
+±2.32
+±14.70
+Xi N8
+FT
+1.783
+−1.082
+1.233
+−5.919
+±13.72
+±9.89
+±3.25
+±10.13
+FC
+−2.929
+−2.403
+−1.526
+−2.103
+±18.76
+±13.835
+±2.406
+±13.96
+Oppo
+FT
+1.350
+−1.814
+−1.5
+−7.70
+±13.62
+±10.00
+±3.20
+±12.25
+FC
+−2.070
+−0.824
+−1.543
+0.362
+±17.251
+±12.937
+±2.464
+±14.91
+Sm 31
+FT
+1.412
+−1.412
+0.866
+−4.797
+±13.90
+±10.24
+±5.00
+±9.94
+FC
+−1.964
+−3.157
+−1.105
+1.982
+±18.10
+±12.80
+±3.621
+±12.59
+Xi N9: Xiaomi Note 9 Pro; Xi N8:Xiaomi Note 8 Pro; Oppo: Oppo
+A15; Sm 31: Samsung M31; FT: Finger Tip PPG; FC: Face PPG;
+BP(S): Systolic Blood Pressure; BP(D): Diastolic Blood Pressure;
+SpO2: Blood Oxygen Saturation; HR: Heart Rate
+tistically acceptable limits (p > 0.05). Both Systolic and
+Diastolic BP readings did not show statistically significant
+results (p > 0.05) when compared to the measurement by
+an expert using mercury based Sphygmomanometer.
+SpO2 and HR
+The plots in Figure 7 shows the variabil-
+ity in the measurements of both SpO2 and HR when the
+pulseoximeter is clamped on to the index finger of the hand
+according to the variations shown in Fig. 3.
+Both SpO2 and Heart Rate showed statistically signifi-
+cant difference (p < 0.05) between pulseoximeter and car-
+diac monitor readings at all positions. A one way ANOVA
+performed on the pulseoximeter readings at different posi-
+tions of hand and index finger showed no statistically signif-
+icant difference (p > 0.05) between the readings for SpO2.
+Heart Rate on the other hand showed a statistically signifi-
+cant difference (p < 0.05) between the readings at different
+positions.
+3.2. Self monitoring of vitals using mobile devices
+Figure 8 shows the rate of failure in measuring the vitals
+on each of the phones considered for this experiment. It can
+be seen that Face PPG in general has a higher rate of failure
+compared to Finger tip PPG. While Finger tip PPG based
+methods had average failure rates of 14.676 ± 7.383%,
+15.796 ± 4.197% and 15.049 ± 9.088% for BP, SpO2 and
+5
+
+Select wearable and
+Select phone pair
+measure vitals
+Record values
+Measure vitals
++
+N
+Are all wearables
+Finger Tip
+Face PPG
+covered?
+PPG
+N
+Are all phone
+Y
+pairs covered?
+Record values
+Proceed to next stationFigure 6. Variability in self monitoring of Blood Pressure (BP) using digital BP monitors. (a,d,g). Variations in the angular placement of
+the sensor (in the cuff) on the arm near the synovial joint; (b,e,h). Variations in the placement of the sensor (in the cuff) along the arm near
+the synovial joint; (c,f,i). Variations in the selection of the arm [Left/Right] to place the sensor for BP measurement.
+HR, respectively, Face PPG based methods had average fail-
+ure rates of 24.875 ± 15.216%, 24.129 ± 15.941% and
+23.631±16.169%. The plots in Fig. 9 shows the number of
+attempts it took for those subjects where it was possible to
+successfully obtain a measurement. The percentage of users
+who could get successful readings in the 1st, 2nd and 3rd
+attempts for Finger tip PPG, respectively stood at 67.492 ±
+18.201%, 31.451±18.504% and 1.056±1.241%, while for
+Face PPG it was 73.072±15.959%, 25.199±15.324% and
+1.728 ± 1.658% for 1st, 2nd and 3rd attempts, respectively.
+Table 2 shows that the error between the measurements
+obtained from Finger tip PPG based methods and cardiac
+monitor are in general lower compared to those obtained be-
+tween Face PPG and cardiac monitor. However, the results
+of Finger tip PPG based measurements statistically varied
+across the phones for the same user while it remained quite
+similar for Face PPG across the phones as indicated by the
+p-values of Table 3.
+6
+
+Angularvariation
+Transversevariation
+Choice of Hand for Measurement
+Errorin measurement [Systole]
+measurement [Systole]
+measurement [Systole]
+Errorin
+Error
+30
+-2.0
+-1.0
+0.0
+2.0
+3.0
+Left
+Right
+Distance from center (in cm)
+Hand
+Angle (in degrees)
+(b)
+(c)
+(a)
+Error in measurement [Diastole]
+iastole]l
+measurement [Diastole]
+in measurement [Di
+2
+20
+u!
+Error
+Error
+60
+3.0
+2.0
+Distance from center (in cm)
+Left
+Right
+Hand
+Angle (in degrees)
+(e)
+(f)
+(d)
+20.0
+17.5
+.20
+15.0
+.00
+samples
+Number of samples
+12.5
+-2.0
+-1.0
+2.0
+Left
+Right
+Distance from center (in cm)
+Hand
+Angle (in degrees)
+(h)
+(i)
+(g)Figure 7. Positional variability in measuring SpO2 and HR using
+pulseoximeter
+Table 3. Results of ANOVA on measurements of vitals across
+phones
+Method
+BP (S)
+BP (D)
+SpO2
+HR
+FT (p-value)
+0.00002
+0.00002
+0.0
+0.067
+FC (p-value)
+0.967
+0.397
+0.520
+0.04
+BP(S): Systolic Blood Pressure; BP(D): Diastolic Blood Pressure;
+SpO2: Blood Oxygen Saturation; HR: Heart Rate
+Table 4. Error in vitals measurement on wearables when readings
+of cardiac monitor are used as gold standard
+Brand
+BP (S)
+BP (D)
+SpO2
+HR
+(mm/hg)
+(mm/hg)
+(%)
+(/min)
+GOQii
+2.015
+−0.492
+−0.940
+−4.422
+±14.73
+±11.58
+±3.725
+±10.53
+Apple watch
+NA
+NA
+−0.744
+−4.536
+±3.884
+±9.744
+3.3. Self monitoring of vitals using wearable
+There were no failures in measuring BP on GOQii and
+HR on both GOQii and Apple Watch. There was however
+a 2% failure rate while measuring SpO2 on Apple Watch.
+The results of Table 4 show that while GOQii smart watch
+has a lower error rate for HR, Apple Watch has a lower error
+Table 5. Results of ANOVA on measurements of vitals across
+wearables
+Method
+SpO2
+HR
+p-value
+2.14
+0.928
+SpO2: Blood Oxygen Saturation; HR: Heart Rate
+Table 6. p-value of differences in measurements between a device
+and the cardiac monitor using t-test
+Brand
+Method
+BP (S)
+BP (D)
+SpO2
+HR
+(mm/hg)
+(mm/hg)
+(%)
+(/min)
+E
+-
+0.457
+0.457
+0.0009
+0.0009
+DM
+-
+0.537
+0.537
+0.0009
+0.0009
+Xi N9
+FT
+0.017
+0.780
+0.0
+0.011
+FC
+0.009
+0.121
+0.0
+0.00038
+Xi N8
+FT
+0.269
+0.622
+0.0
+0.0001
+FC
+0.237
+0.279
+0.0
+0.784
+Oppo
+FT
+0.940
+0.005
+0.00001
+0.0
+FC
+0.015
+0.080
+0.026
+0.001
+SM31
+FT
+0.646
+0.098
+0.025
+0.002
+FC
+0.085
+0.008
+0.022
+0.751
+GO
+-
+0.033
+0.578
+0.0002
+0.0007
+AW
+-
+-
+-
+0.007
+0.0002
+Xi N9: Xiaomi Note 9 Pro; Xi N8:Xiaomi Note 8 Pro; Oppo: Oppo
+A15; Sm 31: Samsung M31; E: Expert; DM: Digital BP Monitor; GO:
+GOQii Smart watch; AW: Apple Watch; FT: Finger Tip PPG; FC:
+Face PPG; BP(S): Systolic Blood Pressure; BP(D): Diastolic Blood
+Pressure; SpO2: Blood Oxygen Saturation; HR: Heart Rate
+rate for SpO2 when the results of cardiac monitor are used
+as the gold standard for comparison. The measurements
+between the two wearables were not statistically significant
+as is evident from Table 5.
+3.4. Overall Comparison
+The results of Table 6 shows that the inter-observer vari-
+ability was statistically insignificant. While Finger tip PPG
+based measurements were in agreement with the gold stan-
+dard for BP, the rest of the methods for all vitals showed
+statistically significant differences between the device mea-
+surements and gold standard.
+4. Discussion, Conclusion and Future work
+In this study we have shown that statistically significant
+variations exist in self monitoring of vitals using medical
+devices. One potential solution to address this is to sen-
+sitise users to the correct procedure to be followed while
+performing self-monitoring. An alternative to this would be
+to ask the subjects to use mobile based or wearable based
+solutions where the degrees of freedom for variabilities are
+fewer and easily manageable.
+7
+
+Positional variation
+0
+5
+0
+0
+0
+[Sp02]
+0
+-5
+00
+00
+0000
+-10
+8
+0
+0
+-15
+-20
+-25
+0
+0
+0
+-30
+0
+0
+-35
+Resting
+Max Angle
+Air
+Position of index finger
+(a)
+30 -
+8
+8
+20 -
+[HR]
+8
+10
+ measurements
+-10
+m
+-20
+rror
+E -30
+0
+0
+8
+-40
+Resting
+Max Angle
+Air
+Position of index finger
+(b)Figure 8. Failure rates in phones while measuring vitals using Finger tip PPG and Face PPG methods
+Figure 9. The number of attempts required to get a successful
+measurement of at least one vital
+The variabilities in self monitoring and monitoring with
+Mobiles and Wearables are similar. The ground truth mea-
+surements for algorithms on mobiles and wearables are ob-
+tained from experts who either use digital monitors or mer-
+cury based monitors (for BP) to measure vitals. Since an in-
+herent variability existis in the technique used by experts as
+noted in [28,30,34], this variability creeps into the training
+data for mobiles and wearables. What is even more interest-
+ing is the fact that the user variability in self monitoring and
+that by experts is more or less similar and thus the variabili-
+ties in self monitoring and that in mobiles and wearables are
+similar. A potential solution to eliminate this variability in
+training data will be to use the positional variability charts
+of Fig. 2 and Fig. 3 as reference while acquiring ground
+truth data for vitals. From the plots of Fig. 6, it can be
+seen that the position of the sensor in the BP cuff between
+[315o,22.5o] and [0cm,1cm] would result in less variability
+in BP measurements and thus result in consistent Ground
+Truth BP readings. Whereas, measurements taken with the
+hand in resting position with the index finger horizontally
+placed on the table is the ideal position to obtain Ground
+Truth measurements for SpO2 and HR.
+Variabilities in hardware used for imaging within mo-
+bile phones result in statistically significant inconsistencies
+across mobile phones of different brand. Not only are the
+results inconsistent, it also takes multiple attempts on cer-
+tain phones with certain technologies to get the measure-
+ments right. This will lead to bad user experience and loss
+of trust in the solution for vitals monitoring thereby result-
+ing in a loss of adoptability of mobile camera based solu-
+tions for self-monitoring of vitals in the wild. One potential
+solution to reduce this variability will be to use camera cali-
+bration techniques which will normalize the color and white
+balance of the image stream or video that is being acquired
+for PPG signal construction.
+Face PPG in general had a higher success rate compared
+to Finger tip PPG and that the varaibilty across phones for
+Face PPG was statistically insignificant compared to that of
+the Finger tip PPG based methods. This is a direct result of
+the experimental setup, where the phones were placed on a
+mobile holder for Face PPG and the user was asked to hold
+the phone for Finger tip PPG. This indicates the influence
+of the following two environment factors in the consistency
+of results: (a) lighting condition (the study was in a well lit
+room); and (b) motion artifacts. The impact of environmen-
+tal factors as contributors for variability in self monitoring
+will be considered as an extension to this study.
+Conclusion and Future Work
+In this paper we demon-
+strate the various variabilities that exist while performing
+self-monitoring of vitals using smart phones, wearables and
+medical devices and establish the statistical significance of
+the results of each when compared to the gold standard mea-
+surement obtained from a cardiac monitor. The study of en-
+vironmental factors for variability in self monitoring, cam-
+era calibration for minimising hardware variability and PPG
+signal quality improvements will be extensions to our cur-
+rent work on Self monitoring in the wild using camera based
+solutions.
+8
+
+BP Measurement
+SpO2 Measurement
+HR Measurement
+Finger Tip PPG
+Finger Tip PPG
+Finger Tip PPG
+Face PPG
+Face PPG
+Face PPG
+40 
+40
+40 -
+(%)
+(%)
+(%)
+failure
+30
+30 
+of failure (
+30 
+20
+20
+Rate
+Rate
+10
+10 -
+10 -
+0 :
+0
+-0
+Xiaomi N9 Xiaomi N8 Oppo A15 Samsung M31
+Xiaomi N9 Xiaomi N8 Oppo A15 Samsung M31
+Xiaomi N9 Xiaomi N8 Oppo A15 Samsung M31
+Phone brand
+Phone brand
+Phone brandFingertip PPG
+Face PPG
+1st attempt
+1st attempt
+2nd Attempt
+2nd Attempt
+80
+>=3 attempts
+80
+>=3 attempts
+(%)
+60
+Percentage
+60
+40
+40
+20
+20
+XiaomiN9XiaomiN8OppoA15SamsungM31
+XiaomiN9
+9XiaomiN8OppoA15SamsungM31
+Phone brand
+PhonebrandReferences
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+

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+arXiv:2301.01546v1  [math.AP]  4 Jan 2023
+On the first Robin eigenvalue of the Finsler
+p-Laplace operator as p Ñ 1
+Rosa Barbato˚, Francesco Della Pietra˚, Gianpaolo Piscitelli˚
+Abstract. Let Ω be a bounded, connected, sufficiently smooth open set,
+p ą 1 and β P R. In this paper, we study the Γ-convergence, as p Ñ 1`, of
+the functional
+Jppϕq “
+ż
+Ω
+F pp∇ϕqdx ` β
+ż
+BΩ
+|ϕ|p FpνqdHN´1
+ż
+Ω
+|ϕ|p dx
+where ϕ P W 1,ppΩqzt0u and F is a sufficientely smooth norm on Rn. We
+study the limit of the first eigenvalue λ1pΩ, p, βq “ infϕPW 1,ppΩq
+ϕ‰0
+Jppϕq, as
+p Ñ 1`, that is:
+ΛpΩ, βq “
+inf
+ϕPBV pΩq
+ϕı0
+|Du|F pΩq ` mintβ, 1u
+ż
+BΩ
+|ϕ| FpνqdHN´1
+ż
+Ω
+|ϕ| dx
+.
+Furthermore, for β ą ´1, we obtain an isoperimetric inequality for ΛpΩ, βq
+depending on β.
+The proof uses an interior approximation result for BV pΩq functions by
+C8pΩq functions in the sense of strict convergence on Rn and a trace inequal-
+ity in BV with respect to the anisotropic total variation.
+MSC 2020: 28A75, 35J25, 35P15.
+Keywords and phrases:
+Finsler p-Laplace eigenvalues; Γ-convergence;
+Isoperimetric inequalities; Trace inequalities; Strict interior approximation.
+∗Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Universit`a degli studi di Napoli Federico
+II, Via Cintia, Monte S. Angelo - 80126 Napoli, Italia.
+Email: f.dellapietra@unina.it (corresponding author), gianpaolo.piscitelli@unina.it
+1
+
+Contents
+1
+Introduction
+2
+2
+Notation and preliminaries
+4
+2.1
+The Finsler norm . . . . . . . . . . . . . . . . . . . .
+4
+2.2
+Anisotropic curvatures
+. . . . . . . . . . . . . . . . .
+6
+2.3
+The anisotropic total variation
+. . . . . . . . . . . . .
+7
+3
+An anisotropic trace inequality
+8
+4
+Interior approximation
+11
+5
+The first Robin eigenvalue of the Finsler p-Laplacian as p Ñ 1 14
+5.1
+The case p “ 1
+. . . . . . . . . . . . . . . . . . . . .
+15
+5.2
+Γ-convergence of Jp . . . . . . . . . . . . . . . . . . .
+19
+5.3
+An isoperimetric inequality . . . . . . . . . . . . . . .
+22
+1 Introduction
+Let Ω be a bounded, connected, sufficiently smooth open set, p ą 1 and β P R. In
+this paper, we study the asympthotic behaviour, as p Ñ 1`, of the following minimum
+problem
+λ1pΩ, p, βq “
+inf
+ϕPW 1,ppΩq
+ϕ‰0
+Jppϕq
+(1.1)
+where
+Jppϕq “
+ż
+Ω
+F pp∇ϕqdx ` β
+ż
+BΩ
+|ϕ|p FpνqdHN´1
+ż
+Ω
+|ϕ|p dx
+,
+(1.2)
+ν is the outer normal to BΩ and F is a sufficiently smooth norm on Rn. If u P W 1,ppΩq
+is a minimizer of (1.1), then it solves the following Robin eigenvalue problem
+#
+´Qpu “ λ1pΩ, p, βq |u|p´2 u
+in Ω
+F p´1p∇uqFξp∇uq ¨ ν ` βFpνq |u|p´2 u “ 0
+on BΩ,
+where Qpu is the anisotropic p´Laplace operator
+Qpu :“ div
+ˆ1
+p∇ξrF psp∇uq
+˙
+.
+From the point of view of finding optimal domains for λ1pΩ, pβq, there is a significant
+difference from the case of β ą 0 to the case β ă 0. It is known that the optimal shape
+2
+
+with a volume constraint for (1.1) depends on the sign of β. For positive values of the
+Robin parameter, the so-called Wulff shape (see Section 2 for details) is a minimizer [9]:
+λ1pΩ, p, βq ě λ1pW, p, βq
+where |W| “ |Ω|.
+If β ă 0, the problem is not completely solved, even in the Euclidean case (that is when
+Fpξq “
+bř
+i ξ2
+i ). Indeed, in 1977 Bareket [5] conjectured that, the first eigenvalue is
+maximized by a ball in the class of the smooth bounded domains of given volume. In
+[13] it has been showed that it is true for domains close, in certain sense, to a ball.
+Subsequently, the authors in [15] (see [19] for the p-Laplacian case) have disproved the
+conjecture for |β| large enough and have showed that it is true for small values of |β|
+in suitable class of domains. In the Finsler setting, this problem has been addressed in
+[23].
+Our final aim is to obtain optimal shapes for the limiting functional of (1.1), as
+p Ñ 1`. To do that, we first study the limit of λ1pΩ, pβq. In particular, we prove that
+when β ą ´1, the functional Jp, defined in (1.2), Γ´converges, as p Ñ 1`, to
+Jpϕq “
+|Dϕ|F pΩq ` mint1, βu
+ż
+BΩ
+|ϕ| FpνqdHN´1
+ż
+Ω
+|ϕ| dx
+,
+where |Dϕ|F pΩq is the anisotropic total variation of ϕ (see Section 2 for the precise
+definition). This will imply that
+lim
+pÑ1` λ1pΩ, p, βq “ Λpβ, Ωq :“
+inf
+ϕPBV pΩq Jpϕq.
+(1.3)
+Then, we prove an isoperimetric inequality for ΛpΩ, βq. In particular, we obtain that
+keeping the volume of Ω fixed, the Wulff shape minimises ΛpΩ, βq when β ě 0, and
+maximises it when ´1 ă β ă 0.
+The proof of the convergence result, and then of the two isoperimetric inequalities,
+relies on two results on the anisotropic total variation, which are also of independent
+interest. The first one is a trace inequality in the BV space:
+ż
+BΩ
+|u| Fpνq dHN´1 ď c1 |Du|F pΩq ` c2
+ż
+Ω
+|u| dx,
+@u P BV pΩq,
+(1.4)
+where c1 and c2 are two constants which depend on the geometry of the domain. This
+inequality has been revealed very useful in capillarity problems, and it has been studied
+for example in [4, 16, 17].
+The second key result is an interior approximation for BV functions by smooth func-
+tions with compact support. It is well known that if Ω is an open set, then the total
+variation of a function u P BV pΩq can be approximated with the corresponding total
+variation of a sequence in C8pΩq.
+Actually, an analogous result is not true in gen-
+eral if one need to axpproximate |Du| with a sequence of C8 function with compact
+3
+
+support in Ω. In order to do that, more regularity is needed on Ω. In the Euclidean
+setting, this problem has been addressed in [20, 24]. In this paper, we show that for any
+u P BV pΩq X LppΩq, for some p P r1, 8q, there exists a sequence tukukPN Ď C8
+0 pΩq such
+that, for any q P r1, ps,
+uk Ñ u in LqpΩq
+and
+|Duk|F pRNq Ñ |Du|F pRNq.
+We finally stress that the problem we deal with is strictly related to capillarity prob-
+lems. We refer the reader, for example, to [16, 17] for the Euclidean case and to [8] for
+the anisotropic case.
+The structure of the paper is the following. In Section 2, we review some useful tools
+on the Finsler norm, the anisotropic curvature and functions of bounded variation. In
+Section 3 we prove the anisotropic trace inequality for general domains, and for smooth
+domains. In Section 4, we give the strict approximation result and finally, in Section 5
+we prove the Γ´convergence results and the isoperimetric inequality for ΛpΩ, βq.
+2 Notation and preliminaries
+In this Section we give several definitions and properties related the Finsler norm. In
+particular, we review some basic facts on the anisotropic total variation of a BV function,
+and on the anisotropic curvatures.
+2.1 The Finsler norm
+Throughout the paper we will assume that F is a convex, even, 1´homogeneous function
+ξ P RN ÞÑ Fpξq P r0, `8r,
+such that
+Fptξq “ |t|Fpξq,
+t P R, ξ P RN,
+(2.1)
+and such that
+a|ξ| ď Fpξq,
+ξ P RN,
+(2.2)
+for some constant a ą 0. It is easily seen that this hypothesis assure the existence of a
+positive constant b ě a such that
+Fpξq ď b|ξ|,
+ξ P RN.
+Throughout the paper, we will also assume that F belongs to C2pRNzt0uq and that
+∇2
+ξrF 2spξq is positive definite in RNzt0u.
+(2.3)
+The assumption (2.3) on F ensures that the operator
+Qpu :“ div
+ˆ1
+p∇ξrF psp∇uq
+˙
+4
+
+is elliptic, therefore there exists a positive constant γ such that
+nÿ
+i,j“1
+∇2
+ξiξjrF pspηqξiξj ě γ|η|p´2|ξ|2
+@η P RNzt0u, @ξ P RN.
+The polar function F o : RN Ñ r0, `8r of F is
+F opvq “ sup
+ξ‰0
+ξ ¨ v
+Fpξq.
+It is easily seen that also F o is a convex function satisfying the properties (2.1) and (2.2).
+Furthermore, we have
+Fpvq “ sup
+ξ‰0
+ξ ¨ v
+F opξq,
+and from this follows that
+|ξ ¨ η| ď FpξqF opηq
+@ξ, η P RN.
+(2.4)
+The Wulff shape centered at the origin is the set denoted by
+W “ tξ P RN : F opξq ă 1u.
+We denote κN “ |W|, where |W| is the Lebesgue measure of W. More generally,
+the set Wrpx0q indicates rW ` x0, that is the Wulff shape centered at x0 with measure
+κNrN. If no ambiguity occurs, we will write Wr instead of Wrp0q.
+The functions F and F o enjoy the following properties:
+Fξpξq ¨ ξ “ Fpξq,
+F o
+ξ pξq ¨ ξ “ F opξq
+@ξ P RNzt0u,
+(2.5)
+FpF o
+ξ pξqq “ F opFξpξqq “ 1
+@ξ P RNzt0u,
+(2.6)
+F opξqFξpF o
+ξ pξqq “ FpξqF o
+ξ pFξpξqq “ ξ
+@ξ P RNzt0u,
+(2.7)
+where Fξ “ ∇Fpξq.
+Given a bounded domain Ω, the anisotropic distance of x P Ω to BΩ is defined as
+dF pxq :“ inf
+yPBΩ F opx ´ yq,
+x P Ω.
+We highlight that, when Fpξq “
+bř
+i ξ2
+i , then dF “ dE is the Euclidean distance
+function from the boundary.
+The function dF is a uniform Lipschitz function in Ω, and
+Fp∇dF pxqq “ 1
+a.e. in Ω.
+We have that dF P W 1,8
+0
+pΩq. Many properties of the anisotropic distance function are
+studied in [7].
+Finally, the anisotropic inradius of Ω is
+RF pΩq “ maxtdF pxq, x P Ωu,
+that is the radius of the largest Wulff shape Wrpxq contained in Ω.
+5
+
+2.2 Anisotropic curvatures
+Here we recall some properties of the anisotropic mean curvature, as well as an integra-
+tion formula in anisotropic normal coordinates. We refer to [7] for further details.
+If Ω has a C2 boundary, the anisotropic outer normal to BΩ is defined as
+nF pyq “ Fξpνpyqq,
+y P BΩ,
+where νpyq is the Euclidean outer normal to BΩ at y. Moreover, by (2.6) it holds that
+F opnFpyqq “ 1.
+Let us denote by TyBΩ the tangent space to BΩ at y; the anisotropic Weingarten map is
+defined as
+dnF : TyBΩ Ñ TnpyqW.
+The eigenvalues κF
+1 ď κF
+2 ď . . . ď κF
+N´1 of this map are called the anisotropic principal
+curvatures at y (see also [25]). The anisotropic mean curvature of BΩ at a point y is
+defined as
+HF pyq “ κF
+1 pyq ` . . . κF
+N´1pyq,
+y P BΩ.
+The anisotropic distance dF is a C2 function in a tubular neighborhood of BΩ; hence
+we are in position to define the matrix-valued function
+Wpyq “ ´Fξξp∇dF pyqq∇2dF pyq,
+y P BΩ.
+Based on this function, it is possible to give a different definition of the anisotropic
+principal curvatures [7, Remark 5.9]. Since Wpyqv P TyBΩ, for any v P Rn, it remains
+defined the map Wpyq: Ty Ñ Ty, as Wpyqw “ Wpyqw, w P Ty. The matrix Wpyq (that is,
+in general, non-symmetric) admits the real eigenvalues κF
+1 pyq ď κF
+2 pyq ď . . . ď κF
+N´1pyq.
+Actually, the definition is equivalent to the preceding one. Moreover, it holds that
+HF pyq “ div rFξ p´∇dFpyqqs “ TrpWpyqq
+(see also [25, Sec. 3]).
+To state the change of variable formula in anisotropic normal coordinates, we need
+some preliminary definitions.
+Let
+Φpy, tq “ y ´ tFξpνpyqq,
+y P BΩ,
+t P R
+and for y P BΩ,
+ℓpyq “ suptdF pzq, z P Ω and y P Πpzqqu
+where
+Πpzq “ tη P BΩ: dF pzq “ F opz ´ ηqu
+(2.8)
+is the set of the anisotropic projections of a point z P Ω on BΩ.
+Then, we recall the following
+6
+
+Theorem 2.1. ([7, Theorem 7.1]) For every h P L1pΩq, it holds
+ż
+Ω
+hpxqdx “
+ż
+BΩ
+Fpνpyqq
+ż ℓpyq
+0
+hpΦpy, tqqJpy, tq dt dHN´1pyq,
+where
+Jpy, tq “
+N´1
+ź
+i“1
+p1 ´ tκF
+i pyqq.
+(2.9)
+Since 1 ´ tκF
+i pyq ą 0 for any i “ 1, . . . , N ´ 1 ([7, Lemma 5.4]), Jpy, tq is positive.
+Moreover it holds that
+´ d
+dt rJpy, tqs
+Jpy, tq
+“
+N´1
+ÿ
+i“1
+κF
+i pyq
+1 ´ tκF
+i pyq.
+(2.10)
+Finally, we conclude this section, by recalling that, for any x P Ω such that Πpxq “ tyu,
+it holds that ([7, Lemma 4.3]):
+∇dF pxq “ ´
+νpyq
+Fpνpyqq.
+(2.11)
+2.3 The anisotropic total variation
+Let u P BV pΩq, the total variation of u with respect to F is defined as
+|Du|F pΩq “ sup
+"ż
+Ω
+u divpgq dx
+: g P C1
+0pΩ; RNq, F opgq ď 1
+*
+and the perimeter of a set E with respect to F is:
+PF pE; Ωq “ |DχE|F pΩq “ sup
+"ż
+E
+divpgq dx
+: g P C1
+0pΩ; RNq, F opgq ď 1
+*
+.
+Moreover,
+PF pE; Ωq “
+ż
+ΩXB˚E
+FpνEqdHN´1
+where B˚E is the reduced boundary of Ω and νE is the Euclidean normal to BE.
+Let us fix u P BV pΩq and assume that u ” 0 in RNzΩ. Then u P BV pRNq and
+|Du|F pRNq “ |Du|F pΩq `
+ż
+BΩ
+|u|FpνqdHN´1
+(2.12)
+(see for example [6, Lemma 3.9]).
+For the anisotropic perimeter, an isoperimetric inequality holds. More precisely,
+PF pΩq ě PF pWRq,
+(2.13)
+where WR is the Wulff shape with the same measure of Ω (see for example [14, Theorem
+2.10]).
+The following approximation results in BV hold (refer to [4], [18, Theorem 1.17] for
+the Euclidean case and to [1, Proposition 2.1] for the Finsler case).
+7
+
+Proposition 2.2. Let f P BV pΩq, then there exists a sequence tfkukPN Ď C8pΩq such
+that:
+lim
+kÑ`8
+ż
+Ω
+|fk ´ f| dx “ 0
+and
+lim
+kÑ`8 |Dfk|F pΩq “ |Df|F pΩq.
+Proposition 2.3. Let E be a set of finite perimeter in Ω. A sequence of C8 sets tEkuk
+exists, such that:
+lim
+kÑ`8
+ż
+Ω
+|χEk ´ χE| dx “ 0
+and
+lim
+kÑ`8 |DχEk|F pΩq “ PF pE; Ωq.
+3 An anisotropic trace inequality
+In this section we prove a trace inequality in BV with respect to the anisotropic total
+variation. We first give the result in a general case (Proposition 3.1), then we refine the
+constants involved in the inequality by requiring more regularity on the boundary of Ω
+(Proposition 3.2).
+Firstly, let us set
+qpyq “ lim
+ρÑ0` sup
+$
+’
+’
+&
+’
+’
+%
+ż
+BΩ
+χAFpνqdHN´1
+|DχA|F pΩq
+: A Ă Ω X Bρpyq, |A| ą 0, PF pA; Ωq ă `8
+,
+/
+/
+.
+/
+/
+-
+and Q “ supyPBΩ qpyq. The following inequality generalizes the trace inequality given in
+[4, Theorem 4].
+Proposition 3.1. Let Ω be a bounded open set with HN´1pΩq ă `8, and let u be a
+function in BV pΩq. Then for any ε ą 0, it holds
+ż
+BΩ
+|u|FpνqdHN´1 ď pQ ` εq|Du|F pΩq ` cpΩ, εq
+ż
+Ω
+|u|dx,
+(3.1)
+where cpΩ, εq does not depend on u.
+Proof. Let us fix y P BΩ and ρpyq ą 0 such that
+ż
+BΩ
+χBFpνqdHN´1 ď pQ ` εq|DχB|F pΩq,
+8
+
+for any B Ă Ω X Bρpyqpyq with PF pB; Ωq ă `8.
+If sptpuq Ă Bρpyqpyq, then
+ż
+BΩ
+|u|FpνqdHN´1 “
+ż
+BΩ
+Fpνq
+ˆż `8
+0
+χt|u|ątupyqdt
+˙
+dHN´1
+“
+ż `8
+0
+ˆż
+BΩ
+Fpνqχt|u|ątupyqdHN´1
+˙
+dt ď pQ ` εq
+ż `8
+0
+|Dχt|u|ątu|F pΩqdt.
+By using the coarea formula, we have
+ż
+BΩ
+|u|FpνqdHN´1 ď pQ ` εq|D|u||F pΩq ď pQ ` εq|Du|F pΩq.
+Let tBρpyquyPBΩ be a cover of BΩ and let us extract a finite sub-cover B1, . . . , Bk.
+Now, considering a partition of unity ϕ1, . . . , ϕk such that
+0 ď ϕi ď 1,
+ϕi P C1
+0pBiq,
+kÿ
+i“1
+ϕipyq “ 1
+if y P BΩ.
+If f P BV pΩq, then
+ż
+BΩ
+|u| F pνq dHN´1 ď pQ ` εq
+ˇˇˇˇˇD
+˜ kÿ
+i“1
+ϕiu
+¸ˇˇˇˇˇ
+F
+pΩq
+ď pQ ` εq
+kÿ
+i“1
+p|ϕiDu|F pΩq ` |uDϕi|F pΩqq
+“ pQ ` εq
+kÿ
+i“1
+ˆż
+Ω
+ϕid|Du|F `
+ż
+Ω
+ud |Dϕi|F pΩq
+˙
+ď pQ ` εq |Du|F pΩq ` cpΩ, εq
+ż
+Ω
+|u| dx.
+If the boundary of Ω is sufficilently smooth, we can show that Q can be taken equal
+to 1 and ε “ 0. More precisely, we have the following.
+Proposition 3.2. Let Ω be a bounded open connected set of class C2. Then there exists
+a positive constant c such that
+ż
+BΩ
+|u| Fpνq dHN´1 ď |Du|F pΩq ` c
+ż
+Ω
+|u| dx,
+@u P BV pΩq.
+(3.2)
+Proof. Since Ω is C2, then a uniform sphere condition of radius r ą 0 holds, in the sense
+that for every point y P BΩ there exists z P Ω such that y P Brpzq Ă Ω. Let R Ps0, `8r
+be the maximum of the principal radii of curvature of BW. Such maximum exists being
+9
+
+F (and F o) strongly convex. If κF
+1 , . . . , κF
+N´1 are the anisotropic principal curvatures,
+we have that
+κiF pyq ď 1
+µ,
+i “ 1, . . . , N ´ 1,
+with µ “
+r
+R ([7, Lemma 5.4]). Therefore, in the set Ω µ
+2 :“
+␣
+x P Ω : dF pxq ă µ
+2
+(
+, it
+holds that dF is C2 ([7, Lemma 4.1 and Theorem 4.16]) and
+κF
+i pyq
+1 ´ κF
+i pyqdF pxq ď
+$
+&
+%
+0
+if κF
+i ď 0
+2
+µ
+if 0 ă κF
+i ď 1
+µ,
+where y P BΩ is the anisotropic projection of x P Ω on BΩ. Then
+N´1
+ÿ
+i“1
+κiFpyq
+1 ´ κiFpyqdF pxq ď 2N ´ 1
+µ
+(3.3)
+for x P Ω µ
+2 . We may restrict ourselves to the case u is nonnegative and smooth. Inte-
+grating by parts and recalling that Fp∇dF pxqq “ 1 in Ω, it holds that
+ż
+Ω
+´∆FdF pxq upxq
+´µ
+2 ´ dF pxq
+¯`
+dx
+“
+ż
+Ω
+Fξp∇dF pxqq ¨ ∇upxq
+´µ
+2 ´ dF pxq
+¯`
+dx
+´
+ż
+Ω µ
+2
+upxqFξp∇dF pxqq ¨ ∇dF pxq dx ´ µ
+2
+ż
+BΩ
+upxqFξp∇dF pxqq ¨ ν dx
+“
+ż
+Ω
+Fξp∇dF pxqq¨∇upxq
+´µ
+2 ´ dF pxq
+¯`
+dx´
+ż
+Ω µ
+2
+upxq dx`µ
+2
+ż
+BΩ
+upyq FpνqdHN´1.
+Now, we estimate the term
+ż
+Ω
+Fξp∇dF pxqq ¨ ∇upxq
+´µ
+2 ´ dF pxq
+¯`
+dx.
+By (2.4) and (2.6) it holds that
+ż
+Ω
+Fξp∇dF pxqq ¨ ∇upxq
+´µ
+2 ´ dF pxq
+¯`
+dx ě ´µ
+2
+ż
+Ω
+Fp∇upxqq dx.
+(3.4)
+On the other hand, the change of variable formula (2.1) gives that
+ż
+Ω
+Fξp∇dF pxqq ¨ ∇upxq
+´µ
+2 ´ dF pxq
+¯`
+dx
+“
+ż
+BΩ
+Fpνq
+ż µ
+2
+0
+´µ
+2 ´ t
+¯ d
+dtrupφpy, tqqsJpy, tqdt dHN´1.
+10
+
+Integrating by parts and using the fact that Jpy, 0q “ 1, the above integral becomes
+´ µ
+2
+ż
+BΩ
+upyqFpνqdHN´1pyq ´
+ż
+BΩ
+Fpνq
+ż µ
+2
+0
+upφpy, tqq
+´µ
+2 ´ t
+¯ dJ
+dt dt dHN´1pyq
+`
+ż
+BΩ
+Fpνq
+ż µ
+2
+0
+upφpy, tqqJpy, tqdt dHN´1pyq
+ď ´µ
+2
+ż
+BΩ
+upyqFpνqdHN´1pyq ` pN ´ 1q
+ż
+BΩ
+Fpνq
+ż µ
+2
+0
+upyqJpy, tq dt dHN´1pyq
+`
+ż
+BΩ
+Fpνq
+ż µ
+2
+0
+upφpy, tqqJpy, tqdt dHN´1pyq
+“ ´µ
+2
+ż
+BΩ
+upyqFpνqdHN´1pyq ` N
+ż
+Ω
+u dx
+where in the inequality we have used (2.10) and the bound (3.3). Hence, joining with
+(3.4) it holds that
+ż
+BΩ
+upyqFpνqdHN´1 ď |Du|F pΩq ` 2N
+µ
+ż
+Ω
+upxq dx.
+Remark 3.3. Using the notation of the above theorem, we explicitly observe that the
+constant in (3.2) is
+c “ 2N
+µ .
+4 Interior approximation
+Now we provide an approximation result for BV -functions by smooth functions with
+compact support in Ω.
+We preliminary state two useful lemmas. Firstly, we recall from [20, Lemma 3.2] the
+following result on diffeomorphic perturbations of sets Ω with Lipschitz boundary. We
+denote by ι and I the identical vector and matrix function, respectively.
+Lemma 4.1. Let Ω Ă RN be a bounded open set with Lipschitz boundary. Then there
+exists τ0 ą 0 and, for 0 ď τ ď τ0, a family of C8´diffeomorphisms Φτ : RN Ñ RN with
+inverses Ψτ such that
+• Φ0 “ Ψ0 “ ι;
+• Φτ Ñ ι and Ψτ Ñ ι as τ Ñ 0 uniformly on RN;
+• ∇Φτpxq Ñ I and ∇Ψτpxq Ñ I as τ Ñ 0 uniformly with respect to x on RN;
+11
+
+• ΦτpΩq Ť Ω for all τ P p0, τ0s.
+Now, we give the anisotropic version of the change of coordinates formula for BV -
+functions, stated in [18, Lemma 10.1].
+Lemma 4.2. Let u be a function in BVlocpΩq, Φ : RN Ñ RN be a diffeomorphism and
+A Ť Ω. Then
+ˇˇDpu ˝ Φ´1q
+ˇˇ
+F pΦpAqq “ |HDu|F pAq,
+(4.1)
+where H “ | det ∇Φ|r∇Φs´1.
+Proof. Let us consider u P C1pΩq and g P C1
+0pA; RNq, then the following change of area
+formula holds
+ż
+ΦpAq
+pg ˝ Φ´1q ¨ ∇pu ˝ Φ´1qdx “
+ż
+ΦpAq
+pg ˝ Φ´1q ¨ pp∇u ˝ Φ´1q∇Φ´1qdx
+“
+ż
+A
+g ¨ p∇up∇Φ´1 ˝ Φqq| det ∇Φ|dz
+“
+ż
+A
+g ¨ pH∇uqdz.
+(4.2)
+Thus, the thesis (4.1) holds for u in C1pΩq, that is
+ż
+ΦpAq
+Fp∇pu ˝ Φ´1qqdx “
+ż
+A
+FpH∇uqdx.
+Suppose now that u P BVlocpΩq.
+By Proposition 2.2 we can approximate u by a
+sequence tuiu Ă C8. Moreover, the corresponding functions ui˝Φ´1 converge to u˝Φ´1
+in L1pAq.
+Hence, we can pass to the limit in (4.2), obtaining
+ż
+ΦpAq
+pg ˝ Φ´1q ¨ dDpu ˝ Φ´1q “
+ż
+A
+g ¨ H dDu “
+ż
+A
+g ¨ pHνqd |Du|
+(4.3)
+where ν is obtained by differentiating Du with respect to |Du|.
+If F opgq ď 1, then also F opg ˝ Φ´1q ď 1 and sptpg ˝ Φ´1qq Ď ΦpAq. Therefore, by
+definition of total variation with respect to F, we have
+ż
+A
+g ¨ pHνqd |Du| ď
+ˇˇDpu ˝ Φ´1q
+ˇˇ
+F pΦpAqq,
+(4.4)
+The inequality (2.4) implies that
+sup
+F opgqď1
+ż
+A
+g ¨ pHνq d |Du| “ |HDu|F pAq.
+Hence, taking the supremum on the left hand side in (4.4), it holds
+|HDu|F pAq ď
+ˇˇDpu ˝ Φ´1q
+ˇˇ
+F pΦpAqq.
+12
+
+For the reverse inequality, we consider g “ γ ˝ Φ P C1
+0pA; RNq, where γ P C1
+0pΦpAq; RNq
+and F opγq ď 1. Therefore g ˝ Φ´1 “ γ and by (4.3), we have
+ż
+ΦpAq
+γ ¨ dDpu ˝ Φ´1q “
+ż
+A
+pγ ˝ Φq ¨ pHνqd |Du| pAq
+ď
+ż
+A
+F opγ ˝ Φq|H|d|Du|F ď
+ż
+A
+|H| d |Du|F .
+Hence, we have
+ˇˇDpu ˝ Φ´1q
+ˇˇ
+F pΦpAqq ď |HDu|F pAq.
+At this stage, we are in position to state the main approximation result.
+Theorem 4.3. Let Ω Ă RN be an open bounded set with Lipschitz boundary and let
+u P BV pΩq X LppΩq for some p P r1, 8q. Then there exists a sequence tukukPN Ď C8
+0 pΩq
+such that, for any q P r1, ps,
+uk Ñ u in LqpΩq
+and
+|Duk|F pRNq Ñ |Du|F pRNq.
+Proof. Let us fix a family pΦτq0ďτďτ0 of diffeomorphisms fron RN to RN with inverses
+pΨτq0ďτďτ0 according to Lemma 4.1, and consider
+uτ :“ u ˝ Ψτ
+for
+τ P r0, τ0s.
+By construction uτ “ 0 a.e. outside a compact subset of Ω. By [20, Theorem 3.1], we
+know that uτ P LppΩq for all τ P r0, τ0s, uτ Ñ u in LppΩq and also in LppRNq.
+The change of coordinates formula in Lemma 4.2 implies that
+|Duτ|F pRNq “
+ż
+RN
+ˇˇp∇ΨτqT ˇˇ |detp∇Φτq| d |Du|F .
+Thus uτ P BV pRNq and, since the integrand on the right hand side uniformly converges
+to 1,
+lim
+τÑ0` |Duτ|F pRNq “ |Du|F pRNq
+It remains only to prove that there exists vτ P C8
+0 pΩq such that
+||uτ ´ vτ||p ă τ
+and
+ˇˇ|Duτ|F pRNq ´ |Dvτ|F pRNq
+ˇˇ ă τ.
+The first convergence (in Lp) has been proved in [20, Theorem 3.1]; meanwhile the
+convergence of the total variation is based on the following argument.
+For any ε ą 0, let us consider the mollification uε :“ uτ ˚ ηε, where ηεpxq “:
+ε´nηpε´1xq, for the standard mollifier η. Hence uε Ñ uτ in LppΩq and for the zero
+extensions, in L1pRNq [3, Proposition 3.2.c].
+13
+
+Then, by the lower semicontinuity of the anisotropic total variation, we have
+|Duτ|F pRNq ď lim
+εÑ0` inf |Duε|F pRNq.
+Therefore, it remains to prove the opposite inequality
+lim
+εÑ0` sup |Duε|F pRNq ď |Duτ|F pRNq.
+(4.5)
+Let us choose ϕ P C8
+0 pRN, RNq with F opϕq ď 1 and calculate
+ż
+RN uε ˚ divpϕqdx
+ż
+RN uτpηε ˚ div ϕqdx “
+ż
+RN uτ divpηε ˚ ϕqdx ď |Duτ|F pRNq, (4.6)
+where the inequality in the last term holds since F opηε ˚ ϕq ď 1. Indeed, by using the
+1-homogeneity of F o and Jensen’s Inequality (see, for instance, [22, Lemma 1.8.2]), we
+gain that
+F o
+ˆż
+RN ηǫpx ´ yqϕpyqdy
+˙
+ď
+ż
+RN F opηǫpx´yqϕpyqqdy “
+ż
+RN ηεpx´yqF opϕpyqqdy ď 1.
+Hence, by passing to the limit in (4.5), we reach the inequality (4.6) by the arbitrariness
+of ϕ.
+5 The first Robin eigenvalue of the Finsler p-Laplacian as p Ñ 1
+In this Section, we give an application of the results proved above to a Robin eigenvalue
+problem. More precisely, our aim is analyze the Γ-limit of the functional
+Jppϕq “
+ż
+Ω
+F pp∇ϕqdx ` β
+ż
+BΩ
+|ϕ|p FpνqdHN´1
+ż
+Ω
+|ϕ|p dx
+,
+ϕ P W 1,ppΩqzt0u,
+(5.1)
+where Ω is a bounded, connected, sufficiently smooth open set, p ą 1 and β P R, and
+prove an isoperimetric inequality for the limit, as p Ñ 1`, of the first eigenvalue
+λ1pΩ, p, βq “
+inf
+ϕPW 1,ppΩq
+ϕ‰0
+Jppϕq,
+(5.2)
+depending on the value of the parameter β. A key point for proving this result is the
+convergence of the functional Jp.
+We first recall the following existence result for (5.2) holds.
+Theorem 5.1 ([9, 12]). Let p ą 1, β P R and Ω bounded Lipschitz domain. Then there
+exists a minimum u P C1,αpΩq X CpΩq of (5.2) that satisfies
+#
+´Qpu “ λpΩ, p, βq |u|p´2 u
+in Ω
+F p´1p∇uqFξp∇uq ¨ ν ` βFpνq |u|p´2 u “ 0
+on BΩ.
+(5.3)
+Moreover, u does not change sign in Ω. Finally, λ1pΩ, p, βq is positive if β ą 0, while is
+negative if β ă 0.
+14
+
+5.1 The case p “ 1
+In order to study the limit case of Jp as p goes to 1, we consider the functional
+Jpϕq “
+|Dϕ|F pΩq ` mintβ, 1u
+ż
+BΩ
+|ϕ| FpνqdHN´1
+ż
+Ω
+|ϕ| dx
+,
+(5.4)
+where ϕ P BV pΩq and u ı 0. Hence, we study the associated minimum problem
+ΛpΩ, βq “
+inf
+ϕPBV pΩq
+ϕı0
+Jpϕq.
+(5.5)
+Depending on β, we will impose different assumptions on the regularity of the domain.
+Indeed:
+• if β ě 0, we will suppose that BΩ is Lipschitz;
+• if ´1 ă β ă 0, we will assume that BΩ is C2.
+In particular, this difference depends on the fact that in the case β ă 0 we use the trace
+inequality, studied in Section 3.
+Finally, if β ď ´1 the problem is not well posed; indeed if β ă ´1, then ΛpΩ, βq “ ´8
+while if β “ ´1, Λ is finite but can be not achieved, also in the case of smooth domains.
+For further details, we refer the reader to the Euclidean case treated in [10].
+Let us discuss the presence of the term mintβ, 1u in (5.4). For any value of β, it could
+seem more natural to study the problem
+λpΩ, 1, βq “
+inf
+ϕPBV pΩq
+ϕ‰0
+|Dϕ|F pΩq ` β
+ż
+BΩ
+|ϕ| FpνqdHN´1
+ż
+Ω
+|ϕ| dx
+.
+(5.6)
+Actually, we have that for β ě 1 it holds
+λpΩ, 1, βq “ ΛpΩ, βq “ hF pΩq,
+where hF pΩq is the first Cheeger constant of Ω in the Finsler setting (see e.g. [6]):
+hF pΩq “
+inf
+ϕPBV pΩq
+ϕı0
+|Dϕ|F pRNq
+ż
+Ω
+|ϕ| dx
+“ inf
+EĎΩ
+PF pE; Ωq
+|E|
+.
+(5.7)
+Indeed, in this case, it is immediate to see that
+λpΩ, 1, βq ě hF pΩq.
+15
+
+On the other hand, if u is a minimizer of (5.6), then, by Theorem 4.3, there exists
+uk P C8
+0 pΩq such that
+uk
+Lq
+ÝÑ u,
+||∇uk||L1pΩq
+L1
+ÝÑ |Du|F pRNq,
+for any q ď
+N
+N ´ 1. Therefore
+λpΩ, 1, βq ď
+lim
+kÑ`8
+ż
+Ω
+Fp∇ukqdx
+ż
+Ω
+|uk| dx
+“ hF pΩq.
+Now we focus on the possibility of studying the minimization problem (5.5) restricting
+our analysis to characteristic functions. Hence if E Ď Ω, we have
+JpχEq “
+PF pE; Ωq ` mint1, βu
+ż
+BΩXB˚E
+FpνEqdHN´1
+|E|
+.
+By denoting
+RpE, βq :“ JpχEq,
+we consider the minimization problem
+ℓpΩ, βq “ inf
+EĎΩ RpE, βq.
+(5.8)
+Before proving the equivalence between problems (5.5) and (5.8), we need the following
+result on the lower semicontinuity of the numerator of the functional J.
+Lemma 5.2. Let β ě ´1. The functional
+Gpuq “ |Du|F pΩq ` mint1, βu
+ż
+BΩ
+|u|FpνqdHN´1
+is lower semicontinuous on BV pΩq with respect to the topology of L1pΩq.
+Proof. If β ě 0, the lower semicontinuity (with Ω Lipschitz) follows immediately by the
+lower semicontinuity of each term. Then we assume β ă 0 (and Ω in C2). The proof is
+an adaptation of [21, Proposition 1.2] to the Finsler case.
+Let u P BV pΩq, and let us consider a sequence tukukPN Ď BV pΩq converging to u in
+L1pΩq; we have the following estimate
+Gpuq ´ Gpukq ď |Du|F pΩq ´ |Duk|F pΩq `
+ż
+BΩ
+|u ´ uk|FpνqdHN´1.
+(5.9)
+Now, for a fixed δ ą 0, let us define Ωδ “ tx P Ω : dEpxq ă δu, where dE is the standard
+Euclidean distance to the boundary of Ω ; moreover let us consider vδ “ p1´χδqpu´ukq,
+16
+
+where χδ is a cut-off function such that χδ “ 1 in ΩzΩδ and |∇χδ| ď 2
+δ in Ω. The trace
+inequality (3.2) applied to vδ gives
+ż
+BΩ
+|u´uk|FpνqdHN´1 ď |Dpu ´ ukq|F pΩδq` 2b
+δ
+ż
+Ωδ
+|u´uk|dx`c
+ż
+Ωδ
+|u´uk|dx. (5.10)
+Moreover, we have
+|Dpu ´ ukq|F pΩδq ď |Du|F pΩδq ` |Duk|F pΩδq ` |Dpu ´ ukq|F pBpΩzΩδqq,
+(5.11)
+but last term is zero on a set of δ’s of positive measure because u ´ uk P BV pΩq, for all
+k P N. Hence, by (5.9)-(5.10)-(5.11), we gain:
+Gpuq ´ Gpukq ď |Du|F pΩq ` |Du|F pΩδq ´ |Duk|F pΩzΩδq `
+ˆ2b
+δ ` c
+˙ ż
+Ωδ
+|u ´ uk|dx.
+By the lower semicontinuity of the functional |Duk|F pΩzΩδq in L1pΩzΩδq, we have that
+lim sup
+kÑ`8
+rGpuq ´ Gpukqs ď 2|Du|F pΩδq.
+The conclusion follows by sending δ Ñ 0`.
+At this stage, we state the main existence result of the minimum problem (5.5).
+Theorem 5.3. For any β ą ´1, there exists a minimum to problem (5.5). In particular,
+it holds
+ΛpΩ, βq “ ℓpΩ, βq.
+Moreover, if u P BV pΩq is a minimum of (5.5), then
+ΛpΩ, βq “ Rptu ą tu, βq,
+for some t P R.
+Proof. Let un be a minimizing sequence in BV pΩq of (5.5), such that }un}L1pΩq “ 1. If
+β ą 0, then un is bounded in BV pΩq and hence
+un
+˚á u in BV pΩq
+and
+un
+L1
+ÝÑ u.
+In particular, if β ě 1, Jpunq “ |Dun|F pRNq, by using the lower semicontinuity of the
+anisotropic total variation [2], we obtain that
+Jpuq ď lim inf
+n
+Jpunq.
+Hence u is the minimum of the functional J.
+If 0 ă β ă 1, let Ωδ “ tx P Ω | dEpxq ă δu, with δ ą 0. We have
+|Dun|F pΩq “ |Dun|F pΩzΩδq ` |Dun|F pΩδq ě |Dun|F pΩzΩδq ` β |Dun|F pΩδq
+17
+
+and hence
+Jpunq ě |Dun|F pΩzΩδq ` β
+„
+|Dun|F pΩδq `
+ż
+BΩ
+|un| FpνqdHN´1
+
+.
+Moreover, by the lower semicontinuity of J, we have
+lim inf
+n
+Jpunq ě |Dun|F pΩzΩδq ` β |Dun|F pRNzpΩzΩδqq.
+By using the fact that u P BV pΩq, we obtain that
+lim inf
+n
+Jpunq ě Jpuq,
+as δ Ñ 0.
+Now, let us take ´1 ă β ă 0. It easily seen that Jpunq ď C. Using the trace inequality
+(3.2), we obtain
+Jpunq ě p1 ` βq |Dun|F pΩq ` cβ ě cβ
+and
+|Dun|F pΩq ď
+C
+1 ` β ´
+βc
+1 ` β .
+Being un P BV pΩq and by the fact that the functional J is lower semicontinuous (proved
+in Lemma 5.2), we have that u is a minimum of J.
+Now, we want to prove last part of the Theorem. Obviously, we have
+ΛpΩ, βq ď ℓpΩ, βq.
+To prove the reverse inequality, we take u P BV pΩq a minimizer of (5.5). By using the
+coarea formula
+|Du|F pΩq “
+ż `8
+´8
+PF ptu ą tu, Ωqdt,
+we have
+ΛF pΩ, βq “
+ż `8
+´8
+PFptu ą tu, Ωqdt ` mintβ, 1u
+ż `8
+´8
+HN´1pBΩ X Btu ą tuqFpνqdt
+ż `8
+´8
+|tu ą tu| dt
+“
+ż `8
+´8
+Rptu ą tu, βq |tu ą tu| dt
+ż `8
+´8
+|tu ą tu| dt
+ě inf
+EĎΩ RpE, βq
+“ ℓpΩ, βq.
+18
+
+This shows that ΛpΩ, βq “ ℓpΩ, βq and, in particular, we have that
+ż `8
+´8
+tRptu ą tu, βq ´ ℓpΩ, βqu |tu ą tu| dt “ 0
+and using the definition of ℓpΩ, βq we observe that the integrand is nonnegative.
+In
+particular, u ı 0 and we have that ΛpΩ, βq “ Rptu ą tu, βq.
+5.2 Γ-convergence of Jp
+Now we will prove that the functional Jp Γ´converges to the functional J, as p Ñ 1`.
+Definition 5.4. A functional Jp Γ-converges to J as p Ñ 1` in the weak˚ topology of
+BV pΩq if, for any u P BV pΩq, the following hold:
+(i) For any sequence up P BV pΩq which converges to u weak˚ in BV pΩq as p Ñ 1`,
+then
+lim inf
+pÑ1` Jppupq ě Jpuq.
+(5.12)
+(ii) There exists a sequence up P W 1,ppΩq which converges to u weak˚ in BV pΩq as
+p Ñ 1`, such that
+lim sup
+pÑ1` Jppupq ď Jpuq.
+(5.13)
+Now, we are in position to prove the convergence theorem for the functional Jp.
+Theorem 5.5. Let β ą ´1, then Jp Γ-converges to J as p Ñ 1`.
+Proof. Let us suppose up P W 1,ppΩq and ||up||LppΩq “ 1.
+We give the proof by dis-
+tinguishing the possible values of β.
+In any cases, we will have to prove (5.12) and
+(5.13).
+The case β ě 1. Let us fix a sequence up P W 1,ppΩq weak˚ converging to u in BV pΩq,
+as p Ñ 1`. By using the H¨older-type inequality contained for example in [11, Proposition
+A.1], we have:
+ˆż
+Ω
+Fp∇upqdx `
+ż
+BΩ
+|up|FpνqdHN´1
+˙p
+ď
+ˆż
+Ω
+Fp∇upqpdx `
+ż
+BΩ
+|up|pFpνqdHN´1
+˙
+p|Ω| ` PF pΩqqp´1 .
+Hence we have
+lim inf
+pÑ1` Jppupq ě |Du|F pΩq `
+ż
+BΩ
+|u|FpνqdHN´1 “ |Du|F pRNq “ Jpuq.
+This proves (5.12).
+19
+
+To give the proof of (5.13), we observe that, by Theorem 4.3, there exists a se-
+quence tukukPN Ď C8
+0 pΩq such that, for any q P r1, ps, uk converges to u in LqpΩq
+and ||Fp∇ukq||L1pΩq converges to |Du|F pRNq as k Ñ `8. Moreover, it easily seen that
+that ||Fp∇ukq||LppΩq converges to ||Fp∇ukq||L1pΩq and hence we have that there exists a
+subsequence pk Ñ 1`, as k Ñ `8, such that ||FpDukq||pk
+Lpk pΩq converges to |Du|F pRNq
+as k Ñ `8. This implies that lim supkÑ`8 Jpkpukq ě Jpuq, that concludes the proof of
+(5.13).
+The case 0 ď β ă 1. Let us consider a sequence up weak˚ converging to u in BV pΩq,
+as p Ñ 1`. A simple application of the Young inequality ap ě pab ´ pp ´ 1qb
+p
+p´1 with
+b “ 1
+p, yields to
+Jppupq “
+ż
+Ω
+F pp∇upqdx ` β
+ż
+BΩ
+|up|pFpνqdHN´1
+ě
+ż
+Ω
+Fp∇upqdx ` β
+ż
+BΩ
+|up|FpνqdHN´1 ´ p ´ 1
+p
+p|Ω| ` PF pΩqq ,
+Therefore, the conclusion (5.12) follows by applying the Proposition 5.2.
+In order to get the second claim, Proposition 2.2 assures the existence of a sequence
+uk P C8pΩq strongly converging to u in L1pΩq and }FpDukq}L1pΩq converges to |Du|F pΩq,
+as k Ñ `8. Moreover ukFpνq converges to uFpνq in L1pBΩ, HN´1q. An argument sim-
+ilar to the previous case leads us to say that }FpDukq}LpkpΩq converges to |Du|F pΩq and
+ż
+BΩ
+|uk|pFpνqdHN´1 converges to
+ż
+BΩ
+|u|FpνqdHN´1, as k Ñ `8. Hence the sequence
+tukukPN satisfies (5.13).
+The case ´1 ă β ă 0. Let us consider a sequence up P W 1,ppΩq weak˚ converging to
+u in BV pΩq, as p Ñ 1`.
+For any δ ą 0, let us set Ωδ “ tx P Ω : dEpxq ă δu and consider a smooth function ψ
+equal to zero in ΩzΩδ and to one on BΩ, such that |∇ψ| ď 2
+δ .
+The trace inequality (3.2) applied to the function v “ pu ´ |up|p´1upqψ gives
+ż
+BΩ
+|u ´ |up|p´1up|FpνqdHN´1
+ď |Dpu ´ |up|p´1upq|F pΩδq `
+ˆ2b
+δ ` c
+˙ ż
+Ωδ
+|u ´ |up|p´1up|dx
+ď |Du|F pΩδq `
+ż
+Ωδ
+Fp∇p|up|p´1upqqdx `
+ˆ2b
+δ ` c
+˙ ż
+Ωδ
+|u ´ |up|p´1up|dx,
+(5.14)
+where we have used that |Dpu´|up|p´1upq|F pBΩδq “ 0 for a set of δ’s of positive measure
+because u ´ |up|p´1up P BV pΩq.
+20
+
+By using (5.14), we have
+Jpuq ´ Jppupq “ |Du|F pΩq ´
+ż
+Ω
+F pp∇upqdx ` β
+ż
+BΩ
+p|u| ´ |up|p´1upqFpνqdHN´1
+ď |Du|FpΩq ´
+ż
+Ω
+F pp∇upqdx ` |β||Du|F pΩδq
+` |β|
+ż
+Ωδ
+Fp∇p|up|p´1upqqdx ` |β|
+ˆ2b
+δ ` c
+˙ ż
+Ωδ
+|u ´ |up|p´1up|dx :“ A.
+(5.15)
+Since
+1
+|β| ą 1, we have
+A ď 2|Du|F pΩδq ` |Du|F pΩzΩδq ´
+ż
+Ω
+Fp∇upqpdx `
+ż
+ΩzΩδ
+Fp∇p|up|p´1upqqdx
+`
+ˆK
+δ ` c
+˙ ż
+Ωδ
+|u ´ |up|p´1up|dx.
+(5.16)
+The Young inequality gives that
+ż
+Ω
+Fp∇p|up|p´1upqqdx “
+ż
+Ω
+p|up|p´1Fp∇upqdx
+ď
+ż
+Ω
+F pp∇upqdx ` pp ´ 1q
+ż
+Ω
+|up|pdx.
+(5.17)
+Furthermore by (5.15), (5.16) and (5.17), we have that
+Jpuq ´ Jppupq ď 2|Du|F pΩδq ` |Du|F pΩzΩδq ´
+ż
+ΩzΩδ
+Fp∇p|up|p´1upqqdx
+` pp ´ 1q
+ż
+Ω
+|up|pdx `
+ˆK
+δ ` c
+˙ ż
+Ωδ
+|u ´ |up|p´1up|dx.
+Since up converges to u in LqpΩq, then |up|p´1up converges to u in L1pΩq, as p Ñ 1`.
+Hence, by taking p Ñ 1`, we have
+lim sup
+pÑ1` rJpuq ´ Jppupqs ď 2|Du|F pΩδq.
+By sending δ Ñ 0`, we obtain (5.12).
+Finally, the inequality (5.13) is obtained as in the previous case.
+The proof of the Γ-convergence of the functional Jp is useful to prove the convergence
+of the eigenvalues and eigenfunction, as p Ñ 1`.
+Proposition 5.6. For any β ą ´1, it holds
+lim
+pÑ1` λ1pΩ, p, βq “ ΛpΩ, βq.
+Moreover, the minimizers up P W 1,ppΩq of (5.1), with }u}LppΩq “ 1, weak˚ converge to
+a minimizer u P BV pΩq of (5.4) as p Ñ 1`.
+21
+
+Proof. The Theorem 5.5 assures the existence of a sequence wp converging to a fixed
+minimizer ¯u of (5.4). Let us consider the sequence of minimizers up of (5.4), we have:
+lim sup
+pÑ1` Jppupq ď lim sup
+pÑ1` Jppwpq ď Jp¯uq “ ΛpΩ, βq.
+(5.18)
+This means that Jppupq is upper bounded for any p ą 1. If β ă 0, by the trace inequality
+(3.2), we have that
+ż
+Ω
+F pp∇upqdx ď ΛpΩ, βq ´ β|Dpup
+pq|F pΩq ´ βc,
+and, by (5.17) and (2.1), that
+p1 ` βqap
+ż
+Ω
+|∇up|pdx ď ΛpΩ, βq ´ βc ´ βpp ´ 1q.
+Hence, by the compactness, we have that up is upper bounded in BV pΩq. If β ě 0, this
+directly follows from (5.18). Therefore up weak˚ converges to u in BV pΩq.
+Finally, by (5.12) and (5.18), we have that
+ΛpΩ, βq “ Jp¯uq ď Jpuq ď lim inf
+pÑ1` Jppupq ď lim sup
+pÑ1` Jppupq ď ΛpΩ, βq,
+and hence the conclusion by observing that λ1pΩ, p, βq “ Jppupq.
+5.3 An isoperimetric inequality
+Here we treat the shape optimization problem for ΛpΩ, βq. To this aim, we briefly recall
+the properties of the eigenvalue problem λ1pWR, p, βq and then we prove an explicit
+computation for ΛpWR, βq. By Theorem 5.1, a minimizer of (5.2) solves the following
+problem:
+#
+´Qpu “ λ1pWR, p, βq |u|p´2 u
+in WR
+pFp∇uqqp´1Fξp∇uq ¨ ν ` βFpνq |u|p´2 u “ 0
+on BWR.
+(5.19)
+In particular, the following result holds (refer to in [9] for the positive values of the Robin
+parameter).
+Theorem 5.7. If up P C1,αpWRq X CpWRq is a positive solution of (5.19), then there
+exists a monotone function ϕp “ ϕpprq, r P r0, Rs, such that ϕp P C8p0, RqXC1pr0, Rsq,
+and
+$
+’
+&
+’
+%
+uppxq “ ϕppF opxqq
+in WR
+ϕ
+1
+pp0q “ 0
+|ϕ
+1
+ppRq|p´2ϕ1
+ppRq ` βϕppRqp´1 “ 0.
+(5.20)
+Moreover, ϕp is decreasing if β ą 0, while is increasing if β ă 0.
+22
+
+We first compute ΛpWR, βq.
+Proposition 5.8. If β ą ´1, then
+ΛpWR, βq “ ˆβhF pWRq “ ˆβ N
+R ,
+(5.21)
+where ˆβ “ mintβ, 1u.
+Proof. If β ě 0, we recall that
+ΛpWR, βq “
+inf
+EĎWR JpχEq.
+By using Theorem 5.3 and the isoperimetric inequality, we have
+RpE, βq “ JpχEq
+“
+PF pE, WRq ` ˆβ
+ż
+BWRXBE
+FpνEqdHN´1
+|E|
+ě ˆβ PF pEq
+|E|
+ě ˆβ PF pWrq
+|Wr|
+ě ˆβ PF pWRq
+|WR|
+“ ˆβ N
+R ,
+where Wr is the wulff shape of radius r ă R, with |Wr| “ |E|.
+This proves ΛpWR, βq ě ˆβ N
+R . For the reverse inequality we take E “ WR, hence
+ΛpWR, βq “ ℓpWR, βq ď RpWR, βq “ ˆβ N
+R .
+Now, we study the case ´1 ă β ă 0 and we will make use of the Γ-convergence.
+Hence, let up P W 1,ppWRq a minimizer of (5.2). We know, thanks to the Proposition
+5.6, that
+lim
+pÑ1` λ1pWR, p, βq “ ΛpWR, βq
+and we take up “ ϕp as in (5.20). So, the minimizer converges strongly in L1pWRq to
+u P BV pWRq, almost everywhere in WR and up
+˚á u in BV pΩq for p Ñ 1`. Moreover,
+up is radially increasing, hence u is nondecreasing and this implies that its superlevel
+sets tu ą tu are concentric Wulff shapes tr ă F opxq ă Ru and, by Theorem 5.3, it holds
+that
+ΛpWR, βq “ N
+R
+p r
+RqN´1 ` β
+1 ´ p r
+RqN´1
+for some r P r0, Rr. Therefore, by minimizing the function
+fptq “ tN´1 ` β
+1 ´ tN
+t P r0, 1r,
+we observe that the minimum is attained at t “ 0. Hence, the thesis follows.
+23
+
+Finally, we prove an isoperimetric inequality for ΛpΩ, βq when a volume constraint
+holds: if β ě 0, the Wulff shape is a minimizer and, if β ă 0, it is a maximizer for
+ΛpΩ, βq.
+Proposition 5.9. If β ě 0 and WR is the wulff shape of radius R and |WR| “ |Ω|, then
+ΛpWR, βq ď ΛpΩ, βq.
+If ´1 ă β ă 0, then
+ΛpWR, βq ě ΛpΩ, βq.
+Proof. If β ě 0, by using the same argument of the Proposition 5.8, we have that then
+RpE, βq “ JpχEq ě ˆβ PF pWRq
+|WR|
+“ ˆβ N
+R “ ΛpWR, βq,
+for any E Ď Ω. The conclusion follows by passing to the infimum on the set E Ď Ω and
+using Theorem 5.3.
+If ´1 ă β ă 0, then using the isoperimetric inequality [14] and (5.21), we have that
+ΛpΩ, βq ď β PF pΩq
+|Ω|
+ď β PF pWRq
+|WR|
+“ ΛpWR, βq.
+Acknowledgement
+This work has been partially supported by the MIUR-PRIN 2017 grant “Qualitative
+and quantitative aspects of nonlinear PDE’s”, by GNAMPA of INdAM, by the FRA
+Project (Compagnia di San Paolo and Universit`a degli studi di Napoli Federico II)
+000022--ALTRI_CDA_75_2021_FRA_PASSARELLI.
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+26
+

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+Naturality of the ∞-Categorical Enriched Yoneda Embedding
+Shay Ben Moshe
+Abstract
+We make Hinich’s ∞-categorical enriched Yoneda embedding natural. To do so, we exhibit
+it as the unit of a partial adjunction between the functor taking enriched presheaves and
+Heine’s functor taking a tensored category to an enriched category. Furthermore, we study a
+finiteness condition of objects in a tensored category called being atomic, and show that the
+partial adjunction restricts to a (non-partial) adjunction between taking enriched presheaves
+and taking atomic objects.
+Contents
+1
+Introduction
+2
+1.1
+Overview
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+2
+1.2
+Relation to Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+1.3
+Further Questions
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+1.4
+Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2
+Generalities on Enriched Categories
+6
+2.1
+Enriched Categories and Tensored Categories . . . . . . . . . . . . . . . . . . . . . .
+7
+2.2
+Enriched Yoneda Lemma and Weighted Colimits . . . . . . . . . . . . . . . . . . . .
+7
+3
+Partial Adjunction
+10
+4
+2-Categorical Structures
+11
+4.1
+Adjunctions in O-Monoidal Categories . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+4.2
+Tensored Categories
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+4.3
+Tensored Categories to Enriched Categories . . . . . . . . . . . . . . . . . . . . . . .
+16
+4.4
+Evaluation and Enriched Hom in Tensored Categories
+. . . . . . . . . . . . . . . . .
+17
+5
+Atomic Objects, Presheaves and Yoneda
+19
+5.1
+Internally Left Adjoints
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+20
+5.2
+Atomic Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+20
+5.3
+Atomics–Presheaves Adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+23
+6
+Heine’s Enriched Yoneda Embedding
+24
+References
+25
+1
+arXiv:2301.00601v1  [math.CT]  2 Jan 2023
+
+1
+Introduction
+1.1
+Overview
+1.1.1
+Partial Adjunction
+The main goal of this paper is to make Hinich’s enriched Yoneda embedding in the ∞-categorical
+setting from [Hin20] into a natural transformation. We work in the framework of enriched ∞-
+categories as developed in [Hin20, Hin21, GH15, Hei20]. These source differ in notation, and we
+introduce our notation along the paper. For brevity, henceforth we use the term (enriched) category
+to mean an (enriched) ∞-category, and a 2-category to mean an (∞, 2)-category. Throughout this
+paper we fix a presentably monoidal category V ∈ Alg(PrL).
+For every V-enriched category C0 ∈ CatV and a presentably V-tensored category D ∈ LModV(PrL),
+Hinich defined an (unenriched) category of V-functors FunV(C0, D) ∈ PrL. Using this, he defined a
+presentably V-tensored category of enriched presheaves PV(C0) := FunVrev(Cop
+0 , V) ∈ LModV(PrL).
+One of the main results of [Hin20] is the construction of a V-enriched Yoneda embedding
+よV : C0 → PV(C0),
+namely, an object of FunV(C0, PV(C0)) ∈ PrL, satisfying the enriched Yoneda lemma. However,
+this map is not shown to be natural in C0, which is the central question of the present paper.
+In a sequel paper [Hin21], Hinich shows that his enriched Yoneda embedding enjoys the following
+universal property:
+Theorem 1.1 ([Hin21, 6.4.7]). Let C0 ∈ CatV. Then, for every D ∈ LModV(PrL), composition
+with the enriched Yoneda embedding induces an equivalence
+(よV)∗ : FunL
+V(PV(C0), D) ∼
+−→ FunV(C0, D).
+This almost exhibits よV as a unit of an adjunction, but there are two problems:
+(1) FunV(C0, D) is not the hom in any category (note that C0 is V-enriched while D is V-tensored).
+(2) C0 is small while D is large.
+To solve Problem (1), we use the fact that presentably V-tensored categories produce V-enriched
+categories, as was first proven in [GH15, Corollary 7.4.9], and made functorial in the work of [Hei20],
+a special case of which reads as follows:
+Theorem 1.2 ([Hei20, Corollary 7.16]). There is an functor
+χ: LModV(PrL) → �
+CatV
+witnessing the source as a (non-full non-wide) subcategory of the target.
+We shall also use [Hei20, Theorem 5.3], showing that for every C0 ∈ CatV and D ∈ LModV(PrL)
+there is an equivalence FunV(C0, D)≃ ∼= homV(C0, χ(D)), where the right hand side denotes the
+space of morphisms in �
+CatV. Combining these result, in Proposition 2.12 we conclude that there is
+an isomorphism of spaces
+homL
+V(PV(C0), D) ∼
+−→ homV(C0, χ(D)),
+2
+
+where the left hand side denotes the underlying space of FunL
+V(PV(C0), D).
+To deal with Problem (2), we recall that one can define adjoints partially, namely an adjoint
+and a (co)unit map defined only on a full subcategory (or, more generally, relative to a functor), see
+Definition 3.1. Furthermore, we recall the folklore result that (partial) adjoints can be constructed
+point-wise, as we show in Proposition 3.5.
+With these results in mind, we deduce our first main result:
+Theorem A (Theorem 3.6). The functor χ: LModV(PrL) → �
+CatV has a partial left adjoint
+PV : CatV → LModV(PrL) with partial unit agreeing with the enriched Yoneda embedding.
+In particular, for every f : C0 → D0 in CatV, we get an induced V-linear left adjoint f! : PV(C0) →
+PV(D0), and an isomorphism f!よV ∼= よVf, as explained in Corollary 3.8.
+1.1.2
+Atomics–Presheaves Adjunction
+In a somewhat different direction, we study atomic objects, a finiteness condition on objects of
+presentably V-tensored categories. We show that this condition is closely related to the enriched
+Yoneda embedding, and, in particular, gives another solution for Problem (2), leading to a (non-
+partial) adjunction.
+Let C ∈ LModV(PrL), and let X ∈ C. The functor − ⊗ X : V → C is a V-linear left adjoint
+functor. We say that X is atomic if the right adjoint homV(X, −) is itself a left adjoint and the
+canonical lax V-linear structure on it is strong (see Definition 5.4). These objects form a small
+full V-subcategory Cat ⊂ χ(C). We show that the construction C �→ Cat is functorial in internally
+left adjoint functors, that is, V-linear left adjoint functors L: C → D ∈ LModV(PrL) whose right
+adjoint is itself a left adjoint and the canonical lax V-linear structure on it is strong. Namely, we
+obtain a functor
+(−)at : LModV(PrL)iL → CatV.
+As a key example, in Proposition 5.17 we show that for any C0 ∈ CatV and X ∈ C0, the
+image よV(X) ∈ PV(C0) is atomic.
+To see this, recall that by the enriched Yoneda lemma
+homV(よV(X), −) is given by evaluation at X, which preserves (co)limits and is V-linear, as we
+show in Proposition 4.24. Therefore, we get a factorization of the enriched Yoneda embedding
+through the atomics よV : C0 → PV(C0)at.
+Furthermore, we show that the functoriality of PV
+obtained in Theorem A sends V-functors to internally left adjoints, namely, restricts to a functor
+PV : CatV → LModV(PrL)iL.
+Our second main result is that the partial adjunction of Theorem A restricts accordingly:
+Theorem B (Theorem 5.20). There is an adjunction
+PV : CatV ⇄ LModV(PrL)iL :(−)at
+with unit agreeing with the enriched Yoneda embedding.
+In particular, for every f : C0 → D0 in CatV, the V-linear left adjoint f! : PV(C0) → PV(D0),
+is internally left adjoint, and thus admits a right adjoint f ⊛ : PV(D0) → PV(C0) which is itself
+a V-linear left adjoint. Using this, in Corollary 5.23 we show that f ⊛(G)(X) ∼= G(f(X)). We
+warn the reader that this does not imply that f ⊛ is the functor given by pre-composition, see
+Subsection 1.3.1 for further discussion.
+3
+
+1.1.3
+2-Categorical Structures
+In our discussion of atomic objects and their relation to the enriched Yoneda embedding, we use
+certain 2-categorical aspects of the theory of enriched categories and tensored categories. Notably,
+we have considered the condition for a V-linear left adjoint functor to be internally left adjoint,
+which we show has a particularly simple 2-categorical interpretation.
+In Definition 4.12 we recall that the category of V-tensored categories and lax V-linear functors
+enhances to a 2-category LModlax
+V . We show that a 1-morphism in that 2-category is a left adjoint, if
+and only if it is strong V-linear and left adjoint (on the underlying category). This result is a special
+case of a more general result we deduce from Lurie’s work on relative adjunctions and [HHLN20a],
+concerning the 2-category Monlax
+O
+of O-monoidal categories and lax O-monoidal functors for some
+operad O, which may be of independent interest.
+Corollary C (Corollary 4.11). The left adjoints in Monlax
+O
+are the lax O-monoidal functors that
+are strong and fiber-wise left adjoint.
+The category LModV(PrL) also enhances into a 2-category, which by the above result is a full 2-
+subcategory LModV(PrL) ⊂ LModlax
+V ( �
+Cat)L of the left adjoints in the 2-category LModlax
+V ( �
+Cat).
+Furthermore, using the above result again, we see that internally left adjoints are precisely the left
+adjoint morphisms in the 2-category LModV(PrL).
+As explained in [Hei20, 7.10], by [GH15, Proposition 5.7.16], the category of V-enriched cate-
+gories also enhances to a 2-category CatV. Furthermore, by [Hei20, Theorem 7.11], the functor χ
+of Theorem 1.2 enhances to a 2-functor. Using these results we also deduce the following result
+which may be of independent interest. Here a 2-functor is called 2-fully faithful if it induces an
+isomorphism on hom categories, i.e. if it exhibits the source as a full 2-subcategory of the target.
+Corollary D (Corollary 4.19, Corollary 5.3). The functor χ enhances to a 2-fully faithful 2-functor
+χ: LModV(PrL) → ( �
+CatV)L.
+Taking left adjoints again we get the 2-fully faithful 2-functor
+χ: LModV(PrL)iL → ( �
+CatV)LL.
+In particular, for f : C0 → D0 in CatV, since f! : PV(C0) → PV(D0) is internally left adjoint, we
+get a corresponding double adjunction χ(f!) ⊣ χ(f ⊛) ⊣ χ(f⊛) in �
+CatV.
+1.2
+Relation to Previous Work
+This paper shares many of the ideas on atomic objects developed in our previous paper with
+Tomer Schlank, most notably [BMS21, Theorem D]. To avoid the usage of enriched categories,
+in our previous paper we work over a mode M, that is, an idempotent algebra in PrL. The key
+feature of modes is that LModM(PrL) is a full subcategory of PrL, namely, a left adjoint functor is
+automatically (and uniquely) M-linear. This allowed us, for example, to simplify the definition of
+X being atomic to having homM(X, −): C → M commute with colimits (and thus automatically
+M-linear). Particularly, in the previous paper we have ignored the M-enriched structure on the
+category of atomic objects. In addition, we worked with unenriched presheaves and the unenriched
+Yoneda embedding, further tensored with M.
+In particular, [BMS21, Theorem D] is a weaker
+statement than the main results of the present paper in these respects.
+4
+
+On the other hand, in the unenriched context, the two different functorialities of presheaves
+and naturalities of the Yoneda embedding were shown coincide, contrary to the enriched case of
+the present paper (see the discussion in Subsection 1.3.1). In addition, our goal in the previous
+paper was different. The unenriched presheaves functor is endowed with a symmetric monoidal
+structure, and through the adjunction, this makes the atomics functor lax symmetric monoidal,
+and the Yoneda a symmetric monoidal natural transformation. In the present paper we do not deal
+with the multiplicative structure (though see the discussion in Subsection 1.3.4).
+1.3
+Further Questions
+We now list several further questions left open, which we expect have a positive answer, but we do
+not know how to approach.
+1.3.1
+Naturality for Pre-composition
+The first question is closely related to [Ram22] and [HHLN20b, Corollary F], which deal with the
+case V = S.
+Recall that using the universal property of PV(C0), we deduced Theorem A, assembling PV into
+a functor, and the enriched Yoneda embedding into a natural transformation. In addition, we saw
+that f! has a right adjoint f ⊛, and we showed that f ⊛(G)(X) ∼= G(f(X)).
+The construction of PV(C0) as enriched functors from Cop
+0
+to V shows that it admits an-
+other functoriality in C0. More specifically, [Hin20, 6.1.4] implies that it assembles into a functor
+PV : CatV → (PrR)op, sending f to f ∗ : PV(D0) → PV(C0) given by pre-composition, admitting a
+left adjoint f?, commonly spelled “f lower what”. Note that this functoriality does not a priori
+give the V-tensored structure. Also note that f ⊛(G)(X) ∼= G(f(X)) ∼= f ∗(G)(X), however it is not
+clear that this holds naturally in X, G or C0.
+Furthermore, Hinich’s construction of the enriched Yoneda embedding, as described for example
+in [Hin21, 7.3.1 and 7.3.2], seems to interact with the f ∗, and thus f?, functoriality, but we don’t
+know how to extract naturality from his results.
+Question 1.3. Can the f? functoriality be extended to V-modules?
+Can the enriched Yoneda
+embedding be made natural for the f? functoriality? Do these agree with the f! functoriality and
+naturality of the enriched Yoneda embedding?
+1.3.2
+Heine’s and Hinich’s Enriched Yoneda Embeddings
+In [Hei20], Heine also defines an enriched Yoneda embedding よV
+Heine : C0 → χ(PV(C0)) by different
+means. As we explain in Section 6, the main results of this paper hold for this version as well,
+producing an adjunction
+PV
+Heine : CatV ⇄ LModV(PrL)iL :(−)at
+with unit agreeing with Heine’s enriched Yoneda embedding. The uniqueness of adjoints implies that
+there is a natural isomorphism ψ: PV
+Heine
+∼
+−→ PV, together with an isomorphism ψよV
+Heine ∼= よV. On
+the other hand, by the construction of the adjunction, for every C0 ∈ CatV there is an isomorphism
+PV
+Heine(C0) ∼= PV(C0). We thus get an automorphism PV(C0)
+ψC0
+−−→ PV
+Heine(C0) ∼= PV(C0). It is not
+clear to us that this automorphism is the identity. Showing this is equivalent to showing that these
+two versions of the enriched Yoneda embedding coincide.
+5
+
+Question 1.4. Do the enriched Yoneda embeddings constructed by Hinich and Heine coincide?
+Namely, is the above automorphism the identity?
+1.3.3
+�
+Cat-Enrichment
+Recall that χ enhances to a 2-functor. Furthermore, the universal property of the enriched Yoneda
+embedding of Theorem 1.1 is originally stated for functor categories, though one of the sides is not
+constructed as the �
+Cat-enriched hom in a category. This leads to the following question:
+Question 1.5. Can the (partial) adjunctions be made �
+Cat-enriched?
+1.3.4
+Multiplicative Structure
+For simplicity, assume that V is presentably symmetric monoidal. [Hei20, Corollary 7.16] enhances
+χ into a symmetric monoidal functor. Via the adjunction, it may be possible to endow PV with an
+oplax symmetric monoidal structure, either by making the subfunctor (−)at lax symmetric monoidal
+and using the main result of [HHLN20a], or by proving a version of it for partial adjunctions
+and directly applying to PV. Analogously to the case of V = S, one would expect the resulting
+oplax symmetric monoidal structure on PV to be strong, and thus make the adjunction symmetric
+monoidal. Assuming this, for any operad O, we get that if C0 is O-monoidal, then PV(C0) and the
+enriched Yoneda embedding are endowed with an O-monoidal structure.
+Separately, in [Hin21, 7.2], Hinich studies the multiplicative structure of enriched presheaves and
+the enriched Yoneda embedding. In particular, under the above assumptions, he endows PV(C0) and
+the enriched Yoneda embedding with an O-monoidal structure. He further proves an O-monoidal
+version of the universal property of Theorem 1.1.
+Question 1.6. Can the adjunction be made symmetric monoidal? Does the induced O-monoidal
+structure on PV(C0) and the enriched Yoneda embedding agree with those constructed by Hinich?
+1.4
+Acknowledgments
+We would like to thank Lior Yanovski for numerous useful conversations about atomic objects,
+enriched categories and weighted colimits. We also thank Shai Keidar, Shaul Ragimov, Maxime
+Ramzi and Noam Zimhoni for comments on earlier drafts of this paper. Particularly, we thank
+Maxime for pointing several subtle missing components in earlier drafts, and suggesting corrections
+for some of them.
+2
+Generalities on Enriched Categories
+In this section we review some generalities on enriched categories, their relationship to tensored
+categories, and the enriched Yoneda embedding. We shall not delve into the details of the con-
+structions, as most of them will not play a role in the present paper, but rather only the formal
+properties of the resulting objects.
+6
+
+2.1
+Enriched Categories and Tensored Categories
+Definition 2.1. We denote the (large) category of V-enriched categories, defined in [Hin20, 7.1.2],
+by CatV. For C0, D0 ∈ CatV, we denote the space of V-functors between them by homV(C0, D0) ∈ S.
+Similarly, we let �
+CatV be the (huge) category of large V-enriched categories.
+V-enriched categories are closely related to categories tensored over V, as was first proven in
+[GH15, Corollary 7.4.9], and made functorial in the work of [Hei20].
+Indeed, Heine constructs
+the category ωLModlax
+V
+(denoted ωLModV there) of weakly V-tensored categories and lax V-linear
+functors, and a full subcategory ωLModcl,lax
+V
+thereof on the closed weakly V-tensored categories.
+Theorem 2.2 ([Hei20, Theorem 7.3 and Proposition 7.8]). There is an equivalence
+χ: ωLModcl,lax
+V
+∼
+−→ CatV.
+Considering the large version of this equivalence, one can restrict the source to the subcategory
+with objects presentable categories with V-action commuting with colimits (which are automatically
+closed) and morphisms the (strong) V-linear left adjoint functors.
+Definition 2.3. We define the category of presentably V-tensored categories to be PrL
+V := LModV(PrL).
+For C, D ∈ PrL
+V, we denote the space of V-linear left adjoint functors between them by homL
+V(C, D) ∈
+�S, which is the space of objects of the category FunL
+V(C, D) ∈ �
+Cat.
+Corollary 2.4 ([Hei20, Corollary 7.16]). There is a functor
+χ: PrL
+V → �
+CatV.
+witnessing the source as a (non-full non-wide) subcategory of the target.
+For C ∈ PrL
+V, this constructs χ(C) ∈ �
+CatV, both of which have the same underlying category
+and thus space of objects.
+2.2
+Enriched Yoneda Lemma and Weighted Colimits
+In [Hin20, Hin21] Hinich constructs enriched presheaves and the enriched Yoneda embedding, which
+we recall in this subsection.
+We begin with Hinich’s model for the category of V-functors from a V-enriched category to a
+V-tensored category. In Hinich’s model, a V-enriched category C0 ∈ CatV is an algebra in some
+operad constructed from the space of objects C≃
+0 (see [Hin20, 3.1.1]). For a presentably V-tensored
+category D ∈ PrL
+V, Hinich endows the (unenriched) category of functors Fun(C≃
+0 , D) with a left
+module structure over this operad (see [Hin20, 6.1.1]). In particular, one can take left C0-modules
+inside.
+Definition 2.5 ([Hin20, 6.1.3]). Let C0 ∈ CatV and D ∈ PrL
+V, then the category of V-functors from
+C0 to D is defined to be
+FunV(C0, D) := LModC0(Fun(C≃
+0 , D)) ∈ PrL.
+We note that FunV(C0, D) is indeed presentable. To see this, note that Fun(C≃
+0 , D) is an (unen-
+riched) presheaf category, and thus is presentable by [Lur09, Proposition 5.5.3.6]. Then, the module
+category FunV(C0, D) is also presentable by [Lur17, Corollary 4.2.3.7].
+7
+
+Remark 2.6. We warn the reader that in [Hei20], FunV(C0, D) is denoted by FunV(C0, χ(D)).
+Definition 2.7 ([Hin20, 6.2.2]). Let C0 ∈ CatV. The V-tensored category of V-enriched presheaves
+is defined to be
+PV(C0) := FunVrev(Cop
+0 , V) ∈ PrL
+V.
+Remark 2.8. V is a V-V-bimodule in PrL. One of the V-module structures is used to define the
+presentable category of Vrev-enriched functors, and the other is used to endow the resulting category
+with a V-module structure.
+We now record the main results about Hinich’s enriched Yoneda embedding. We remark that
+in [Hin20, 6.2.7], Hinich proves his version of the enriched Yoneda lemma, very closely related to
+(1) of Theorem 2.9.
+However, the form described below is somewhat different, and relies on a
+definition of homV and the evaluation at X that were not introduced thus far. We postpone these
+definitions to Definition 4.20 and Definition 4.23, and the proof to Proposition 4.25, but include
+the statement here for completeness of the exposition. Note that (1) (as well as (2)) will not be
+used before Section 5.
+Theorem 2.9 ([Hin20, Hin21]). Let C0 ∈ CatV. Then, there is a V-enriched Yoneda embedding
+V-functor
+よV : C0 → PV(C0),
+that is, an object of FunV(C0, PV(C0)). For every D ∈ PrL
+V, there is a weighted colimit functor
+colim(−)
+C0 (−): PV(C0) × FunV(C0, D) → D.
+These satisfy the following properties:
+(1) For every X ∈ C0, the functor homV(よV(X), −): PV(C0) → V agrees with evaluation at X
+as V-linear functors.
+(2) colim(−)
+C0 (−) commutes with colimits in both arguments separately, and commutes with the
+V-action in the first argument.
+(3) There is an equivalence
+(よV)∗ : FunL
+V(PV(C0), D) ⇄ FunV(C0, D) :colim(−)
+C0 (−).
+Proof. よV is constructed in [Hin20, 6.2.4]. (1) is deferred to Proposition 4.25, though see [Hin20,
+6.2.7]. colim(−)
+C0 (−) is constructed in [Hin21, 6.2.2], where (2) is explained. (3) is [Hin21, 6.3.3 and
+6.4.6].
+In [Hei20, Theorem 5.3, see also the discussion in the beginning of Section 5], Heine shows that
+FunV(C0, D) is closely related to the hom in V-enriched categories via χ (see also [Hin20, 6.3.6]).
+We recall a special case of Heine’s results as follows:
+Proposition 2.10 ([Hei20, Theorem 5.3]). Let C0 ∈ CatV and D ∈ PrL
+V, then there is an equiva-
+lence
+FunV(C0, D)≃ ∼= homV(C0, χ(D))
+8
+
+natural in C0 and D. Namely, there is a natural isomorphism between the functors
+(CatV)op × PrL
+V
+i×χ
+−−→ (�
+CatV)op × �
+CatV
+homV
+−−−−→ �S
+and
+(CatV)op × PrL
+V
+FunV(−,−)
+−−−−−−−→ Cat
+(−)≃
+−−−→ S ⊂ �S
+We use this to transform the enriched Yoneda embedding to a morphism in �
+CatV, and deduce a
+universal property similar to (3) of Theorem 2.9 where both the source and the target are the hom
+in some category.
+Definition 2.11. We denote by the same notation the enriched Yoneda embedding V-functor
+よV : C0 → χ(PV(C0)), corresponding to the enriched Yoneda embedding under the equivalence
+FunV(C0, PV(C0))≃ ∼= homV(C0, χ(PV(C0))).
+Proposition 2.12. Let C0 ∈ CatV and D ∈ PrL
+V, then the composition
+homL
+V(PV(C0), D)
+χ−→ homV(χ(PV(C0)), χ(D))
+(よV)∗
+−−−−→ homV(C0, χ(D))
+is an equivalence.
+Proof. Naturality in the PrL
+V coordinate of Proposition 2.10 shows that for C0 ∈ CatV and D, E ∈
+PrL
+V we get a commutative square:
+homL
+V(E, D)
+homV(χ(E), χ(D))
+hom(FunV(C0, E)≃, FunV(C0, D)≃)
+hom(homV(C0, χ(E)), homV(C0, χ(D)))
+χ
+∼
+Using the exponential adjunction we get a commutative square:
+homL
+V(E, D) × FunV(C0, E)≃
+homV(χ(E), χ(D)) × homV(C0, χ(E))
+FunV(C0, D)≃
+homV(C0, χ(D))
+χ
+◦
+◦
+∼
+Taking E = PV(C0), and picking the point よV ∈ FunV(C0, PV(C0))≃, we get a commutative square:
+homL
+V(PV(C0), D)
+homV(χ(PV(C0)), χ(D))
+FunV(C0, D)≃
+homV(C0, χ(D))
+χ
+(よV)∗
+(よV)∗
+∼
+The left morphism is an equivalence by applying (−)≃ to (3) of Theorem 2.9. Thus, the left-bottom
+composition is an equivalence, and by the commutativity of the diagram, so is the upper-right
+composition, concluding the proof.
+9
+
+3
+Partial Adjunction
+In this section we prove Theorem A, namely the naturality of the enriched Yoneda embedding.
+To do so, we first define (unenriched) partial adjunctions. Then, we show the folklore result that
+(partial) adjoints can be constructed point-wise, from which the naturality of the enriched Yoneda
+embedding follows immediately.
+Definition 3.1. Let L: C0 → D0 be a functor, and i: E0 → D0 another functor. The data of
+a partial right adjoint (of L relative to i) is a functor R: E0 → C0 and a natural transformation
+ε: LR ⇒ i called the partial counit, such that for every X ∈ C0 and Y ∈ E0 the composition
+homC0(X, RY )
+L−→ homD0(LX, LRY )
+εY ◦−
+−−−→ homD0(LX, iY )
+is an isomorphism. A partial left adjoint with a partial unit is defined dually, by taking (−)op.
+Remark 3.2. What we call a partial adjunction is typically called a relative adjunction. However, in
+Section 4 we use the distinct concept of relative adjunctions developed by Lurie. To avoid confusion,
+we call what is typically called a relative adjunction a partial adjunction.
+Remark 3.3. In this paper we only use the notion of a partial adjunction when i: E0 → D0 is an
+inclusion of a full subcategory. In this case, one can think of R as an adjoint defined only partially
+on D0, justifying the name.
+Remark 3.4. Unlike in a standard adjunction, a partial adjunction is not a symmetric concept.
+Particularly, note that there is no partial unit (in fact, L and R can not be composed in the other
+direction).
+Proposition 3.5. Let L: C0 → D0 be a functor, and i: E0 → D0 another functor. Assume that
+for every Y ∈ E0 we are given an object RY ∈ C0 and a morphism εY : LRY → iY , such that for
+every X ∈ C0 the composition
+homC0(X, RY )
+L−→ homD0(LX, LRY )
+εY ◦−
+−−−→ homD0(LX, iY )
+is an isomorphism. Then, R and ε assemble into the data of a partial right adjoint.
+Proof. Consider the functor ˜R: D0 → P(C0) given by the composition
+D0
+よD0
+−−−→ P(D0)
+L∗
+−−→ P(C0),
+which sends Z ∈ D0 to homD0(L(−), Z): Cop
+0 → S.
+Let Y ∈ E0. Consider the natural transformation
+homC0(−, RY )
+L−→ homD0(L(−), LRY )
+εY ◦−
+−−−→ homD0(L(−), iY ) = ˜R(iY ).
+(1)
+By assumption, it is an isomorphism at every X ∈ C0, and thus it is a natural isomorphism.
+Namely, the presheaf ˜R(iY ) ∈ P(C0) is representable by RY ∈ C0. In other words, the composition
+˜Ri: E0 → P(C0) lands in the essential image of the (unenriched) Yoneda embedding よC0 : C0 →
+P(C0). As the Yoneda embedding is fully faithful, we get an induced functor R: E0 → C0, together
+with a natural isomorphism よC0R ∼= ˜Ri = L∗よD0i of functors E0 → P(C0).
+10
+
+By construction, R agrees with RY at every Y ∈ E0. Furthermore, the natural isomorphism at
+Y is given by (1), constructed via εY . We now extract the partial counit ε: LR ⇒ i. Consider the
+following composition:
+よD0LR
+(1)
+∼= (LR)!よE0
+(2)
+⇒ (LR)!R∗よC0R
+(3)
+∼= (LR)!R∗L∗よD0i
+∼= (LR)!(LR)∗よD0i
+(4)
+⇒ よD0i.
+(1) is naturality of the (unenriched) Yoneda embedding. (2) is the fact that R is a functor, giving
+homE0(−, −) ⇒ homC0(R(−), R(−)), which by exponential adjunction is the same as よE0 ⇒
+R∗よC0R. (3) is the isomorphism よC0R ∼= L∗よD0i. (4) is the counit of the adjunction (LR)! ⊣
+(LR)∗. Thus, we have constructed a natural transformation よD0LR ⇒ よD0i. Since よD0 is fully
+faithful, this induces a natural transformation ε: LR ⇒ i. Recall that the isomorphism used at
+step (3) is given at Y ∈ E0 by (1), which shows that ε indeed agrees with εY .
+Theorem 3.6. The functor χ: PrL
+V → �
+CatV has a partial left adjoint PV : CatV → PrL
+V with partial
+unit よV : id|CatV → χ PV, agreeing with enriched presheaves and enriched Yoneda embedding.
+Proof. This follows immediately from Proposition 2.12 and (the dual statement to) Proposition 3.5.
+Definition 3.7. For f : C0 → D0 a morphism in CatV, we denote the induced morphism in PrL
+V
+by f! : PV(C0) → PV(D0).
+An instance of the naturality of the enriched Yoneda embedding is the following:
+Corollary 3.8. Let f : C0 → D0. Then, よV
+D0f ∼= χ(f!)よV
+C0 in homV(C0, χ(PV(D0))). Similarly,
+よV
+D0f ∼= f!よV
+C0 in FunV(C0, PV(D0)) of Definition 2.5.
+Proof. The first part is an immediate application of the adjunction of Theorem 3.6. For the sec-
+ond part, recall from Proposition 2.10 that there is a natural isomorphism homV(−, χ(−)) ∼=
+FunV(−, −)≃.
+Recall that in Definition 2.11 we have defined よV
+C0 ∈ homV(C0, χ(PV(C0))) to
+be the map corresponding to よV
+C0 ∈ FunV(C0, PV(C0)) via this isomorphism, and similarly for
+D0. By applying the naturality of the isomorphism in the target to the morphism f!, we get that
+χ(f!)よV
+C0 ∈ homV(C0, χ(PV(D0))) corresponds to f!よV
+C0 ∈ FunV(C0, PV(D0)). Similarly, by ap-
+plying the naturality of the isomorphism in the source to the morphism f, we get that よV
+D0f ∈
+homV(C0, χ(PV(D0))) corresponds to よV
+D0f ∈ FunV(C0, PV(D0)). Thus, under the natural isomor-
+phism, the isomorphism よV
+D0f ∼= χ(f!)よV
+C0 corresponds to an isomorphism よV
+D0f ∼= f!よV
+C0.
+4
+2-Categorical Structures
+In this section we study certain 2-categorical aspects of enriched categories and tensored categories.
+The first main result of this section is Corollary 4.11, showing that a lax O-monoidal functor between
+11
+
+O-monoidal categories is a left adjoint in the 2-category Monlax
+O , if and only if it is strong O-monoidal
+and fiber-wise left adjoint. This is proven by applying Lurie’s [Lur17, §7.3.2 Relative Adjunctions] to
+the 2-categories constructed in [HHLN20a]. From this, we deduce Corollary 4.13, the corresponding
+result for lax V-linear functors between V-tensored categories. Then, after recalling that in [Hei20,
+Theorem 7.11] Heine shows that χ is a 2-functor, we deduce Corollary 4.19, our second main
+result, which says that χ enhances to a 2-fully faithful 2-functor χ: PrL
+V → ( �
+CatV)L. Finally, in
+Proposition 4.25 we finish the proof of (1) of Theorem 2.9.
+4.1
+Adjunctions in O-Monoidal Categories
+Definition 4.1 ([HHLN20a, 3.1.7]). Let B be a a category. Denote by Cocartlax
+B
+the 2-category
+of cocartesian fibrations over B and functors over B.
+We shall not recall the precise definition of this 2-category (for which we refer the reader to
+[HHLN20a, 3.1.7]), rather, let us recall its objects, 1-morphisms and 2-morphisms.
+Proposition 4.2. Cocartlax
+B
+has
+• objects: cocartesian fibrations q: C → B,
+• 1-morphisms: functors over B, namely, a 1-morphism from q: C → B to p: D → B is a
+functor F : C → D and a natural isomorphism pF ∼= q,
+• 2-morphisms: natural transformations over B, namely, a 2-morphism from F to G is a natural
+transformation α: F ⇒ G and an identification of the natural transformation q ∼= pF
+pα
+==⇒
+pG ∼= q with idq, i.e. exhibiting the following square as commutative:
+pF
+pG
+q
+q
+pα
+≀
+≀
+Proof. By construction, the underlying category of the 2-category Cocartlax
+B is the full subcategory
+of Cat/B on the cocartesian fibrations, which explains the objects and the 1-morphisms. For the 2-
+morphisms, we recall that as explained in [HHLN20a, 3.1.9], a 2-morphism is commutative diagram
+of the form:
+C × [1]
+D
+B
+q
+p
+¯α
+Under the exponential adjunction, ¯α: C × [1] → D corresponds to the natural transformation
+α: F ⇒ G, and the isomorphism p¯α ∼= q corresponds to the identification pα ∼= idq.
+Lemma 4.3. A 2-morphism in Cocartlax
+B
+given by α: F ⇒ G together with an identification
+pα ∼= idq is invertible if and only if α is invertible.
+12
+
+Proof. Clearly, if the 2-morphism is invertible then in particular α is.
+For the other direction,
+assume that α has an inverse α−1 : G ⇒ F.
+We shall enhance it to an inverse 2-morphism in
+Cocartlax
+B . Consider the following diagram:
+pF
+pG
+pF
+q
+q
+q
+≀
+≀
+≀
+pα
+pα−1
+The identification pα ∼= idq is precisely an (invertible) 3-morphism in Cat making the left square
+commute. The outer square commutes as pα−1pα ∼= idpF ∼= idq. Thus, by composing the outer
+square with the inverse of the left square, we get commutativity data for the right square. This
+makes α−1 into a 2-morphism in Cocartlax
+B , which by construction is the required inverse.
+Our next goal is to understand left adjoints in the 2-category Cocartlax
+B , achieved in Proposi-
+tion 4.8. To that end, we shall employ Lurie’s [Lur17, §7.3.2 Relative Adjunctions]. The definitions
+and results in that section are phrased for cartesian fibrations and right adjoints, which by taking
+(−)op correspond to our case of cocartesian fibrations and left adjoints, as we present here.
+First, we recall that in a 2-category, we say that a 1-morphism L: X → Y is a left adjoint
+if there exists a 1-morphism R: Y → X and two 2-morphisms u: idX ⇒ RL and c: LR ⇒ idY
+satisfying the zigzag identities, namely
+(R
+uR
+==⇒ RLR
+Rc
+==⇒ R) ∼= idR,
+(L
+Lu
+==⇒ LRL
+cL
+==⇒ L) ∼= idL.
+We also recall that to check that L is a left adjoint, we can check a weaker condition:
+Lemma 4.4 ([RV22, 2.1.11]). In any 2-category, a 1-morphism L: X → Y is a left adjoint if and
+only if there exist a 1-morphism R: Y → X and 2-morphisms u: idX ⇒ RL and c: LR ⇒ idY such
+that the zigzag morphisms
+R
+uR
+==⇒ RLR
+Rc
+==⇒ R,
+L
+Lu
+==⇒ LRL
+cL
+==⇒ L
+are invertible (in which case there exists a possibly different 2-morphism ˜c: LR ⇒ idY for which
+the zigzag identities hold).
+Proof. The proof of [RV22, 2.1.11] works in an arbitrary 2-category, although stated in the context
+of functors between (∞-)categories.
+Definition 4.5 ([Lur17, Definition 7.3.2.2]). Let
+C
+D
+B
+q
+p
+L
+be a commutative diagram of categories. We say that L admits a relative right adjoint if L has
+a (non-relative) right adjoint R: D → C, and the counit map c: LR ⇒ idD satisfies the condition
+that pc is equivalent to idp (and in particular qR ∼= p).
+13
+
+Remark 4.6. The definition of [Lur17, Definition 7.3.2.2] is via the two equivalent conditions of
+[Lur17, Proposition 7.3.2.1]. First, we note that these are phrased for admitting a relative left
+adjoint, but the theory is symmetric via taking (−)op. Second, Lurie assumes that p and q are
+categorical fibrations, but every functor is equivalent to a categorical fibration, so this assumption
+can be dropped. Finally, and most importantly, we note that our definition clearly implies condition
+(1) and is implied by condition (2) in [Lur17, Proposition 7.3.2.1], and thus is also equivalent to
+them.
+Proposition 4.7. A 1-morphism L: C → D over B in Cocartlax
+B
+is a left adjoint if and only if L
+admits a relative right adjoint.
+Proof. If L is a left adjoint in Cocartlax
+B then it clearly admits a relative right adjoint. For the other
+direction, assume that L admits a relative right adjoint, and we shall show that it is a left adjoint
+in Cocartlax
+B . Let R: D → C, u: idC ⇒ RL and c: LR ⇒ idD be the (non-relative) adjunction
+data, satisfying that pc ∼= idp. The identification pc ∼= idp makes c into a 2-morphism in Cocartlax
+B .
+This also identifies qR ∼= p, making R into a 1-morphism in Cocartlax
+B . In the rest of the proof we
+make u into a 2-morphism as well, and show that L is a left adjoint using Lemma 4.4.
+We begin by making u into a 2-morphism. Consider the following diagram:
+pL
+pLRL
+pL
+q
+q
+q
+pLu
+pcL
+≀
+≀
+≀
+The upper composition is p(−) applied the zigzag morphism, which is equivalent to idpL, thus
+making the outer square commute. The assumption that pc is equivalent to idp makes the right
+square commute. Thus, by composing the outer square with the inverse of the right square, we get
+that the left square commutes as well. In other words, we obtained an identification of pLu with
+idq. The identification pL ∼= q thus shows that we got an identification of qu with idq, making u
+into a 2-morphism.
+By Lemma 4.4, to see that L is a left adjoint in Cocartlax
+B
+it suffices to check that the zigzag
+morphisms corresponding to these 2-morphisms are invertible. By Lemma 4.3, a 2-morphism in
+Cocartlax
+B is invertible if and only if the underlying natural transformations are invertible. Indeed,
+the underlying natural transformations are simply
+R
+uR
+==⇒ RLR
+Rc
+==⇒ R,
+L
+Lu
+==⇒ LRL
+cL
+==⇒ L
+which are invertible (in fact, equivalent to the identities) by the assumption that u and c are a unit
+and a counit for a (non-relative) adjunction L ⊣ R.
+The following is a recast of [Lur17, Proposition 7.3.2.6] in an appropriate 2-category.
+Proposition 4.8. The left adjoints in Cocartlax
+B
+are those functors over B that preserve cocarte-
+sian morphisms and are fiber-wise left adjoint
+(Cocartlax
+B )L = CocartL-fw
+B
+.
+Proof. By Proposition 4.7, a 1-morphism L between two cocartesian fibrations is a left adjoint if
+and only if it admits a relative right adjoint. Thus by (the (−)op of) [Lur17, Proposition 7.3.2.6],
+14
+
+L is a left adjoint if and only if it preserves locally cocartesian morphisms and is fiber-wise left
+adjoint. By [Lur09, Proposition 2.4.2.8], locally cocartesian morphisms and cocartesian morphisms
+coincide, concluding the proof.
+Remark 4.9. There is a possible alternative route to Proposition 4.8.
+Recall that [HHLN20a,
+Theorem E] shows that Cocartlax
+B
+∼= Funlax(B, Cat), where the right hand side denotes the 2-
+category of functors and lax natural transformations. Additionally, [Hau20, Theorem 4.6] shows that
+for any 2-categories X, Y, the left adjoints in Funlax(X, Y) are those lax natural transformations
+which are strong and point-wise left adjoint. Combined, this would imply Proposition 4.8. However,
+the definition of Funlax in [HHLN20a] and [Hau20] rely on two different models of the Gray tensor
+product, which to the best of our knowledge were not shown to be equivalent so far, and thus this
+does not constitute a complete proof.
+In the rest of the subsection we follow the notations of [HHLN20a, 3.4] for operads. Namely, an
+operad O is a functor O → Fin∗ (satisfying certain properties), whereas in [Lur17], it is typically
+denoted by O⊗.
+Definition 4.10 ([HHLN20a, 3.4.1]). Let O be an operad. Let Monlax
+O
+⊂ Cocartlax
+O
+be the 1-
+full 2-subcategory on the O-monoidal categories and lax O-monoidal functors (i.e. functors over O
+preserving cocartesian morphisms lying over inert morphisms).
+The following is a recast of [Lur17, Corollary 7.3.2.7] in an appropriate 2-category.
+Corollary 4.11. The left adjoints in Monlax
+O
+are those lax O-monoidal functors that are strong
+and fiber-wise left adjoint
+(Monlax
+O )L = MonL-fw
+O
+.
+Proof. For the first direction, note that a left adjoint in Monlax
+O
+is in particular a left adjoint in
+Cocartlax
+O , thus it is strong O-monoidal (preserves all cocartesian morphisms) and fiber-wise left
+adjoint.
+For the second direction, let L: C → D over O in MonL-fw
+O
+. As MonL-fw
+O
+⊂ CocartL-fw
+O
+=
+(Cocartlax
+O )L, there is a right adjoint R: D → C over O in Cocartlax
+O
+with some unit and counit.
+Since Monlax
+O
+is a 1-full 2-subcategory of Cocartlax
+O , all we need to show is that the right adjoint
+is in Monlax
+O , whence the unit and counit are also automatically there. Namely, we need to know
+that R: D → C is lax O-monoidal, which follows from the construction, as explained in [Lur17,
+Corollary 7.3.2.7].
+4.2
+Tensored Categories
+As explained in [Hei20, 7.10], the category ωLModlax
+V
+is enhanced to a 2-category, and thus so is its
+full subcategory LModlax
+V . We repeat the construction in more detail, building on the definitions
+above. See also [Bar22, Definition 2.14] for a closely related discussion.
+We recall the operad Assoc classifying an associative algebra, and the operad LM classifying
+an associative algebra and a left module over it. Consider the monoidal category V as an object in
+Mon(Cat) ∼= MonAssoc ⊂ Monlax
+Assoc. This leads us to the following:
+Definition 4.12. We define the 2-category of V-tensored categories and lax V-linear functors to
+be LModlax
+V
+:= {V} ×Monlax
+Assoc Monlax
+LM.
+15
+
+As a particular case of Corollary 4.11, we deduce the following, which is an “if and only if”
+version of [Lur17, Example 7.3.2.8 and Remark 7.3.2.9] phrased in an appropriate 2-category.
+Corollary 4.13. The left adjoints in LModlax
+V
+are those lax V-linear functors that are strong
+V-linear and left adjoint (on the underlying category)
+(LModlax
+V )L = LModL-fw
+V
+.
+Proof. The result follows from Corollary 4.11 by taking O = LM, and pulling back over V. In more
+detail, consider the following diagram:
+(LModlax
+V )L
+(Monlax
+LM)L
+{V}L
+(Monlax
+Assoc)L
+{V}
+Monlax
+Assoc
+≀
+The bottom square is a pullback square because the lower right morphism is an inclusion of a
+2-subcategory, and the lower left morphism is the map from a point to itself. The upper square
+is a pullback square, because (−)L commutes with limits, as it is given by hom from the walking
+adjunction. Therefore, the outer square is also a pullback square. We thus finish by Corollary 4.11.
+Remark 4.14. Definition 4.12 and Corollary 4.13 work for an arbitrary monoidal category V, without
+needing to assume that it is presentably monoidal.
+We now enhance PrL
+V of Definition 2.3 to a 2-category.
+Definition 4.15. We define the 2-category of presentably V-tensored categories to be the 1-full
+2-subcategory PrL
+V := LModV(PrL) ⊂ LModlax
+V ( �
+Cat) on the objects and morphisms of PrL
+V.
+Proposition 4.16. There is a full 2-subcategory inclusion
+PrL
+V ⊂ (LModcl,lax
+V
+( �
+Cat))L.
+Proof. First, we claim that there is a full 2-subcategory inclusion
+PrL
+V = LModV(PrL) ⊂ LModcl,L-fw
+V
+( �
+Cat).
+It is indeed an inclusion, since a presentably V-tensored category is automatically closed, and mor-
+phism in both categories are the V-linear functors that are left adjoint on the underlying category.
+Second, by Corollary 4.13, the target is (LModcl,lax
+V
+( �
+Cat))L, which finishes the argument.
+4.3
+Tensored Categories to Enriched Categories
+We recall that CatV can be enhanced into a 2-category, as explained for example in [Hei20, 7.10].
+Indeed, by [GH15, Proposition 5.7.16], the functor Alg(PrL) → PrL given by V �→ CatV is lax
+symmetric monoidal. Since V is a S-module in PrL, we get that CatV is a Cat-module in PrL as
+well, which indeed models a 2-category (for example via χ).
+16
+
+Definition 4.17. We denote the 2-category of V-enriched categories by CatV.
+Heine shows that χ of Theorem 2.2 enhances to a 2-equivalence.
+Theorem 4.18 ([Hei20, Theorem 7.11]). The equivalence of Theorem 2.2 enhances to a 2-equivalence
+χ: ωLModcl,lax
+V
+∼
+−→ CatV.
+Using this we deduce the following:
+Corollary 4.19. The functor from Corollary 2.4 enhances to a 2-fully faithful 2-functor
+χ: PrL
+V → ( �
+CatV)L.
+Proof. Consider the large version of Theorem 4.18, namely ωLModcl,lax
+V
+( �
+Cat)
+∼
+−→ �
+CatV, and re-
+strict it to the full 2-subcategory LModcl,lax
+V
+( �
+Cat). Taking left adjoints and using Proposition 4.16,
+we get the 2-fully faithful 2-functor PrL
+V ⊂ (LModcl,lax
+V
+( �
+Cat))L → ( �
+CatV)L.
+4.4
+Evaluation and Enriched Hom in Tensored Categories
+In this subsection, we introduce homV in the presentably V-tensored context and show that it satis-
+fies expected properties for adjunction in Proposition 4.21 (see also Remark 4.22 for the connection
+to adjunction in V-enriched categories). With this definition in mind, we finish the proof of (1) of
+Theorem 2.9 in Proposition 4.25.
+Let C ∈ PrL
+V. We have an equivalence of categories FunL
+V(V, C) ∼
+−→ C given by evaluation at 1V.
+Recall from Proposition 4.16 that there is a full 2-subcategory inclusion PrL
+V ⊂ (LModcl,lax
+V
+( �
+Cat))L.
+Thus, passing to right adjoint, gives us a map FunL
+V(V, C) → Funlax-V(C, V)op. Composing the two
+and taking (−)op, we get Cop → Funlax-V(C, V).
+Definition 4.20. We let homV(−, −): Cop × C → V to be the functor corresponding to the above
+under the exponential adjunction. By construction, it is lax V-linear in the second argument.
+Namely, this construction sends X to the lax V-linear functor homV(X, −): C → V, right adjoint
+to the V-linear left adjoint − ⊗ X : V → C.
+Proposition 4.21. Let L: C → D ∈ PrL
+V be a V-linear left adjoint functor, and let R: D → C
+denote its right adjoint. Then there is a natural isomorphism
+homV(L(−), −) ∼= homV(−, R(−))
+of functors Cop × D → V lax V-linear in the second coordinate.
+Proof. Indeed, consider the following diagram:
+C
+FunL
+V(V, C)
+Funlax-V(C, V)
+D
+FunL
+V(V, D)
+Funlax-V(D, V)
+L
+L◦−
+−◦R
+∼
+∼
+17
+
+The left square commutes by passing to the right adjoints and noting that evaluation at 1V com-
+mutes with post-composition. To see that the right square commutes, recall that the horizontal
+morphisms were constructed by the full 2-subcategory inclusion PrL
+V ⊂ (LModcl,lax
+V
+( �
+Cat))L and
+passing to the right adjoints, and the vertical morphisms are also adjoints in the same category, so
+the commutativity is the fact that the right adjoint of a composition is the composition of the right
+adjoints in reverse order.
+Remark 4.22. We note that Proposition 4.21 is in the context of presentably V-tensored categories,
+and not that of V-enriched categories. One of the main features of χ is that the V-enrichment
+of χ(C) is given by homV(X, Y ), as is shown in [GH15, Corollary 7.4.9]. By Corollary 4.19, the
+adjunction L ⊣ R produces an adjunction χ(L) ⊣ χ(R) in �
+CatV.
+One might expect that this
+adjunction provides a natural isomorphism similar to the one of Proposition 4.21 for the V-enriched
+hom’s of χ(C) and χ(D). However, we do not provide such a natural isomorphism, nor show its
+compatibility with the one constructed above.
+Let C0 ∈ CatV, X ∈ C0 and D ∈ PrL
+V.
+Recall from Definition 2.5 that FunV(C0, D) :=
+LModC0(Fun(C≃
+0 , D)). Consider i: pt → C≃
+0 choosing X, and the following diagram:
+D
+Fun(pt, D)
+Fun(C≃
+0 , D)
+LModC0(Fun(C≃
+0 , D))
+i!
+i∗
+free
+forget
+Here the dashed arrows are the left adjoints of the solid arrows. Also recall from Definition 2.7 that
+if we replace C0 by Cop
+0
+and let D = V, the V-functor category is PV(C0). Moreover, in this case
+the three categories in the diagram are presentably V-tensored and the two left adjoint functors are
+V-linear, as in [Hin20, 6.2.3]. Via Proposition 4.16, the right adjoints are canonically lax V-linear.
+Definition 4.23. Let C0 ∈ CatV, X ∈ C0 and D ∈ PrL
+V be as above. We define the evaluation at
+X functor evalX : FunV(C0, D) → D by evalX := i∗ ◦ forget. For the case evalX : PV(C0) → V, it is
+canonically lax V-linear.
+Proposition 4.24. Let C0 ∈ CatV and D ∈ PrL
+V, and consider FunV(C0, D) ∈ PrL. Then evalX
+commutes with (co)limits (i.e. (co)limits are computed level-wise), and are jointly conservative over
+all X.
+For PV(C0) ∈ PrL
+V, the lax V-linear structure on evalX is strong (i.e., the V-action is
+level-wise).
+Proof. Recall that evalX = i∗ ◦ forget. The functor forget commutes with all (co)limits by [Lur17,
+Corollary 4.2.3.3 and Corollary 4.2.3.5] as the forgetful from modules, while i∗ commutes with
+(co)limits as they are computed level-wise in (unenriched) functor categories.
+The forgetful from modules is always conservative by [Lur17, Corollary 4.2.3.2], and the evalu-
+ation at X functors Fun(C≃
+0 , D) → D are jointly conservative.
+For the case of PV(C0), the V-action is by construction given level-wise, as explained in [Hin20,
+6.2.3].
+Finally, we are in position to prove our variant of Hinich’s enriched Yoneda lemma, appearing
+as (1) of Theorem 2.9.
+Proposition 4.25. Let C0 ∈ CatV, then homV(よV(X), −) ∼= evalX as lax V-linear functors, and
+in particular homV(よV(X), −) is also strong V-linear.
+18
+
+Proof. We need to show that homV(よV(X), −) ∼= evalX as lax V-linear functors. By construction,
+this is equivalent to showing that −⊗よV(X) ∼= free ◦i! as V-linear functors. Consider the following
+diagram:
+V
+Fun(Cop,≃
+0
+, V)
+LModCop
+0 (Fun(Cop,≃
+0
+, V))
+S
+Fun(Cop,≃
+0
+, S)
+−⊗1V
+hom(1V,−)
+−⊗1V
+hom(1V,−)
+i!
+i∗
+i!
+i∗
+free
+forget
+Here the dashed arrows are the left adjoints of the solid arrows. Clearly, the solid square commutes,
+and thus, the dashed square obtained by passing to left adjoints also commutes.
+By [Hin20, 6.2.6], we have that よV(X) ∼= free((−⊗1V)◦よ(X)), where here よ(X) ∈ Fun(Cop,≃
+0
+, S)
+is the image of X under the (unenriched) Yoneda embedding. Applying the naturality of the (un-
+enriched) Yoneda embedding to i: pt → S, and noting that よ: pt → S sends よ(pt) = pt, we get
+that よ(X) = よ(i(pt)) ∼= i!(よ(pt)) ∼= i!(pt). Using the commutativity of the dashed square above,
+we conclude that
+よV(X) ∼= free((− ⊗ 1V) ◦ よ(X)) ∼= free((− ⊗ 1V) ◦ i!(pt)) ∼= free(i!1V).
+Since free and i! are V-linear functors, we finally get that
+− ⊗ よV(X) ∼= − ⊗ free(i!1V) ∼= free ◦i!(− ⊗ 1V) ∼= free ◦i!
+as V-linear functors, concluding the proof.
+5
+Atomic Objects, Presheaves and Yoneda
+In this section, building on the 2-categorical results of the previous section, we study internally left
+adjoint functors and atomic objects, and connect them to the enriched Yoneda embedding.
+We begin in Definition 5.1 by defining internally left adjoints between presentably V-tensored
+categories. Namely, V-linear left adjoint functors, whose right adjoints are also left adjoint and their
+canonical lax V-linear structure is strong. Following this, in Definition 5.4 we define a finiteness
+condition called being atomic. We say that X in C ∈ PrL
+V is atomic if − ⊗ X : V → C is internally
+left adjoint, that is, if homV(X, −) commutes with colimits and is V-linear. For instance, in the case
+V = Sp, this coincides with condition of being compact. From the definition, internally left adjoints
+send atomic objects to atomic objects. The key technical result of this section is Proposition 5.16,
+giving a converse result under the assumption that the source category is generated from the atomics
+under weighted colimits.
+Next, in Proposition 5.17 we show that よV(X) is atomic in PV(C0), and in Proposition 5.18 we
+show that together they generate PV(C0) under weighted colimits. This allows us to use Proposi-
+tion 5.16 to prove Theorem 5.20, the main result of this section. This result says that the partial
+adjunction of Theorem 3.6 restricts to a (non-partial) adjunction between the enriched presheaves
+functor and taking the atomics, with the unit being (the factorization through the atomics of) the
+enriched Yoneda embedding.
+19
+
+5.1
+Internally Left Adjoints
+Definition 5.1. A V-linear left adjoint functor is called internally left adjoint if it is left adjoint
+in the 2-category PrL
+V. We denote the wide 2-subcategory on the internally left adjoint functors by
+PriL
+V := (PrL
+V)L. For C, D ∈ PriL
+V , we denote the category of V-linear internally left adjoint functors
+between them by FuniL
+V (C, D) ∈ �
+Cat and the corresponding homiL
+V (C, D) ∈ �S.
+Note that for any V-linear left adjoint functor L: C → D, Proposition 4.16 shows that it admits
+a lax V-linear right adjoint R: D → C.
+Proposition 5.2. A V-linear left adjoint functor L: C → D is internally left adjoint if and only if
+the lax V-linear right adjoint R: D → C is strong and itself a left adjoint.
+Proof. As above, R is a morphism in LModcl,lax
+V
+( �
+Cat) between objects of the 2-subcategory PrL
+V,
+and the condition is that it is in fact a morphism in that 2-subcategory. By Proposition 4.16, PrL
+V
+is a full 2-subcategory of (LModcl,lax
+V
+( �
+Cat))L, thus the condition is that R is a morphism in that
+2-subcategory. The result then follows from Corollary 4.13.
+Taking left adjoints in Corollary 4.19, we immediately get:
+Corollary 5.3. χ restricts to a 2-fully faithful 2-functor
+χ: PriL
+V → ( �
+CatV)LL.
+5.2
+Atomic Objects
+Definition 5.4. Let C ∈ PrL
+V. We say that X ∈ C is atomic if the functor − ⊗ X : V → C is
+internally left adjoint. We denote the full V-subcategory on the atomic objects by Cat ⊂ χ(C).
+Example 5.5. The unit 1V ∈ V is always atomic, because − ⊗ 1V is the identity functor.
+Example 5.6 ([BMS21, Proposition 2.6]). In the case V = Sp, atomic objects coincide with compact
+objects.
+Recall that for C ∈ PrL
+V, there is an equivalence FunL
+V(V, C)
+∼
+−→ C given by evaluation at 1V,
+with inverse sending X to − ⊗ X : V → C. Thus, immediately from the definition, we deduce the
+following:
+Proposition 5.7. Evaluation at 1V induces an equivalence FuniL
+V (V, C) ∼
+−→ Cat.
+Since the right adjoint of − ⊗ X : V → C is homV(X, −): C → V, the following follows immedi-
+ately from Proposition 5.2.
+Proposition 5.8. An object X ∈ C is atomic if and only if homV(X, −): C → V preserves colimits
+and the lax V-linear structure is strong.
+Proposition 5.9. The atomics are a small V-enriched category.
+Proof. The argument is identical to [BMS21, Proposition 2.7]. We repeat the details for the conve-
+nience of the reader.
+Let κ be a regular cardinal such that the unit 1V is κ-compact, namely hom(1V, −): V → S
+commutes with κ-filtered colimits. We show that all atomic objects are κ-compact. Let C ∈ PrL
+V
+20
+
+and let X ∈ C be atomic.
+By Proposition 5.8, homV(X, −): C → V commutes with κ-filtered
+colimits. Since hom(X, −) ∼= hom(1V ⊗ X, −) ∼= hom(1V, homV(X, −)), we get that hom(X, −)
+also commutes with κ-filtered colimits, i.e. X is κ-compact. We have shown that Cat ⊆ Cκ, the
+latter being a small category, concluding the proof.
+Proposition 5.10. Let L: C → D be an internally left adjoint, i.e. a morphism in PriL
+V , then
+it sends atomics to atomics.
+Thus, the V-functor χ(L): χ(C) → χ(D) factors to a V-functor
+L: Cat → Dat.
+Proof. Let X ∈ Cat. Then, since L is V-linear, we have a natural isomorphism −⊗LX ∼= L(−⊗X)
+of functors V → D. Since both − ⊗ X and L are internally left adjoints, so is − ⊗ LX, i.e. LX is
+atomic as required.
+Recall that Cat is a full V-subcategory of χ(C), thus Cat → χ(C) is a subobject (indeed, the
+space of maps to Cat is exactly the subspace of maps to χ(C) that land in Cat). Furthermore, by
+Proposition 5.10, the restriction of internally left adjoint functors to the atomics factor through
+the atomics. Thus, by [Ram22, Proposition A.1], this assembles into an induced subfunctor, which
+furthermore lands in CatV ⊂ �
+CatV.
+Definition 5.11. We denote the induced subfunctor of atomics by
+(−)at : PriL
+V → CatV,
+equipped with a natural transformation (−)at ⇒ χ|PriL
+V of functors PriL
+V → �
+CatV.
+Definition 5.12. Let C ∈ PrL
+V. We say that a collection of atomic objects B ⊆ Cat are atomic
+generators, if C is generated from B under weighted colimits. If such B exists, we say that C is
+molecular.
+Remark 5.13. We note that this definition a priori differs from our definition of molecular in [BMS21]
+(where we work over a mode). We expect that closure under weighted colimits coincides with closure
+under colimits and the V-action, but we are unaware of a proof of this statement. Such a result
+would make the connection transparent.
+We now wish to prove a converse to Proposition 5.10 under the assumption that the source is
+molecular. To that end, we begin with the following.
+Definition 5.14. Let I ∈ CatV and C ∈ PrL
+V, and fix f ∈ FunV(I, C). By (2) of Theorem 2.9, the
+functor colim(−)
+I
+(f): PV(I) → C is colimit preserving, i.e. a left adjoint, and V-linear. We denote
+the lax V-linear right adjoint by
+homV(f(−), −): C → PV(I).
+Lemma 5.15. There is a natural isomorphism of lax V-linear functors
+evali ◦ homV(f(−), −) ∼= homV(f(i), −).
+Proof. Observe that we have an equivalence of V-linear left adjoint functors
+colim(−)
+I
+(f) ◦ (− ⊗ よV(i)) = colim(−⊗よV(i))
+I
+(f)
+(1)
+∼= − ⊗ colimよV(i)
+I
+(f)
+(2)
+∼= − ⊗ f(i),
+21
+
+where (1) follows from (2) of Theorem 2.9, and (2) follows from (3) of Theorem 2.9. Passing to the
+lax V-linear right adjoints, we get
+homV(よV(i), homV(f(−), −)) ∼= homV(f(i), −).
+We finish by recalling that homV(よV(i), −) is the evaluation at i by (1) of Theorem 2.9.
+Proposition 5.16. Let L: C → D be in PrL
+V. If C is molecular and L sends a collection of atomics
+generators B ⊂ Cat to atomic objects of D, then L is internally left adjoint.
+Proof. We adapt the proof of [BMS21, Proposition 2.18]. We wish to show that R: D → C, the
+right adjoint of L, is itself a left adjoint and that the lax V-action is strong. Thus, it suffices to
+show that for any v ∈ V and Y : I → D, the canonical map v ⊗ colimI RYi → R(v ⊗ colimI Yi) is an
+isomorphism. By the (unenriched) Yoneda lemma in the category C, this is equivalent to checking
+that for every X ∈ C the map
+hom(X, v ⊗ colim
+I
+RYi) → hom(X, R(v ⊗ colim
+I
+Yi))
+(2)
+is an isomorphism. We will in fact show the stronger statement that
+homV(X, v ⊗ colim
+I
+RYi) → homV(X, R(v ⊗ colim
+I
+Yi))
+(3)
+is an isomorphism, which implies the previous statement by taking hom(1V, −). Let A ⊂ C be the
+collection of objects X for which it is an isomorphism, and we will show that A = C.
+We first show that B ⊂ A. Let X ∈ B. By Proposition 4.21, the following diagram commutes,
+and both vertical maps are isomorphisms:
+v ⊗ colimI homV(X, RYi)
+homV(X, v ⊗ colimI RYi)
+homV(X, R(v ⊗ colimI Yi))
+v ⊗ colimI homV(LX, Yi)
+homV(LX, v ⊗ colimI Yi)
+≀
+≀
+The upper-left morphism is an isomorphism because X is atomic, and similarly the bottom mor-
+phism is an isomorphism because LX is atomic since X ∈ B and L sends B to atomic objects. This
+shows that the upper-right morphism is an isomorphism as well.
+We now show that A is closed under weighted colimits.
+Let J ∈ CatV, W ∈ PV(J), and
+f ∈ FunV(J, C) a V-functor landing in A ⊂ C. We show that X := colimW
+J (f) is in A. By definition,
+homV(f(−), −): C → PV(J) is the lax V-linear right adjoint of colim(−)
+J
+(f): PV(J) → C. Thus, by
+Proposition 4.21 we have a natural isomorphism of functors C → V
+homV
+C(X, −) = homV
+C(colimW
+J (f), −) ∼= homV
+PV(J)(W, homV(f(−), −)).
+Thus it suffices to check that the map
+homV
+C(f(−), v ⊗ colim
+I
+RYi) → homV
+C(f(−), R(v ⊗ colim
+I
+Yi))
+∈ PV(J)
+is an isomorphism. By Proposition 4.24, the evaluation at j ∈ J are jointly conservative, so it
+suffices to check that the map is an isomorphism after evaluation at every j. By Lemma 5.15, this
+means that we need to check that
+homV(f(j), v ⊗ colim
+I
+RYi) → homV(f(j), R(v ⊗ colim
+I
+Yi))
+22
+
+is an isomorphism, which holds since by assumption f lands in A.
+Recall that B are atomic generators of C, and we have shown that B ⊂ A and that A is closed
+under weighted colimits, thus A = C, as required.
+5.3
+Atomics–Presheaves Adjunction
+Proposition 5.17. Let C0 ∈ CatV, then the enriched Yoneda embedding lands in the atomic objects,
+yielding a V-functor よV : C0 → PV(C0)at.
+Proof. Recall from Proposition 5.8 that よV(X) is atomic if and only if homV(よV(X), −): PV(C0) →
+V preserves colimits and the lax V-linear structure is strong. By (1) of Theorem 2.9, this V-functor
+is the evaluation at X, concluding the proof by Proposition 4.24.
+We thus get an induced natural transformation よV : id ⇒ PV(−)at of functors CatV → CatV.
+Proposition 5.18. The category PV(C0) is molecular, with the image of the enriched Yoneda
+embedding as atomic generators.
+Proof. By Proposition 5.17, the image of よV are atomic in PV(C0).
+The fact that PV(C0) is
+generated from the image of enriched Yoneda embedding under weighted colimits is [Hin21, 6.3.1],
+recalled as (3) of Theorem 2.9.
+Proposition 5.19. The functor PV : CatV → PrL
+V of Theorem 3.6 lands in PriL
+V .
+Proof. Let f : C0 → D0 be a V-functor, and we need to show that f! : PV(C0) → PV(D0) is internally
+left adjoint.
+Recall from Proposition 5.18 that PV(C0) is molecular with the image of よV
+C0 as
+atomic generators. As in Corollary 3.8, naturality of the enriched Yoneda embedding says that
+よV
+D0f ∼= f!よV
+C0.
+Thus, f! sends the image of よV
+C0 to the image of よV
+D0 which are atomic in
+PV(D0). Thus, f! sends a collection of atomic generators to atomic objects, so it is internally left
+adjoint by Proposition 5.16.
+Theorem 5.20. The partial adjunctions of Theorem 3.6 restricts to an adjunction
+PV : CatV ⇄ PriL
+V :(−)at
+with unit the enriched Yoneda embedding よV : id ⇒ PV(−)at.
+Proof. We need to check that for any C0 ∈ CatV and D ∈ PriL
+V , the map
+homiL
+V (PV(C0), D)
+(−)at
+−−−→ homV(PV(C0)at, Dat)
+(よV)∗
+−−−−→ homV(C0, Dat)
+(4)
+is an equivalence. Recall that
+homL
+V(PV(C0), D)
+χ−→ homV(χ(PV(C0)), χ(D))
+(よV)∗
+−−−−→ homV(C0, χ(D))
+(5)
+is an equivalence. Furthermore, both the first and last spaces in (4) are a collection of connected
+components of the first and last spaces in (5), showing that the composition in (4) is an inclusion
+of connected components.
+To finish the argument, we need to show that the composition in (4) hits every connected
+component.
+To that end, let f : C0 → Dat be a V-functor.
+We can post-compose it with the
+23
+
+inclusion V-functor Dat → D, and using (5) we get ˜f : PV(C0) → D in PrL
+V. It is left to show that ˜f
+is internally left adjoint. Recall that the image of the enriched Yoneda embedding forms a collection
+of atomic generators by Proposition 5.18. Furthermore, by construction, ˜f(よV(X)) = f(X) ∈ Dat
+is atomic. Thus, we have shown that ˜f sends a collection of atomic generators to atomics, so it is
+indeed internally left adjoint by Proposition 5.16.
+We extend our notations from Definition 3.7:
+Definition 5.21. For f : C0 → D0 a morphism in CatV, consider the internally left adjoint functor
+f! : PV(C0) → PV(D0). It has a right adjoint in PrL
+V, i.e. a V-linear left and right adjoint functor
+denoted f ⊛ : PV(D0) → PV(C0). This functor thus has a further lax V-linear right adjoint denoted
+f⊛ : PV(C0) → PV(D0).
+Using Corollary 5.3 we get the composition
+χ PV : CatV
+PV
+−−→ PriL
+V
+χ−→ (�
+CatV)LL,
+allowing to pass the (double) adjunction to �
+CatV.
+Corollary 5.22. For f : C0 → D0 a morphism in CatV, we get a double adjunction χ(f!) ⊣ χ(f ⊛) ⊣
+χ(f⊛) in �
+CatV.
+Corollary 5.23. Let f : C0 → D0 in CatV, then f ⊛(G)(X) ∼= G(f(X)).
+Proof. By Corollary 3.8 we have f!よV ∼= よVf in FunV(C0, PV(D0)). We thus get
+f ⊛(G)(X) ∼= homV(よV(X), f ⊛(G)) ∼= homV(f!(よV(X)), G) ∼= homV(よV(f(X)), G) ∼= G(f(X)),
+where the first and last steps follow from the enriched Yoneda lemma of (1) of Theorem 2.9, the
+second step is by Proposition 4.21, and the third step is by the isomorphism above.
+6
+Heine’s Enriched Yoneda Embedding
+Independently of Hinich’s enriched Yoneda embedding, in [Hei20], Heine defines an enriched Yoneda
+embedding as well, which satisfies the exact same universal property appearing in Proposition 2.12.
+Theorem 6.1 ([Hei20, Corollary 6.2]). Let C0 ∈ CatV.
+There is a V-natural transformation
+よV
+Heine : C0 → χ(PV(C0)). For every D ∈ PrL
+V, the composition
+homL
+V(PV(C0), D)
+χ−→ homV(χ(PV(C0)), χ(D))
+(よV
+Heine)∗
+−−−−−−→ homV(C0, χ(D))
+is an equivalence.
+We also note these versions of the enriched Yoneda embedding agree point-wise. In particular,
+Heine’s version also satisfies the enriched Yoneda lemma.
+Proposition 6.2. For every X ∈ C0, there is an isomorphism よV
+Heine(X) ∼= よV(X). In particular,
+homV(よV
+Heine(X), −): PV(C0) → V is given by evaluation at X.
+24
+
+Proof. Consider the composition
+C≃
+0
+よ
+−→ Fun(Cop,≃
+0
+, S)
+1V⊗−
+−−−−→ Fun(Cop,≃
+0
+, V)
+free
+−−→ LModCop
+0 (Fun(Cop,≃
+0
+, V)) = PV(C0).
+By [Hin20, 6.2.6] and [Hei20, Proposition 6.5], よV(X) and よV
+Heine(X) (respectively) are the image
+of X under this composition.
+The second part then follows from (1) of Theorem 2.9.
+Note that these two results do not show that Hinich’s and Heine’s enriched Yoneda embeddings
+agree as functors, but rather only up to an automorphism of PV(C0) (which may not be the identity).
+See Subsection 1.3.2 for further discussion. Nevertheless, we now explain that the main results of
+this paper hold for Heine’s enriched Yoneda embedding as well.
+The proofs in Section 3 and
+Section 5 have relied on the the universal property of Proposition 2.12 and the enriched Yoneda
+lemma of (1) of Theorem 2.9 for よV. These two results also hold for よV
+Heine by the above. We
+also used weighted colimits, and their relationship with the enriched Yoneda embedding appearing
+in (2) and (3) of Theorem 2.9. However, this was only used in the proof of Proposition 5.18, whose
+statement is about the image of the enriched Yoneda embedding which is the same for よV and
+よV
+Heine by Proposition 6.2, and in the proof of Lemma 5.15, whose statement does not involve
+the enriched Yoneda embedding. Thus, we see that the results of Section 3 and Section 5 hold
+for Heine’s enriched Yoneda embedding as well. Notably, the analogue of Theorem 5.20 gives an
+adjunction
+PV
+Heine : CatV ⇄ PriL
+V :(−)at
+with unit Heine’s enriched Yoneda embedding よV
+Heine : id ⇒ PV
+Heine(−)at.
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