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+DOT: Fast Cell Type Decomposition of Spatial
+Omics by Optimal Transport
+Arezou Rahimi, Luis A. Vale-Silva, Maria F¨alth Savitski, Jovan Tanevski, Julio Saez-Rodriguez
+!
+Abstract
+Single-cell RNA sequencing (scRNA-seq) and spatially-resolved imaging/sequencing technologies have revolutionized biomedical
+research. On one hand, scRNA-seq data provides for individual cells information about a large portion of the transcriptome, but does not
+include the spatial context of the cells. On the other hand, spatially resolved measurements come with a trade-off between resolution,
+throughput and gene coverage. Combining data from these two modalities can provide a spatially resolved picture with enhances
+resolution and gene coverage. Several methods have been recently developed to integrate these modalities, but they use only the
+expression of genes available in both modalities. They don’t incorporate other relevant and available features, especially the spatial
+context. We propose DOT, a novel optimization framework for assigning cell types to tissue locations. Our model (i) incorporates ideas
+from Optimal Transport theory to leverage not only joint but also distinct features, such as the spatial context, (ii) introduces scale-
+invariant distance functions to account for differences in the sensitivity of different measurement technologies, and (iii) provides control
+over the abundance of cells of different types in the tissue. We present a fast implementation based on the Frank-Wolfe algorithm and
+we demonstrate the effectiveness of DOT on correctly assigning cell types or estimating the expression of missing genes in spatial data
+coming from two areas of the brain, the developing heart, and breast cancer samples.
+Index Terms
+Optimal Transport, optimization, Frank-Wolfe, single-cell, biology, spatial, tissue, decomposition, deconvolution
+Arezou Rahimi is with the Institute for Computational Biomedicine, Faculty of Medicine, Heidelberg University, Heidelberg University Hospital, and Cellzome
+GmbH, GlaxoSmithKline, Heidelberg, Germany (e-mail: [email protected]).
+Luis A. Vale-Silva is with Cellzome GmbH, GlaxoSmithKline, Heidelberg, Germany (e-mail: [email protected])
+Maria F¨alth Savitski is with Cellzome GmbH, GlaxoSmithKline, Heidelberg, Germany (e-mail: [email protected])
+Jovan Tanevski is with the Institute for Computational Biomedicine, Faculty of Medicine, Heidelberg University and Heidelberg University Hospital, Heidelberg,
+Germany, and Department of Knowledge Technologies, Joˇzef Stefan Institute, Ljubljana, Slovenia (e-mail: [email protected]).
+Julio Saez-Rodriguez is with the Institute for Computational Biomedicine, Faculty of Medicine, Heidelberg University and Heidelberg University Hospital,
+Heidelberg, Germany (e-mail: [email protected]).
+Jovan Tanevski and Julio Saez-Rodriguez cosupervised this work.
+arXiv:2301.01682v1  [cs.CE]  4 Jan 2023
+
+1
+DOT: Fast Cell Type Decomposition of Spatial
+Omics by Optimal Transport
+1
+INTRODUCTION
+The organization of cells within human tissues, their molec-
+ular programs and their response to perturbations are cen-
+tral to better understand physiology, disease progression
+and to eventual identification of targets for therapeutic in-
+tervention [1], [2]. Cell types are distinct subpopulations of
+cells with unique trasncriptional signatures, which are often
+identified by known markers and/or by data-driven tech-
+niques, most commonly clustering based on transcriptomic
+profiles [3]. Single-cell RNA sequencing can profile the
+entire transcriptome (mRNA expression of the full range of
+genes) of large portions of individual (single) cells. This has
+made scRNA-seq an essential tool for revealing distinct cell
+types in complex tissues and has profoundly impacted our
+understanding of biological processes and the underlying
+mechanisms that control cellular functions [4], [5], [6], [7].
+However, scRNA-seq requires dissociation of the tissue [8],
+losing the information about the spatial context and physical
+relationship between cells.
+To overcome these limitations, there has been recent ad-
+vancements in spatially resolved transcriptomics methods
+[9]. Spatial transcriptomics methods measure gene expres-
+sion in locations, hereafter referred to as spots, coupled with
+their two- or three-dimensional position. These methods
+vary in two axes: spatial resolution and gene throughput.
+On one hand, technologies such as Multiplexed Error-
+Robust Fluorescence In-Situ Hybridization (MERFISH) and
+In-Situ Sequencing (ISS), achieve cellular or even subcellular
+resolution [10] through cell segmentation [11], [12], but
+are limited to measuring up to a couple of hundred pre-
+selected genes. On the other hand, spatially resolved RNA
+sequencing, such as Spatial Transcriptomics [13], commer-
+cially available as 10x’s Visium, and Slide-seq [14], enable
+high-throughput gene profiling by capturing mRNAs in-situ
+at the cost of spots with the size of tens of cells. Thus, there
+is a trade-off between the resolution and the richness of the
+data.
+A strategy to overcome these limitations is to combine
+scRNA-seq data with high resolution spatial data to map
+dissociated cells to spatial locations or more generally to
+combine it with low-resolution spatial data to estimate the
+composition of cell types and expression in each spot. We
+refer to this task as decomposition. Alternatively, we can
+attempt to enrich high-resolution data by predicting the
+expression of unmeasured genes. As the latter requires
+extrapolation to various degrees, machine learning and opti-
+mization methods are generally suited to the decomposition
+task. We will show that our tailored Optimal Transport
+formulation is capable of tackling both decomposition and
+enrichment tasks in high- and low-resolution spatial data.
+Since the initial efforts to bridge this gap [15], there has
+been an increased interest in improvement and new method
+development (see Section 2). However, so far the methods
+rely on the genes that are captured both by scRNA-seq and
+spatial data without using the remaining genes captured in
+each modality, do not use the spatial relationships between
+spots in the spatial data, and usually come with high com-
+putation cost for large instances. Neglecting the spatial con-
+text is equivalent to assuming random placement of spots in
+the space, which is in contrast to the established structure-
+function relationship of tissues. Considering only a subset
+of genes limits the applicability of these methods to cases
+where the two data sets share several informative genes,
+which might not be the case when different technologies are
+used for profiling, or when few genes are measured in the
+spatial data (e.g., in MERFISH).
+We address these limitations by incorporating ideas
+from the Optimal Transport (OT) theory and adapting
+a Gromov-Wasserstein (GW) distance [16], [17] between
+scRNA-seq and spatial data. We present DOT (Fast Cell
+Type Decomposition by Optimal Transport), a fast and
+scalable optimization framework to integrate scRNA-seq
+and spatial data for cell type localization by solving a multi-
+criteria probabilistic matching problem. We summarize the
+main contributions of our work as follows:
+(i) We propose a novel formulation for mapping cell types
+from scRNA-seq to spots in spatial data by casting
+this problem to a multi-objective probabilistic matching
+problem. Our model is applicable to both high- and
+low-resolution spatial data, in the form of inferring
+membership probabilities for the former and relative
+abundance of cell types in the latter, and is capable of
+estimating the expression of genes that are missing in
+the spatial data but present in the scRNA-seq data.
+(ii) We adapt a generalization of OT with a Gromov-
+Wasserstein objective to leverage spatial information
+and to go beyond the use of genes common to the two
+modalities.
+(iii) We introduce a scale-invariant metric based on cosine-
+similarity to account for differences in the scale of gene
+expressions in different technologies.
+(iv) We present a very fast implementation for our model
+based on the Frank-Wolfe algorithm, ensuring scalabil-
+ity and efficient solvability in large-scale datasets.
+2
+RELATED WORK
+Cell type decomposition. Several decomposition methods
+(also known as deconvolution methods) have been pro-
+posed in recent years. As cell type decomposition, partic-
+ularly in the high-resolution spatial data, is inherently a
+multiclass classification task, classification methods, such as
+Random Forests [18], can be used for tackling this problem.
+
+2
+However, because of the domain-specific properties of this
+problem, including differences in gene coverage, resolution,
+measurement sensitivity, and modality-specific characteris-
+tics, tailored approaches are needed.
+While most of these models are designed specifically for
+low-resolution spatial data, some are also applicable to high-
+resolution spatial data. [19] proposed SPOTlight, which es-
+timates relative abundance of cell types in spots using non-
+negative matrix factorization regression and non-negative
+least squares. Robust cell type decomposition (RCTD) [20]
+fits a statistical model by maximum-likelihood estimation,
+assuming a Poisson distribution for the expression of each
+gene at each spot. cell2location assumes a two-step Bayesian
+model for inferring cell type composition of spots [21].
+Tangram [22] proposes a deep learning model to find the
+best placement of single cells in spots using a designed
+loss function and can thus carry cell type information as a
+byproduct. Seurat V3 workflow [23] is a widely-used toolkit
+for analyzing scRNA-seq data, which offers an “anchoring”
+technique based on mutual nearest neighbours classifier for
+aligning two modalities in the PCA space.
+Optimal Transport. Optimal Transport (OT) [24] is a way
+to match, with minimal cost, data points/histograms be-
+tween two domains embedded in possibly different spaces
+using different variants of the Wasserstein distance [25], [26],
+[27]. Over the past years, OT has been applied to various
+machine learning problems in a wide variety of contexts,
+including but not limited to generative modeling [28], [29],
+Wasserstein auto-encoders [30], feature aggregation [31],
+generalization error prediction [32], dataset denoising [33],
+graph matching/classification [34], and domain adaptation
+[35], [36], [37].
+Recently, OT has been employed in biology, in particular
+to analyse single-cell data. For example, [38] model cellu-
+lar dynamics as an unbalanced dynamic transport, with
+the goal of transporting entities from one cross sectional
+measurement to the next. [39] use OT for studying devel-
+opmental time courses and understanding the molecular
+programs that guide differentiation during development
+by incorporating temporal information and modeling cell
+growth over time. Similarly, [40] employ graphical models
+and OT to reconstruct developmental trajectories from time
+courses with snapshots of cell states and lineages.
+3
+MODEL
+3.1
+Preliminaries
+Given a reference scRNAseq data (R for short), which is
+a collection of single cells each annotated with a cell type
+c ∈ C, and a target spatially resolved transcriptomics data
+(S for short), which consists of a set I of spots without cell
+type annotations, the goal of decomposition is to determine
+the composition of cell types in spots of S. Note that the
+term “spot” can refer to one or a group of cells in certain
+spatial contexts. We denote by ni the given size (number of
+cells) of spot i ∈ I. When such information is not available,
+or when spots are at single-cell resolution, we set ni = 1 to
+compute the proportion or probability of cell types in each
+spot rather than computing the number of cells of each type.
+Let XR
+c,g denote the mean expression of gene g ∈ GR in
+cell type c ∈ C, where GR is the set of genes measured in R.
+Each spot i ∈ I of S consists of spatial coordinates xi ∈ R2 or
+R3 and gene expressions XS
+i,g for g ∈ GS, where GS is the set
+of genes that are measured in S. Further, if prior information
+about the expected abundance of cell types in S is available
+(e.g., estimated from a neighboring single-cell level tissue),
+we denote the expected abundance of cell type c ∈ C in S
+by rc. Note that r is scaled such that �
+i∈I ni = �
+c∈C rc.
+For convenience, we also define G = GR ∩ GS as the set of
+genes that are common between R and S. In the following,
+unless otherwise mentioned, vectors of gene expressions are
+assumed to be in the space of common genes.
+To assess dissimilarity between expression vectors a and
+b, we also introduce the distance function
+dcos(a, b) :=
+�
+1 − cos (a, b),
+(1)
+where cos (a, b) =
+1
+∥a∥∥b∥⟨a, b⟩. Note that dcos is convex for
+positive vectors a and b, and is scale-invariant, in the sense
+that it is indifferent to the magnitudes of the vectors. This
+is by design, since we want to assess dissimilarity between
+expression vectors regardless of the measurement sensitiv-
+ities of different technologies. We also note the following
+important property of dcos (proofs given in Appendix C).
+Proposition 1. Unlike cosine dissimilarity (i.e., 1 − cos(·, ·)),
+dcos is a metric distance function.
+3.2
+High-Level Model
+Our model relies on determining a “many-to-many” map-
+ping Y of cell types in R to spots in S, with Yc,i denoting the
+proportion (or probability when ni = 1) of spot i ∈ I that is
+of cell type c ∈ C. A high-quality mapping should naturally
+match the expression of common genes across R and S. We
+ensure this by considering the following genomic criteria:
+(i) Expression of genes in each spot of S should match
+expression of genes mapped to that spot via Y .
+(ii) Centroid of each cell type in R should match the cen-
+troid of that cell type in S as determined via Y .
+(iii) Distribution of expression of each gene across spots of S
+should be similar to the distribution of that gene across
+spots as mapped from R to S via Y .
+Additionally, we may incorporate prior knowledge in the
+form of spatial location of spots as well as expected abun-
+dance of cell types using the following auxiliary criteria:
+(iv) Spots that are both adjacent in space and have similar
+expression profiles should attain similar cell type pro-
+files.
+(v) If prior information about abundance of cell types in S
+is available (e.g., when R and S correspond to adjacent
+tissues), abundance of cell types mapped to S should
+match with the given abundances.
+The genomic objectives naturally take precedence over
+the auxiliary objectives, especially when a large number
+of genes are common between R and S, but the auxiliary
+objectives are useful when the common genes are limited.
+Note that objective (v) is meant to provide additional control
+over the abundance of cell types in the spatial data, but can
+be ignored if prior information about the abundance of cell
+types is not available. We elaborate on these objectives in
+the following.
+
+3
+3.3
+Formulation
+Objective (i) ensures that the vector of gene expressions in
+spot i ∈ I (i.e., XS
+i,:) is most similar to the vector of gene ex-
+pressions mapped to spot i through Y (i.e., �
+c∈C Yc,iXR
+c,:).
+To achieve this objective, we minimize dissimilarity between
+these vectors by using
+di(Y ) := dcos(XS
+i,:,
+�
+c∈C Yc,iXR
+c,:).
+(2)
+Objective (ii) is in nature similar to objective (i). Here,
+we would like to minimize dissimilarity between centroid
+of cell type c ∈ C in R (i.e., XR
+c,:) and centroid of cell type
+c in S as determined via Y (i.e.,
+1
+ρc
+�
+i∈I Yc,iXS
+i,:). Given
+the scale-invariance property of dcos, we can drop 1/ρc and
+measure the dissimilarity between these centroids using the
+following distance function
+dc(Y ) := dcos(XR
+c,:, ρ−1
+c
+�
+i∈I Yc,iXS
+i,:)
+= dcos(XR
+c,:,
+�
+i∈I Yc,iXS
+i,:).
+(3)
+Our goal in objective (iii) is to match distribution of
+expression of gene g ∈ G in S (i.e., XS
+:,g) with the one
+mapped to S through Y (i.e., �
+c∈C Yc,:XR
+c,g). Hence, we
+minimize dissimilarity between these vectors by using
+dg(Y ) := dcos(XS
+:,g,
+�
+c∈C Yc,:XR
+c,g).
+(4)
+To achieve objective (iv), we borrow ideas from Optimal
+Transport theory and the Gromov-Wasserstein metric. Let
+M R and M S be metrics in R and S, respectively, in that
+M R
+c,k defines distance between cell types c and k, while
+M S
+i,j defines distance between spots i and j. Note that these
+distances are defined for each dataset independently; hence,
+we can use the entire features in each set: the entire genome
+in R, including the genes not measured in S, and the un-
+common/common genes as well as the spatial coordinates
+in S (see Section 4 for how these matrices are computed).
+The 2-Gromov-Wasserstein distance [16] between R and S
+for given mapping Y , denoted dGW(Y ), is defined in (5).
+Minimizing dGW(Y ) ensures that similar pair of spots in
+S (with respect to their locations and expressions) are not
+assigned to dissimilar pair of cell types in R, and vice versa.
+dGW(Y ) :=
+�
+�
+�
+�
+�
+i∈I
+�
+j∈I
+�
+c∈C
+�
+k∈C
+�
+M R
+c,k − M S
+i,j
+�2
+Yc,iYk,j
+(5)
+Let ρc := �
+i∈I Yc,i denote the abundance of cell type c in
+S as determined by mapping Y . As noted by [41], we may
+simplify (5) as stated in Proposition 2 below.
+Proposition 2. Define parameter
+¯mi
+:=
+�
+j∈I(M S
+i,j)2nj
+and auxiliary variables ¯mc := �
+k∈C(M R
+c,k)2ρk and Z :=
+M RY M S. GW distance function in (5) is equivalent to
+dGW(Y ) =
+��
+c∈C
+�
+i∈I Yc,i( ¯mc + ¯mi − 2Zc,i).
+(6)
+Objective (v) provides optional control over abundance
+of cell types mapped to S, when prior information about
+expected abundance of cell types is available. We employ
+Jensen-Shannon divergence between ρ and r to measure
+their dissimilarity
+dA(Y ) := 1
+2DKL
+�
+ρ
+����
+ρ + r
+2
+�
++ 1
+2DKL
+�
+r
+����
+ρ + r
+2
+�
+,
+(7)
+where DKL (p∥q) = �
+j pj log(pj/qj) denotes the Kull-
+back–Leibler divergence [42]. In addition, to avoid overfit-
+ting, we may require that all cell types are at least minimally
+represented in the mapping. To achieve this goal, we define
+dR(Y ) := −
+�
+c∈C log(ρc) = DKL (¯r∥ρ) ,
+(8)
+where ¯rc = 1 for all c ∈ C. Equation (8) is in fact a
+Nash fairness [43] objective and its logarithmic form ensures
+presence of all cell types (i.e., ρc > 0).
+We treat these criteria as objectives in a multi-objective
+optimization problem and to achieve them simultaneously
+(i.e., produce a Pareto-optimal solution), we optimize Y
+against a linear combination of these objectives as formu-
+lated below, hereafter referred to DOT model:
+min
+�
+i∈I
+nidi(Y ) + λC
+�
+c∈C
+ρcdc(Y ) + λG
+�
+g∈G
+dg(Y )
++ λGWdGW(Y ) + λAdA(Y ) + λRdR(Y )
+(9)
+w.r.t.
+Y ∈ R|C|×|I|
++
+, ρ ∈ R|C|
+(10)
+s.t.
+�
+c∈C Yc,i = ni
+∀i ∈ I,
+(11)
+�
+i∈I Yc,i = ρc
+∀c ∈ C,
+(12)
+where λC, λG, λGW, λA and λR are the user-defined penalty
+weights, and coefficients ni and ρc in (9) balance the scales
+of deviations in spots and cell types, respectively.
+Remark 1. Unlike the conventional OT formulations, DOT does
+not require the cell type abundances in S (i.e., ρ) to be strictly
+equal to their expected abundances (i.e., r), and rather penalizes
+their deviation in the objective function.
+4
+ALGORITHM
+We propose a solution to the DOT model based on the
+Frank-Wolfe (FW) algorithm [44], [45], which is a first-order
+method for solving non-linear optimization problems of the
+form minx∈X f(x), where f : Rn → R is a (potentially
+non-convex) continuously differentiable function over the
+convex and compact set X. FW operates by replacing the
+non-linear objective function f with its linear approximation
+˜f(x) = f(x(0))+∇xf(x(0))⊤(x−x(0)) at a trial point x(0) ∈
+X, and solving a simpler problem ˆx = arg minx∈X ˜f(x) to
+produce an “atom” solution ˆx. The algorithm then iterates
+by taking a convex combination of x(0) and ˆx to produce
+the next trial point x(1), which remains feasible thanks to
+convexity of X. The FW algorithm is described in Algorithm
+1, in which f(Y ) is the objective function in (9).
+4.1
+Distance Matrices
+Distance matrices M R and M S incorporate the features that
+are not shared across R and S. To compute M R
+c,k, we calculate
+the dissimilarity between the centroids of cell types c and k
+considering all genes in R (i.e., XR
+c,: = (XR
+c,g)g∈GR for each
+c ∈ C)
+M R
+c,k = dcos(XR
+c,:, XR
+k,:).
+
+4
+Algorithm 1: Frank-Wolfe algorithm for DOT
+1 Initialization: Setup distance matrices M R and M S.
+2 Set t = 0 and find an initial map Y (0) (see Appendix
+A.1).
+3 while not converged do
+4
+Compute gradient ∆(t) = ∇Y f(Y (t)) (see
+Appendix A.2)
+5
+for each spot i ∈ I do
+6
+Find current best cell type
+ˆc = arg minc∈C{∆(t)
+c,i}
+7
+Compute atom solution ˆY (t)
+ˆc,i = ni and
+ˆY (t)
+c,i = 0 for c ̸= ˆc
+8
+Update Y (t+1) = Y (t) +
+2
+2+t( ˆY (t) − Y (t))
+9
+t ← t + 1
+Remark 2. M R is a metric in the domain of R since dcos is a
+metric.
+The matrix M S captures the dissimilarity of S spots
+in terms of their locations and expressions. Let D1
+i,j and
+D2
+i,j represent distance of spots (i, j) with respect to their
+locations and expressions, respectively, as defined below:
+D1
+i,j = 1condition
+�∥xi − xj∥ > ¯d
+�
+D2
+i,j = dcos
+�
+XS
+i,:, XS
+j,:
+�
+,
+where ¯d is a given distance threshold, and D2
+i,j is computed
+with respect to all genes in S (i.e., GS). Finally, we take M S
+to be the average of D1 and D2:
+M S = (D1 + D2)/2
+(13)
+Remark 3. M S is a metric in the domain of S, since both D1
+and D2 are metrics.
+To see why this definition of M S makes sense, we first
+note that cell types, by definition, are distinct subpopula-
+tions in the scRNA-seq data. Therefore, it is reasonable to
+assume that their centroids are dissimilar (i.e., Mc,k ≈ 1 for
+c ̸= k). This yields the following result.
+Proposition 3. Let α = �
+i∈I
+�
+j∈I(1−M S
+i,j)2ninj. Assuming
+that cell types are relatively distinct, so that M R
+c,k ≈ 1, for c, k ∈
+C, c ̸= k, then
+dGW(Y ) ≈
+�
+α +
+�
+i∈I
+�
+j∈I
+�
+2M S
+i,j − 1
+�
+⟨Y:,i, Y:,j⟩
+Remark 4. Observe that ⟨Y:,i, Y:,j⟩ measures similarity between
+cell type profiles of spots i and j. Therefore, dGW (i) rewards
+⟨Y:,i, Y:,j⟩ when 2M S
+i,j −1 ≈ +1 (i.e., encourages adjacent spots
+to attain similar cell types if their expressions are similar) and
+(ii) penalizes ⟨Y:,i, Y:,j⟩ when 2M S
+i,j − 1 ≈ −1 (i.e., prevents
+distant spots from attaining similar cell types if their expressions
+are different). Moreover, (iii) dGW is indifferent to pair (i, j)
+when 2M S
+i,j − 1 ≈ 0 (i.e., if i and j are distant or different
+in expressions, but not both).
+4.2
+Producing an Atom Solution
+While the DOT model is not separable, its linear approxi-
+mation can be decomposed to |I| independent subproblems,
+one for each spot i ∈ I. This is because, unlike conventional
+OT formulations, we do not require the distribution of cell
+types (i.e., ρ) to be equal to their expected distribution
+(i.e., r), but have penalized their deviations in the objective
+function using dA (7). The subproblem i then becomes
+min
+�
+⟨Y:,i, ∆(t)
+:,i ⟩ : Y:,i ∈ R|C|
++ ,
+�
+c∈C Yc,i = ni
+�
+which, in turn, is a simple sorting problem. This property
+of Algorithm 1 enables it to efficiently tackle problems with
+large number of spots in the spatial data.
+4.3
+Convergence
+Under suitable conditions, FW converges to an optimal so-
+lution in linear rate when optimizing a convex function over
+a polytope domain [46]. Given the non-convex objective
+function in (9), Algorithm 1 instead obtains a first-order sta-
+tionary point at a rate of O(1/
+√
+t) [47], [48]. We numerically
+assess the convergence of Algorithm 1 at iteration t using
+the so-called “FW-gap” [45]
+δ(t) :=
+�
+i∈I
+�
+c∈C(Y (t)
+c,i − ˆY (t)
+c,i )∆(t)
+c,i.
+We also implemented acceleration techniques such as av-
+eraging gradients [49] and away steps [46], [50], but did
+not observe practical gains compared to the vanilla FW.
+Moreover, while it is common practice to use entropic or
+other strongly-convex regularizations in OT to facilitate
+producing the atom solutions, we did not incorporate such
+regularizations because an atom solution can be produced
+easily in our formulation.
+5
+PRACTICAL ENHANCEMENTS
+In this section, we introduce practical enhancements to
+incorporate the domain-specific properties of the problems.
+5.1
+Cell Heterogeneity
+While cell types are distinct subpopulations of cells, signif-
+icant variations may naturally exist within each cell type.
+This means, a single vector XR
+c,: may not properly represent
+the distribution of cells within this cell type. Consequently,
+mapping cell types solely based on the centroids of cell types
+can be error-prone. To capture the intrinsic heterogeneity
+of cell types, we cluster each cell type into predefined κ
+smaller groups using an unsupervised learning method,
+and produce a total of κ|C| centroids to replace the original
+|C| centroids. With this definition of centroids, we treat all
+terms except dA and dR as before. For dA and dR, since
+prior information about cell types (and not sub-clusters)
+are available, we keep ρ to represent the abundance of
+original cell types by setting ρc = �
+k∈Kc
+�
+i∈I Yk,i, where
+Kc denotes the set of sub-clusters of cell type c. Finally, once
+Y is obtained, �
+k∈Kc Yk,i determines probability that spot
+i is of cell type c.
+
+5
+5.2
+Sparse Mapping
+As previously discussed, spatial data are either high-
+resolution (single-cell level) or low-resolution (multicell
+level). In the case of high-resolution spatial data, given that
+each spot corresponds to an individual cell (i.e., ni = 1),
+it is desirable to produce sparse allocations, in the sense
+that we prefer Yc,i close to 0 or 1. In general, assuming that
+Yc,i ∈ {0, ni}, then (11) implies that Yc,i = ni for exactly one
+cell type c and is zero for all other cell types. Consequently,
+for binary Y we obtain
+dcos
+�
+XS
+i,:,
+�
+c∈C
+Yc,iXR
+c,:
+�
+= 1
+ni
+�
+c∈C
+Yc,idcos
+�
+XS
+i,:, XR
+c,:
+�
+,
+which is linear in Y . As linear objectives promote sparse (or
+corner point) solutions, we may control the level of sparsity
+of the mapping by introducing a parameter θ ∈ [0, 1] and
+redefining di(Y ) as
+di(Y ) =(1 − θ)dcos
+�
+XS
+i,:,
+�
+c∈C Yc,iXR
+c,:
+�
++ θ
+ni
+�
+c∈C Yc,idcos
+�
+XS
+i,:XR
+c,:
+�
+.
+(14)
+Note that a higher value for θ yields a sparser solution.
+Indeed, with θ = 1 and zero weights assigned to other
+objectives, the optimal mapping will be completely binary.
+6
+RESULTS
+We compared the performance of our method, abbreviated
+DOT, against five state of the art models in the litera-
+ture: SPOTlight [19], RCTD [20], cell2location [21],
+Tangram [22], and Seurat [23]. We designed three exper-
+iments to evaluate the performance of DOT from different
+perspectives. Briefly, in Section 6.2, we evaluate the per-
+formance of models in predicting the cell type of single-
+cell level spots in high-resolution spatial data, followed by
+cell type decomposition in multicell spots in low-resolution
+spatial data in Section 6.3. Finally, in Section 6.4, we evaluate
+capability of DOT in estimating the expression of genes that
+are missing in the spatial data but present in the reference
+single-cell data.
+We performed experiments on data coming from (i) the
+primary motor cortex of the mouse brain, (ii) the primary
+somatosensory cortex of the mouse brain, (iii) the develop-
+ing human heart, and (iv) the human breast cancer, specifics
+of which are presented in Appendix B.
+6.1
+Experimental Setup
+6.1.1
+Parameter Setting
+For DOT, we set penalty weights λC
+= 1 and λG
+=
+|n|/|G| to balance the scales of different objectives, where
+|n| := �
+i∈I ni. This is because both �
+i∈I nidi(Y ) and
+�
+c∈C rcdc(Y ) are in the range of 0 and |n|, while 0 ≤
+�
+g∈G dg(Y ) ≤ |G|. For the GW objective, it is not difficult
+to verify that 0 ≤ dGW(Y ) ≤ |n|. However, although
+spatial information contributes to the accuracy of cell type
+mapping, meaning that λGW > 0 is desirable, a large value
+for λGW may dominate the genomic objectives di(Y ), dc(Y )
+and dg(Y ), thus reduce accuracy. A middle-ground is to
+set a small positive value for λGW. In our computations, we
+found that λGW = 0.1 works best in most cases. Whenever
+prior information about expected abundance of cell types
+is available, we set λA = 1 and λR = 1. We computed
+ρc, the expected abundance of cell type c, based on the
+observed fraction of cell type c in the reference scRNA-
+seq data multiplied by |n|. We set the sparsity parameter
+θ = 1 for high resolution spatial data, and set θ = 0 for
+low resolution spatial data. To capture heterogeneity of cell
+types, we clustered each cell type into κ = 10 clusters.
+The distance threshold ¯d is computed as follows. For each
+spot we computed its Euclidean distance to 8 closest spots
+in space1, yielding 8|I| values. We then took ¯d as the 99th
+percentile of these values.
+For RCTD, SPOTlight, Tangram, and C2L we used the
+default parameters suggested by the authors with the fol-
+lowing exceptions. For RCTD we set the parameter UMI_min
+to 50 to prevent the model from removing too many cells
+from the data. Given the large number of cell types in
+the mouse MOp datasets, for SPOTlight we reduced the
+number of cells per cell type to 100 to enhance the computa-
+tion time. Similarly, as Tangram was not able to produce
+results in a reasonable time for the MOp instances, we
+randomly selected 500 cells per cell type to reduce the com-
+putation time. For C2L, we used 20000 epochs to balance
+computation performance and accuracy. For Seurat and
+SingleR, we followed the package documentations, with
+functions used with default parameters. For RF we used
+the implementation provided in the R package ranger [51]
+with all parameters set at their default values.
+6.1.2
+Performance Metrics
+We compared the predictive performance of DOT against
+the other methods using three metrics. Accuracy in the
+context of high-resolution spatial data (i.e., when each spot
+corresponds to an individual cell) is the proportion of
+correctly classified spots (i.e., sum of the main diagonal in
+the confusion matrix) over all spots. To assess the accuracy
+of membership probabilities produced by each model, we
+compared the models using Brier Score, also known as mean
+squared error:
+Brier Score = |I|−1 �
+i∈I
+�
+c∈C(Yc,i − Pc,i)2,
+where Pc,i = 1 if spot i is of cell type c and Pc,i = 0
+otherwise, and Yc,i is the predicted probability that spot i is
+of cell type c. As Brier Score is a strictly proper scoring rule
+for measuring the accuracy of probabilistic predictions [52],
+a model with lower Brier Score produces better-calibrated
+probabilities.
+Besides the cell type that each spot is annotated with, we
+can produce a cell type probability distribution for each spot
+by considering the cell type of its neighboring spots, using
+a Gaussian smoothing kernel of the form
+˜Pc,i =(
+�
+j∈I Ki,j)−1 �
+j∈I Ki,jPc,j,
+where Ki,j = exp
+�−∥xi − xj∥2/2σ2�
+and σ is the kernel
+width parameter which we set to 0.5 ¯d. Note that as spot j
+becomes closer to spot i, its label contributes more to the
+1. We used 8 closest neighbors to mimic the number of adjacent tiles
+in a 2D regular grid.
+
+6
+probability distribution at spot i. Using these probabilities,
+we also introduce the Spatial Jensen-Shannon (SJS) divergence
+to compare the probability distributions assigned to spots
+(i.e., Y ) with the smoothed probabilities (i.e., ˜P )
+SJS = |I|−1 �
+i∈I JS(Y:,i, ˜P:,i),
+where JS(Y:,i, ˜P:,i) is the Jensen-Shannon divergence be-
+tween probability distributions Y:,i and ˜P:,i with base 2
+logarithm [42], also defined in (7).
+6.2
+Experiment
+1:
+Cell
+Type
+Prediction
+in
+High-
+Resolution Spatial Data
+Our goal with our first set of experiments is to evaluate
+the performance of different models in determining the
+probability distribution of cell types at each spot. Since the
+identity of the cell type represented by the spot is known in
+our high resolution spatial data, we can use this information
+as ground-truth when evaluating the performance of the
+different models. In addition to deconvolution methods,
+we used SingleR [53], a method to define cell type from
+single-cell resolution data. Given the multiclass classifica-
+tion nature of this task, we also used RF [18] as a multiclass
+classifier baseline.
+We use the high-resolution MERFISH spatial data of the
+primary motor cortex region (MOp) of the mouse brain [54],
+which contains the spatial information of 280,186 cells across
+75 samples (Appendix B.1). With each sample, we created a
+reference scRNA-seq data using all the 280,186 cells, except
+the cells contained in the sample, and the 254 genes to
+estimate the centroids of the 99 reference cell types. We
+further created 15 high resolution spatial datasets for each
+sample (i.e., a total of 1125 spatial datasets) as follows. To
+simulate the effect of number of shared features between the
+spatial and scRNA-seq data, we assumed that only a subset
+of the 254 genes are available in the spatial data by selecting
+the first |G| genes, where |G| ∈ {50, 75, 100, 125, 150} (i.e.,
+20%, 30%, 40%, 50%, 60% of genes). Moreover, to simulate
+the effect of differences in measurement sensitivities of
+different technologies, we introduced random noise in the
+spatial data by multiplying the expression of gene g in spot
+i by 1+βi,g, where βi,g ∼ U(−ϕ, ϕ) with ϕ ∈ {0, 0.25, 0.5}.
+We compare the predictive performance of DOT to
+Seurat, RCTD, Tangram, SingleR and RF in Fig. 1. We
+removed SPOTlight and C2L from these plots due to their
+clear under-performance in the high resolution spatial data.
+We observe that not only does DOT dominate the three alter-
+natives in assigning correct cell types to the spots (Fig. 1a),
+but also produces well-calibrated probabilities (Fig. 1b) and
+better captures the relationships between cell types in space
+(Fig. 1c), owing to its capacity to incorporate the spatial
+information in dGW through the distance matrices. We also
+observe that even with very few genes in common between
+the spatial data and the reference scRNA-seq data (e.g.,
+|G| ≤ 75), DOT is able to reliably determine the cell type
+of spots in the space with high accuracy. In contrast, RCTD
+fails to produce results due to lack of shared information,
+and Seurat and Tangram produce results with low accu-
+racy. Additionally, we observe that DOT is more immune to
+fluctuations in expressions in the spatial data, implying the
+effectiveness of our dcos distance function in accounting for
+ϕ = 0
+ϕ = 0.25
+ϕ = 0.5
+50
+75
+100
+125
+150
+50
+75
+100
+125
+150
+50
+75
+100
+125
+150
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+Accuracy
+(a)
+ϕ = 0
+ϕ = 0.25
+ϕ = 0.5
+50
+75
+100
+125
+150
+50
+75
+100
+125
+150
+50
+75
+100
+125
+150
+0.4
+0.6
+0.8
+Brier Score
+(b)
+ϕ = 0
+ϕ = 0.25
+ϕ = 0.5
+50
+75
+100
+125
+150
+50
+75
+100
+125
+150
+50
+75
+100
+125
+150
+0.5
+0.6
+0.7
+SJS
+(c)
+DOT
+Seurat
+RCTD
+Tangram
+RF
+SingleR
+Fig. 1: Predictive performance of the algorithms in the high-
+resolution spatial data across different number of genes
+in the spatial data (x-axis) and different noise factors (ϕ).
+Points represent the median of 75 values, and the shaded ar-
+eas correspond to their interquartile interval. Note: SingeR
+does not produce probabilities and is compared based on
+Accuracy only.
+differences in measurement scales of different technologies.
+In terms of algorithmic performance (Table 1), DOT takes
+on average 441 seconds to solve each instance, which is an
+order of magnitude faster than RCTD, Tangram, and RF, and
+is comparable to Seurat and SingleR.
+6.3
+Experiment 2: Cell Type Decomposition in Low-
+Resolution Spatial Data
+We evaluated the performance of models on low-resolution
+spatial data. For these experiments, since there is no ground
+truth for real multicell spatial data such as Visium and Slide-
+seq, we resort to producing ground truth low-resolution
+spatial data by pooling the adjacent cells in the high res-
+olution spatial data of primary motor cortex of the mouse
+brain (MOp), primary somatosensory cortex of the mouse
+brain (SSp), and the developing human heart. Fig. A1 in
+Appendix B.1 illustrates a sample low-resolution spatial
+data obtained from a MERFISH MOp tissue. Unlike the
+high-resolution spatial data, the ground truth Pc,i now
+corresponds to relative abundance of cell type c in spot
+i. We can therefore assess the performance of each model
+by comparing the probability distributions P:,i and the
+estimated probabilities (i.e., Y:,i) using Brier Score or Jensen-
+Shannon metrics.
+6.3.1
+Experiments on the Mouse MOp
+To produce ground truth for MOp, using the common
+subclass annotations between MERFISH MOp and scRNA-
+seq MOp [55] (see Appendix B.1), for each of the 75 MER-
+FISH MOp samples, we randomly assigned each cell in the
+MERFISH MOp data to a cell in the scRNA-seq MOp data
+of the same subclass. Next, we lowered the resolution of
+spatial data by splitting each sample into regular grids of
+
+7
+Experiment
+Resolution
+Instances
+DOT
+Seurat
+RCTD
+Tangram
+SPOTlight
+C2L
+SingleR
+RF
+MOp
+High
+1125
+441
+380
+4748
+10141
+7884
+3310
+303
+7427
+MOp
+Low
+75
+457
+1086
+4705
+8250
+52825
+6119
+—
+—
+SSp
+Low
+1
+4
+21
+117
+248
+705
+364
+—
+—
+Heart
+Low
+1
+8
+11
+185
+88
+316
+398
+—
+—
+TABLE 1: Average computation times (in seconds) of different models across different experiments.
+0.1
+0.2
+0.3
+0.4
+0.2
+0.4
+0.6
+0.8
+Jensen−Shannon
+Brier Score
+DOT
+Seurat
+RCTD
+SPOTlight
+C2L
+Tangram
+Fig. 2: Predictive performance of the algorithms in the low-
+resolution spatial data across 75 samples of MOp. Each point
+in the scatter plots denoting the average performance across
+all spots in the sample.
+length 100µm to mimic the size and inter-distance of spots
+in low-resolution spatial transcriptomics, such as Visium.
+Finally, we aggregated the expression profiles of cells within
+each tile as the expression profile of the respective spots.
+Fig. 2 compares the performance of DOT against RCTD,
+SPOTlight, cell2location (C2L), Tangram and Seurat in
+determining the cell type composition of the multicell spots.
+We observe that DOT outperforms other models with respect
+to both metrics. As presented in Table 1, DOT took on
+average 457 seconds to solve an instance, which proved to
+be more than twice faster than Seurat, and orders of mag-
+nitude faster than RCTD, SPOTlight, C2L and Tangram,
+further highlighting the superiority of DOT in terms of both
+accuracy and computational efficiency.
+6.3.2
+Experiments on the Mouse SSp and the Developing
+Human Heart
+We also carried out experiments on data from the SSp
+region of mouse brain as well as the developing human
+heart to evaluate the performance of models on tissues of
+different structures. We employed single-cell level spatial
+data coming from osmFISH technology [56] to produce
+multicell data for SSp (Appendix B.2). For the developing
+human heart, we used subcellular spatial data generated
+by the ISS platform [57] (Appendix B.3). We tested the
+performance of the six deconvolution methods on these
+two samples, results of which are illustrated in Fig. 3. Each
+subplot shows the distribution of prediction error based on
+the Jensen-Shannon divergence at each spot in the spatial
+data, with the average value over all spots given on top
+of each plot. DOT outperforms other models in the human
+heart sample and is among the best-performing models in
+the mouse SSp sample. Moreover, performance of DOT is not
+sensitive to different regions/cell type of the tissue (compare
+to Tangram and Seurat in SSp and RCTD in human heart).
+These results further highlight the competitive performance
+of DOT and its robustness in identifying the cell type com-
+position of spots across different tissues.
+6.4
+Experiment 3: Gene Expression Estimation in High-
+Resolution Spatial Data
+Recall that, in high-resolution spatial data, only a few genes
+are measured. Hence, as a result of mapping scRNA-seq to
+spatial data, we can estimate the expression of genes that
+were not measured in the spatial data. Therefore, in our
+final set of experiments, we evaluate the performance of
+DOT in estimating the expression of missing genes in the
+high resolution spatial data. For this experiment, we use
+the spatial data from breast cancer tumor microenvironment
+produced by the 10X Xenium In Situ technology [58]. The
+dataset is unique in that it contains both high-resolution
+(Xenium) and low-resolution (Visium) spatial data of serial
+sections from the same tissue. The high-resolution data con-
+tains spatial information of 313 genes across 167,782 single-
+cell spots, while the low-resolution data contains the spatial
+information of around 18,000 genes across 4,992 multicell
+spots. The dataset also contains the dissociated scRNA-seq
+data coming from a tissue adjacent to the tissues used for
+high- and low-resolution spatial data, which contains the
+measurements of 18,000 genes across 30,365 cells. Therefore,
+we can use Visium as a proxy for ground truth to validate
+the distribution of genes that are not measured in Xenium,
+and are mapped by DOT from scRNA-seq.
+For this experiment, we matched the common capture
+areas of high- and low-resolution spatial data using the
+Hematoxylin-Eosin (H&E) images accompanying these spa-
+tial data (Fig. A2), which corresponded to 134,664 cells in
+the high-resolution and 3,928 spots in the low-resolution
+spatial data. Given that the task here is to complete the
+expression of missing genes in the high-resolution spatial
+data, we first performed community detection on the graph
+of shared nearest neighbors of cells in scRNA-seq using the
+Leiden implementation in [23], which is common practice
+in single-cell analysis and is used as a first step towards
+cell type identification. This resulted in 218 clusters, and
+we then mapped the centroids of these clusters to the high-
+resolution spatial data. (We also tried as high as 1000 fine-
+grained clusters but got essentially the same results.) We
+also turned off the cell-type-related objectives since we are
+not mapping cell types. Although the datasets are extremely
+
+8
+DOT (0.308)
+SPOTlight (0.456)
+Tangram (0.484)
+C2L (0.294)
+RCTD (0.298)
+Seurat (0.404)
+0.00
+0.25
+0.50
+0.75
+1.00
+JS
+DOT (0.392)
+SPOTlight (0.548)
+Tangram (0.408)
+C2L (0.541)
+RCTD (0.607)
+Seurat (0.468)
+0.25
+0.50
+0.75
+1.00
+JS
+Mouse SSp
+Human Heart
+Fig. 3: Distribution of performance of models on each individual spot in the low-resolution spatial data of Mouse SSp (top)
+and developing human heart (bottom).
+large, DOT was able to perform the mapping in less than
+two hours.
+We start by evaluating the performance of DOT on genes
+that are present in the high-resolution spatial data as a true
+ground truth. The qualitative comparisons of gene maps of
+eight genes associated with breast cancer [59] produced by
+DOT with those of high-resolution (ground truth) and low-
+resolution data (approximate ground truth) can be seen in
+Fig. 4. As can be observed, the expression maps produced by
+DOT match almost perfectly with the ground truth expres-
+sion maps. Both DOT and the ground truth high-resolution
+spatial data also match the low resolution gene expression
+maps almost perfectly, which further validate the quality
+of mapping produced by DOT. Note that due to the single-
+cell resolution of the high-resolution spatial data, expression
+levels are higher at high resolution, thus the brighter colors.
+Nonetheless, the spatial patterns match between all three
+rows.
+Next, we compare the expression map of genes that
+are not present in the high-resolution spatial data but are
+estimated by DOT. Fig. 5a illustrates the expression maps
+of five genes associated with breast cancer that are not
+measured in the high-resolution spatial data. For a quanti-
+tative comparison of expression maps, given that there is no
+one-to-one mapping between single-cell spots in the high-
+resolution and multicell spots in the low-resolution spatial
+data, we split the tissue into a 10 by 10 grid, and aggregated
+the expression of each gene within each tile. Consequently,
+we obtained two 100 by 18,000 matrices, one for the low-
+resolution spatial data and another for DOT, with rows
+corresponding to tiles and columns corresponding to genes.
+Fig. 5b compares the column-wise cosine similarities across
+different genes. These results further confirm the ability of
+DOT in reliably estimating the expression of missing genes
+in high-resolution spatial data.
+7
+CONCLUSION
+Single-cell
+RNA-seq
+and
+spatially-resolved
+imag-
+ing/sequencing technologies, the cutting edge technologies
+in transcriptomic data generation, each provide a partial
+picture in understanding the organization of complex
+tissues. To obtain a full picture, computational methods aim
+at combining data from these two modalities. We present
+DOT, a fast and scalable optimization framework based
+on Optimal Transport theory, for assigning cell types to
+tissue locations by leveraging the spatial information as
+well as both joint and distinct genes across scRNA-seq and
+spatial data. Using experiments on data from mouse brain
+and human heart, we show that DOT predicts the cell type
+composition of spots in spatial data with high accuracy,
+outperforming the state of the art methods both in terms of
+predictive performance and computation time.
+APPENDIX A
+IMPLEMENTATION DETAILS OF THE FW ALGORITHM
+A.1
+Initial Solution
+A good quality initial solution plays a critical role in fast
+convergence of FW. Given the multi-objective nature of our
+model, we produce an initial solution as convex combina-
+tion of two solutions. In the first solution, for each spot
+i we first find cell type ˆc = arg minc∈C{dcos
+�
+XS
+i,:, XR
+c,:
+�
+}
+and set Yc,i = ni if c = ˆc and Yc,i = 0 otherwise. Note
+that this solution is optimal for the sparse case when di is
+the only objective. In the second solution, we simply set
+Yc,i = niρc/|I| for each i and c. Note that this solution
+is optimal for dA. We then set the initial solution as the
+convex combination of these two solutions, with 0.9 weight
+assigned to the first solution.
+A.2
+Derivatives
+To find the derivatives of di(Y ) and dc(Y ), defined in (2)
+and (3), we introduce auxiliary quantities ¯
+XS := Y ⊤XR
+and ¯
+XR := Y XS to denote the expressions mapped through
+Y to spots and cell types, respectively. Derivatives for di(Y )
+and dc(Y ) can then be calculated as:
+∂di
+∂Yc,i
+=
+1
+∥XS
+i,:∥⟨XR
+c,:, T S
+i,:⟩,
+∂dc
+∂Yc,i
+=
+1
+∥XRc,:∥⟨XS
+i,:, T R
+c,:⟩,
+
+9
+CEACAM6 (Xenium) POSTN (Xenium)
+ITGAX (Xenium)
+VWF (Xenium)
+KRT15 (Xenium)
+FOXA1 (Xenium)
+GATA3 (Xenium)
+TACSTD2 (Xenium)
+CEACAM6 (DOT)
+POSTN (DOT)
+ITGAX (DOT)
+VWF (DOT)
+KRT15 (DOT)
+FOXA1 (DOT)
+GATA3 (DOT)
+TACSTD2 (DOT)
+CEACAM6 (Visium)
+POSTN (Visium)
+ITGAX (Visium)
+VWF (Visium)
+KRT15 (Visium)
+FOXA1 (Visium)
+GATA3 (Visium)
+TACSTD2 (Visium)
+Fig. 4: Expression map of eight breast cancer markers measured in both Xenium (ground truth; top) and Visium (low-
+resolution proxy; bottom), and as mapped from scRNA-seq to Xenium using DOT (estimated; middle). Brighter means
+higher expression.
+SCGB2A2 (DOT)
+KRT17 (DOT)
+CDH2 (DOT)
+SFRP2 (DOT)
+MT−ND1 (DOT)
+SCGB2A2 (Visium)
+KRT17 (Visium)
+CDH2 (Visium)
+SFRP2 (Visium)
+MT−ND1 (Visium)
+(a)
+0%
+10%
+20%
+30%
+40%
+50%
+0.00
+0.25
+0.50
+0.75
+1.00
+Expression map similarity
+% Genes
+(b)
+Fig. 5: (a) Expression map of five breast cancer markers that are measured in Visium (bottom) but are missing in Xenium
+and are mapped from scRNA-seq using DOT (top). (b) Cosine similarity between expression maps of Visium and DOT for
+the genes that are not measured in Xenium.
+where
+T S
+i,g =
+−1
+2di(Y )
+�
+XS
+i,g
+∥ ¯
+XS
+i,:∥ −
+¯XS
+i,g
+∥ ¯
+XS
+i,:∥3 ⟨XS
+i,:, ¯
+XS
+i,:⟩
+�
+,
+T R
+c,g =
+−1
+2dc(Y )
+�
+XR
+c,g
+∥ ¯
+XRc,:∥ −
+¯XR
+c,g
+∥ ¯
+XRc,:∥3 ⟨XR
+c,:, ¯
+XR
+c,:⟩
+�
+.
+Derivative of ρcdc(Y ) then can be computed using the
+product rule. Similarly, we may derive the derivative for
+dg(Y ) via
+∂dg
+∂Yc,i
+=
+−1
+2dg(Y )
+XR
+c,g
+∥XS:,g∥
+�
+XS
+i,g
+∥ ¯
+XS:,g∥ −
+Yc,i
+∥ ¯
+XS:,g∥3 ⟨XS
+:,g, ¯
+XS
+:,g⟩
+�
+Taking into account the simplification from Proposi-
+tion 2, noting that ¯mc and Zc,i are functions of Y while
+¯mi is constant, we can show that
+∂dGW
+∂Yc,i
+=
+1
+2dGW(Y )(2 ¯mc + ¯mi − 4Zc,i).
+Finally, the derivatives for dA and dR, defined in (7) and
+(8) respectively, can be calculated as:
+∂dA
+∂Yc,i
+= 1
+2 log
+�
+2ρc
+ρc + rc
+�
+,
+∂dR
+∂Yc,i
+= − 1
+ρc
+.
+APPENDIX B
+DATASETS
+B.1
+Mouse Primary Motor Cortex (MOp)
+MERFISH. For high-resolution spatial transcriptomics, we
+used the spatially resolved cell atlas of the MOp recently
+generated using multiplexed error-robust fluorescence in
+situ hybridization (MERFISH) technology and made pub-
+licly available by [54]. The processed dataset contains nor-
+malized RNA counts of 254 genes and coordinates of the
+boundaries of a total of 280,186 segmented cells across 75
+samples in the MOp of two adult mice, with the number of
+cells within each sample ranging from 1000 to 7500 cells. We
+computed the (x, y) coordinates of the center of each cell by
+taking the average of the coordinates of its boundary. The
+study also identifies 99 trasncriptionally distinct cell types
+
+10
+0
+500
+1000
+1500
+2000
+2500
+0
+500
+1000
+1500
+Fig. A1: Synthetic multicell spatial data from MERFISH.
+Dots show individual cells and tiles represent multicell
+spots (darker tile means denser spot).
+by community detection applied on a cell similarity graph.
+The clustering resulted in 39 excitatory neuronal cell types
+(clusters), 42 inhibitory neuronal cell types, 14 non-neuronal
+cell types, and four other cell types.
+scRNA-seq. The corresponding scRNA-seq data comes
+from a cell atlas of the MOp [55], which contains the
+mRNA expression of the full range of genes for more than
+500,000 individual cells across several omics layers. We used
+the scRNA-seq dataset scRNA_10x_v2_A generated at the
+Allen Institute, which contains 145,748 cells and 100 cell
+types. After removing the unannotated cells and low quality
+cell types (as categorized in the study), we retrieved 124,330
+cells and 90 distinct cell types. For computational efficiency,
+we also selected the top 5,000 variable genes according to
+their means and variances [23].
+Fig. A1 illustrates a sample low-resolution spatial data
+obtained from a MERFISH MOp tissue.
+B.2
+Mouse Primary Somatosensory Cortex (SSp)
+Similar to MOp, another well-studied tissue area is the pri-
+mary somatosensory cortex area (SSp). Here, we used high-
+resolution spatial data coming from the osmFISH platform
+[56], which contains measurements of 33 genes across 4,837
+cells, as well as annotations based on 11 major cell types.
+For reference scRNA-seq data with matched cell types, we
+used the annotations independently generated by [60] using
+5,392 single cells in the same SSp region.
+B.3
+Developing Human Heart
+For the developing human heart, we used subcellular spatial
+data generated by the ISS platform [57], which contains
+tissue sections from human embryonic cardiac samples
+collected at different times. We selected the PCW6.5 slide
+which contains measurements of 69 genes across 17,454 cells
+as well as annotations of 12 major cell types. The same
+study also provides scRNA-seq data for similar slide, which
+contains matched cell types for 3,253 cells.
+Fig. A2: Common region (cyan) in the capture areas of
+Visium (dashed blue lines) and Xenium (dark orange). The
+pink region is the H&E image accompanying Visium.
+B.4
+Human Breast Cancer
+Breast cancer is a complex disease with significant cel-
+lular and molecular heterogeneity. The breast cancer tu-
+mor microvenvironment dataset generated in [58] contains
+both single-cell and multicell data. The single-cell reso-
+lution spatial dataset is produced by the recent 10X Xe-
+nium In Situ technology, and contains two replications.
+We used Xenium_FFPE_Human_Breast_Cancer_Rep1,
+which contains the spatial information of 313 genes for
+167,782 cells. The low-resolution multicell spatial dataset
+is produced by the 10X Visium Spatial Transcriptomics
+technology, which contains the spatial information of 18,000
+genes for 4,992 multicell spots. For the reference scRNA-
+seq data, we used the Single Cell Gene Expression
+Flex (FRP) data generated from a tissue section adjacent
+to the tissue sections used for Visium and Xenium work-
+flows, which contains expression of 18,000 genes across
+30,365 cells.
+APPENDIX C
+C.1
+Proof of Proposition 1
+Proof: Note that
+����
+a
+∥a∥ −
+b
+∥b∥
+����
+2
+= ∥a∥2
+∥a∥2 + ∥b∥2
+∥b∥2 − 2 ⟨a, b⟩
+∥a∥∥b∥ = 2 − 2 cos(a, b)
+⇒ dcos(a, b) =
+�
+1 − cos(a, b) =
+√
+2∥a/∥a∥ − b/∥b∥∥.
+This shows that dcos is a metric since ∥ · ∥ is a metric. We
+can easily show that cosine dissimilarity (i.e., 1 − cos(·, ·))
+is not a metric. For instance, consider a = (1, 0, 0), b =
+(0, 1, 0) and c = (x, x,
+√
+1 − 2x2) for arbitrary x ∈ ( 1
+2,
+1
+√
+2),
+and let f denote the cosine dissimilarity. Then f(a, b) =
+1 − cos(a, b) = 1, and f(a, c) = f(c, b) = 1 − x, which
+violates the triangular inequality since f(a, c) + f(c, b) =
+2−2x < 1 = f(a, b). It is not difficult to see that dcos(a, c)+
+dcos(c, b) > 1.
+
+11
+C.2
+Proof of Proposition 2
+Proof: Rewrite g(Y ) as
+g(Y ) =
+�
+i∈I
+�
+j∈I
+�
+c∈C
+�
+k∈C
+�
+M R
+c,k − M S
+i,j
+�2
+Yc,iYk,j
+=
+�
+c∈C
+�
+i∈I
+Yc,i
+�
+k∈C
+�
+j∈I
+Yk,j
+�
+(M R
+c,k)2 + (M S
+i,j)2 − 2M R
+c,kM S
+i,j
+�
+.
+Expanding the inner summations we obtain:
+�
+k∈C
+�
+j∈I
+Yk,j(M R
+c,k)2 =
+�
+k∈C
+(M R
+c,k)2 �
+j∈I
+Yk,j = ¯mc
+�
+k∈C
+�
+j∈I
+Yk,j(M S
+i,j)2 =
+�
+j∈I
+(M S
+i,j)2 �
+k∈C
+Yk,j = ¯mi
+�
+k∈C
+�
+j∈I
+Yk,jM R
+c,kM S
+i,j =
+�
+M RY M S�
+c,i = Zc,i,
+where we have used �
+j∈I
+Yk,j = ρk and �
+k∈C
+Yk,j = nj.
+Substituting these equations into g(Y ) gives the result.
+C.3
+Proof of Proposition 3
+Proof: Provided that M R
+c,k = 1 for c ̸= k and M R
+c,c = 0,
+we obtain
+g(Y ) =
+�
+i∈I
+�
+j∈I
+�
+c∈C
+�
+M S
+i,j
+�2
+Yc,iYc,j
++
+�
+i∈I
+�
+j∈I
+�
+c∈C
+�
+k∈C,k̸=c
+�
+1 − M S
+i,j
+�2
+Yc,iYk,j
+=
+�
+i∈I
+�
+j∈I
+�
+c∈C
+��
+M S
+i,j
+�2
+−
+�
+1 − M S
+i,j
+�2�
+Yc,iYc,j
++
+�
+i∈I
+�
+j∈I
+�
+c∈C
+�
+k∈C
+�
+1 − M S
+i,j
+�2
+Yc,iYk,j
+=
+�
+i∈I
+�
+j∈I
+�
+2M S
+i,j − 1
+�
+⟨Y:,i, Y:,j⟩ + α,
+where
+α =
+�
+i∈I
+�
+j∈I
+�
+1 − M S
+i,j
+�2 �
+c∈C
+�
+k∈C
+Yc,iYk,j
+=
+�
+i∈I
+�
+j∈I
+�
+1 − M S
+i,j
+�2
+ninj
+since �
+c∈C
+Yc,i = ni and �
+k∈C
+Yk,j = nj.
+ETHIC STATEMENT
+The human biological samples were sourced ethically and
+their research use was in accord with the terms of the
+informed consents under an IRB/EC approved protocol.
+All animal studies were ethically reviewed and carried
+out in accordance with European Directive 2010/63/EEC
+and the GSK Policy on the Care, Welfare and Treatment of
+Animals.
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+

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+Impact of random and targeted disruptions on information diffusion during outbreaks
+Impact of random and targeted disruptions on information diffusion
+during outbreaks
+Hosein Masoomy,1, a) Tom Chou,2, b) and Lucas Böttcher3, c)
+1)Dept. of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran
+2)Depts. of Computational Medicine and Mathematics, UCLA, Los Angeles, CA 90095
+3)Centre for Human and Machine Intelligence, Frankfurt School of Finance and Management, 60322 Frankfurt am Main,
+Germany
+(Dated: 3 January 2023)
+Outbreaks are complex multi-scale processes that are impacted not only by cellular dynamics and the ability of
+pathogens to effectively reproduce and spread, but also by population-level dynamics and the effectiveness of miti-
+gation measures. A timely exchange of information related to the spread of novel pathogens, stay-at-home orders, and
+other containment measures can be effective at containing an infectious disease, particularly during in the early stages
+when testing infrastructure, vaccines, and other medical interventions may not be available at scale. Using a multiplex
+epidemic model that consists of an information layer (modeling information exchange between individuals) and a spa-
+tially embedded epidemic layer (representing a human contact network), we study how random and targeted disruptions
+in the information layer (e.g., errors and intentional attacks on communication infrastructure) impact outbreak dynam-
+ics. We calibrate our model to the early outbreak stages of the SARS-CoV-2 pandemic in 2020. Mitigation campaign
+can still be effective under random disruptions, such as failure of information channels between a few individuals.
+However, targeted disruptions or sabotage of hub nodes that exchange information with a large number of individuals
+can abruptly change outbreak characteristics such as the time to reach the peak infection. Our results emphasize the
+importance of using a robust communication infrastructure that can withstand both random and targeted disruptions.
+Online communication platforms and exposure notifica-
+tion apps can help slow down and contain the spread of
+an infectious disease1. Individuals who have been made
+aware of an outbreak are likely to adapt their behavior to
+reduce their risk of being infected. To study the interplay
+between infectious disease outbreaks and corresponding
+changes in individual contact behaviors, Granell et al.2 in-
+troduced an epidemic model that accounts for the spread
+of awareness through an information layer that is coupled
+to a human contact network. Building upon their model of
+awareness diffusion, our work studies the impact of ran-
+dom and targeted disruptions in the information layer on
+the overall outbreak dynamics.
+I.
+INTRODUCTION
+The study of epidemic processes in networks has pro-
+vided many insights into the interplay between structure and
+dynamics.3,4 The aim of many works in this area has been
+to analyze the impact of different structural features such as
+clustering5, community structure6,7, hub nodes, and scale-
+free degree distributions8 on the evolution of susceptible-
+infected-susceptible (SIS) and susceptible-infected-recovered
+(SIR) models and their extensions.9–11 Connections between
+epidemic processes and percolation contributed to the devel-
+opment of analytical methods that are useful to analyze epi-
+demic transitions and determine outbreak size.12–16 Along
+a)[email protected]
+b)[email protected]
+c)[email protected]
+with progress in understanding epidemic processes in static
+single-layer networks, developments in the study of tempo-
+ral networks17, multilayer networks18,19, and other structures
+describing higher-order interactions20–23 have allowed for the
+integration of time-varying and non-binary interactions.
+Before research turned to epidemic models in multilayer
+networks, interactions between disease and behavioral dy-
+namics have been studied mainly in single-layer networks24
+and well-mixed populations.25–28 In an extension of the
+classical SIS model, the so-called susceptible-infected-alert-
+susceptible (SIAS) model, a new compartment was used to
+study the effect of “alert” individuals that are surrounded by
+a certain number of infecteds on disease dynamics.29,30 The
+SIAS model has been implemented using a two-layer net-
+work31 with a contact layer and an information-dissemination
+layer to find optimal information dissemination strategies that
+help contain an outbreak.
+The interplay between behavioral effects and network dy-
+namics has also been analyzed in terms of a multiplex struc-
+ture where information on an outbreak diffuses in an infor-
+mation layer.2,32 In a multiplex network, all of the interlayer
+edges are edges between nodes and their counterparts in other
+layers. As in the SIAS model, individuals in the information
+layer can be either aware or unaware of a disease. Aware-
+ness then translates into a reduced infection rate. The origi-
+nal awareness model has been modified in various ways. One
+study used a threshold model in the information layer and
+identified awareness cascades.33 Other research investigated
+the effects of dynamically varying transmission rates34, cou-
+pled SIR and unaware-aware-unaware (UAU) dynamics with
+and without latency35,36, SIS and UAU dynamics that propa-
+gate at different speeds37, and higher-order interactions38. For
+a detailed overview of models of coevolving spreading pro-
+cesses in networks, we refer the reader to Ref. 39.
+arXiv:2301.00748v1  [physics.soc-ph]  2 Jan 2023
+
+Impact of random and targeted disruptions on information diffusion during outbreaks
+2
+In this work, we study coevolving susceptible-exposed-
+infected-recovered-deceased (SEIRD) and UAU dynamics on
+a multiplex network that consists of an epidemic layer and an
+information layer. The exposed compartment in our model
+accounts for latency (i.e., the time difference between infec-
+tion and becoming infectious). Different variants of SEIRD
+models have been used to mechanistically describe the spread
+of an infectious disease for which the latency period between
+time of infection to time of becoming infectious cannot be
+neglected9,40–42. Examples of such infectious diseases include
+measles, smallpox, and SARS-CoV-2.
+One of the main goals of this work is to provide insight into
+the impact of disruptions in the information diffusion layer
+on the overall outbreak dynamics. We therefore study differ-
+ent edge removal protocols that describe random and targeted
+disruptions. In Sec. II, we define the disease and awareness
+model, develop a heterogeneous mean-field model, define ran-
+dom and targeted edge removal protocols, and briefly describe
+the structure of the considered networks. In Sec. III, we first
+discuss a baseline simulation that uses model parameters that
+are aligned with empirical data on the outbreak of SARS-
+CoV-2 in early 2020. We then use this baseline simulation
+as a reference to study the impact of disruptions in the in-
+formation diffusion layer on three disease severity measures:
+(i) final outbreak size, (ii) maximum proportion of infectious
+nodes on a given day (i.e., the height of the infection peak),
+and (iii) the time until the infection peak is reached.
+II.
+METHODS
+A.
+Epidemic model with information diffusion
+We study the interplay between information diffusion and
+epidemic dynamics in a multiplex network with two layers
+[see Fig. 1(a)].
+In the first layer, individuals exchange information (e.g.,
+through online social media or messaging services) on the
+prevalence of a certain disease in the overall population ac-
+cording to the unaware-aware-unaware (UAU) model.2 Indi-
+viduals in the “information layer” (IL) can be in two states.
+They are either unaware (U) or aware (A) of the disease and
+do not necessarily have to be in close proximity (in terms of
+connectivity) to exchange information. Unaware nodes can
+become aware in two ways. First, if an unaware node is in
+contact with an aware node, it becomes aware at rate λ. Sec-
+ond, nodes that have been infected and experience symptoms
+become aware at rate κ. Given that certain individuals forget
+or do not adhere to intervention measures after a certain time,
+we also account for transitions from aware to unaware at rate
+δ. A schematic of UAU dynamics is shown in Fig. 1(b).
+In the second layer, we model an epidemic outbreak
+using the susceptible-exposed-infected-recovered-deceased
+(SEIRD) model. In the “epidemic layer” (EL), nodes can be in
+states S (susceptible), E (exposed), I (infected), R (recovered),
+and D (deceased). We distinguish between two infection rates,
+β u and β a, that describe the rates at which susceptible nodes
+become infected if they are unaware and aware, respectively.
+The disease transmission rate associated with aware individu-
+als is assumed to be strictly lower than the disease transmis-
+sion rate associated with unaware individuals (i.e., β a < β u),
+accounting for the decreased likelihood of an aware individ-
+ual to become infected. We assume a latent rate σ, resolu-
+tion rate γ, and infection fatality ratio f that are independent
+of the awareness status. This assumption is valid for infec-
+tious diseases for which no medication is available that pos-
+itively affects recovery, even if a person is aware of an in-
+fection before developing symptoms. For example, during the
+early outbreak stages of SARS-CoV-2, there was very little in-
+formation available on how to medically support patients that
+were aware of their infection, but did not show symptoms yet.
+Non-pharmaceutical interventions such as contact restrictions,
+mask mandates, and quarantine are often the only possibility
+to combat novel pathogens.1
+According to the described UAU and SEIRD dynamics,
+nodes can be in the following states: (U,S), (A,S), (U,E),
+(A,E), (U,I), (A,I), (U,R), (A,R), and (U,D). The first en-
+try in each tuple describes the awareness state (either U or A)
+while the second entry describes vital and disease states (S, E,
+I, R, and D). Deceased nodes are not aware.
+B.
+Heterogeneous mean-field theory
+In accordance with Ref. 43, we formulate a heterogeneous
+mean-field theory of SEIRD-UAU dynamics. We use xjyk ≡
+xjyk(t) (x ∈ {u,a},y ∈ {s,e,i,r,d}) to denote the proportion of
+nodes in state XjYk (X ∈ {U,A},Y ∈ {S,E,I,R,D}) with de-
+grees j and k in the IL and EL at time t, respectively. For ex-
+ample, ujsk ≡ u jsk(t) denotes the proportion of unaware and
+susceptible nodes with degrees j and k in the IL and EL at time
+t, respectively. Henceforth, we will not explicitly include the
+time dependence in the notation xjyk for the sake of notational
+brevity.
+The proportions of susceptible, exposed, infected, recov-
+ered, and deceased nodes are
+sk =
+J
+∑
+j=1
+(ujsk +ajsk),
+(1)
+ek =
+J
+∑
+j=1
+(ujek +ajek),
+(2)
+ik =
+J
+∑
+j=1
+(u jik +ajik),
+(3)
+rk =
+J
+∑
+j=1
+(u jrk +ajrk),
+(4)
+dk =
+J
+∑
+j=1
+ujdk ,
+(5)
+where J is the maximum (or cut-off) degree in the IL. Simi-
+larly, we find that the proportions of unaware and aware nodes
+
+Impact of random and targeted disruptions on information diffusion during outbreaks
+3
+FIG. 1. Model schematic. (a) Information layer and epidemic layer. Nodes in the information layer are either unaware (U) or aware (A) while
+nodes in the epidemic layer can be in one of five different states: susceptible (S), exposed (E), infected (I), recovered (R), and deceased (D).
+Edge removal that is caused by disruptions in the information layer is indicated by the scissor symbol. (b) Unaware nodes become aware at rate
+λ if they are adjacent to an aware node. If unaware nodes are infected, they can also become aware at rate κ. Aware nodes transition back to
+an unaware state at rate δ. (c) Infectious nodes transmit a disease to unaware and aware susceptible nodes at rates β u and β a, respectively. To
+account for a reduction in infectiousness risk of aware nodes, we assume the value of the disease transmission rate β u associated with unaware
+nodes is strictly larger than the value of the disease transmission rate β a associated with aware nodes (β u > β a). Once susceptible nodes have
+been infected, they enter an exposed state and become infectious at rate σ. The characteristic time scale σ−1 corresponds to the latency period
+of the disease. Infected nodes either die or recover at rates fγ and (1− f)γ, respectively.
+are
+uj =
+K
+∑
+k=1
+(ujsk +ujek +u jik +ujrk +dk),
+(6)
+aj =
+K
+∑
+k=1
+(ajsk +ajek +a jik +ajrk),
+(7)
+where K is the maximum (or cut-off) degree in the EL. These
+quantities satisfy the normalization conditions
+K
+∑
+k=1
+(sk +ek +ik +rk +dk) = 1,
+(8)
+J
+∑
+j=1
+(uj +aj) = 1.
+(9)
+Assuming an uncorrelated network44, the rate equations of
+the heterogeneous mean-field model are
+dujsk
+dt
+=−λ ju jsk
+⟨˜k⟩ ∑
+j′
+j′aj′ −β u ku jsk
+⟨k⟩ ∑
+k′
+k′ik′ +δajsk , (10)
+dajsk
+dt
+=λ ju jsk
+⟨˜k⟩ ∑
+j′
+j′aj′ −β a ka jsk
+⟨k⟩ ∑
+k′
+k′ik′ −δajsk ,
+(11)
+dujek
+dt
+=−λ ju jek
+⟨˜k⟩ ∑
+j′
+j′aj′ +β u ku jsk
+⟨k⟩ ∑
+k′
+k′ik′
+(12)
+−σujek +δajek
+and
+dajek
+dt
+=λ jujek
+⟨˜k⟩ ∑
+j′
+j′aj′ +β a kajsk
+⟨k⟩ ∑
+k′
+k′ik′
+(13)
+−σajek −δa jek ,
+dujik
+dt
+=−λ ju jik
+⟨˜k⟩ ∑
+j′
+j′aj′ +σu jek −γujik
+(14)
+−κu jik +δajik ,
+da jik
+dt
+=λ jujik
+⟨˜k⟩ ∑
+j′
+j′aj′ +σajek −γajik
+(15)
++κu jik −δajik ,
+dujrk
+dt
+=−λ ju jrk
+⟨˜k⟩ ∑
+j′
+j′aj′ +(1− f)γujik +δa jrk ,
+(16)
+dajrk
+dt
+=λ jujrk
+⟨˜k⟩ ∑
+j′
+j′aj′ +(1− f)γajik −δajrk ,
+(17)
+dujdk
+dt
+=fγ(uj +aj)ik ,
+(18)
+where ⟨k⟩ and ⟨˜k⟩ denote the mean degrees of the EL and IL,
+respectively.
+C.
+Networks
+In our numerical experiments, we use a Barabási–Albert
+(BA) network45 to model the information layer of the two-
+layer structure underlying SEIRD-UAU dynamics.
+Such
+networks exhibit scale-free degree distributions p(k) ∝ k−γ
+
+(a)
+(b)
+Information layer
+aware neighbor
+already infected
+C
+E
+Epidemic layerImpact of random and targeted disruptions on information diffusion during outbreaks
+4
+(a)
+(b)
+FIG. 2. Multiplex networks. Information layer (top layer) with BA structure and and epidemic layer (bottom layer) with GIRG structure
+determined by exponents α = 2, τ = 2.5 (a) and α = 2, τ = 3.5 (b). In the BA network, each new node has m = 2 edges that connect it to
+existing nodes using linear preferential attachment. We use blue and orange edges in the epidemic layer to indicate short-range and long-range
+connections, respectively. An edge connecting two nodes i, j is considered a short-range connection if the corresponding positions xi,x j satisfy
+∥xi − xj∥ < 7. Otherwise, it is considered a long-range connection. The numbers of nodes in panels (a) and (b) are N = 921 and N = 973,
+respectively.
+(γ > 0), and are often found in social and technological
+systems.46–50 Note that other distributions such as log-normal
+distributions may also provide good descriptions of empirical
+degree distributions in seemingly scale-free networks.51 In the
+epidemic layer, we use a geometric inhomogeneous random
+graph (GIRG)52, a spatial network that has found applications
+in representing spatially embedded metapopulation structures
+in COVID-19 models.53
+1.
+Barabási–Albert network
+Barabási–Albert networks45 are constructed using a prefer-
+ential attachment procedure in which new nodes that are itera-
+tively added to an existing network have a higher likelihood of
+being attached to nodes that have higher numbers of connec-
+tions. A mean field analysis of the BA model and correspond-
+ing numerical results show that the exponent of the power-law
+degree distribution is γ ≈ 3.54
+To construct the BA network that we will use in our sim-
+ulations, we start with a star graph with one root node and
+two leave nodes and iteratively add new nodes until we reach
+N nodes. Each new node has m = 2 edges that connect it to
+existing nodes using linear preferential attachment. A visual-
+ization of such a BA information layer network with N ≈ 103
+is given in the top row of Fig. 2. In our simulations, we use
+a BA network with a larger node number of N ≈ 104 that is
+constructed in the same way as the ILs in Fig. 2.
+2.
+Geometric inhomogeneous random graph
+The GIRG model52,55 produces a spatially embedded scale-
+free random network. In this model, N points are first se-
+lected uniformly at random in the n-dimensional hypercube
+Kn = [0,1]n.
+We denote the randomly selected point po-
+sitions by xi ∈ Kn (1 ≤ i ≤ N) and assign each of them a
+weight wi whose value is drawn from a power-law distribution
+˜p(w) = (τ −2)w−τ (w ≥ 1,τ ≥ 2).52,55 Note that the distribu-
+tion ˜p(w) is normalized such that its mean value is equal to
+1. Pairs of nodes i, j with positions xi,xj are adjacent with
+probability
+Πij = 1−exp
+�
+−
+�
+wi w j
+∥xi −xj∥n
+�α�
+,
+(19)
+where ∥xi − xj∥ denotes the Euclidean distance between
+points xi and x j. The resulting degrees ki (1 ≤ i ≤ N) are
+also distributed according to a power law with exponent τ.
+According to Eq. 19, the exponent α tunes the distance and
+weight dependence of Πij. For α = 0, the probability that
+two nodes i, j are adjacent is independent of their distance
+|xi − xj|. That is, Πij = 1 − e−1 for all i, j. By increasing α,
+the distance-dependence of Πij strongly influences the struc-
+ture of the network so that only nearby nodes are likely to be
+adjacent. The bottom row of Fig. 2 shows GIRGs for various
+parameters.
+For small exponents τ ≥ 2, the number of nodes with large
+weight values increases. According to Eq. 19, nodes with
+large weights are more likely to be connected than nodes with
+
+Impact of random and targeted disruptions on information diffusion during outbreaks
+5
+small weights. The abundance of these large-weight nodes,
+which are the hubs of the underlying scale-free network, im-
+pacts the global structure of GIRG. By decreasing τ, many
+long-range connections are added to the GIRG. In the bottom
+row of Fig. 2, we observe that smaller values of τ are associ-
+ated with a larger proportion of long-range connections.
+D.
+Edge removal
+To model disruptions in the IL, we consider two different
+edge-removal protocols: (i) random edge removal and (ii) tar-
+geted edge removal. In both protocols, we select ˜N ≤ N nodes
+and denote the proportion of selected nodes by q = ˜N/N. For
+each selected node, we remove each of its edges with proba-
+bility p. Values of p,q > 0 correspond to disruptions in the
+IL that slow down the information spread. For p = q = 1,
+there are no awareness dynamics and the epidemic progresses
+without interference from the information layer.
+In random edge removal, ˜N nodes are selected uniformly
+at random while we select ˜N hub nodes (i.e., nodes with the
+largest degrees) in targeted edge removal. Such random and
+targeted disruptions have been studied to provide insight into
+the ability of different types of networks to withstand errors
+and intentional attacks.65 It has been shown that structural fea-
+tures of scale-free networks such as the size of the largest con-
+nected component are very sensitive to intentional attacks (or
+sabotage).66,67
+We next explore how variations in p,q ∈ [0,1] impact the
+total proportion of infections i∗ = 1 − s∗, peak infection (i.e.,
+the maximum proportion of the population that was infected
+on any day), and the time between the beginning of the out-
+break until peak infection is reached.
+III.
+RESULTS
+First consider a baseline case of SEIRD-UAU dynamics
+without edge removal (i.e., pq = 0) in two different multi-
+plex networks. Both multiplex networks are connected and
+have the same BA information layer (see Sec. II C 1). In the
+epidemic layer, we set τ = 3.5 and τ = 2.5 to model contact
+networks with different proportions of long-range connections
+(see Fig. 2). In the remainder of this work, we will refer to the
+networks with τ = 2.5 and τ = 3.5 as long-range and short-
+range networks, respectively. In both networks, we set α = 2
+[see Eq. (19)]. All stochastic simulations are implemented us-
+ing Gillespie’s algorithm.68–70
+A.
+Baseline
+We have chosen the model parameters that we use in the
+baseline simulation in accordance with empirical data on the
+outbreak of SARS-CoV-2 in the beginning of 2020. For ex-
+ample, for the two multiplex networks that we use in our sim-
+ulations, we have set the infection rate of unaware nodes to
+β u = 0.17,0.6 day−1 to obtain a basic reproduction number R0
+FIG. 3.
+Stochastic simulation of baseline scenario without
+information-layer disruption (i.e., pq = 0). (a,b) Proportions of sus-
+ceptible (s(t)), exposed (e(t)), infected (i(t)), recovered (r(t)), and
+deceased (d(t)) nodes at time t. The exponent τ in the epidemic
+layer in panels (a,c) and (b,d) is set to 3.5 (short range) and 2.5
+(long range), respectively. The corresponding numbers of nodes are
+N = 10049 and N = 10025. Solid colored lines represent mean val-
+ues that are based on 10 i.i.d. realizations (thin grey lines).
+FIG. 4. Heterogeneous mean-field solution of baseline scenario with-
+out information-layer disruption (i.e., pq = 0). (a,b) Proportions of
+susceptible (s(t)), exposed (e(t)), infected (i(t)), recovered (r(t)),
+and deceased (d(t)) nodes at time t. The exponent τ in the epi-
+demic layer in panels (a,c) and (b,d) is set to 3.5 (short range) and
+2.5 (long range), respectively. The corresponding numbers of nodes
+are N = 10049 and N = 10025.
+
+1.0
+a
+(b)
+0.8
+0.6
+0.4
+0.2
+0.0
+0
+50
+100
+150
+0
+50
+100
+150
+1.0
+(d)
+(c)
+0.8
+u(t)
+Proportion
+a(t)
+0.6
+s(t)
+(+)a
+0.4
+i(t)
+r(t)
+30 × d(t)
+0.2
+0.0
+0.0
+2.5
+5.0
+7.5
+10.0
+0.0
+2.5
+5.0
+7.5
+10.0
+Time [days]
+Time [days]1.0
+b
+0.8
+Proportion
+0.6
+0.4
+0.2
+0.0
+0
+50
+100
+150
+0
+50
+100
+150
+1.0
+(d)
+0.8
+u(t)
+Proportion
+a(t)
+0.6
+s(t)
+e(t)
+0.4
+i(t)
+r(t)
+30 × d(t)
+0.2
+0.0
+0.0
+2.5
+5.0
+7.5
+10.0
+0.0
+2.5
+5.0
+7.5
+10.0
+Time [days]
+Time [days]Impact of random and targeted disruptions on information diffusion during outbreaks
+6
+Parameter
+Symbol
+Value
+Units
+Comments/references
+Infection rate (unaware)
+β u
+0.17,0.6
+day−1
+inferred from R0 ≈ 2−4 for a given γ56,57
+Infection rate (aware)
+β a
+0.2β u
+day−1
+58
+Latent rate
+σ
+1/5
+day−1
+59
+Resolution rate
+γ
+1/14
+day−1
+60,61
+Infection fatality ratio
+f
+1%
+...
+62,63
+Awareness rate (infected)
+κ
+1
+day−1
+64
+Base awareness rate
+λ
+0.5κ
+day−1
+64
+Unawareness rate
+δ
+1/30
+day−1
+64
+TABLE I. Overview of model parameters. We use infection rates β u = 0.17 day−1 and β u = 0.6 day−1 for GIRG networks with τ = 2.5 (long
+range) and τ = 3.5 (short range), respectively.
+of about 2−4.56,57 Given a latency period of about 5 days59,
+we set the latent rate to σ = 1/5 day−1. The resolution rate is
+set to γ = 1/14 day−1, and we use an infection fatality ratio f
+of 1%.60–63 Other model parameters that are associated with
+UAU dynamics are as in Ref. 64. We provide an overview of
+all parameters and corresponding references in Tab. I.
+Figure 3 shows the stochastic evolution of the proportions
+of susceptible s(t), exposed e(t), infected i(t), recovered r(t),
+and deceased d(t) nodes in the EL and of unaware u(t) and
+aware a(t) nodes in the IL. Initially, 10 nodes are infectious
+and 1 node is aware. For networks of about N = 10000 nodes
+that are used in our stochastic simulations, these initial condi-
+tions correspond to i(0) ≈ 10−3 and a(0) ≈ 10−4. The simu-
+lation results shown in Figs. 3(a,c) and Figs. 3(b,d) are based
+on short-range (τ = 3.5) and long-range (τ = 2.5) GIRGs, re-
+spectively. The evolution of the UAU dynamics in the IL is
+very similar for both GIRGs. However, structural differences
+between the ELs directly impact the evolution of SEIRD dy-
+namics. The infected fraction peaks at ∼ 0.17 after about 38
+days in the long-range EL but peaks at ∼ 0.21 at about 51 days
+in the short-range EL. Figure 3 also shows that the final epi-
+demic size 1−s(t → ∞) in both networks differs significantly.
+To understand what causes the different outbreak character-
+istics in both networks, we examined the degree distribution
+of susceptible nodes at T = 150: there are substantially more
+susceptible low-degree nodes in the long-range GIRG where
+τ = 2.5 compared to the short-range GIRG with τ = 3.5. Al-
+though, there are more hub nodes with large degree in the
+long-range GIRG, the proportion of low-degree nodes is also
+larger. Hence, there are more low-degree nodes in the long-
+range GIRG that are less exposed to the outbreak dynamics.
+To complement the stochastic simulation results, we nu-
+merically solve the heterogeneous mean-field model (10)-(18)
+for the same networks and model parameters (see Tab. I).
+We set the degree cut-offs to J = 210, K = 400 (τ = 2.5)
+and J = 210, K = 164 (τ = 3.5). In the multiplex network
+with short-range IL with τ = 3.5, the degree cut-offs corre-
+spond to the maximum degrees. In the long-range EL where
+τ = 2.5, the maximum degree is 856, and to keep the solu-
+tion of the mean-field model computationally feasible we set
+the cut-off K = 400. Initially, we set ajik(0) = pj ˜pka(0)/2,
+ajsk(0) = pj ˜pka(0)/2, ujsk(0) = pj ˜pk(1 − i(0) − a(0)/2),
+ujik(0) = pj ˜pk(i(0) − a(0)/2), where pj and ˜pk denote the
+degree distributions in the IL and EL, respectively. Both de-
+gree distributions are normalized according to ∑J
+j=1 pj = 1
+and ∑K
+k=1 ˜pk = 1.
+Note that these initial conditions satisfy
+s(0) = ∑
+j,k
+�
+u jsk(0)+ajsk(0)
+�
+(20)
+= ∑
+j,k
+pj ˜pk
+�
+1−i(0)
+�
+= 1−i(0),
+(21)
+i(0) = ∑
+j,k
+�
+u jik(0)+ajik(0)
+�
+= ∑
+j,k
+pj ˜pki(0),
+(22)
+u(0) = ∑
+j,k
+�
+u jsk(0)+ujik(0)
+�
+(23)
+= ∑
+j,k
+pj ˜pk
+�
+1−a(0)
+�
+= 1−a(0),
+(24)
+a(0) = ∑
+j,k
+�
+a jsk(0)+ajik(0)
+�
+= ∑
+j,k
+pj ˜pka(0).
+(25)
+In accordance with the initial conditions that we used in the
+stochastic simulations, we set i(0) = 10−3 and a(0) = 10−4.
+Figure 4 shows the corresponding numerical results. Compar-
+ing Figs. 3 to 4, we observe that the heterogeneous mean-field
+model captures characteristic features that arise in the evolu-
+tion of stochastic SEIRD-UAU dynamics. Examples of such
+features include (i) the rapid spread of awareness in the IL
+and (ii) differences between both ELs in the final epidemic
+size 1 − s(t → ∞). In the heterogeneous mean-field model
+(10)-(18), we account only for differences in node degree and
+neglect other structural features of the considered multiplex
+networks. Subpopulations interact in a well-mixed manner
+and susceptible nodes of the same degree have the same risk
+of being infected at any given time. As a consequence of these
+approximations, the mean-field model overestimates both the
+number of new infections and final outbreak size compared to
+the stochastic simulation results in Fig. 3.
+
+Impact of random and targeted disruptions on information diffusion during outbreaks
+7
+FIG. 5. Random edge removal. The impact of random edge removal in the IL on disease dynamics in the EL. Epidemic size 1 − s(t → ∞)
+(left column), peak infection (middle column), and time to peak infection (right column) as a function of the proportion of selected nodes q
+and the corresponding edge removal probability p. The exponent τ in the ELs in top row and bottom row is set to 3.5 (short range) and 2.5
+(long range), respectively. The corresponding numbers of nodes are N = 10049 and N = 10025. Simulation results are based on 230 i.i.d.
+realizations.
+FIG. 6. Targeted edge removal. The impact of random edge removal in the IL on disease dynamics in the EL. Epidemic size 1 − s(t → ∞)
+(left panel), peak infection (middle panel), and time to peak infection (right panel) as a function of the proportion of selected nodes q and the
+corresponding edge removal probability p. The exponent τ in the ELs in top row and bottom row is set to 3.5 (short range) and 2.5 (long range),
+respectively. The corresponding numbers of nodes are N = 10049 and N = 10025. Simulation results are based on 230 i.i.d. realizations.
+
+(a) Epidemic size (T = 3.5)
+(b) Peak infection (T = 3.5)
+(c) Time to peak infection (↑ = 3.5)
+1.00
+0.40
+50
+0.98
+0.35
+0.96
+0.30
+40
+0.94
+0.25
+0.92
+0.20
+30
+0.00
+0.00
+0.00
+.1.00
+0.25
+1.00
+0.25
+1.00
+0.25
+0.75
+0.75
+0.75
+0.50
+0.50
+0.50
+0.50
+0.50
+0.50
+0.75
+0.75
+0.75
+0.25
+0.25
+0.25
+p
+p
+p
+1.00
+1.00
+1.00
+0.00
+0.00
+q
+0.00
+q
+q
+(d) Epidemic size (T = 2.5)
+(e) Peak infection (T = 2.5)
+(f) Time to peak infection (T = 2.5)
+1.0
+0.4
+40
+0.9
+35
+0.3
+0.8
+30
+0.7
+0.2
+0.6
+25
+0.00
+0.00
+0.00
+0.25
+1.00
+0.25
+1.00
+0.25
+1.00
+0.75
+0.75
+0.75
+0.50
+0.50
+0.50
+0.50
+0.50
+0.50
+0.75
+0.75
+0.75
+0.25
+0.25
+0.25
+p
+p
+p
+1.00
+0.00
+1.00
+0.00
+1.00
+0.00
+q
+q
+q(a) Epidemic size (π = 3.5)
+(b) Peak infection (T = 3.5)
+(c) Time to peak infection (↑ = 3.5)
+1.00
+0.40
+50
+0.98
+0.35
+40
+0.96
+0.30
+0.94
+0.25
+30
+0.92
+0.20
+0.00
+0.00
+0.00
+0.25
+1.00
+0.25
+1.00
+0.00
+0.25
+0.75
+0.75
+0.25
+0.50
+0.50
+0.50
+0.50
+0.50
+0.50
+0.75
+0.75
+0.75
+0.25
+0.25
+0.75
+p
+p
+p
+1.00
+1.00
+1.00
+0.00
+0.00
+q
+q
+q
+1.00
+(d) Epidemic size (T = 2.5)
+(e) Peak infection (T = 2.5)
+(f) Time to peak infection (T = 2.5)
+40
+1.0
+0.4
+35
+0.9
+0.3
+0.8
+30
+0.7
+0.2
+25
+0.6
+0.00
+0.00
+0.00
+0.25
+1.00
+0.25
+1.00
+0.00
+0.25
+0.75
+0.75
+0.25
+0.50
+0.50
+0.50
+0.50
+0.50
+0.50
+0.75
+0.75
+0.75
+0.25
+0.25
+0.75
+p
+p
+p
+1.00
+0.00
+1.00
+0.00
+1.00
+q
+q
+q
+1.00Impact of random and targeted disruptions on information diffusion during outbreaks
+8
+B.
+Impact of edge removal
+We now study the impact of random and targeted edge re-
+moval in the IL (see Sec. II D) on SEIRD dynamics in terms
+of three disease severity measures: (i) final epidemic size, (ii)
+peak infection, and (iii) time to peak infection.
+1.
+Random edge removal
+In random edge removal, we first select a proportion of q =
+˜N/N nodes in the IL uniformly at random. For each of the
+selected nodes, each of its edges are removed with probability
+p.
+Figure 5(a,d) shows the epidemic size as a function of p,q
+for both short-range and long-range GIRGs. The epidemic
+size increases with p and q because larger values of p,q are
+associated with fewer edges in the IL, leading to a smaller pro-
+portion of aware nodes. Hence, the proportion of nodes with
+a reduced infection rate β u also decreases. For the long-range
+GIRG (τ = 2.5), the final epidemic size undergoes a transition
+from about 0.6 for p,q ≈ 0 to about 0.9 for p,q ≈ 1. Because
+the final epidemic size in the short-range GIRG (τ = 3.5) is
+already about 0.9, random edge removal has relatively little
+impact on this quantity.
+As with the impact on final epidemic size 1−s(t → ∞), ran-
+dom edge removal generates a similar-looking p,q-dependent
+infection peak, as shown in Fig. 5(b,e). The time to reach peak
+infection decreases with p,q since higher p,q are associated
+with smaller proportions of aware nodes. Thus, the proportion
+of nodes with a reduced infection rate β u also decreases, and
+the epidemic spreads faster through the network.
+2.
+Targeted edge removal
+For targeted edge removal where the ˜N selected nodes cor-
+respond to the hubs (i.e., largest-degree nodes) of the IL, we
+find that the overall dependence of epidemic size, peak infec-
+tion, and time to peak infection on p,q is qualitatively simi-
+lar to random edge removal (see Fig. 6). As in random edge
+removal, the impact of targeted edge removal on the final epi-
+demic size is smaller for the short-range GIRG compared to
+the long-range one. A key difference in targeted edge removal
+is that all studied quantities are more sensitive to variations in
+q, the proportion of selected hub nodes. For example, the tran-
+sition of the epidemic size for p = 1 as a function of q in tar-
+geted edge removal [see Fig. 6(a,d)] is steeper than the corre-
+sponding transition in random edge removal [see Fig. 5(a,d)].
+Targeted edge removal selects nodes based on their degree
+rather than uniformly, and leads to more significant changes
+in epidemic size, peak infection, and time to peak infection
+as p ≥ 0.5. These findings are in accordance with previous
+work that showed that scale-free networks break down more
+easily under intentional attacks than under uniform random
+failure.67 Our work provide insights into how such disruptions
+in information diffusion translate into differences in disease
+severity measures.
+IV.
+DISCUSSION
+In this work, we studied the impact of disruptions in com-
+munication networks on information diffusion and subse-
+quently disease outcome during an outbreak. To do so, we
+constructed a multiplex network that consists of two lay-
+ers. The first layer, called information layer (IL), is used to
+model communication between individuals (e.g., online in-
+formation exchange via a social media platform). The sec-
+ond layer, called epidemic layer (EL), is used to represent a
+spatially embedded human contact network in which infec-
+tious individuals can transmit a disease to susceptible indi-
+viduals. We use this multiplex network to simulate coevolv-
+ing unaware-aware-unaware (UAU) and susceptible-exposed-
+infected-recovered-deceased (SEIRD) dynamics. The model
+parameters that we use in our simulations have been selected
+in accordance with empirical data on the early outbreak stages
+of SARS-CoV-2 in the beginning of 2020.
+We studied two different epidemic layers with different pro-
+portions of long-range connections, representing human con-
+tact networks with different contact characteristics. To illus-
+trate the impact of disruptions in the IL on the evolution of an
+outbreak, we utilized two different edge removal protocols: (i)
+random edge removal and (ii) targeted edge removal. In both
+protocols, we select a proportion q of nodes and then remove
+corresponding edges with probability p. In random edge re-
+moval, we select nodes in the IL uniformly at random while
+we select nodes with the largest degree (i.e., hub nodes) in
+targeted edge removal. Although edge removal may render
+the IL disconnected, the EL is always connected in our sim-
+ulations such that all nodes in the EL can potentially become
+infected. Previous work has shown that scale-free networks
+such as the IL in our multiplex network are more robust to
+random than targeted disruptions.65–67 The reason for this ef-
+fect is that by removing hub nodes of a scale-free network, a
+large number of all edges in the network is being removed,
+strongly impacting the connectivity properties of such a net-
+work. We observe that targeted edge removal can abruptly
+change outbreak characteristics such as time to peak infection,
+even for small proportions of selected nodes. Our results ex-
+tend those presented in previous work on random and targeted
+disruptions65–67 by establishing a connection to coevolving in-
+formation and epidemic diffusion.
+DATA AVAILABILITY
+Our
+source
+codes
+are
+publicly
+available
+at
+https://gitlab.com/ComputationalScience/
+information-epidemic.
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+Inigo, G. L. Wagner, S. Pei, C. Daraio, R. Ferrari, et al., “Epidemic manage-
+ment and control through risk-dependent individual contact interventions,”
+PLOS Computational Biology 18, e1010171 (2022).
+2C. Granell, S. Gómez, and A. Arenas, “Dynamical interplay between aware-
+ness and epidemic spreading in multiplex networks,” Physical Review Let-
+ters 111, 128701 (2013).
+3J. P. Gleeson, “Binary-state dynamics on complex networks: Pair approxi-
+mation and beyond,” Physical Review X 3, 021004 (2013).
+
+Impact of random and targeted disruptions on information diffusion during outbreaks
+9
+4R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani,
+“Epidemic processes in complex networks,” Reviews of Modern Physics
+87, 925 (2015).
+5M. E. J. Newman, “Properties of highly clustered networks,” Physical Re-
+view E 68, 026121 (2003).
+6W. Huang and C. Li, “Epidemic spreading in scale-free networks with com-
+munity structure,” Journal of Statistical Mechanics: Theory and Experiment
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+

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@@ -0,0 +1,681 @@
+Diagonalization Games
+Noga Alon1,5, Olivier Bousquet6, Kasper Green Larsen4,
+Shay Moran2,3,6, and Shlomo Moran2
+1Departments of Mathematics and Computer Science, Tel Aviv University
+2Department of Computer Science, Technion, Israel
+3Department of Mathematics, Technion, Israel
+4Department of Computer Science, Aarhus University
+5Department of Mathematics, Princeton University
+6Google Research
+January 6, 2023
+Abstract
+We study several variants of a combinatorial game which is based on Cantor’s diagonal
+argument. The game is between two players called Kronecker and Cantor. The names of the
+players are motivated by the known fact that Leopold Kronecker did not appreciate Georg
+Cantor’s arguments about the infinite, and even referred to him as a “scientific charlatan”.
+In the game Kronecker maintains a list of m binary vectors, each of length n, and Cantor’s
+goal is to produce a new binary vector which is different from each of Kronecker’s vectors, or
+prove that no such vector exists. Cantor does not see Kronecker’s vectors but he is allowed
+to ask queries of the form
+“What is bit number j of vector number i?”
+What is the minimal number of queries with which Cantor can achieve his goal? How much
+better can Cantor do if he is allowed to pick his queries adaptively, based on Kronecker’s
+previous replies?
+The case when m = n is solved by diagonalization using n (non-adaptive) queries. We
+study this game more generally, and prove an optimal bound in the adaptive case and nearly
+tight upper and lower bounds in the non-adaptive case.
+1
+Introduction
+The concept of infinity has been fascinating philosophers and scientists for hundreds, perhaps
+thousands of years. The work of Georg Cantor (1845 – 1918) played a pivotal role in the
+mathematical treatment of the infinite. Cantor’s work is based on a simple notion which asserts
+that two (possibly infinite) sets have the same size whenever their elements can be paired
+in one-to-one correspondence with each other [Can74]. Despite being simple, this notion has
+counter-intuitive implications: for example, a set can have the same size as a proper subset of it1;
+this phenomena is nicely illustrated by Hilbert’s paradox of the Grand Hotel, see e.g. [Wik22b].
+This simple notion led Cantor to develop his theory of sets, which forms the basis of modern
+mathematics. Alas, Cantor’s set theory was controversial at the start, and only later became
+widely accepted:
+1E.g. the natural numbers and the even numbers, via the correspondence “n �→ 2n”.
+1
+arXiv:2301.01924v1  [math.CO]  5 Jan 2023
+
+Figure 1: Georg Cantor (1845 – 1918)
+Figure 2: Leopold Kronecker (1823 – 1891)
+The objections to Cantor’s work were occasionally fierce: Leopold Kronecker’s public opposition
+and personal attacks included describing Cantor as a ”scientific charlatan”, a ”renegade” and a
+”corrupter of youth”. Kronecker objected to Cantor’s proofs that the algebraic numbers are
+countable, and that the transcendental numbers are uncountable, results now included in a
+standard mathematics curriculum. [Wik22a]
+1.1
+Diagonalization
+One of the most basic and compelling results in set theory is that not all infinite sets have the same
+size. To prove this result, Cantor came up with a beautiful argument, called diagonalization.
+This argument is routinely taught in introductory classes to mathematics, and is typically
+presented as follows. Let N denote the set of natural numbers and let {0, 1}N denote the set of
+all infinite binary vectors. Clearly both sets are infinite, but it turns out that they do not have
+the same size: assume towards contradiction that there is a one-to-one correspondence j �→ vj,
+where vj = (vj(1), vj(2), . . .) is the infinite binary vector corresponding to j ∈ N. Define a vector
+u = (1 − v1(1), 1 − v2(2), . . .).
+That is, u is formed by letting its j’th entry be equal to the negation of the j’th entry of vj.
+Notice that this way the resulting vector u disagrees with vj on the j’th entry, and hence
+u ̸= vj for all j. Thus, we obtain a binary vector which does not correspond to any of the natural
+numbers via the assumed correspondence – a contradiction.
+Rather than reaching a contradiction, it is instructive to take a positivist perspective according
+to which diagonalization can be seen as a constructive procedure that does the following:
+Given binary vectors v1, v2, . . ., find a binary vector u such that u ̸= vj for all j.
+Moreover, notice that Cantor’s diagonal argument involves querying only a single entry per
+each of the input vectors vj (i.e. the “diagonal” entries vj(j)). Thus, it is possible to construct u
+while using only a little information about the input vectors vi’s (a single bit per vector).
+In this manuscript we study a finite variant of the problem in which m binary vectors
+v1, . . . , vm of length n are given and the goal is to produce a vector u which is different from all
+of the vi’s, or to report that no such vector exists, while querying as few as possible entries of
+the vi’s. We first study the case when m < 2n whence such a u is guaranteed to exist, and the
+goal boils down to finding one, and later the case when m ≥ 2n.
+2
+
+v1 = 0, 1, 1, 0, 1, 0
+v2 = 1, 0, 0, 1, 1, 1
+v3 = 1, 1, 1, 0, 0, 0
+v4 = 0, 1, 0, 1, 1, 0
+v5 = 1, 1, 0, 1, 0, 1
+v6 = 0, 1, 1, 1, 1, 1
+u = 1, 1, 0, 0, 1, 0
+Figure 3: An illustration of Cantor’s diagonalization: the vector u at the bottom is not equal to
+any of the vi’s at the top.
+2
+The Cantor-Kronecker Game
+Consider a game between two players called Kronecker and Cantor. In the game there are
+two parameters m and n, where m, n are positive integers. Kronecker maintains a set V =
+{v1, v2, . . . , vm} of m binary vectors, each of length n. Cantor’s goal is to produce a binary
+vector u, also of length n, which differs from each vi, or to report that no such vector exists. To
+do so, he is allowed to ask queries, where each query is of the form
+“What is bit number j of vector number i?”,
+where 1 ≤ j ≤ n, 1 ≤ i ≤ m. Kronecker is answering each query being asked. The objective
+of Cantor is to minimize the number of queries enabling him to produce u, whereas Kronecker
+tries to maximize the number of queries. We distinguish between two versions of the game:
+• In the adaptive version Cantor presents his queries to Kronecker in a sequential manner,
+and may decide on the next query as a function of Kronecker’s answers to the previous
+ones.
+• In the oblivious version Cantor must declare all of his queries in advance, before getting
+answers to any of them.
+For m ≤ n the smallest number of queries, both in the adaptive and oblivious versions, is m.
+Indeed, Cantor can query bit number i of vi for all 1 ≤ i ≤ m and return a vector u whose i’th
+bit differs from the i’th bit of vi, for all i. The lower bound is even simpler: if Cantor asks less
+than m queries then there is some vector vi about which he has no information at the end of the
+game. In this case he cannot ensure that his vector u will not be equal to this vi.
+Organization.
+We begin with the case where m < 2n: in the next section (Section 3) we
+derive nearly tight bounds both in the adaptive and oblivious cases. We do so by exhibiting and
+analyzing near optimal strategies for Cantor. Then, in Section 4 we consider the case where
+m ≥ 2n and derive an optimal bound of m·n in this case (for both the oblivious and the adaptive
+versions). We do so by exhibiting and analyzing an optimal strategy for Kronecker. Finally, in
+Section 5 we discuss some algorithmic aspects, and conclude with some suggestions to future
+research.
+3
+
+3
+The Cantor-Kronecker Game with m < 2n
+3.1
+Adaptive Version
+Theorem 3.1. Let g(n, m) denote the smallest number of queries that suffices for Cantor when
+he is allowed to use adaptive strategies. Then,
+g(n, m) =
+�
+m
+m ≤ n,
+2m − n
+n < m < 2n.
+The case 1 ≤ m ≤ n is proved in the previous section so we assume n ≤ m < 2n.
+Upper Bound.
+We present a strategy for Cantor which combines diagonalization with another
+simple idea. To illustrate this idea let us first consider the case m = n + 1. This special case
+appeared as a question in the 2022 Grossman Math Olympiad for high-school students, and so
+perhaps the reader might enjoy trying to solve it before continuing reading.
+Let v1, . . . , vn+1 be the input vectors. Cantor begins with querying the first bit of v1, v2, and
+of v3. Getting the answers, there is a bit ε so that at least two vectors among v1, v2, v3 have
+their first bit equals to ε. Cantor now defines the first bit of u to be u(1) = 1 − ε and can remove
+the two vectors among v1, v2, v3 whose first bit equals ε. Now Cantor is left with at most n − 1
+vectors and can therefore set the last n − 1 coordinates of u according to the diagonalization
+construction.
+The general case is handled similarly by induction on n: for n = 1 since n ≤ m < 2n, also m
+must be 1 and the result is trivial.
+Assuming the result for n − 1, let v1, . . . , vm be the m vectors of Kronecker. First, note that
+there is an integer x satisfying 1 ≤ x ≤ ⌈m/2⌉ so that n − 1 ≤ m − x < 2n−1: indeed, for x = 1
+we have m − x ≥ n − 1 and for x = ⌈m/2⌉ we have m − x ≤ m/2 < 2n−1. Starting from x = 1
+keep increasing it in steps, where in each step it is increased by 1, until it reaches ⌈m/2⌉. As
+m − x changes by 1 in each step we can take the smallest x ≥ 1 that satisfies m − x < 2n−1, and
+it will clearly be at most ⌈m/2⌉ and satisfy m − x ≥ n − 1 as well.
+Having x as above, Cantor first queries the first bit of each of the vectors v1, v2, . . . , v2x−1.
+(Note that 2x − 1 ≤ m hence this is possible). Getting the answers, there is a bit ε ∈ {0, 1} so
+that at least x of the vectors have their first bit equal to ε. Cantor now defines the first bit of
+his vector u to be 1 − ε, removes from the set V exactly x of the vectors whose first bit is ε,
+and defines as V ′ the set of all restrictions of the remaining m − x vectors to their last n − 1
+coordinates. Note that n − 1 ≤ m − x < 2n−1.
+By the induction hypothesis, Cantor can now play the game for the set V ′ producing an
+appropriate vector u′ by asking at most 2(m − x) − (n − 1) additional queries. The total number
+of queries is thus (2x − 1) + 2(m − x) − (n − 1) = 2m − n, as needed. The vector u obtained
+by concatenating the 1-bit vector 1 − ε and the vector u′ is clearly different from each member
+of V . This completes the induction step argument and finishes the proof of the upper bound.
+Lower Bound.
+For the lower bound, we present a strategy for Kronecker which essentially
+mirrors Cantor’s strategy from the upper bound. Suppose Cantor manages to produce the
+required vector u after making exactly bj queries in coordinate number j of some of the vectors vi.
+Kronecker chooses his answers ensuring that for each such j, the answers for bits in the j’th
+location are balanced, that is, at most ⌈bj/2⌉ of the answers are 0 and at most ⌈bj/2⌉ of the
+answers are 1.
+Consider the vector u produced by Cantor. For every 1 ≤ j ≤ n, there are at most ⌈bj/2⌉
+vectors vi known to be different than u in coordinate number j. Thus altogether there are at
+4
+
+most
+n
+�
+j=1
+�bj
+2
+�
+≤
+n
+�
+j=1
+bj + 1
+2
+.
+vectors vi that are known to Cantor to be different than u. In order to ensure u is indeed
+different from each vi this number has to be at least m and hence
+m ≤
+n
+�
+j=1
+bj + 1
+2
+.
+By rearranging, this implies that the total number of queries �n
+j=1 bj must be at least 2m − n,
+as stated.
+3.2
+Oblivious Version
+Theorem 3.2. Let f(n, m) denote the smallest number of queries that suffices for Cantor when
+he is restricted to use oblivious strategies. Then,
+f(n, m) =
+�
+m
+m ≤ n
+m
+�
+log
+� m
+n
+�
++ o
+�
+log
+� m
+n
+���
+n < m < 2n.
+Quantitatively, for all n < m < 2n
+m ·
+�
+log
+�
+m
+n − log m + 1
+�
+− 1
+�
+≤ f(n, m) ≤ m
+�
+log
+�2m
+n
+�
++ 2 log
+�
+log
+�2m
+n
+��
++ 1
+�
+,
+The case 1 ≤ m ≤ n is proved above so we assume n < m < 2n.
+Upper Bound.
+Like in the adaptive case, we present a strategy for Cantor which combines
+diagonalization with another simple idea. We first illustrate this idea by handling the case m =
+n + 1, and again, we encourage the reader to try and handle this case before continuing reading.
+Let v1, . . . , vn+1 be the input vectors. Cantor begins with querying the first two bits of each
+of v1, v2, and v3 (for a total of 6 queries). Notice that there are 22 = 4 possible combinations of
+0/1 patterns on the first two bits, but at most three of them are realized by v1, v2, v3. Hence,
+there must be a pair of bits ε1, ε2 which is not realized by v1, v2, nor v3:
+(ε1, ε2) /∈
+��
+v1(1), v1(2)
+�
+,
+�
+v2(1), v2(2)
+�
+,
+�
+v3(1), v3(2)
+��
+.
+Thus, by setting u(1) = ε1 and u(2) = ε2, Cantor rules out v1, v2, v3 and is left with n−2 vectors
+v3, . . . , vn+1 which can be obliviously ruled out with the last n − 2 using diagonalization.
+For the general case, let d be an integer (to be determined later). Pick mutually disjoint
+subsets of coordinates J1, . . . , J⌊n/d⌋ ⊆ [n], each of size d, and pick a partition of the m vectors to
+⌊n/d⌋ subsets V1, . . . , V⌊n/d⌋ such that the partition is as balanced as possible (i.e. the difference
+between each pair of sizes is ≤ 1). Thus, each set has size
+|Vi| ≤
+�
+m
+⌊n/d⌋
+�
+≤ 2md
+n .
+Cantor queries (obliviously) as follows.
+For each i and each vector in Vi query all the coordinates in Ji.
+5
+
+Thus, the total number of queries is exactly m · d. Now, notice that if d satisfies
+2d > 2md
+n ,
+(1)
+then there must exist an assignment fi : Ji → {0, 1} such that fi disagrees with each of the
+vectors in Vi on at least one coordinate in Ji. Hence Cantor can output the vector u, which
+agrees with each of the fi on Ji. Note that Equation 1 is satisfied iff 2d
+d > 2m
+n ; since m > n,
+it can be verified that this inequality holds when d ≥ log( 2m
+n ) + 2 log(log( 2m
+n )) + 1. Thus for
+d =
+�
+log
+� 2m
+n
+�
++ 2 log
+�
+log
+� 2m
+n
+��
++ 1
+�
+, the total number of queries is at most
+m · d = m
+�
+log
+�2m
+n
+�
++ 2 log
+�
+log
+�2m
+n
+��
++ 1
+�
+.
+Lower Bound.
+The lower bound proof is based on the following simple idea. Let Ji denote
+the set of coordinates of vi which Cantor queries. Thus, the total number of queries Cantor uses
+is |J1| + . . . + |Jm|. Now, let fi : Ji → {0, 1} denote Kronecker’s answers for the queries on vi.
+The crucial observation is that the vector u that Cantor outputs must satisfy
+(∀i) : u|Ji ̸= fi.
+Indeed, if u|Ji = fi for some i then Kronecker can fail Cantor by picking his i’th vector vi to be
+equal to Cantor’s output u (which would be consistent with Kronecker’s answers).
+We summarize the above consideration with a definition that characterizes the winning (or
+losing) strategies of Cantor in the oblivious case.
+Definition 3.3 (Covering Assignments). We say that a sequence of sets J1, . . . , Jm ⊆ [n]
+has a covering assignment if there are m functions fi : Ji → {0, 1} such that every binary
+vector v ∈ {0, 1}n agrees with one of the fi on Ji (i.e. v|Ji = fi).
+Thus, Kronecker has a winning strategy if and only if the sequence of sets J1, . . . , Jm that
+Cantor queries has a covering assignment. The following lemma establishes the lower bound.
+Lemma 3.4. Let J1, . . . , Jm ⊆ [n] such that
+|J1| + . . . + |Jm| < m ·
+�
+log
+�
+m
+n − log m + 1
+�
+− 1
+�
+.
+(2)
+Then, J1, . . . , Jm has a covering assignment.
+Equivalently, if for each vector vi Cantor queries its entries in Ji and Equation 2 holds, then
+Kronecker has a winning strategy.
+Proof. Let ti = |Ji| and let t = �
+i ti. Assume, without loss of generality, that t1 ≤ t2 ≤ . . . ≤ tm.
+To prove a lower bound of the form md for t, where d will be specified later, we show that if t is
+smaller than md then there are m functions fi : Ji → {0, 1} so that for every possible vector
+v ∈ {0, 1}n there is i ≤ m so that v|Ji = fi.
+We do so by explicitly constructing the fi’s (which corresponds to describing a winning
+strategy for Kronecker). Starting with the set V = {0, 1}n of all possible potential vectors z,
+go over the vectors vi in order. In step i we choose the function fi : Ji → {0, 1} such that
+|{v ∈ V : v|Ji = fi}| is maximized. Since there are 2ti possible choices for fi, the maximizing
+choice satisfies
+���{v ∈ V : v|Ji = fi}
+��� ≥ |V |
+2ti .
+6
+
+After picking fi, we remove all the vectors of V that agree with fi and proceed to the next step.
+Therefore, after the first i steps, the size of the set V of the remaining vectors is at most
+2n
+i�
+j=1
+(1 − 1/2tj).
+We can continue with this analysis until the size of the set V becomes smaller than 1, namely
+the set becomes empty. It is a bit better, however, to apply a simpler reasoning once the size
+of V becomes smaller than 2d, and only argue that at least one vector from V is eliminated
+in each step. (Continuing the same analysis as before would only guarantee that V shrinks by
+a factor of (1 − 1/2ti) which by the choice of d would be roughly 1 − 1/2d < 1). To simplify
+the computation it is not too wasteful to apply the simpler analysis already when the size of
+V becomes smaller than m/2. If this happens in the first m/2 steps then by removing a single
+vector in each of the remaining steps we will eliminate all of the vectors. This means that if
+2n
+m/2
+�
+j=1
+�
+1 − 1/2tj�
+≤ m
+2
+then the sequence J1. . . . , Jm has a covering assignment. Since d is such that the total number of
+queries is m · d, the above amounts to �m/2
+j=1 tj ≤ md/2; that is, the average tj for 1 ≤ j ≤ m/2
+is at most d. This implies that
+2n
+m
+2
+�
+j=1
+�
+1 − 1
+2tj
+�
+≤ 2n
+m
+2
+�
+j=1
+exp
+�
+− 1
+2tj
+�
+(1 + x ≤ exp(x) for all x ∈ R)
+= 2n exp
+�
+−
+m
+2
+�
+j=1
+1
+2tj
+�
+≤ 2n exp
+�
+− m
+2d+1
+�
+,
+where the last inequality follows because exp(−x) is decreasing and because
+m
+2
+�
+j=1
+1
+2tj ≥ m
+2 ·
+1
+2
+1
+m/2
+�m/2
+j=1 tj
+≥ m
+2 · 1
+2d ,
+which follows by convexity of the function f(x) = 2x and because t1 ≤ t2 ≤ . . . ≤ tm.
+We have thus shown that if |J1| + . . . + |Jm| = m · d such that
+2n exp
+�
+− m
+2d+1
+�
+≤ m
+2
+then the sequence J1, . . . , Jm has a covering assignment. The last inequality surely holds provided
+m
+2d+1 ≥ n + 1 − log m.
+That is, provided
+2d+1 ≤
+m
+n + 1 − log m,
+or
+d ≤ log
+�
+m
+n + 1 − log m
+�
+− 1
+completing the proof.
+7
+
+4
+The Cantor-Kronecker Game with m ≥ 2n
+Assume now that Kronecker’s list V consists of m ≥ 2n binary vectors of length n. In this case
+V may contain all the binary vectors of length n and there is no vector Cantor can output that
+is different from each vector on Kronecker’s list. In this regime it is more natural to first focus
+on the decision problem in which Cantor’s goal is to decide whether V contains {0, 1}n, and if
+this is not the case, to provide a vector which is not in V .2 Clearly Cantor can achieve this if he
+queries all mn possible queries. Can he do better?
+We first observe that mn queries are in fact needed in the oblivious case: assume that Cantor
+submits only mn − 1 queries, and leaves the j’th bit of vi unqueried. Then Kronecker may set vi
+to be the unique occurrence of the all ones vector 1n, and set the remaining m − 1 vectors in V
+to include all 2n − 1 vectors that are different from the all ones vector. Clearly, it is necessary
+for Cantor to query also the last bit of vi in order to see whether vi is the all ones vector or not.
+Consequently, Cantor must query all mn queries in the oblivious case.
+How about the adaptive case? A similar argument shows that for m = 2n, Kronecker can
+force mn = 2nn queries also in the adaptive case, by using a list which contains each binary
+vector of length n exactly once: indeed, if only mn − 1 bits are queried, then the last, yet
+unqueried bit, belongs to a vector which occurs only once in V . Hence it is necessary to get the
+value of this bit in order to verify that V contains all 2n vectors.
+The case when m > 2n turns out to be more subtle. Nevertheless, we prove that mn queries
+are necessary even in this case. We start with introducing some notation.
+Notation.
+Each step of the game consists of a query by Cantor followed by a response by
+Kronecker. The status of the game after each such step is given by an m × n matrix L, where
+L(i, j) denotes the status of the j’th bit of vi, that is: L(i, j) ∈ {0, 1, ⋆}, where L(i, j) = ⋆ means
+that the j’th bit of vi was not queried yet, and otherwise L(i, j) equals the value of this bit as
+answered by Kronecker.
+Definition 4.1. FIXED(L) =
+�
+v ∈ L : v ∈ {0, 1}n�
+. That is, FIXED(L) is the set of all vectors
+in L that were fully queried by Cantor.
+Definition 4.2. L is complete if FIXED(L) = {0, 1}n.
+Definition 4.3. A subset S of 2n rows of L is useful if it either contains all the 2n binary vectors
+of length n, or it can be converted to this set by replacing each ⋆-entry in S by 0 or 1.
+Definition 4.4. A matrix L is unblocked if it can be completed; that is, if L has a useful subset.
+Otherwise L is called blocked.
+Notice that for m ≥ 2n, the m by n matrix all whose entries are ⋆ is unblocked.
+As a warmup, and to get used to the definitions, let us assume first that Cantor’s queries
+the vectors one by one according to their order; i.e. he first queries all the bits of v1 from left
+to right, then all the bits of v2 from left to right, and so on. We use the following strategy for
+Kronecker: when Cantor queries the j’th bit of vi (i.e. the value of L(i, j)), Kronecker replies
+according to the following “0 first” strategy:
+modified value of L(i, j) =
+�
+1
+If setting L(i, j) to 0 blocks L
+0
+otherwise
+(3)
+It is not hard to verify that since Cantor queries the vectors one by one, and from left (most
+significant bit) to right, the following matrix is produced: each of the first m − 2n + 1 rows will
+2Later we will see that the decision and search variants are in fact equivalent.
+8
+
+be set to the all-zeros vector, and the last 2n − 1 rows will be set to the 2n − 1 non zero vectors
+in increasing lexicographical order: starting with 0n−11 and ending with 1n. Hence Cantor is
+forced to query all mn entries as in the oblivious case.
+Interestingly, it turns out that, for any strategy of Cantor, the above “0 first” strategy of
+Kronecker forces Cantor to make mn queries.
+Theorem 4.5. Let m > 2n. Then for any strategy of Cantor, the “0 first” strategy of Kronecker
+forces Cantor to make mn queries in order to determine if L contains {0, 1}n.
+In the following we consider an arbitrary execution of the game, where Kronecker follows the
+“0 first” strategy (and Cantor’s strategy is arbitrary). We denote by Lt the m × n matrix L after
+t steps of the game; thus L0 is the initial matrix which is filled only with ⋆’s.
+By the fact that if L is unblocked and L(i, j) = ⋆, then it is possible to set L(i, j) to 0 or
+to 1 without blocking L, we get:
+Observation 4.6. If Lt is unblocked, so is Lt+1. Hence Lmn is complete; i.e. it contains {0, 1}n.
+Definition 4.7. We say that a row L(i) is essential for an unblocked matrix L if every useful
+subset of L’s rows contains L(i).
+Note that if Lt(i) is essential for Lt, then Ls(i) is essential for Ls for all s ≥ t. Also, if Lmn(i)
+is essential for Lmn, then Lmn(i) is equal to a unique vector in {0, 1}n which is different from all
+other rows of Lmn.
+Lemma 4.8. Assume that Lt(i) is not essential for Lt and Lt(i, j) = ⋆. If Lt(i, j) is queried at
+time t + 1, then it is set to 0, i.e. Lt+1(i, j) = 0.
+Proof. By the “0 first” strategy, and the fact that if L(i) is not essential for an unblocked
+matrix L, then setting L(i, j) to 0 does not block L.
+By a straightforwards induction Lemma 4.8 implies:
+Corollary 4.9. If Lt(i) is not essential for Lt, then Lt(i) contains no 1’s (only 0’s or ⋆’s).
+Specifically, if Lmn(i) is not essential for Lmn, then Lmn(i) is the zero vector 0n. Hence, every
+row of Lmn which is not the zero vector is essential, and thus it is different from all other rows
+of Lmn.
+Lemma 4.10. Let Lmn−1(i, j) be the last bit queried in the game. Then Lmn−1(i) is an essential
+row of Lmn−1.
+Proof. To simplify notation, we assume without loss of generality that j = 1. Assume towards
+contradiction that Lmn−1(i) is not essential for Lmn−1. By Corollary 4.9, this implies that
+Lmn−1(i) = ⋆0n−1 and Lmn(i) = 0n. (i.e. Kronecker sets Lmn−1(i, 1) to 0 at Cantor’s mn’th
+query). Since Lmn is complete (Observation 4.6), this implies that Lmn−1 contains a distinct
+occurrence of each of the 2n − 1 nonzero vectors of {0, 1}n, and in particular for some k ̸= i,
+Lmn−1(k) is the unique row of Lmn−1 which equals 10n−1. Then, any subset S of Lmn−1 which
+contains
+• the row Lmn−1(i),
+• the 2n − 2 non zero rows of Lmn−1 excluding Lmn−1(k), and
+• some zero row of Lmn−1 (by Corollary 4.9 there are m − 2n > 0 such rows in Lmn−1),
+9
+
+is a useful subset of Lmn−1 which does not contain Lmn−1(k). Hence Lmn−1(k) is not essential
+for Lmn−1, and by Lemma 4.8 Lmn−1(1) = 0 ̸= 1, which stands in contradiction with Lmn−1(1) =
+10n−1.
+Proof of Theorem 4.5. Let Lmn−1(i, j) be the last query in the game. By Lemma 4.10, Lmn−1(i),
+and hence also Lmn(i), is essential, meaning that Lmn(i) is different from all other rows of Lmn.
+Thus Cantor must get the value of Lmn−1(i, j) in order to reach a decision.
+A remark on computational complexity.
+A naive implementation of the “0 first” strategy
+might take exponential time: indeed, it requires checking whether setting the queried bit to 0
+blocks the current matrix, which involves checking a potentially exponential list of constraints.
+Nevertheless, we next show that this strategy in fact admits a polynomial time implementation.
+Firstly, notice that the first m − 2n steps are trivially efficient, because setting L(i, j) to any
+value cannot block L (since at least 2n rows of L are not queried yet).
+Thus it suffices to show that in each later step, deciding whether setting L(i, j) to 0 blocks
+the matrix, can be performed in time which is polynomial in mn, the size of L. Let Lt be the
+matrix L after t steps of the game, t > m − 2n. Consider the bipartite graph Gt = (At, B, Et),
+where At = {Lt(i) : 1 ≤ i ≤ m} is the set of rows of Lt, B = {0, 1}n, and (Lt(i), u) ∈ Et if
+and only if Lt(i) can be converted to the binary vector u by replacing the ⋆’s in Lt(i) (if any)
+by binary digits. Then, a subset S of Lt is useful for Lt if and only if Gt contains a perfect
+matching between the vertices in At which correspond to S and B.
+Assume now that we are given the graph Gt, and the corresponding matching, and let Lt(i, j)
+be the entry queried by Cantor at step t+1. To check if setting Lt(i, j) to 0 blocks Lt, we remove
+from Gt all the edges (Lt(i), u) in which u(j) = 0, and check if the resulted graph contains a
+perfect matching. Since we are given a perfect matching Mt for Gt, and removing these edges
+eliminates at most one edge from Mt, this checking can be done by executing one phase in some
+classical algorithm for bipartite matching, which can be done in O(|Et|) = O(m2n) = O(m2)
+time (see e.g. [Eve11]).
+5
+Concluding Remarks and Future Research
+We studied the Cantor-Kronecker game for different values of m and n: when m ≤ n the trivial
+lower bound of m is tight (a lower bound of m follows because Cantor must query at least one
+bit in each vector); when m ≥ 2n, the trivial upper bound of mn is tight (an upper bound of
+mn follows because querying all the bits is clearly sufficient); when n < m < 2n the landscape
+is more interesting, and in particular the bounds depend on whether Cantor is adaptive or
+oblivious.
+Further Research.
+We conclude with suggestions for possible future research:
+1. Study the Cantor-Kronecker game when there are r rounds of adaptivity: i.e. there are
+r rounds in which Cantor can submit queries, and in each round the submitted queries
+may depend on Kronecker’s answers to queries from previous rounds. How does the query
+complexity change as a function of r? Note that r = 1 is the oblivious case and r = ∞ is
+the adaptive case. (In fact r = n is already equivalent to r = ∞.)
+2. Consider the following generalization of the game. Let k ≤ m, ℓ ≤ n be positive integers.
+Kronecker maintains an m×n binary matrix, and Cantor queries the entries of Kronecker’s
+matrix. Cantor’s goal is to find a k × ℓ matrix which does not appear as a submatrix of
+Kronecker’s m × n matrix, or to decide that one does not exist. So, the original game
+10
+
+is when k = 1, ℓ = n. What is the query complexity as a function of k, ℓ, m, n in the
+adaptive/oblivious case? For which values does Cantor have a strategy that uses strictly
+less than m · n queries?
+3. Find tighter bounds for the oblivious case. Specifically, notice that Cantor’s original
+diagonalization provides tight bound on the number of queries needed for the oblivious
+case when m ≤ n. It will be interesting to derive tight bounds and optimal strategies in
+the remaining cases. As we exemplify below, this question has connections with natural
+combinatorial problems.
+Consider the case when m is at the other end of the scale, namely 2n−1 ≤ m < 2n. Then,
+Cantor can win the game by querying nm − d bits, where d = 2n − m − 1. In fact, it
+suffices that Cantor chooses his queries such that each of the d unqueried entries belongs
+to a different vector: in this case any assignments of values to the unqueried entries covers
+(in the sense of Definition 3.3) the m − d fully queried vectors, and at most two additional
+vectors per each of the remaining d vectors (each of which contains one unqueried entry):
+altogether at most (m − d) + 2d = m + d vectors. Hence, Cantor is guaranteed to win the
+game provided that m + d < 2n (equivalently d ≤ 2n − m − 1).
+Is the above strategy optimal? i.e., can Kronecker win the game when Cantor queries only
+mn − (2n − m) bits? Informally, Kronecker has a winning strategy if, for any distribution
+of the 2n − m unqueried entries, there is an assignment which covers sufficiently many
+vectors. This is formalized below.
+Definition 5.1 (cube(v), J-cube). Let v be a vector with possibly some unqueried entries.
+cube(v) is the set of binary vectors which can be obtained by replacing the unqueried
+entries in v by zeros or ones. In particular, cube(v) = {v} if v is fully queried. The cube
+cube(v) is called a J-cube if J = {j : the j′th bit of v is not queried}. For j ∈ [n], a
+{j}-cube is denoted by j-edge.
+Assume that Cantor distributes the (2n − m) unqueried entries among vectors v1, . . . , vq.
+Then Kronecker answers to the queried entries define a cube C(vi) for each vector vi.
+Kronecker wins if and only if those cubes cover {0, 1}n. Hence Kronecker has a winning
+strategy when Cantor uses mn − (2n − m) queries (2n−1 + 1 ≤ m < 2n) if and only if the
+following holds:
+Conjecture 5.2. Let d = 2n − m < 2n−1. For any collection J1, J2, . . . , Jq of nonempty
+subsets of [n] satisfying �q
+i=1 |Ji| = d, there are cubes C1, . . . , Cq s.t. Ci is a Ji-cube, and
+|�q
+i=1 Ci| ≥ d + q.
+The following result of [FHK93] proves Conjecture 5.2 for the case that each Ji-cube is a
+ji-edge.
+Theorem 5.3 ([FHK93]). Let d < 2n−1. For any multiset D = {j1, j2, . . . , jd} of elements
+of [n], {0, 1}n contains a matching {e1, . . . , ed} s.t. for i = 1, . . . , d, ei is a ji-edge.
+It is also shown in [FHK93] that Conjcture 5.2 does not hold when d = 2n−1: in this case
+a corresponding matching exists if and only if each element in [n] occurs an even number
+of times in D. This implies that when m = 2n−1 Cantor has a winning strategy with only
+mn − (2n − m) = mn − 2n−1 queries: he may query n − 1 entries per each vector, so that
+at least one dimension is left unqueried in an odd number of vectors.
+11
+
+References
+[Can74]
+Georg Cantor. Ueber eine Eigenschaft des inbegriffs aller reellen algebraischen Zahlen.
+Journal f¨ur die reine und angewandte Mathematik (Crelles Journal), 1(77):258–262,
+1874.
+[Eve11]
+Shimon Even. Graph Algorithms. Cambridge University Press, New York, NY, USA,
+2nd edition, 2011.
+[FHK93] Alexander Felzenbaum, Ron Holzman, and Daniel J. Kleitman. Packing lines in a
+hypercube. Discrete Mathematics, 117(1):107–112, 1993.
+[Wik22a] Wikipedia contributors. Georg Cantor — Wikipedia, the free encyclopedia. https:
+//en.wikipedia.org/wiki/Georg_Cantor, 2022. [Online; accessed 20-November-
+2022].
+[Wik22b] Wikipedia contributors. Hilbert’s paradox of the Grand Hotel — Wikipedia, the
+free encyclopedia. https://en.wikipedia.org/wiki/Hilbert’s_paradox_of_the_
+Grand_Hotel, 2022. [Online; accessed 20-November-2022].
+12
+

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+page_content=' Technion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
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+page_content=' Aarhus University  5Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Princeton University  6Google Research  January 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' 2023  Abstract  We study several variants of a combinatorial game which is based on Cantor’s diagonal  argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' The game is between two players called Kronecker and Cantor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' The names of the  players are motivated by the known fact that Leopold Kronecker did not appreciate Georg  Cantor’s arguments about the infinite, and even referred to him as a “scientific charlatan”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  In the game Kronecker maintains a list of m binary vectors, each of length n, and Cantor’s  goal is to produce a new binary vector which is different from each of Kronecker’s vectors, or  prove that no such vector exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Cantor does not see Kronecker’s vectors but he is allowed  to ask queries of the form  “What is bit number j of vector number i?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  What is the minimal number of queries with which Cantor can achieve his goal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' How much  better can Cantor do if he is allowed to pick his queries adaptively, based on Kronecker’s  previous replies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  The case when m = n is solved by diagonalization using n (non-adaptive) queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' We  study this game more generally, and prove an optimal bound in the adaptive case and nearly  tight upper and lower bounds in the non-adaptive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  1  Introduction  The concept of infinity has been fascinating philosophers and scientists for hundreds, perhaps  thousands of years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' The work of Georg Cantor (1845 – 1918) played a pivotal role in the  mathematical treatment of the infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Cantor’s work is based on a simple notion which asserts  that two (possibly infinite) sets have the same size whenever their elements can be paired  in one-to-one correspondence with each other [Can74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Despite being simple, this notion has  counter-intuitive implications: for example, a set can have the same size as a proper subset of it1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  this phenomena is nicely illustrated by Hilbert’s paradox of the Grand Hotel, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' [Wik22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  This simple notion led Cantor to develop his theory of sets, which forms the basis of modern  mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Alas, Cantor’s set theory was controversial at the start, and only later became  widely accepted:  1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' the natural numbers and the even numbers, via the correspondence “n �→ 2n”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='01924v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='CO]  5 Jan 2023  Figure 1: Georg Cantor (1845 – 1918)  Figure 2: Leopold Kronecker (1823 – 1891)  The objections to Cantor’s work were occasionally fierce: Leopold Kronecker’s public opposition  and personal attacks included describing Cantor as a ”scientific charlatan”, a ”renegade” and a  ”corrupter of youth”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Kronecker objected to Cantor’s proofs that the algebraic numbers are  countable, and that the transcendental numbers are uncountable, results now included in a  standard mathematics curriculum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' [Wik22a]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='1  Diagonalization  One of the most basic and compelling results in set theory is that not all infinite sets have the same  size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' To prove this result, Cantor came up with a beautiful argument, called diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  This argument is routinely taught in introductory classes to mathematics, and is typically  presented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Let N denote the set of natural numbers and let {0, 1}N denote the set of  all infinite binary vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Clearly both sets are infinite, but it turns out that they do not have  the same size: assume towards contradiction that there is a one-to-one correspondence j �→ vj,  where vj = (vj(1), vj(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=') is the infinite binary vector corresponding to j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Define a vector  u = (1 − v1(1), 1 − v2(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  That is, u is formed by letting its j’th entry be equal to the negation of the j’th entry of vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Notice that this way the resulting vector u disagrees with vj on the j’th entry, and hence  u ̸= vj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Thus, we obtain a binary vector which does not correspond to any of the natural  numbers via the assumed correspondence – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Rather than reaching a contradiction, it is instructive to take a positivist perspective according  to which diagonalization can be seen as a constructive procedure that does the following:  Given binary vectors v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=', find a binary vector u such that u ̸= vj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Moreover, notice that Cantor’s diagonal argument involves querying only a single entry per  each of the input vectors vj (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' the “diagonal” entries vj(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Thus, it is possible to construct u  while using only a little information about the input vectors vi’s (a single bit per vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  In this manuscript we study a finite variant of the problem in which m binary vectors  v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , vm of length n are given and the goal is to produce a vector u which is different from all  of the vi’s, or to report that no such vector exists, while querying as few as possible entries of  the vi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' We first study the case when m < 2n whence such a u is guaranteed to exist, and the  goal boils down to finding one, and later the case when m ≥ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  2  v1 = 0, 1, 1, 0, 1, 0  v2 = 1, 0, 0, 1, 1, 1  v3 = 1, 1, 1, 0, 0, 0  v4 = 0, 1, 0, 1, 1, 0  v5 = 1, 1, 0, 1, 0, 1  v6 = 0, 1, 1, 1, 1, 1  u = 1, 1, 0, 0, 1, 0  Figure 3: An illustration of Cantor’s diagonalization: the vector u at the bottom is not equal to  any of the vi’s at the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  2  The Cantor-Kronecker Game  Consider a game between two players called Kronecker and Cantor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' In the game there are  two parameters m and n, where m, n are positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Kronecker maintains a set V =  {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , vm} of m binary vectors, each of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Cantor’s goal is to produce a binary  vector u, also of length n, which differs from each vi, or to report that no such vector exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' To  do so, he is allowed to ask queries, where each query is of the form  “What is bit number j of vector number i?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=',  where 1 ≤ j ≤ n, 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Kronecker is answering each query being asked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' The objective  of Cantor is to minimize the number of queries enabling him to produce u, whereas Kronecker  tries to maximize the number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' We distinguish between two versions of the game:  In the adaptive version Cantor presents his queries to Kronecker in a sequential manner,  and may decide on the next query as a function of Kronecker’s answers to the previous  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  In the oblivious version Cantor must declare all of his queries in advance, before getting  answers to any of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  For m ≤ n the smallest number of queries, both in the adaptive and oblivious versions, is m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Indeed, Cantor can query bit number i of vi for all 1 ≤ i ≤ m and return a vector u whose i’th  bit differs from the i’th bit of vi, for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' The lower bound is even simpler: if Cantor asks less  than m queries then there is some vector vi about which he has no information at the end of the  game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' In this case he cannot ensure that his vector u will not be equal to this vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  We begin with the case where m < 2n: in the next section (Section 3) we  derive nearly tight bounds both in the adaptive and oblivious cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' We do so by exhibiting and  analyzing near optimal strategies for Cantor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Then, in Section 4 we consider the case where  m ≥ 2n and derive an optimal bound of m·n in this case (for both the oblivious and the adaptive  versions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' We do so by exhibiting and analyzing an optimal strategy for Kronecker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Finally, in  Section 5 we discuss some algorithmic aspects, and conclude with some suggestions to future  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  3  3  The Cantor-Kronecker Game with m < 2n  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='1  Adaptive Version  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Let g(n, m) denote the smallest number of queries that suffices for Cantor when  he is allowed to use adaptive strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Then,  g(n, m) =  �  m  m ≤ n,  2m − n  n < m < 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  The case 1 ≤ m ≤ n is proved in the previous section so we assume n ≤ m < 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Upper Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  We present a strategy for Cantor which combines diagonalization with another  simple idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' To illustrate this idea let us first consider the case m = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' This special case  appeared as a question in the 2022 Grossman Math Olympiad for high-school students, and so  perhaps the reader might enjoy trying to solve it before continuing reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Let v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , vn+1 be the input vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Cantor begins with querying the first bit of v1, v2, and  of v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Getting the answers, there is a bit ε so that at least two vectors among v1, v2, v3 have  their first bit equals to ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Cantor now defines the first bit of u to be u(1) = 1 − ε and can remove  the two vectors among v1, v2, v3 whose first bit equals ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Now Cantor is left with at most n − 1  vectors and can therefore set the last n − 1 coordinates of u according to the diagonalization  construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  The general case is handled similarly by induction on n: for n = 1 since n ≤ m < 2n, also m  must be 1 and the result is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Assuming the result for n − 1, let v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , vm be the m vectors of Kronecker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' First, note that  there is an integer x satisfying 1 ≤ x ≤ ⌈m/2⌉ so that n − 1 ≤ m − x < 2n−1: indeed, for x = 1  we have m − x ≥ n − 1 and for x = ⌈m/2⌉ we have m − x ≤ m/2 < 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Starting from x = 1  keep increasing it in steps, where in each step it is increased by 1, until it reaches ⌈m/2⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' As  m − x changes by 1 in each step we can take the smallest x ≥ 1 that satisfies m − x < 2n−1, and  it will clearly be at most ⌈m/2⌉ and satisfy m − x ≥ n − 1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Having x as above, Cantor first queries the first bit of each of the vectors v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , v2x−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  (Note that 2x − 1 ≤ m hence this is possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Getting the answers, there is a bit ε ∈ {0, 1} so  that at least x of the vectors have their first bit equal to ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Cantor now defines the first bit of  his vector u to be 1 − ε, removes from the set V exactly x of the vectors whose first bit is ε,  and defines as V ′ the set of all restrictions of the remaining m − x vectors to their last n − 1  coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Note that n − 1 ≤ m − x < 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  By the induction hypothesis, Cantor can now play the game for the set V ′ producing an  appropriate vector u′ by asking at most 2(m − x) − (n − 1) additional queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' The total number  of queries is thus (2x − 1) + 2(m − x) − (n − 1) = 2m − n, as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' The vector u obtained  by concatenating the 1-bit vector 1 − ε and the vector u′ is clearly different from each member  of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' This completes the induction step argument and finishes the proof of the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Lower Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  For the lower bound, we present a strategy for Kronecker which essentially  mirrors Cantor’s strategy from the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Suppose Cantor manages to produce the  required vector u after making exactly bj queries in coordinate number j of some of the vectors vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Kronecker chooses his answers ensuring that for each such j, the answers for bits in the j’th  location are balanced, that is, at most ⌈bj/2⌉ of the answers are 0 and at most ⌈bj/2⌉ of the  answers are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Consider the vector u produced by Cantor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' For every 1 ≤ j ≤ n, there are at most ⌈bj/2⌉  vectors vi known to be different than u in coordinate number j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Thus altogether there are at  4  most  n  �  j=1  �bj  2  �  ≤  n  �  j=1  bj + 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  vectors vi that are known to Cantor to be different than u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' In order to ensure u is indeed  different from each vi this number has to be at least m and hence  m ≤  n  �  j=1  bj + 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  By rearranging, this implies that the total number of queries �n  j=1 bj must be at least 2m − n,  as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='2  Oblivious Version  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Let f(n, m) denote the smallest number of queries that suffices for Cantor when  he is restricted to use oblivious strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Then,  f(n, m) =  �  m  m ≤ n  m  �  log  � m  n  �  + o  �  log  � m  n  ���  n < m < 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Quantitatively, for all n < m < 2n  m ·  �  log  �  m  n − log m + 1  �  − 1  �  ≤ f(n, m) ≤ m  �  log  �2m  n  �  + 2 log  �  log  �2m  n  ��  + 1  �  ,  The case 1 ≤ m ≤ n is proved above so we assume n < m < 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Upper Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Like in the adaptive case, we present a strategy for Cantor which combines  diagonalization with another simple idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' We first illustrate this idea by handling the case m =  n + 1, and again, we encourage the reader to try and handle this case before continuing reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Let v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , vn+1 be the input vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Cantor begins with querying the first two bits of each  of v1, v2, and v3 (for a total of 6 queries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Notice that there are 22 = 4 possible combinations of  0/1 patterns on the first two bits, but at most three of them are realized by v1, v2, v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Hence,  there must be a pair of bits ε1, ε2 which is not realized by v1, v2, nor v3:  (ε1, ε2) /∈  ��  v1(1), v1(2)  �  ,  �  v2(1), v2(2)  �  ,  �  v3(1), v3(2)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Thus, by setting u(1) = ε1 and u(2) = ε2, Cantor rules out v1, v2, v3 and is left with n−2 vectors  v3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , vn+1 which can be obliviously ruled out with the last n − 2 using diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  For the general case, let d be an integer (to be determined later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Pick mutually disjoint  subsets of coordinates J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , J⌊n/d⌋ ⊆ [n], each of size d, and pick a partition of the m vectors to  ⌊n/d⌋ subsets V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , V⌊n/d⌋ such that the partition is as balanced as possible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' the difference  between each pair of sizes is ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Thus, each set has size  |Vi| ≤  �  m  ⌊n/d⌋  �  ≤ 2md  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Cantor queries (obliviously) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  For each i and each vector in Vi query all the coordinates in Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  5  Thus, the total number of queries is exactly m · d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Now, notice that if d satisfies  2d > 2md  n ,  (1)  then there must exist an assignment fi : Ji → {0, 1} such that fi disagrees with each of the  vectors in Vi on at least one coordinate in Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Hence Cantor can output the vector u, which  agrees with each of the fi on Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Note that Equation 1 is satisfied iff 2d  d > 2m  n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' since m > n,  it can be verified that this inequality holds when d ≥ log( 2m  n ) + 2 log(log( 2m  n )) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Thus for  d =  �  log  � 2m  n  �  + 2 log  �  log  � 2m  n  ��  + 1  �  , the total number of queries is at most  m · d = m  �  log  �2m  n  �  + 2 log  �  log  �2m  n  ��  + 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Lower Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  The lower bound proof is based on the following simple idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Let Ji denote  the set of coordinates of vi which Cantor queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Thus, the total number of queries Cantor uses  is |J1| + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' + |Jm|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Now, let fi : Ji → {0, 1} denote Kronecker’s answers for the queries on vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  The crucial observation is that the vector u that Cantor outputs must satisfy  (∀i) : u|Ji ̸= fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Indeed, if u|Ji = fi for some i then Kronecker can fail Cantor by picking his i’th vector vi to be  equal to Cantor’s output u (which would be consistent with Kronecker’s answers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  We summarize the above consideration with a definition that characterizes the winning (or  losing) strategies of Cantor in the oblivious case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='3 (Covering Assignments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' We say that a sequence of sets J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , Jm ⊆ [n]  has a covering assignment if there are m functions fi : Ji → {0, 1} such that every binary  vector v ∈ {0, 1}n agrees with one of the fi on Ji (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' v|Ji = fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Thus, Kronecker has a winning strategy if and only if the sequence of sets J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , Jm that  Cantor queries has a covering assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' The following lemma establishes the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Let J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , Jm ⊆ [n] such that  |J1| + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' + |Jm| < m ·  �  log  �  m  n − log m + 1  �  − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  (2)  Then, J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , Jm has a covering assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Equivalently, if for each vector vi Cantor queries its entries in Ji and Equation 2 holds, then  Kronecker has a winning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Let ti = |Ji| and let t = �  i ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Assume, without loss of generality, that t1 ≤ t2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' ≤ tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  To prove a lower bound of the form md for t, where d will be specified later, we show that if t is  smaller than md then there are m functions fi : Ji → {0, 1} so that for every possible vector  v ∈ {0, 1}n there is i ≤ m so that v|Ji = fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  We do so by explicitly constructing the fi’s (which corresponds to describing a winning  strategy for Kronecker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Starting with the set V = {0, 1}n of all possible potential vectors z,  go over the vectors vi in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' In step i we choose the function fi : Ji → {0, 1} such that  |{v ∈ V : v|Ji = fi}| is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Since there are 2ti possible choices for fi, the maximizing  choice satisfies  ���{v ∈ V : v|Ji = fi}  ��� ≥ |V |  2ti .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  6  After picking fi, we remove all the vectors of V that agree with fi and proceed to the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Therefore, after the first i steps, the size of the set V of the remaining vectors is at most  2n  i�  j=1  (1 − 1/2tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  We can continue with this analysis until the size of the set V becomes smaller than 1, namely  the set becomes empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' It is a bit better, however, to apply a simpler reasoning once the size  of V becomes smaller than 2d, and only argue that at least one vector from V is eliminated  in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' (Continuing the same analysis as before would only guarantee that V shrinks by  a factor of (1 − 1/2ti) which by the choice of d would be roughly 1 − 1/2d < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' To simplify  the computation it is not too wasteful to apply the simpler analysis already when the size of  V becomes smaller than m/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' If this happens in the first m/2 steps then by removing a single  vector in each of the remaining steps we will eliminate all of the vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' This means that if  2n  m/2  �  j=1  �  1 − 1/2tj�  ≤ m  2  then the sequence J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , Jm has a covering assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Since d is such that the total number of  queries is m · d, the above amounts to �m/2  j=1 tj ≤ md/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' that is, the average tj for 1 ≤ j ≤ m/2  is at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' This implies that  2n  m  2  �  j=1  �  1 − 1  2tj  �  ≤ 2n  m  2  �  j=1  exp  �  − 1  2tj  �  (1 + x ≤ exp(x) for all x ∈ R)  = 2n exp  �  −  m  2  �  j=1  1  2tj  �  ≤ 2n exp  �  − m  2d+1  �  ,  where the last inequality follows because exp(−x) is decreasing and because  m  2  �  j=1  1  2tj ≥ m  2 ·  1  2  1  m/2  �m/2  j=1 tj  ≥ m  2 · 1  2d ,  which follows by convexity of the function f(x) = 2x and because t1 ≤ t2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' ≤ tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  We have thus shown that if |J1| + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' + |Jm| = m · d such that  2n exp  �  − m  2d+1  �  ≤ m  2  then the sequence J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , Jm has a covering assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' The last inequality surely holds provided  m  2d+1 ≥ n + 1 − log m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  That is, provided  2d+1 ≤  m  n + 1 − log m,  or  d ≤ log  �  m  n + 1 − log m  �  − 1  completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  7  4  The Cantor-Kronecker Game with m ≥ 2n  Assume now that Kronecker’s list V consists of m ≥ 2n binary vectors of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' In this case  V may contain all the binary vectors of length n and there is no vector Cantor can output that  is different from each vector on Kronecker’s list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' In this regime it is more natural to first focus  on the decision problem in which Cantor’s goal is to decide whether V contains {0, 1}n, and if  this is not the case, to provide a vector which is not in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='2 Clearly Cantor can achieve this if he  queries all mn possible queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Can he do better?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  We first observe that mn queries are in fact needed in the oblivious case: assume that Cantor  submits only mn − 1 queries, and leaves the j’th bit of vi unqueried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Then Kronecker may set vi  to be the unique occurrence of the all ones vector 1n, and set the remaining m − 1 vectors in V  to include all 2n − 1 vectors that are different from the all ones vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Clearly, it is necessary  for Cantor to query also the last bit of vi in order to see whether vi is the all ones vector or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Consequently, Cantor must query all mn queries in the oblivious case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  How about the adaptive case?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' A similar argument shows that for m = 2n, Kronecker can  force mn = 2nn queries also in the adaptive case, by using a list which contains each binary  vector of length n exactly once: indeed, if only mn − 1 bits are queried, then the last, yet  unqueried bit, belongs to a vector which occurs only once in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Hence it is necessary to get the  value of this bit in order to verify that V contains all 2n vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  The case when m > 2n turns out to be more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Nevertheless, we prove that mn queries  are necessary even in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' We start with introducing some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Each step of the game consists of a query by Cantor followed by a response by  Kronecker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' The status of the game after each such step is given by an m × n matrix L, where  L(i, j) denotes the status of the j’th bit of vi, that is: L(i, j) ∈ {0, 1, ⋆}, where L(i, j) = ⋆ means  that the j’th bit of vi was not queried yet, and otherwise L(i, j) equals the value of this bit as  answered by Kronecker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' FIXED(L) =  �  v ∈ L : v ∈ {0, 1}n�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' That is, FIXED(L) is the set of all vectors  in L that were fully queried by Cantor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' L is complete if FIXED(L) = {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' A subset S of 2n rows of L is useful if it either contains all the 2n binary vectors  of length n, or it can be converted to this set by replacing each ⋆-entry in S by 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' A matrix L is unblocked if it can be completed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' that is, if L has a useful subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Otherwise L is called blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Notice that for m ≥ 2n, the m by n matrix all whose entries are ⋆ is unblocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  As a warmup, and to get used to the definitions, let us assume first that Cantor’s queries  the vectors one by one according to their order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' he first queries all the bits of v1 from left  to right, then all the bits of v2 from left to right, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' We use the following strategy for  Kronecker: when Cantor queries the j’th bit of vi (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' the value of L(i, j)), Kronecker replies  according to the following “0 first” strategy:  modified value of L(i, j) =  �  1  If setting L(i, j) to 0 blocks L  0  otherwise  (3)  It is not hard to verify that since Cantor queries the vectors one by one, and from left (most  significant bit) to right, the following matrix is produced: each of the first m − 2n + 1 rows will  2Later we will see that the decision and search variants are in fact equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  8  be set to the all-zeros vector, and the last 2n − 1 rows will be set to the 2n − 1 non zero vectors  in increasing lexicographical order: starting with 0n−11 and ending with 1n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Hence Cantor is  forced to query all mn entries as in the oblivious case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Interestingly, it turns out that, for any strategy of Cantor, the above “0 first” strategy of  Kronecker forces Cantor to make mn queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Let m > 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Then for any strategy of Cantor, the “0 first” strategy of Kronecker  forces Cantor to make mn queries in order to determine if L contains {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  In the following we consider an arbitrary execution of the game, where Kronecker follows the  “0 first” strategy (and Cantor’s strategy is arbitrary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' We denote by Lt the m × n matrix L after  t steps of the game;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' thus L0 is the initial matrix which is filled only with ⋆’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  By the fact that if L is unblocked and L(i, j) = ⋆, then it is possible to set L(i, j) to 0 or  to 1 without blocking L, we get:  Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' If Lt is unblocked, so is Lt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Hence Lmn is complete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' it contains {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' We say that a row L(i) is essential for an unblocked matrix L if every useful  subset of L’s rows contains L(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Note that if Lt(i) is essential for Lt, then Ls(i) is essential for Ls for all s ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Also, if Lmn(i)  is essential for Lmn, then Lmn(i) is equal to a unique vector in {0, 1}n which is different from all  other rows of Lmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Assume that Lt(i) is not essential for Lt and Lt(i, j) = ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' If Lt(i, j) is queried at  time t + 1, then it is set to 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Lt+1(i, j) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' By the “0 first” strategy, and the fact that if L(i) is not essential for an unblocked  matrix L, then setting L(i, j) to 0 does not block L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  By a straightforwards induction Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='8 implies:  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' If Lt(i) is not essential for Lt, then Lt(i) contains no 1’s (only 0’s or ⋆’s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Specifically, if Lmn(i) is not essential for Lmn, then Lmn(i) is the zero vector 0n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Hence, every  row of Lmn which is not the zero vector is essential, and thus it is different from all other rows  of Lmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Let Lmn−1(i, j) be the last bit queried in the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Then Lmn−1(i) is an essential  row of Lmn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' To simplify notation, we assume without loss of generality that j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Assume towards  contradiction that Lmn−1(i) is not essential for Lmn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='9, this implies that  Lmn−1(i) = ⋆0n−1 and Lmn(i) = 0n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Kronecker sets Lmn−1(i, 1) to 0 at Cantor’s mn’th  query).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Since Lmn is complete (Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='6), this implies that Lmn−1 contains a distinct  occurrence of each of the 2n − 1 nonzero vectors of {0, 1}n, and in particular for some k ̸= i,  Lmn−1(k) is the unique row of Lmn−1 which equals 10n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Then, any subset S of Lmn−1 which  contains  the row Lmn−1(i),  the 2n − 2 non zero rows of Lmn−1 excluding Lmn−1(k), and  some zero row of Lmn−1 (by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='9 there are m − 2n > 0 such rows in Lmn−1),  9  is a useful subset of Lmn−1 which does not contain Lmn−1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Hence Lmn−1(k) is not essential  for Lmn−1, and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='8 Lmn−1(1) = 0 ̸= 1, which stands in contradiction with Lmn−1(1) =  10n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Let Lmn−1(i, j) be the last query in the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='10, Lmn−1(i),  and hence also Lmn(i), is essential, meaning that Lmn(i) is different from all other rows of Lmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Thus Cantor must get the value of Lmn−1(i, j) in order to reach a decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  A remark on computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  A naive implementation of the “0 first” strategy  might take exponential time: indeed, it requires checking whether setting the queried bit to 0  blocks the current matrix, which involves checking a potentially exponential list of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Nevertheless, we next show that this strategy in fact admits a polynomial time implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Firstly, notice that the first m − 2n steps are trivially efficient, because setting L(i, j) to any  value cannot block L (since at least 2n rows of L are not queried yet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Thus it suffices to show that in each later step, deciding whether setting L(i, j) to 0 blocks  the matrix, can be performed in time which is polynomial in mn, the size of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Let Lt be the  matrix L after t steps of the game, t > m − 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Consider the bipartite graph Gt = (At, B, Et),  where At = {Lt(i) : 1 ≤ i ≤ m} is the set of rows of Lt, B = {0, 1}n, and (Lt(i), u) ∈ Et if  and only if Lt(i) can be converted to the binary vector u by replacing the ⋆’s in Lt(i) (if any)  by binary digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Then, a subset S of Lt is useful for Lt if and only if Gt contains a perfect  matching between the vertices in At which correspond to S and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Assume now that we are given the graph Gt, and the corresponding matching, and let Lt(i, j)  be the entry queried by Cantor at step t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' To check if setting Lt(i, j) to 0 blocks Lt, we remove  from Gt all the edges (Lt(i), u) in which u(j) = 0, and check if the resulted graph contains a  perfect matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Since we are given a perfect matching Mt for Gt, and removing these edges  eliminates at most one edge from Mt, this checking can be done by executing one phase in some  classical algorithm for bipartite matching, which can be done in O(|Et|) = O(m2n) = O(m2)  time (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' [Eve11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  5  Concluding Remarks and Future Research  We studied the Cantor-Kronecker game for different values of m and n: when m ≤ n the trivial  lower bound of m is tight (a lower bound of m follows because Cantor must query at least one  bit in each vector);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' when m ≥ 2n, the trivial upper bound of mn is tight (an upper bound of  mn follows because querying all the bits is clearly sufficient);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' when n < m < 2n the landscape  is more interesting, and in particular the bounds depend on whether Cantor is adaptive or  oblivious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Further Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  We conclude with suggestions for possible future research:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Study the Cantor-Kronecker game when there are r rounds of adaptivity: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' there are  r rounds in which Cantor can submit queries, and in each round the submitted queries  may depend on Kronecker’s answers to queries from previous rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' How does the query  complexity change as a function of r?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Note that r = 1 is the oblivious case and r = ∞ is  the adaptive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' (In fact r = n is already equivalent to r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=')  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Consider the following generalization of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Let k ≤ m, ℓ ≤ n be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Kronecker maintains an m×n binary matrix, and Cantor queries the entries of Kronecker’s  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Cantor’s goal is to find a k × ℓ matrix which does not appear as a submatrix of  Kronecker’s m × n matrix, or to decide that one does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' So, the original game  10  is when k = 1, ℓ = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' What is the query complexity as a function of k, ℓ, m, n in the  adaptive/oblivious case?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' For which values does Cantor have a strategy that uses strictly  less than m · n queries?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Find tighter bounds for the oblivious case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Specifically, notice that Cantor’s original  diagonalization provides tight bound on the number of queries needed for the oblivious  case when m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' It will be interesting to derive tight bounds and optimal strategies in  the remaining cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' As we exemplify below, this question has connections with natural  combinatorial problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Consider the case when m is at the other end of the scale, namely 2n−1 ≤ m < 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Then,  Cantor can win the game by querying nm − d bits, where d = 2n − m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' In fact, it  suffices that Cantor chooses his queries such that each of the d unqueried entries belongs  to a different vector: in this case any assignments of values to the unqueried entries covers  (in the sense of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='3) the m − d fully queried vectors, and at most two additional  vectors per each of the remaining d vectors (each of which contains one unqueried entry):  altogether at most (m − d) + 2d = m + d vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Hence, Cantor is guaranteed to win the  game provided that m + d < 2n (equivalently d ≤ 2n − m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Is the above strategy optimal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=', can Kronecker win the game when Cantor queries only  mn − (2n − m) bits?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Informally, Kronecker has a winning strategy if, for any distribution  of the 2n − m unqueried entries, there is an assignment which covers sufficiently many  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' This is formalized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='1 (cube(v), J-cube).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Let v be a vector with possibly some unqueried entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  cube(v) is the set of binary vectors which can be obtained by replacing the unqueried  entries in v by zeros or ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' In particular, cube(v) = {v} if v is fully queried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' The cube  cube(v) is called a J-cube if J = {j : the j′th bit of v is not queried}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' For j ∈ [n], a  {j}-cube is denoted by j-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Assume that Cantor distributes the (2n − m) unqueried entries among vectors v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Then Kronecker answers to the queried entries define a cube C(vi) for each vector vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Kronecker wins if and only if those cubes cover {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Hence Kronecker has a winning  strategy when Cantor uses mn − (2n − m) queries (2n−1 + 1 ≤ m < 2n) if and only if the  following holds:  Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Let d = 2n − m < 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' For any collection J1, J2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , Jq of nonempty  subsets of [n] satisfying �q  i=1 |Ji| = d, there are cubes C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , Cq s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Ci is a Ji-cube, and  |�q  i=1 Ci| ≥ d + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  The following result of [FHK93] proves Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='2 for the case that each Ji-cube is a  ji-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='3 ([FHK93]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Let d < 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' For any multiset D = {j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , jd} of elements  of [n], {0, 1}n contains a matching {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , ed} s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' , d, ei is a ji-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  It is also shown in [FHK93] that Conjcture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='2 does not hold when d = 2n−1: in this case  a corresponding matching exists if and only if each element in [n] occurs an even number  of times in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' This implies that when m = 2n−1 Cantor has a winning strategy with only  mn − (2n − m) = mn − 2n−1 queries: he may query n − 1 entries per each vector, so that  at least one dimension is left unqueried in an odd number of vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  11  References  [Can74]  Georg Cantor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Ueber eine Eigenschaft des inbegriffs aller reellen algebraischen Zahlen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  Journal f¨ur die reine und angewandte Mathematik (Crelles Journal), 1(77):258–262,  1874.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  [Eve11]  Shimon Even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Graph Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Cambridge University Press, New York, NY, USA,  2nd edition, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  [FHK93] Alexander Felzenbaum, Ron Holzman, and Daniel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Kleitman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Packing lines in a  hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Discrete Mathematics, 117(1):107–112, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  [Wik22a] Wikipedia contributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Georg Cantor — Wikipedia, the free encyclopedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' https:  //en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='org/wiki/Georg_Cantor, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' accessed 20-November-  2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  [Wik22b] Wikipedia contributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' Hilbert’s paradox of the Grand Hotel — Wikipedia, the  free encyclopedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='org/wiki/Hilbert’s_paradox_of_the_  Grand_Hotel, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content=' accessed 20-November-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}
+page_content='  12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'}

4NE4T4oBgHgl3EQf0w0Q/content/tmp_files/2301.05284v1.pdf.txt

@@ -0,0 +1,1123 @@
+Numerical Study of the Rate of Convergence of Chernoff
+Approximations to Solutions of the Heat Equation
+K.A. Dragunova, A.A. Garashenkova, I.D. Remizov
+Research and educational group “Evolution semigroups and their applications”
+International Laboratory of Dynamical Systems and Applications
+National Research University Higher School of Economics
+Email: [email protected]
+MSC2020: 65M12, 47D06, 35K05, 35E15, 35C99
+Keywords: heat equation, initial value problem, operator semigroups, Chernoff approximations,
+rate of convergence.
+Abstract. Chernoff approximations are a flexible and powerful tool of functional analysis, which
+can be used, in particular, to find numerically approximate solutions of some differential equations with
+variable coefficients. For many classes of equations such approximations have already been constructed,
+however, the speed of their convergence to the exact solution has not been properly studied.
+We
+developed a program in Python 3 that allows to model a wide class of Chernoff approximations to
+a wide class of evolution equations on the real line.
+After that we select the heat equation (with
+already known exact solutions) as a simple yet informative model example for the study of the rate
+of convergence of Chernoff approximations. Examples illustrating the rate of convergence of Chernoff
+approximations to the solution of the Cauchy problem for the heat conduction equation are constructed
+in the paper. Numerically we show that for initial conditions that are smooth enough the order of
+approximation is equal to the order of Chernoff tangency of the Chernoff function used.
+We also
+consider not smooth enough initial conditions and show how H¨older class of initial condition is related
+to the rate of convergence. This method of study can be applied to general second order parabolic
+equation with variable coefficients by a slight modification of our Python 3 code, the full text of it is
+provided in the appendix to the paper.
+Contents
+1
+Introduction
+2
+2
+Preliminaries
+2
+3
+Numerical simulation results
+4
+3.1
+Problem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+3.2
+Approximations for initial condition u0(x) = sin(x) . . . . . . . . . . . . . . . . . . . .
+6
+3.3
+Approximations for initial condition u0(x) = | sin(x)|3/2
+. . . . . . . . . . . . . . . . .
+7
+3.4
+Approximations for initial condition u0(x) = e−|x| . . . . . . . . . . . . . . . . . . . . .
+8
+3.5
+Approximations for initial condition u0(x) = | sin(x)|5/2
+. . . . . . . . . . . . . . . . .
+9
+3.6
+Approximations for initial condition u0(x) = | sin(x)|7/2
+. . . . . . . . . . . . . . . . .
+9
+3.7
+Approximations for initial condition u0(x) = | sin(x)|9/2
+. . . . . . . . . . . . . . . . .
+10
+4
+Discussion
+11
+Appendix: Python 3 code
+13
+References
+14
+1
+arXiv:2301.05284v1  [math.NA]  12 Jan 2023
+
+1
+Introduction
+Chernoff approximations are a flexible and powerful tool of functional analysis [3, 4, 5], which
+can be used, in particular, to find numerically approximate solutions of some differential equa-
+tions with variable coefficients, see [2, 14] for an introduction to this topic, and also Preliminaries
+section of the paper. For given linear evolution equation the method of Chernoff approximation
+generates a sequence of functions un(t, x) that converge to the exact solution u(t, x) of the equa-
+tion studied. For arbitrary fixed moment of time t functions x �−→ u(t, x) and x �−→ un(t, x) are
+elements of some Banach space, and Chernoff’s theorem guarantees that ∥u(t, ·)−un(t, ·)∥ → 0
+as n → ∞.
+To our current knowledge all contributions to a very young “theory of rates of convergence
+in Chernoff’s theorem” can be found in [8, 19, 20, 7, 6] and references therein. These papers
+provide estimates for the rate of convergence under some conditions but if these conditions are
+not satisfied then one can say nothing about the quality of Chernoff approximations. There
+are also very few “practical” research papers [10, 16] that measure the speed of convergence in
+particular cases obtained via numerical simulations. In our research we continue contributions
+to this field of study.
+We consider initial value problem for the heat equation
+� u′
+t(t, x) = u′′
+xx(t, x) for t > 0, x ∈ R1
+u(0, x) = u0(x) for x ∈ R1
+(1)
+which is a good model example because its bounded solution u(t, x) is already known and given
+by the formula
+u(t, x) =
+�
+R
+Φ(x − y, t)u0(y)dy, where Φ(x, t) = (2
+√
+πt)−1 exp
+�−x2
+4t
+�
+.
+Then we obtain Chernoff approximations un(t, x) to the exact solution u(t, x) for n =
+1, 2, . . . , 11 and fixed time t = 1/2, and via numerical simulation and linear regression (ordinary
+least squares method) discover that
+sup
+x∈R
+|u(t, x) − un(t, x)| ≈
+�1
+n
+�β
+with a reasonable accuracy (R2 > 0.98). Coefficient β > 0 depends on the smoothness of initial
+condition u0 and of the way of constructing the Chernoff approximations.
+P.S.Prudnikov in 2020 studied [16] this question in a similar setting, but his approach does
+not allow a direct generalization. Meanwhile the simulation method that we use allows to study
+not only heat equation, but also equations with variable coefficients. Also we consider more
+initial conditions than were studied in [16].
+Now let us provide necessary background on the topic to explain the notion of Chernoff
+tangency and Chernoff operator-valued function that are important to understand how we
+obtain Chernoff approximations un(t, x).
+2
+Preliminaries
+Let F be a Banach space. Let L (F) be a set of all bounded linear operators in F. Suppose we
+have a mapping V : [0, +∞) → L (F), i.e. V (t) is a bounded linear operator V (t): F → F for
+each t ≥ 0. The mapping V is called [5] a C0-semigroup, or a strongly continuous one-parameter
+semigroup of operators iff it satisfies the following conditions:
+1) V (0) is the identity operator I, i.e. ∀ϕ ∈ F : V (0)ϕ = ϕ;
+2
+
+2) V maps the addition of numbers in [0, +∞) into the composition of operators in L (F),
+i.e. ∀t ≥ 0, ∀s ≥ 0 : V (t + s) = V (t) ◦ V (s), where for each ϕ ∈ F the notation (A ◦ B)(ϕ) =
+A(B(ϕ)) = ABϕ is used;
+3) V is continuous with respect to the strong operator topology in L (F), i.e. ∀ϕ ∈ F
+function t �−→ V (t)ϕ is continuous as a mapping [0, +∞) → F.
+The definition of a C0-group is obtained by the substitution of [0, +∞) by R in the paragraph
+above.
+It is known [5] that if (V (t))t≥0 is a C0-semigroup in Banach space F, then the set
+�
+ϕ ∈ F : ∃ lim
+t→+0
+V (t)ϕ − ϕ
+t
+�
+denote
+=
+Dom(L)
+is a dense linear subspace in F. The operator L defined on the domain Dom(L) by the equality
+Lϕ = lim
+t→+0
+V (t)ϕ − ϕ
+t
+is called an infinitesimal generator (or just generator to make it shorter) of the C0-semigroup
+(V (t))t≥0, and notation V (t) = etL is widely used.
+One of the reasons for the study of C0-semigroups is their connection with differential
+equations. If Q is a set, then the function u: [0, +∞) × Q → R, u: (t, x) �−→ u(t, x) of two
+variables (t, x) can be considered as a function u: t �−→ [x �−→ u(t, x)] of one variable t with
+values in the space of functions of the variable x. If u(t, ·) ∈ F then one can define Lu(t, x) =
+(Lu(t, ·))(x). If there exists a C0-semigroup (etL)t≥0 then the Cauchy problem for a linear
+evolution equation
+� u′
+t(t, x) = Lu(t, x) for t > 0, x ∈ Q
+u(0, x) = u0(x) for x ∈ Q
+(2)
+has a unique (in sense of F, where u(t, ·) ∈ F for every t ≥ 0) solution
+u(t, x) = (etLu0)(x)
+depending on u0 continuously. Compare also different meanings of the solution [5], including
+mild solution which solves the corresponding integral equation. Note that if there exists a
+strongly continuous group (etL)t∈R then in the Cauchy problem the equation u′
+t(t, x) = Lu(t, x)
+can be considered not only for t > 0, but for t ∈ R, and the solution is provided by the same
+formula u(t, x) = (etLu0)(x).
+Definition 1 (Introduced in [13]).
+Let us say that C is Chernoff-tangent to L iff the
+following conditions of Chernoff tangency (CT) hold:
+(CT0). Let F be a Banach space, and L (F) be a space of all linear bounded operators
+in F. Suppose that we have an operator-valued function C: [0, +∞) → L (F), or, using other
+words, we have a family (C(t))t≥0 of linear bounded operators in F. Closed linear operator
+L: Dom(L) → F is defined on the linear subspace Dom(L) ⊂ F which is dense in F.
+(CT1) Function t �−→ C(t)f ∈ F is continuous for each f ∈ F.
+(CT2) C(0) = I, i.e. C(0)f = f for each f ∈ F.
+(CT3) There exists such a dense subspace D ⊂ F that for each f ∈ D there exists a limit
+C′(0)f = lim
+t→0
+C(t)f − f
+t
+.
+(CT4) The closure of the operator (C′(0), D) is equal to (L, Dom(L)).
+Remark 1. Let us consider one-dimensional example F = L (F) = R. Then g: [0, +∞) →
+R is Chernoff-tangent to l ∈ R iff g(t) = 1 + tl + o(t) as t → +0.
+Theorem 1 (P. R. Chernoff (1968), see [5, 3]). Let F and L (F) be as above. Suppose
+3
+
+that the operator L: F ⊃ Dom(L) → F is linear and closed, and function C takes values in
+L (F). Suppose that these assumptions are fulfilled:
+(E) There exists a C0-semigroup (etL)t≥0 with the infenitesimal generator (L, Dom(L)).
+(CT) C is Chernoff-tangent to (L, Dom(L)).
+(N) There exists such a number ω ∈ R, that ∥C(t)∥ ≤ eωt for all t ≥ 0.
+Then for each f ∈ F we have (C(t/n))nf → etLf as n → ∞ with respect to norm in F
+uniformly with respect to t ∈ [0, T] for each T > 0, i.e.
+lim
+n→∞ sup
+t∈[0,T]
+��etLf − (C(t/n))nf
+�� = 0.
+Remark 2. In our one-dimensional example (F = L (F) = R) the Chernoff theorem says
+that etl = limn→∞ g(t/n)n = limn→∞(1 + tl/n + o(t/n))n, which is a simple fact of calculus.
+Definition 2. Let F, L (F), L be as above. If C is Chernoff-tangent to L and the equation
+limn→∞ supt∈[0,T]
+��etLf − (C(t/n))nf
+�� = 0 holds, then C is called a Chernoff function for the
+operator L, and the (C(t/n))nf is called a Chernoff approximation expression to etLf.
+Remark 3. If L is a linear bounded operator in F, then etL = �+∞
+k=0(tL)k/k! where the
+series converges in the usual operator norm topology in L (F). When L is not bounded (such
+as Laplacian and many other differential operators), expressing (etL)t≥0 in terms of L is not
+an easy problem that is equivalent to the problem of finding (for each u0 ∈ F) the F-valued
+function U that solves the Cauchy problem U ′(t) = LU(t); U(0) = u0. If one finds this solution,
+then etL is obtained for each u0 ∈ F and each t ≥ 0 in the form etLu0 = U(t).
+Remark 4. In the definition of the Chernoff tangency the family (C(t))t≥0 usually does
+not have a semigroup composition property, i.e. C(t1 + t2) ̸= C(t1)C(t2), while (etL)t≥0 has it:
+et1Let2L = e(t1+t2)L. However, each C0-semigroup (etL)t≥0 is Chernoff-tangent to its generator
+L and appears to be it’s Chernoff function. When coefficients of the operator L are variable,
+usually there is no simple formula for etL due to the remark 3. On the other hand, even in this
+case one can find rather simple formula to construct Chernoff function C for the operator L,
+because there is no need to worry about the composition property, and then obtain etL in the
+form etL = limn→∞ C(t/n)n via the Chernoff theorem.
+3
+Numerical simulation results
+3.1
+Problem setting
+Definition 3.
+We say that operator-valued function C is Chernoff-tangent of order k to
+operator L iff C is Chernoff-tangent to L in the sense of definition 1 and the following condition
+(CT3-k) holds:
+There exists such a dense subspace D ⊂ F that for each f ∈ D we have
+C(t)f =
+�
+I + tL + 1
+2t2L2 + . . . + 1
+k!tkLk
+�
+f + o(tk) as t → 0.
+(CT3 − k)
+Remark 5. It is clear that for k = 1 condition (CT3-k) becomes just (CT3). For the semigroup
+C(t) = etL condition (CT3-k) holds for all k = 1, 2, 3, . . . So one can expect that the bigger k
+is the better rate of convergence C(t/n)nf → etLf as n → ∞ will be, if f belongs to the space
+D. This idea was proposed in [12], where two conjectures about the convergence speed were
+formulated explicitly, and one of them were recently proved in [7, 6]. For initial conditions that
+are good enough and t fixed, Chernoff function with Chernoff tangency of order k by conjecture
+should provide ∥u(t, ·) − un(t, ·)∥ = O(1/nk) as n → ∞. However, if f ̸∈ D then nothing is
+known on the rate of convergence. In the present paper we are starting to fill this gap for
+4
+
+operator L given by (Lf)(x) = f ′′(x) for all x ∈ R and all bounded, infinitely smooth functions
+f: R → R, and k = 1, 2.
+Problem setting. In the initial value problem (2) consider Q = R, and Banach space
+F = UCb(R) of all bounded, uniformly continuous functions f: R → R endowed with the
+uniform norm ∥f∥ = supx∈R |f(x)|. Consider operator L given by (Lf)(x) = f ′′(x) for all
+x ∈ R and all f ∈ D = C∞
+b (R) of all infinitely smooth functions R → R that are bounded with
+all the derivatives. Then (2) reads as (1). Cauchy problem (1) is a constant (one, zero, zero)
+coefficients particular case of the Cauchy problem considered in [15], and the corresponding
+Chernoff function was found in [15]. The particular case of this Chernoff function reads as
+(G(t)f)(x) = 1
+2f(x) + 1
+4f(x + 2
+√
+t) + 1
+4f(x − 2
+√
+t)
+where we write G(t) instead of C(t) in order to show that C(t) is a general abstract Chernoff
+function for some operator L, meanwhile G(t) is this particular above-given Chernoff function
+for operator d2/dx2. It was proved in [15] that G(t) is first order Chernoff-tangent to d2/dx2.
+A.Vedenin (see [19]) proposed another Chernoff function for operator L considered in [15],
+and the constant coefficient particular case of this operator is d2/dx2. The particular case of
+the Chernoff function obtained by A.Vedenin reads as
+(S(t)f)(x) = 2
+3f(x) + 1
+6f(x +
+√
+6t) + 1
+6f(x −
+√
+6t),
+and it was proved by A.Vedenin that S(t) is second order Chernoff-tangent to d2/dx2.
+In the paper we study how supx∈R |u(t, x) − un(t, x)| depends on n while t = 1/2 is fixed
+and un(t, x) is given by
+un(t, x) = (C(t/n)nu0)(x)
+where C ∈ {G, S}, C(t/n) is obtained by substitution of t by t/n in the formula that defines
+C(t), and C(t/n)n = C(t/n)C(t/n) . . . C(t/n) is a composition of n copies of linear bounded
+operator C(t/n). We consider several initial conditions u0 that are all H¨older continuous (hence
+all belong to the UCb(R) space) but have different H¨older exponents. Then we remark how the
+rate of tending of supx∈R |u(t, x) − un(t, x)| to zero depends on these H¨older exponents and the
+order of Chernoff tangency (which is 1 for G(t), and 2 for S(t)).
+Comments on computational techniques. Calculations were performed in the Python
+3 environment using a program we wrote and which is available in the Appendix. All measure-
+ments, for the sake of reducing computational complexity, for each value of n (varying from 1 to
+11) were carried out for 1000 points uniformly dividing the segment [−π, π] or [−2π, 2π]. Initial
+conditions of the form u0(x) = | sin x|α for various α ∈ {9/2, 7/2, 5/2, 3/2, 1, 3/4, 1/2, 1/4}, like
+any of Chernoff approximations based on them, are periodic functions. So, the standard norm
+in UCb(R), namely
+d = ∥un(t, ·) − u(t, ·)∥ = sup
+x∈R
+|un(t, x) − u(t, x)|,
+where u is the exact solution of (1) and un is the Chernoff approximation, is reached at the
+interval corresponding to the period.
+The program code was written with the possibility to set any operator and any initial con-
+dition, i.e. without simplifying Chernoff functions and using binomial coefficients, in contrast
+to the work [16] published earlier. Moreover, the initial condition does not necessarily have
+to be a smooth function. The number of iterations is not limited to 11, the value n can be
+changed, both upward and downward. We have chosen the optimal value n since the program
+is very time consuming: via Jupyter Notebook 6.1.4 Anaconda 3 Python 3.8.3 set on personal
+computer with Windows 10, CPU Intel Core i5-1035G1, 1.0-3.6 GHz, 8 Gb RAM it takes about
+20 minutes to complete the program for all initial conditions with construction of graphs for
+5
+
+them. At the research stage of the new method (Chernoff approximations) this is acceptable,
+but in the future, of course, the code will be optimized for a better speed, since this is important
+in practice. Our goal is to continue research and in the future write a library that allows to
+solve partial derivative equations in this way.
+3.2
+Approximations for initial condition u0(x) = sin(x)
+Let us first analyze the approximations for the initial condition u0(x) = sin x.
+fig. 1.1, n = 1, u0(x) = sin x, t = 1
+2
+Figure 1.1 shows the exact solution, which coincides with the graph of the function y =
+e−1/2 sin x, and approximate solutions for the functions S(t) (left) and G(t) (right) at n = 1.
+The initial condition u0 = sin x is very good, since its derivatives of any order exist, have
+no discontinuities and are bounded. And already at n = 1 the function S(t) gives a good
+approximation.
+Figure 1.2 below shows plots of the decreasing error of Chernoff approximations as a function
+of n, where 1 ≤ n ≤ 11. On the left are plots of decreasing error for Chernoff functions S(t)
+(in blue) and G(t) (in green) in regular scale, and on the right – the same plots in logarithmic
+scale. The graph in the logarithmic scale allows us to estimate how much the convergence rate
+for the function G(t) is less than the convergence rate for the function S(t). Here and through
+all the paper we use the following notation:
+d = ∥un(t, ·) − u(t, ·)∥ = sup
+x∈R
+|un(t, x) − u(t, x)|.
+fig. 1.2, 1 ≤ n ≤ 11, u0(x) = sin x, t = 1
+2
+6
+
+Approximetion of the sohution via Chemoff fmctin S(t)
+Approximetion of the sohution via Chemoff fmcticm G(t)
+0.6
+0.6
+0.4
+0.4
+0.2
+0.2
+0.0
+-2
+2
+-2
+-1
+0.0
+.3
+13
+i
+2
+3
+02
+7
+0.4
+0.4
+Chernoff approximation
+Chernoff approximation
+0.6
+Solution
+0.6
+SolutionNorm decay
+Logarithmic dependence
+R2=0,9949
+Discrepancy (S(t)f)(x)
+-4
+Discrepancy (G(t)f)(x)
+0.025
+-5
+R"=0.9949
+0.020
+-6
+0.015
+Discrepancy (S(t)f)(x)
+Discrepancy (G(t)f)(x)
+-7
+0.010
+-8
+0.005
+6-
+0.000
+.
+10
+2
+6
+8
+10
+0'0
+0.5 
+10
+1'5
+2.0
+2.5
+n
+In nYou can see that the points on the right graph lie on the straight lines with good accuracy.
+Using the method of least squares (in Excel) we found the equations of these lines. Rounding
+off the coefficients, we see that for the blue line the equation is as follows:
+ln(d) = −2.092 ln(n) − 5.0671, i.e. d = n−2.092e−5.0671 = 0.0063
+n2.092 .
+Similarly, for the green line, the equation ln(d) = −1.0416 ln(n) − 3.5796, i.e.
+d = n−1.0416e−3.5796 = 0.0279
+n1.0416.
+Using the same approach, we study the behavior of the error for other initial conditions.
+3.3
+Approximations for initial condition u0(x) = | sin(x)|3/2
+fig. 5.1, n = 4, u0(x) = | sin(x)|3/2, t = 1
+2
+fig. 5.2, 1 ≤ n ≤ 11, u0(x) = | sin(x)|3/2, t = 1
+2
+The line (green) corresponding to the decreasing error of the function G(t) in the logarithmic
+scale was constructed without taking into account n = 1.
+For the green line (see Fig. 5.2, right) the equation ln(d) = −0.9785 ln(n) − 2.8973, i.e.
+d = n−0.9785e−2.8973 = 0.0552
+n0.9785.
+Similarly, for the blue line (see Figure 5.2), the equation is as follows: ln(d) = −1.5109 ln(n)−
+1.8234, i.e. d = n−1.5109e−1.8234 = 0.1615
+n1.5109.
+As can be seen from Figure 5.2, the difference between the error decay rates using Chernoff
+functions S(t) and G(t) for u0(x) = | sin(x)|3/2 is larger than for u0(x) = | sin x|. This is due to
+the greater smoothness of u0(x) = | sin(x)|3/2.
+7
+
+Approximetion of the sohution via Chemoff fmcticn S(t)
+Approximetion of the sohution via Chemoff fmcticn G(t)
+0.62
+0.62
+0.60 
+0.60
+0.58
+0.58
+0.56
+0.56
+0.54
+0.54
+4152
+0.5b
+0.48
+Chernoff approximation
+0.50
+Chernoff approximation
+Solution
+Solution
+-3
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3Norm decay
+Logarithmic dependence
+0.16
+=0.9976
+Discrepancy (S(t)f)(x)
+2.0
+Discrepancy (G(t)f)(x)
+0.14 
+2.5
+0.12
+3.0
+0.10 
+R2=0,9903
+Discrepancy (S(t)f)(x)
+ 3.5
+0.08
+Discrepancy (G(t)f)(x)
+4.0
+0.06
+0.04
+4.5
+0.02
+5.0
+0.00
+5.5
+2
+4
+6
+8
+10
+0'0
+0.5
+10
+15
+2.0
+2.5
+n
+In n3.4
+Approximations for initial condition u0(x) = e−|x|
+Let us consider a non-smooth and non-periodic function e−|x| as an initial condition.
+fig. 6.1, n = 4, u0(x) = e−|x|, t = 1
+2
+fig. 6.2, 1 ≤ n ≤ 11, u0(x) = e−|x|, t = 1
+2
+Figures 6.1 and 6.2 show plots of the exact solution, approximations to the solution, and
+rates of convergence of the error to zero. As can be seen, the result is similar: the conver-
+gence rate of the function S(t) is higher than that of G(t), but the order of convergence is
+approximately the same, as can be seen from the fact that the lines are almost parallel.
+For the green line (see Fig. 6.2, right), the equation is as follows: ln(d) = −0.9294 ln(n) −
+2.3832, i.e. d = n−0.9294e−2.3832 = 0.0923
+n0.9294.
+Similarly, for the blue line (see Figure 6.2) the equation is as follows: ln(d) = −1.056 ln(n)−
+1.5543, i.e. d = n−1.056e−1.5543 = 0.2113
+n1.5543.
+8
+
+Approximetion of the sohution via Chemoff fmcticn S(t)
+Approximetion of the sohution via Chemoff fumcticn G(t)
+- 9'0
+0.5/
+0.5
+Q4
+0/4
+0.3
+0.3
+0.2
+0.2
+0.1
+0.1
+Chernoff approximation
+Chernoff approximation
+0.0
+Solution
+0.0 
+Solution
+10.0
+7.5
+5.0
+2.5
+0.0
+25
+5.D
+7.5
+1±.0
+-10.0
+7.5
+0'5-
+2.5
+0.D
+25
+5.D
+7.5
+1±.0Norm decay
+Logarithmic dependence
+0.200
+1.5
+Discrepancy (S(t)f)(x)
+6660-_
+Discrepancy (G(t)f)(x)
+0.175
+2.0
+0.150
+2.5
+0.125
+Discrepancy (S(t)f)(x)
+R
+=0.9979
+ 0.100
+Discrepancy (G(t)f)(x)
+0.075
+3.5
+0.050 
+4.0
+0.025
+4.5
+2
+6
+8
+10
+0'0
+0.5 
+10
+1'5
+2.0
+2.5
+n
+In n3.5
+Approximations for initial condition u0(x) = | sin(x)|5/2
+fig. 7.1, n = 4, u0(x) = | sin(x)|5/2, t = 1
+2
+fig. 7.2, 1 ≤ n ≤ 11, u0(x) = | sin(x)|5/2, t = 1
+2
+The lines (green and blue) corresponding to the decreasing error of the functions G(t) and
+S(t) in the logarithmic scale was constructed without taking into account n = 1 and n = 2.
+3.6
+Approximations for initial condition u0(x) = | sin(x)|7/2
+fig. 8.1, n = 4, u0(x) = | sin(x)|7/2, t = 1
+2
+9
+
+Approximetion of the sohution via Chemoff fmcticn S(t)
+Approximetion of the soution via Chemoff fumcticn G(t)
+0.52
+0.52
+0.50 
+0.50
+0.48 
+0.48
+0.46
+0.46
+0.44
+0.44
+1.42
+442
+00
+Chernoff approximation
+0.40
+Chernoff approximation
+Solution
+Solution
+E-
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3Norm decay
+Logarithmic dependence
+0.14
+2
+Discrepancy (S(t)f)(x)
+Discrepancy (G(t)f)(x)
+0.12
+E-
+0.10 
+0.08
+-4
+Discrepancy (S(t)f)(x)
+2
+p ul
+=0.9963
+Discrepancy (G(t)f)(x)
+0.06 
+-5
+0.04
+-6
+0.02
+00°0
+2
+6
+-7
+4
+8
+10
+0'0
+0.5
+10
+15
+2.0
+2.5
+n
+In nApproximetion of the sohution via Chemoff fmcticn S(t)
+Approximetion of the soution via Chemoff fumcticn G(t)
+0.46
+0.46
+0.44
+0.44
+0.42
+0.42
+0.40 
+0.40
+0.38
+0.38
+1.36
+436
+0.14
+Chernoff approximation
+0.34
+Chernoff approximation
+Solution
+Solution
+E-
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3fig. 8.2, 1 ≤ n ≤ 11, u0(x) = | sin(x)|7/2, t = 1
+2
+The line (green) corresponding to the decreasing error of the function G(t) in the logarithmic
+scale was constructed without taking into account n = 1 and n = 2.
+3.7
+Approximations for initial condition u0(x) = | sin(x)|9/2
+fig. 9.1, n = 4, u0(x) = | sin(x)|9/2, t = 1
+2
+fig. 9.2, 1 ≤ n ≤ 11, u0(x) = | sin(x)|9/2, t = 1
+2
+The line (green) corresponding to the decreasing error of the function G(t) in the logarithmic
+scale was constructed without taking into account n = 1 and n = 2.
+10
+
+Norm decay
+Logarithmic dependence
+0.200
+R=0.9864
+:
+Discrepancy (S(t)f)(x)
+-2
+Discrepancy (G(t)f)(x)
+0.175
+0.150
+-3
+0.125
+Discrepancy (S(t)f)(x)
+-4
+p ul
+0,9753
+ 0.100
+Discrepancy (G(t)f)(x)
+-5
+0.075
+0.050
+-6 
+0.025
+-7
+0.000
+4
+6
+8
+10
+0'0
+0.5 
+10
+1'5
+2.0
+2.5
+n
+In nApproximetion of the sohution via Chemoff fmcticn S(t)
+Approximetion of the soution via Chemoff fumcticn G(t)
+0.42
+0.42
+0.40
+0.40
+0.38 
+0.38
+0.36
+0.36
+0.34 
+0.34
+.32
+432
+0.30
+Chernoff approximation
+Chernoff approximation
+Solution
+Solution
+0.28
+E-
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3Norm decay
+Logarithmic dependence
+0.25
+Discrepancy (S(t)f)(x)
+-1
+R
+=0.9777
+Discrepancy (G(t)f)(x)
+0.20
+-2
+E-
+0.15
+Discrepancy (S(t)f)(x)
+-4
+R°=0,9309
+Discrepancy (G(t)f)(x)
+0.10 
+-5
+-6
+0.05
+-7 
+0.00
+-8 
+2
+6
+10
+0'0
+0.5
+10
+1'5
+2.0
+2.5
+n
+In n4
+Discussion
+The table below shows experimentally (using simulation in Python 3) the orders of decreas-
+ing of error depending on the smoothness class of the initial condition and the Chernoff function.
+The smoothness class of the initial
+condition u0
+Order of decreasing er-
+ror
+on
+the
+Chernoff
+function
+G(t),
+which
+has the 1st order of
+the
+Chernoff
+tangent
+to the operator L =
+d2
+dx2
+Order of decreasing er-
+ror
+on
+the
+Chernoff
+function
+S(t),
+which
+has the 2nd order of
+tangency by Chernoff
+to the operator L =
+d2
+dx2
+C∞, i.e.
+all derivatives exist and are
+bounded, u0(x) = sin(x)
+-1.0416
+-2.092
+C4 1
+2 , the first, second, third, and fourth
+derivatives exist and are bounded, and
+the fourth is H¨older with a H¨older ex-
+ponent 1/2, u0(x) = | sin(x)|9/2
+-1.0212, the regression was
+done without n = 1, n = 2
+-3.1219, but the points do
+not fit well on a straight
+line, so the number is un-
+informative
+C3 1
+2 , the first, second, and third deriva-
+tives exist and are bounded, and the
+third is H¨older with H¨older exponent
+1/2, u0(x) = | sin(x)|7/2
+-1.4013,regression
+was
+done without considering
+n = 1, n = 2, but the
+points do not lie well on
+the line, so the number is
+uninformative
+-2.5045, but the points do
+not lie well on the line, so
+the number is uninforma-
+tive
+C2 1
+2 , the first and second derivative
+exist and are bounded, while the sec-
+ond derivative is H¨older continuous
+with H¨older exponent 1/2, u0(x) =
+| sin(x)|5/2
+-1.1433,
+regression
+was
+done without considering
+n = 1, n = 2
+-1.7923,
+regression
+was
+done without considering
+n = 1, n = 2
+C1 1
+2 ,
+the
+first
+derivative:
+exists,
+is
+bounded
+and
+H¨older
+continuous
+with H¨older exponent 1/2, u0(x) =
+| sin(x)|3/2
+-0.9785, the regression was
+done without considering
+n = 1
+-1.5109
+H1, the H¨older with the H¨older expo-
+nent 1, u0(x) = | sin(x)|
+-1.0508
+-1.0948
+H1,the H¨older with the H¨older expo-
+nent 1, u0(x) = e−|x|
+-0.9294
+-1.056
+H3/4, the H¨older with the H¨older expo-
+nent 3/4, u0(x) = | sin(x)|3/4
+-0.815
+-0.9262
+H1/2, the H¨older with the H¨older expo-
+nent 1/2, u0(x) = | sin(x)|1/2
+-0.6905
+-0.7723
+H1/4, the H¨older with the H¨older expo-
+nent 1/4, u0(x) = | sin(x)|1/4
+-0.6138
+-0.6653
+We see that on the initial condition with high smoothness (first line in the table), the first
+order of Chernoff tangency corresponds to a decreasing error rate of about 1/n, and the second
+order – a decreasing rate of about 1/n2. This is in accordance with the conjecture from [12]
+and theorem from [6].
+As the smoothness is lost (second line in the table and below), theory from [6] stops working,
+and the experimental evidence is the following: the convergence speed gradually decreases and
+the advantages of the Chernoff function with the second order of Chernoff tangency gradually
+vanish. Let us present the results from the table graphically:
+11
+
+fig. 10
+The regression was carried out without taking into account the point (2.5; -1.7923). The
+equation of the approximating line: y = −0.684x − 0.4467.This may be interpreted as follows:
+when the smoothness class α of the initial condition u0 is not greater than the order of Chernoff
+tangency then
+d = ∥un(t, ·) − u(t, ·)∥ = sup
+x∈R
+|un(t, x) − u(t, x)| ≈ const ·
+�1
+n
+�0.68α+0.45
+.
+Meanwhile when the smoothness class α of the initial condition u0 is greater than the order of
+Chernoff tangency then there is no such easy-to-state dependence but still Chernoff function
+S(t) with the second order Chernoff tangency provides better approximations than Chernoff
+function G(t) with the first order Chernoff tangency.
+Conclusion.
+The results of the numerical simulation are generally in agreement with
+and confirm the theory arising from the conjecture in [12]. However, some of the points that
+do not lie on straight lines exactly. This deserve closer attention: n = 11 for some initial
+conditions is not sufficient to derive conclusions about the asymptotic behavior of the calculation
+error. For not smooth initial conditions that we studied numerically there are not known any
+theoretical bounds on the rate of convergence. And, of course, the most interesting case of
+variable coefficients should be considered, understanding them as parameters analogously with
+u0. So the research in this direction is far from ending.
+Acknowledgements. The publication was prepared within the framework of the Academic
+Fund Program at HSE University in 2021 (grant №20-04-022) by the Research and educational
+group ”Evolution semigroups and their applications”. Authors are thankful to Prof. O.E.Galkin
+and other members of the group for discussions of the results presented in the paper.
+12
+
+0.6
+S(t)
+0.8
+(a)9
+1.0
+1.2
+decre
+1.4
+ 1.6
+1.8
+R2 = 0,9854
+2.0
+2.2
+0.5
+1.0
+1.5
+2.0
+2.5
+The smoothness class of the initial conditionAppendix: Python 3 code
+13
+
+L.1Moduliandvariables
+In [1]:
+# importing moduli
+importmatplotlib.pyplotasplt
+from sympy import oo
+from scipy importintegrate
+import numpy as np
+importmath
+In [2]:
+# variables declaration
+tau = 1/2
+n = 11
+In[3]:
+1=[]
+fori inrange(1,n+1):
+1.append(math.log(i))1.2 Functions and operators
+# almost self-contained class
+classclassoffunctions(object):
+def thefunction(self, x):
+return x
+# Chernoff function (s(t)f)(x)=(2/3)f(x) + (1/6)f(x + (6t)^(1/2)) + (1/6)f(x - (6t)^(1/2))
+defoper(inputobject:classoffunctions,t):
+outputobject=inputobject
+f= inputobject.thefunction
+def funct(x):
+return (2/3)*f(×) +(1/6)*f(× + (6*t)**(1/2)) + (1/6)*f(x- (6*t)**(1/2))
+outputobject.thefunction=funct
+returnoutputobject
+# Chernoff function (G(t)f)(x) = (1/4)* f(x+2t^(1/2))+(1/4) f(x-2t^(1/2))+(1/2)f(x)
+defoperl(inputobject:classoffunctions,t):
+outputobject=inputobiect
+f=inputobject.thefunction
+def funct(x):
+return (1/4)*f(x+2*t**(1/2))+(1/4)*f(x-2*t**(1/2))+(1/2)*f(×)
+outputobject.thefunction=funct
+return outputobjectReferences
+[1] Bogachev V.I., Smolyanov O.G. Real and Functional Analysis. — Springer, 2020.
+[2] Butko Ya.A. The method of Chernoff approximation. Springer Proceedings in Mathematics
+and Statistics. Volume 325. — Springer, Cham, 2020. Pp. 19–46.
+[3] Chernoff P.R. Note on product formulas for operator semigroups.// J. Functional Analysis
+2:2 (1968), 238-242.
+[4] Engel K.-J., Nagel R. A. Short Course on Operator Semigroups. — N.Y. Springer Science,
+Business Media, 2006.
+[5] Engel K.-J., Nagel R. One-Parameter Semigroups for Linear Evolution Equations. —
+Springer, 2000.
+[6] Galkin O.E., Remizov I.D. Rate of Convergence of Chernoff Approximations of operator
+C0-semigroups.// Mathematical Notes, to appear (2022)
+[7] Galkin O. E., Remizov I. D. Upper and lower estimates for rate of convergence in Chernoff’s
+product formula for semigroups of operators // https://arxiv.org/abs/2104.01249, 2021
+[8] Gomilko A., Kosowicz S., Tomilov Yu. A general approach to approximation theory of
+operator semigroups. // Journal de Math´ematiques Pures et Appliqu´ees. 127 (2019), 216–
+267.
+14
+
+# composition degree of Chernoff function (s(t)f)(x)
+def degr(g, tau, n):
+y=[]
+obj=classoffunctions()
+for n_p in range(1, n + 1):
+objk=obj
+obj_k.thefunction = g
+for k in range(1, n_p + 1):
+objk=oper(objk,tau/np)
+y.append(obj_k.thefunction)
+return y
+# composition degree of Chernoff function (G(t)f)(x)
+def degrl(g, tau, n):
+y=[]
+obj=classoffunctions()
+for n_p in range(1, n + 1):
+obj_k = obj
+obj_k.thefunction =g
+for k in range(1, n_p + 1):
+obj k = oper1(obj k,tau/n p)
+y.append(obj_k.thefunction)
+return y
+#normcomputation
+def norm(y, sol):
+[] = p
+for n p in range(O, n):
+d.append(np.max(np.abs(sol -y[n_pj(x))))
+return d[9] Hille E., Phillips R. S. Functional Analysis and Semi-Groups. — American Mathematical
+Society, 1975.
+[10] Orlov Yu.N., Sakbaev V.Zh., Smolyanov O.G. Rate of convergence of Feynman approxima-
+tions of semigroups generated by the oscillator Hamiltonian. // Theoret. and Math. Phys.
+172:1 (2012), 987–1000.
+[11] Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations.
+— Springer-Verlag, 1983.
+[12] Remizov I.D. On estimation of error in approximations provided by Chernoff ’s product
+formula.// International Conference "ShilnikovWorkshop-2018" dedicated to the memory of
+outstanding Russian mathematician Leonid Pavlovich Shilnikov (1934-2011), Lobachevsky
+State University of Nizhny Novgorod, December 17-18, 2018 Book of abstracts, pp.38-41
+[13] Remizov I.D. A method of obtaining the evolution operator for the Schr¨odinger equation
+Quasi-Feynman formulas.// J. Funct. Anal. 270:12, (2016), 4540-4557.
+[14] Remizov I.D. Feynman and Quasi-Feynman Formulas for Evolution Equations// Doklady
+Mathematics, 96:2 (2017), 433-437
+[15] Remizov I. D. Approximations to the solution of Cauchy problem for a linear evolution
+equation via the space shift operator (second-order equation example)// Applied Mathe-
+matics and Computation, 328, 243-246, 2018
+[16] Prudnikov P.S. Speed of convergence of Chernoff approximations for two model examples:
+heat equation and transport equation// arXiv:2012.09615 [math.FA] (2020).
+[17] Smolyanov O.G., Tokarev A.G.,Truman A. Hamiltonian Feynman path integrals via the
+Chernoff formula.// J. Math. Phys. 43. 10 (2002), 5161-5171.
+[18] Vedenin A.V. Fast converging Chernoff approximations to solution of a parabolic differ-
+ential equation on a real line.// International school-conference Mathematical spring 2019,
+Russia, Nizhny Novgorod 2-5 May 2019, Book of abstracts, pp 22-23
+[19] Vedenin A.V., Voevodkin V.S., Galkin V.D., Karatetskaya E.Yu., Remizov I.D. . Speed
+of Convergence of Chernoff Approximations to Solutions of Evolution Equations. // Math.
+Notes, 108:3 (2020), 451–456.
+[20] Zagrebnov V.A. Notes on the Chernoff product formula. // J. Funct. Anal. 279:7 (2020),
+108696
+15
+

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+ 
+ 
+ 
+ 
+ 
+Sensors 2022, 22, x. https://doi.org/10.3390/xxxxx 
+www.mdpi.com/journal/sensors 
+Article 
+Multi-Finger Haptics: Analysis of Human Hand Grasp towards 
+a Tripod Three-Finger Haptic Grasp model 
+Jose James 
+Brown University, Providence, RI, 02912, USA.  
+Correspondence: [email protected]. 
+Abstract: Grasping is an incredible ability of animals using their arms and limbs in their daily life. 
+The human hand is an especially astonishing multi-fingered tool for precise grasping, which 
+helped humans to develop the modern world. The implementation of the human grasp to virtual 
+reality and tele robotics is always interesting and challenging at the same time. In this work, au-
+thors surveyed, studied, and analyzed the human hand grasping behavior for the possibilities of 
+haptic grasping in the virtual and remote environment. This work is focused on the motion and 
+force analysis of fingers in human hand grasping scenarios and the paper describes the transition 
+of the human hand grasping towards a tripod haptic grasp model for effective interaction in virtu-
+al reality. 
+Keywords: hand grasp; grasp analysis; multi-finger haptics; haptic grasp interface 
+ 
+1. Introduction 
+The human hand is a highly skilled, prehensile, multi-fingered, perception and ma-
+nipulation organ at the distal end of the arm [1]. Prehensility is the quality of an ap-
+pendage or organ that has adapted for grasping or holding. In the past decade’s, re-
+searchers [2] explored the various aspects of the evolution, morphology, anthropology, 
+and social significance of the human hand as a tool [3], as a symbol [4] and as a weapon 
+[5]. Humans cognitively manipulating a variety of objects in daily life using various 
+hand configurations, resulting from changing the position, orientation and placement of 
+hand and fingers based on the object properties such as its weight, shape, texture, fric-
+tion, hardness etc. Such a variety of grasps is possible because of the dexterity, various 
+degrees of freedom, and the great control strategy. 
+Hands are associated with the aligning capability of the body, kinesthetic percep-
+tion of the limb and the richest tactile sense. Many researchers studied the anatomy, 
+muscles, biomechanics, kinematics, functionalities, and skills of human hand [6,7]. These 
+studies helped to do more focused research on human hand prehension [8] and grasp 
+[9,10]. Based on all these studies hand grasp types are classified and different grasp tax-
+onomies were arising in the literature [11,12], covering a broad range of domains. 
+As technologies like virtual reality and tele-robotics being progressed, humans 
+started interacting with virtual and remote environments. But the integration of hand 
+grasping into virtual and remote environments still challenging because of the complex 
+architecture behind it. The extensive research in the analysis and synthesis of human 
+grasps [13,14] over the past years has provided a basic theoretical framework towards 
+better progress in human-computer interaction [15], robotic grasping [16] and dexterous 
+manipulation and lead to the design of artificial robotics hands and arms for the pros-
+thetic application [17]. Since the last few decades, researchers put effort to mimic the 
+human hand to design robotic grippers [18], etc. But still, these frameworks need more 
+extensive studies for the practical implementation of the direct involvement of humans 
+Citation: James, J.; Title. Sensors 
+2022, 
+22, 
+x. 
+https://doi.org/10.3390/xxxxx 
+Academic Editor: Firstname Last-
+name 
+Received: 27 April 2022 
+Accepted: date 
+Published: date 
+Publisher’s Note: MDPI stays neu-
+tral with regard to jurisdictional 
+claims in published maps and insti-
+tutional affiliations. 
+ 
+Copyright: © 2022 by the authors. 
+Submitted for possible open access 
+publication under the terms and 
+conditions of the Creative Commons 
+Attribution 
+(CC 
+BY) 
+license 
+(https://creativecommons.org/license
+s/by/4.0/). 
+
+sehsorsMDPIBYSensors 2022, 22, x FOR PEER REVIEW 
+2 of 20 
+ 
+ 
+to grasp objects in virtual and remote environments with multi-fingers. 
+The grasping force depends on the orientation of fingers, palm, and wrist [19]. The 
+force output on the fingertip is highly joint dependent and provides stable grasp and 
+precise manipulation of objects. Previous works [20], more focused on muscle activation 
+patterns and resultant positions/forces as a function of the joints as well as subject inde-
+pendent leads to the structural variability in human hands [21]. Motion and force analy-
+sis of fingers in various hand manipulation actions can be observed, learned, and ana-
+lyzed to come up with better framework and devices for virtual and remote manipula-
+tions [22,23] Adapting the gripping functions, manipulation capabilities, kinematics, dy-
+namics, and size of the human hand, will accelerate the design of the human-like artifi-
+cial arms and hands for the direct grasping interaction in the virtual world. 
+As a previous work, authors designed haptic interfaces for tweezer pinch grasp [24] 
+and tripod grasp [25]. Also implemented the hand grasp through augmented haptics by 
+means of custom-made attachments for virtual tools in motor skill training interfaces 
+[26,27]. The aim of this work is to understand human grasping and manipulations, sur-
+veying different types of grasp taxonomies, study the characterization of hand in grasp-
+ing, model a tripod haptic grasp and design an interface for multi-finger haptic grasping 
+which can offer better interactions with tasks in the virtual and remote environments. 
+2. Human Hand 
+The human hand is a prehensile, multi-fingered astonishing organ/tool of complex 
+engineering used to carry and manipulate objects [2]. In view of human grasping, a short 
+description of hand anatomy, mechanisms and kinematics will help to model a multi-
+finger haptic grasp and design a haptic grasping interface. 
+2.1. Hand Anatomy 
+The human hand includes mainly three areas and five digits (fingers): Thumb, In-
+dex finger, Middle finger, Ring finger, and Little finger are numbered 1-5 as shown in 
+Figure 1. The palm with fingers holds most pressure and support for the hand to grasp. 
+Fingers are the densest areas of nerve endings and the richest source of tactile feedback. 
+So, hands are the primary tool for a sense of touch and positioning capability. The hu-
+man hand consists of 27 bones and 45 muscles with at least 23 degrees of freedom at the 
+joints [6] including the wrist as shown in Figure 1(a).  
+ 
+Figure 1. Skeleton system and arches of human hand 
+The human hand can grasp objects and do daily tasks by forming bony arches: 
+Longitudinal arches, transverse arches, and oblique arches as drawn in Figure 1(b). Lon-
+
+(a)HumanHandskeletonsystem
+(b)HumanHandarches
+DIPjoint
+PIP joint.
+MCPjoint
+Distal
+Longitudinalarch
+1
+Middle
+Phalanges
+(Fingers)
+Obliquearch
+Proximal
+Metacarpaltransversearch
+Metacarpals
+Distaltransversearch
+(Palm)
+Hamate
+Trapezium
+Pisiform
+Carpals
+Trapezoid-
+Scaphoid
+-Triquetrum
+(Wrist)
+Capitate
+Lunate
+Carpaltransversearch
+Radius
+UlnaSensors 2022, 22, x FOR PEER REVIEW 
+3 of 20 
+ 
+ 
+gitudinal arches shaped by the finger bones and their associated metacarpal bones, 
+transverse arches by the carpal bones and the distal ends of the metacarpal bones, and 
+oblique arches by the thumb and four fingers [7]. These arches are the basic frames for 
+various grasp patterns. The extrinsic and intrinsic muscles in hand controlled the motion 
+of the fingers and making grasping possible. Thumb together with the index and middle 
+finger forms the dynamic tridactyl configuration responsible for most grips which not 
+requiring force. The ring and little fingers are more static. So, for this multi-finger haptic 
+grasping interface model, authors considered the motion of the first three fingers the 
+thumb, index finger and middle finger. 
+2.2 Hand Kinematics 
+Hand’s numerous patterns of action was resulted from the skeleton mechanical sys-
+tems and the twenty-four muscle groups regulated by the diverse motor and sensory 
+nerve pathways [6]. 
+The MCP and PIP joints exhibit a common rotation pattern. The virtual center of ro-
+tation of hand is the center of curvature of the distal end of the proximal member [28]. 
+The lateral rotation of fingers is small in the MCP joints and decreasing towards the pha-
+langeal hinge joints. The thumb has the greater mobility in the CMC articulation. Other 
+fingers being more arched from index to little finger. The thumb, palm, and fingers to-
+gether permitted to grasp a 1.75-inch cylinder at about 45 degrees to the radioulnar axis. 
+Bunnell [29] considers this "an ancestral position ready for grasping limbs, weapons, or 
+other creatures."  
+The major wrist motions are extension (or dorsiflexion), flexion (or volar flexion), 
+radial flexion and ulnar flexion, based on the angle of rotation of the wrist. The fixation 
+movements and ballistic movements are also major types of movements in the hand [30]. 
+The hand with the fully extended arm can be rotated through almost 360 degrees with 
+the participation of shoulder and elbow. From palm up to palm down, the hand can be 
+rotated through 180 degrees, with the elbow flexed. Thumb can provide a variety of flex-
+ions extension patterns of the phalanges for any given metacarpal position and due to 
+the relative mobility of the CMC joint, which allows the thumb to act in any plane neces-
+sary to oppose the digits. In the principal opposition cases and prehensions, the plane of 
+the thumb action is inclined 45 to 60 degrees to the palmar plane. In lateral prehension, 
+the plane is approximately parallel to the palmar plane. 
+2.3 Hand Dynamics 
+Fick [31] investigated the actions and contractile forces of hand muscles and esti-
+mated the summed forces of the individual muscles participating in the action. But the 
+measured isometric forces are only 10% of the total forces because of the effective small 
+moment arm upon any of the wrist or hand joints. The flexor-extensor forces in the wrist 
+and the prehensile forces in the hand varied with wrist angle and it reaches a maximum 
+at a wrist angle of about 145 degrees. So, for very strong prehensions, wrist likely to at-
+tain this angle [32]. 
+Kamper et.al [33] was analyzed the joint angles and finger trajectories in reach-and-
+grasp tasks which fit the actual finger positions with a mean error of 0.23 ± 0.25 cm and 
+accounted for over 98% of the variance in finger position. The direction of the thumb tra-
+jectories exhibited a greater dependence on object type than the finger trajectories, but 
+still utilized a small percentage (<5%) of the available workspace [34]. Previous studies 
+of musculoskeletal models [35] observed that the role of the intrinsic muscles of the hand 
+as the main force-producing muscles in power grip [36]. These models were used in the 
+commercial software ANYBODY for the thumb and the index finger [37]. However, 
+these models did not address the coupling between the fingers and the interaction with 
+the wrist which limits the investigation of human grasping. 
+
+Sensors 2022, 22, x FOR PEER REVIEW 
+4 of 20 
+ 
+ 
+In [38] authors presented an upper limb musculature model for the full arm includ-
+ing the shoulder, elbow, wrist, thumb, and index finger and provides valuable data on 
+the wrist-finger joint coupling and extrinsic hand muscle anatomy [39]. The force output 
+on the fingertip is highly joint dependent and provides stable grasp and precise manipu-
+lation of objects. That's why the force transfer is important for the human hand in con-
+tact and manipulating purpose to avoid the slipping and deformation of the object. Pre-
+vious works focused more on muscle activation patterns and resultant positions/forces 
+as a function of the joints [20] as well as a subject independent lead to the structural var-
+iability in human hands. 
+2.4 Purpose of Study 
+This study conducted to analyze the human hand grasping to isolate the functional 
+properties with the final goal to optimize haptic grasping by building simpler multi-
+finger haptic grasping interfaces with at least similar grasping and manipulation capa-
+bilities. This work helped to learn more about the complex engineering structure of the 
+human hand and leads to characterizing the multi-finger haptic grasping systems. For 
+instance, the proposed three-finger haptic grasping system has an independent joint ar-
+chitecture for three fingers in the hand, which may be advantageous in virtual grasping. 
+Also, this led to identify the independence of joints needs in normal grasping tasks. 
+Conversely, the control of such an independent architecture is very challenging. By add-
+ing synergies, we can reduce the complexity of control, but we also want to keep a cer-
+tain, currently unquantified, level of dexterity. 
+3. Human Grasp 
+A grasp is a system wherein the desired object is gripped by the fingers of a human 
+(or robot) hand.     
+3.1 Human Grasp Patterns 
+Napier [40] categorized human grasp into two basic grips: power grasps and preci-
+sion grasps (pinch grasps). In power grasp, the object is in the palm of the hand and en-
+closed by the fingers which lead to large area of contact between the palm, the fingers, 
+and the object. In precision grasp (pinch grip), the object is held between the tip of the 
+thumb and finger, which offer more dexterity. Precision grasps become more relevant in 
+robotic and virtual grasping. The power grasp is enhanced by the precision grasp be-
+tween the thumb and the distal finger pads, and it is inherently stable. Pinch grip re-
+quires the six joints between the index finger and the thumb to be stabilized; it requires 
+more activity of the intrinsic finger muscles to maintain this balance. A large variety of 
+prehension patterns are identified from studies of the muscle-bone-joint anatomy and 
+from observation of the postures and motions of the hand. The object-contact pattern 
+furnishes a satisfactory basis for classification of major prehension patterns [41].  
+All the ages, the human hand was a part of most creative arts of every culture [42] 
+to speak and convey human emotions and the hands symbolize cultural behaviors, val-
+ues, and beliefs. A mudra is a symbolic gesture in the spiritual practice of Indian reli-
+gions and traditional art forms performed with the hands with a specific pattern of fin-
+ger configurations [43]. A canonical set of predefined hand postures and modifiers can 
+be used in digital human modelling to develop the standard hand posture libraries and 
+a universal referencing scheme and continuum of hand poses from simple posture to 
+complex one. Researchers [9] have studied features for force-closure grasp by human 
+hands and characterized into four mutually independent properties for robotic arm 
+grasping listed as dexterity, equilibrium, stability, and dynamic behavior. The principal 
+component analysis of static hand posture of several subjects provides information 
+about the finger joint variance and shape of the grasped object and did not consider the 
+hand position/orientation relative to the object placement [44]. 
+
+Sensors 2022, 22, x FOR PEER REVIEW 
+5 of 20 
+ 
+ 
+3.2 Grasp Taxonomy 
+Based on the various studies about human hand in the literature, hand grasp types 
+are classified, and different grasp taxonomies were raised in the literature [11,12] as 
+shown in Table 1. If the grasp object size/shape is not considered, this taxonomy might 
+be lowered to broad range.  
+Grasps are classified based on precision [45], grasped object’s size [9], shape [46], 
+weight, rigidity, force requirement, the position of thumb (adducted or abducted posi-
+tion) and the situation. Based on the level of precision, grasps are classified as precision 
+grasp [10], intermediate grasp [47], and power grasp [48]. The movements of the hand in 
+the power grip are evoked by the arm but in the precision handling, the intrinsic move-
+ments on the hand not evoked by the arm. In intermediate grasp, elements of power and 
+precision grasps are present in roughly the same proportion. Later studies included oth-
+er grasps like hook grasp, flat hand grasp, platform grasp, push grasp [49]. In static and 
+stable grasps [12], the object is in a constant relation to the hand. 
+ Based on the direction of force relative to the hand coordinate frame, applied by 
+the hand on the object to hold it securely [8] opposition type grasps are classified as pad 
+opposition, palm opposition and side opposition grasp [50]. Pad Opposition occurs be-
+tween hand surfaces along a direction generally parallel to the palm, usually occurs be-
+tween volar (palmar) surfaces and the fingers and thumb, near or on the pads. Palm Op-
+position occurs between hand surfaces along a direction generally perpendicular to the 
+palm. Side Opposition occurs between hand surfaces along a direction generally trans-
+verse to the palm. 
+The taxonomy of Cutkosky [9], which is widely used in the field of robotics, lists 15 
+different grasps. Other taxonomies mentioned in works of Kamakura et al. [47], Ed-
+wards et al. [10], Kapandji [45] are listed with 14, 20 and 21 grasps respectively. A simi-
+lar study [51], that used a different categorization which incorporated non-prehensile 
+grasps. Even though there has been a considerable effort in creating statistics of human 
+hand use and grouping of hand grasps [11,12].  The extensive research in human grasp 
+analysis and taxonomies over the past years helped towards better progress in human-
+computer interaction, robotic grasping, and dexterous manipulation and lead to the de-
+sign of artificial robotics hands and arms for the prosthetic application. 
+This helped to classify all hand usage in everyday life situations [52]. Furthermore, 
+the taxonomy could be extended to include non-prehensile “grasps”, or for dynamic 
+within - hand manipulation movements. Kamakura et al. [47] classified the tripod grasps 
+as intermediate grasps, apart from that it was classified as a precision grasp.  Several 
+studies have investigated classifying grasps into a discrete set of types [8,9,47], and oth-
+ers have been aimed at understanding certain aspects of human hand usage [53]. The 
+number of fingers used for grasping increases with the size and mass of the object [54] 
+until a two-handed grasp is required, indicating that object size and mass are strong fac-
+tors in determining the grasp type. The rigidity of the objects also influencing the grasp 
+[55].  
+ 
+
+Sensors 2022, 22, x FOR PEER REVIEW 
+6 of 20 
+ 
+ 
+ 
+Table 1. Taxonomy of grasps and allocation of virtual fingers. 
+The studies in [56,13,14] analyzed the human grasping scenarios and behavior of 
+unstructured tasks and investigated the relationship between grasp types and object 
+properties and the results indicated that three-fingertip precision grasps such as thumb-
+2 finger, tripod, or lateral tripod can be used to handle dexterous manipulation of a wide 
+range of objects. In [12] authors analyzed and compared 33 existing human grasp taxon-
+omies of static and stable grasps performed by one hand and synthesized them into a 
+single new taxonomy called ‘The GRASP Taxonomy’. The grasps are arranged according 
+to opposition type, the virtual finger assignments, type in terms of power, precision or 
+intermediate grasp, and the position of the thumb. The classifications of micro-
+interaction grasp instances [57] helped researchers in defining human hand capabilities 
+[58] and affordances in robotic hand design [59]. 
+The concept of the virtual finger [15] has also incorporated in this taxonomy as an 
+abstract representation through which the human brain plans grasping tasks [60]. The 
+virtual finger is a functional unit of several fingers work together comprised of at least 
+one real physical finger to reduce the degrees of the human hand to perform the grasp-
+ing task. This concept replaces the analysis of the mechanical degrees of freedom of in-
+
+Grasp
+Thumb
+Virtual
+Min. No.
+Grasp
+Thumb
+Virtual
+Min. No.
+No.
+Name
+Picture
+Type
+Opp.Type Position
+Fingers
+Mod. VF
+of Fingers
+No.
+Name
+Picture
+Type
+Opp.Type Position
+Fingers
+Mod. VF
+of Fingers
+VF1:
+VF1: 1
+VF1:
+VF1: 1
+large
+VF2: 2-5
+VF2: 2-5
+Extension
+VF2: 2-5
+VF2: 2-5
+Diameter
+Power
+Palm
+Abducted
+VF3:
+VF3: Palm
+3
+18
+Type
+Power
+Pad
+Abducted VF3:
+VF3:
+3
+VF1:
+VF1: Palm
+VF1:
+VF1: 1
+Small
+心
+VF2: 2-5
+VF2: 2-5
+Distal
+VF2: 2-5
+VF2: 2-5
+2
+Daimeter
+Power
+Palm
+Abducted
+VF3:
+VF3:
+2
+19
+Type
+Power
+Pad
+Abducted VF3:
+VF3:
+2
+VF1:
+VF1: Palm
+VF1:
+VF1: 1
+Medium
+VF2: 2-5
+VF2: 2-5
+Writing
+VF2: 2
+VF2: 2
+m
+Wrap.
+Power
+Palm
+Abducted
+VF3:
+VF3:
+m
+20
+Tripod
+Precision
+Side
+Abducted VF3:
+VF3: 3
+m
+VF1:
+VF1: Palm
+VF1:
+VF1: 1
+Adducted
+VF2: 2-5
+VF2: 2-5
+Tripod
+Intermed
+VF2: 3-4
+VF2: 3-4
+Thumb
+Power
+Palm
+Adducted
+VF3: 1
+VF3: 1
+3
+21
+Variation
+iate
+Side
+AbductedVF3:
+VF3: 2
+VF1:
+VF1: Palm
+VF1:
+VF1: 1
+VF2: 2-5
+VF2: 2-5
+Parallel
+VF2: 2-5
+VF2: 2-5
+5
+Light Tool
+Power
+Palm
+Adducted
+VF3: (1)
+VF3: (1)
+3
+22
+Extension
+Precision
+Pad
+Adducted VF3:
+VF3:
+m
+VF1:
+VF1: 1
+VF1:
+VF1: 2
+Prismatic 4
+VF2: 2-5
+VF2: 2
+Adductio
+Intermed
+VF2: 2
+VF2: 3
+6
+Finger
+Precision
+Pad
+Abducted
+VF3:
+VF3: 3-5
+m
+23
+n Grip
+iate
+Side
+Abducted VF3:
+VF3:
+2
+VF1:
+VF1: 1
+VF1:
+VF1: 1
+Prismatic 3
+VF2: 2-4
+VF2: 2
+VF2: 2
+VF2: 2
+Finger
+Precision
+Pad
+Abducted
+VF3:
+VF3: 3-4
+3
+24
+Tip Pinch
+Precision
+Pad
+Abducted/VF3:
+VF3:
+2
+VF1:
+VF1: 1
+VF1:
+VF1: 1-2
+Prismatic 2
+VF2: 2-3
+VF2: 2
+Lateral
+Intermed
+VF2: 3
+VF2: 3
+Finger
+Precision
+Pad
+Abducted
+VF3:
+VF3: 3
+3
+25
+podul
+iate
+Side
+Adducted VF3:
+VF3:
+3
+VF1:
+VF1: 1
+VF1:
+VF1: 1
+Palmar
+VF2: 2
+VF2: 2
+Sphere4
+VF2: 2-4
+VF2: 2-4
+6
+Pinch
+Precision
+Pad
+Abducted
+VF3:
+VF3:
+2
+26
+Finger
+Power
+Pad
+Abducted/VF3:
+VF3:
+m
+VF1:
+VF1: 1
+VF1:
+VF1: 1
+VF2: 2-5
+VF2: 2-5
+VF2: 2-4
+VF2: 2-4
+10
+Power Disk
+Power
+Palm
+Abducted
+VF3:
+VF3: Palm
+3
+27
+QuadPod
+Precision
+Pad
+Abducted VF3:
+VF3:
+3
+VF1:
+VF1: 1
+VF1:
+VF1: 1
+Power
+VF2: 2-5
+VF2: 2-5
+Sphere 3
+VF2: 2-3
+VF2: 2-3
+11
+Sphere
+Power
+Palm
+Abducted
+VF3:
+VF3: Palm
+3
+28
+Finger
+Power
+Pad
+Abducted VF3:
+VF3:
+3
+VF1:
+VF1: 1
+VF1:
+VF1: 1
+Precision
+VF2: 2-5
+VF2: 2-5
+Intermed
+VF2: 2
+VF2: 2
+12
+Disk
+Precision
+Pad
+Abducted
+VF3:
+VF3:
+29
+Stick
+iate
+Side
+Adducted VF3:
+VF3:3-53
+VF1:
+VF1: 1
+VF1:
+1,Palm
+Precision
+VF2: 2-5
+VF2: 2-5
+VF2: 2-5
+VF2: 2-5
+13
+Sphere
+Precision
+Pad
+Abducted
+VF3:
+VF3:
+3
+30
+Palmar
+Power
+Palm
+Adducted VF3:
+VF3:
+3
+VF1:
+VF1: 1
+VF1:
+VF1: 1
+VF2: 2-3
+VF2: 2-3
+VF2: 2
+VF2: 2
+14
+podul
+Precision
+Pad
+Abducted
+VF3:
+VF3:
+31
+Ring
+Power
+Pad
+Abducted VF3:
+VF3:
+2
+VF1:
+VF1: Palm
+VF1:
+VF1: 1
+VF2: 2-5
+VF2: 2-5
+Intermed
+VF2: 2
+VF2: 2
+15
+Fixed Hook
+Power
+Palm
+Adducted
+VF3:
+VF3:
+3
+32
+Ventral
+iate
+Side
+Adducted VF3:
+VF3: 3-5
+3
+VF1:
+VF1: 1
+VF1:
+VF1: 1
+Intermedia
+VF2: 2
+VF2: 2
+Inferier
+VF2: 2
+VF2: 2
+16
+Lateral
+te
+Side
+Adducted
+VF3:
+VF3: 3-5
+2
+33
+Pincer
+Precision Pad
+Abducted VF3:
+VF3:
+2
+Index
+心
+VF1:
+VF1: 1
+Finger
+VF2: 3-5
+VF2: 3-5
+17
+Extension
+Power
+Palm
+Adducted
+VF3: 2
+VF3: 2
+3Sensors 2022, 22, x FOR PEER REVIEW 
+7 of 20 
+ 
+ 
+dividual fingers by the analysis of the functional roles of forces being applied in a grasp. 
+The virtual fingers oppose each other in the grasp. Virtual fingers are assigned for each 
+grasp in the grasp taxonomy as mentioned in Table 1. Our characterization study re-
+vised the existing virtual fingers allocation and replaced with new as mentioned in Table 
+1. In this work, the authors aim for designing a three-finger haptic grasping interface for 
+virtual grasping the common objects in everyday life and tools in motor skill profes-
+sions. For the proposed three-finger haptic grasping interface, the thumb assigned as 
+𝑉𝐹1, Index finger as 𝑉𝐹2 and other three fingers as a single virtual finger 𝑉𝐹3 as ex-
+plained in section 5. 
+4. Characterization Experiments 
+4.1 Overview 
+For designing a multi-finger haptic grasping interface, it is quintessential to study 
+the characteristics of the motion and force distribution of fingers in grasping activities. 
+This characterization study helped to propose models for multi-finger haptic rendering 
+and grasping haptic devices. Here the authors conducted experiments for calculating the 
+finger movements, trajectories, positions, orientations on different grasping activities. 
+Also tracked the positions and orientations of the skeleton of each finger through motion 
+tracking techniques. A characterization study was carried out to compute the models 
+and the design of the multi-finger haptic device. 
+4.2 Subjects and Methods 
+Six subjects, 3 males and 3 females between the ages of 23 and 35 years were taking 
+part in this grasping experiment study with 10 common objects as listed in Table 2 with 
+possible grasp patterns and minimum number of virtual fingers to execute the grasp. 
+Each subject grasps each object for 5 seconds and repeats for 5 trials results total 300 in-
+stances for the data sets. This study aimed on the force distribution and orientation of 
+wrist, palm, and finger in all grasping scenarios. The experiment set up consists of a leap 
+motion sensor [61] to track the movement of fingers and a wearable glove with Force 
+Sensitive Resistor (FSR) [62] to measure the force on fingers while grasping different ob-
+jects and a computer display with the virtual grasping interface as shown in Figure 2. 
+ 
+Table 2. Selected objects with possible grasp patterns and minimum number of virtual fingers to 
+execute the grasp. 
+
+Index Finger Extension
+Prismatic 4 Finger
+Prismatic 3 Finger
+Prismatic 2 Finger
+n Sphere
+Tripod Variation
+[Parallel Extension
+Small Daimeter
+Medium Wrap.
+[Extension Type
+Adduction Grip
+sphere 4 Finger
+Sphere 3 Finger
+Grasp Pattern
+[large Diameter
+[Power Sphere
+Disk
+Lateral Tripod
+Inferier Pincer
+[Palmar Pinch
+Power Disk
+Fixed Hook
+Distal Type
+Adducted
+Light Tool
+Precision I
+Precision
+Tip Pinch
+padpeno
+[Lateral
+Palmar
+stick
+Ring
+min.num. of VF|3
+3
+2
+3
+3
+3
+3
+3
+3
+2
+3
+3
+3
+3
+3
+3
+2
+3
+3
+2
+3
+3
+3
+2
+2
+3
+3
+3
+3
+3
+3
+2
+3
+2
+Objects
+Ball Pen
+Marker Pen
+Cube box
+Toy Wheel
+1
+Cup
+Plastic bottle
+Tennis ball
+Credit card
+Scissors
+ScrewdriverSensors 2022, 22, x FOR PEER REVIEW 
+8 of 20 
+ 
+ 
+ 
+Figure 2. Experimental setup for characterization study: (a) Motion tracking setup, (b) Force track-
+ing setup for hand fingers during grasping and (c) 3d printed mounting module for FSR. 
+The Leap Motion sensor is a small USB peripheral device which is placed on the ta-
+ble surface and connected to computer interface. Subjects grasped the 10 objects in the 
+hemispherical workspace area of Leap motion sensor and traced the position and orien-
+tation parameters of user’s hand with an average accuracy of 0.7 mm [63]. The experi-
+ment set up for tracking grasping forces in fingers as shown in Figure 2(b), users wore a 
+glove with FSR to measure force and pressure. But the sensing ability of FSR is depend-
+ent on its contact area. To overcome this, a 3d printed mounting module placed on the 
+sensing area of the FSR in the fingertips of gloves as shown in Figure 2(c). The values 
+from FSR processed into multiple linear areas using an Arduino and measured the ex-
+erted forces in Newton(N). Experiment procedure in one trial consist of pick the object, 
+grasp for 5 seconds, and place back the object and repeat for five trials for each object. In 
+each trial subject’s grasp parameters such as the position, speed, orientation, and force 
+are tracked. Further analysis of these primary data leads to the modelling of multi-finger 
+grasping haptics interface.  
+4.3 Results and Discussions 
+The 30 primary parameters were tracked and used for calculating the user's hand 
+grasp movements and forces. Here the x-axis is defined as forward-backward, the y-axis 
+as right-left, and the z-axis as up-down. Roll (γ) is taken to be about the x-axis, pitch (β) 
+about the y-axis and yaw (α) about the z-axis. 
+4.3.1. Direction, Trajectory, and Rotation of hand in grasping 
+ 
+
+(a)Motiontrackingsetup
+(b)Forcetrackingsetup
+(c)3dprintedmountingmoduleforFSR(a)Directionofhand
+(b)Trajectoryofhand
+(c)Rotationofhand
+100
+Direction.x
+-0.4
+Trajectory
+Yaw
+Direction.y
+O
+Position
+Pitch
+Direction.z
+50
+Position (cm)
+0.5
+-0.6
+Angle (degree)
+Roll
+axis
+0
+N-0.8
+&
+-1
+-50
+-0.5
+1
+-100
+0
+0
+0.2
+-1
+-150
+0
+500
+1000
+1500
+Yaxis
+-1
+-0.4
+-0.2
+0
+500
+1000
+1500
+Time (cs)
+Xaxis
+Time (cs)
+(d)Boxplotofyawrotationofhand
+(e)Boxplotofpitchrotationofhand
+(f)Boxplotofroll rotationofhand
+十
+-20
+5
+Angle (degree)
+0
+Angle (degree)
+-40
+-5
+-60
+-10
+Angle
+20
+-80
+-15
+-100
+0
+-20
+-120
+-140
+1
+1
+1Sensors 2022, 22, x FOR PEER REVIEW 
+9 of 20 
+ 
+ 
+Figure 3. Plots for the movement of hand: (a) Direction of hand, (b) Trajectory of hand, (c) Rotation 
+of hand, (d) Box plot of yaw rotation of the hand, (e) Box plot of pitch rotation, (f) Box plot of roll 
+rotation of the hand. 
+The direction, trajectory, and rotation of hand in grasping (based on the experiment 
+data set) is plotted as shown in Figure 3. The direction of the Hand in the 3D co-ordinate 
+axes is shown in Figure 3(a). The x, y and z components of hand direction are spanning 
+from -3 mm to 1 mm, -2 mm to 8 mm, and -10 mm to -6 mm respectively with standard 
+deviation (SD) of 0.11, 0.20 and 0.08. The trajectory of hand movement direction is plot-
+ted in Figure 3(b). The movement in the y-direction is more than x and z-direction. The 
+ellipse region in the plot represents the trajectory of hand exactly in the grasping time. 
+Figure 3(c) plots the rotation of hand in the coordinate frame during grasping scenarios. 
+The span of the roll is more than yaw and pitch. The boxplot of hand rotation clearly ex-
+plains the distribution of the yaw, pitch, and roll of hand in grasping exercises shown in 
+Figure 3(d)-(f). The minimum angle of hand yaw is -22.61° and the maximum is -0.86° 
+with an average of -11.57° and SD of 6.7°. Inter Quartile Range (IQR) of hand yaw is 
+9.18°. The minimum angle of hand pitch is -11.06° and the maximum is 56.04° with an 
+average of 20.04° and SD of 13.09°. Inter Quartile Range (IQR) of hand pitch is 19.21°. 
+The minimum angle of hand roll is -136.69° and the maximum is -12.90° with an average 
+of -53.68° and SD of 32.29°. Inter Quartile Range (IQR) of hand roll is 50.14°.   
+4.3.2. Position and trajectory of wrist and palm in grasping 
+The wrist is acted as the basement for most of the grasping scenarios. The infor-
+mation about the movement of the wrist is helpful for characterizing the basement of 
+multi-finger gripper modules. The Position and trajectory of the wrist in grasping dur-
+ing the experiments were tracked and plotted as shown in Figure 4. Here the position of 
+wrist n the 3D coordinate frame shows the span of position in x, y, and z-axes. It is clear 
+from the plot that the span of the position of the wrist in grasping is average of 50 mm. 
+Also, the plot of the trajectory of wrist shows a minimum workspace of movement for 
+the wrist in grasping is in shape of a square with each side 50 mm. The thick ellipse re-
+gion represents the data exactly during the grasping scenarios after noise filtering. Scat-
+ter plots in Figure 4 gives a clearer picture of the position of the wrist in the 3d space and 
+2D planes (x-y, y-z, and x-z). 
+ 
+Figure 4. Plots for the movement of the wrist: (a) Position of the wrist, (b) Trajectory and scatter 
+plot of wrist, (c) Position of the wrist in the x-y plane, (d) Position of the wrist in y-z plane, and (e) 
+Position of the wrist in x-z plane. 
+
+(a)Positionofwrist
+(b)Trajectoryandscatterplotofwrist
+200
+Position.x
+Trajectory
+Position.y
+200
+Position
+150
+Position.z
+(mm)
+sixe
+150
+Position
+100
+N100
+50
+50
+120
+100
+80
+0
+60
+200
+600
+800
+1000
+1200
+80
+0
+400
+40
+60
+Time (cs)
+Yaxis
+20
+Xaxis
+(c)Position ofwrist inX-Yplane
+(d)PositionofwristinY-Zplane
+(e)PositionofwristinX-Zplane
+120
+160
+160
+O
+110
+00
+O
+140
+140
+100
+120
+O
+120
+axis
+90
+8
+N
+N
+80
+O
+80
+80
+O
+70
+60
+60
+O
+60
+40
+40
+30
+40
+50
+60
+70
+80
+60
+70
+80
+90
+100
+110
+120
+30
+40
+50
+60
+70
+80
+Xaxis
+Yaxis
+XaxisSensors 2022, 22, x FOR PEER REVIEW 
+10 of 20 
+ 
+ 
+Next, to the wrist, the palm is also an important part of better grasping. In some of 
+the grasping types, the palm acts as an additional supportive area for the successful 
+grasp. So, it is important to study the movement factors of palm while grasping. This 
+will help to model the minimal number of virtual fingers (VF) for a general haptic grasp-
+ing interface. The position and trajectory of the palm are plotted in Figure 5. It shows 
+that there is not much deflection in the position of palm while grasping time in three ax-
+es. The trajectory of the palm shows the workspace of palm during the grasping scenari-
+os. After filtering out the noise in the dataset, the maximum span in the 3D space for a 
+grasping activity is 30 mm. 
+ 
+Figure 5. Position and trajectory of palm: (a) Position of palm, (b) Trajectory and scatter plot of 
+palm, (c) Position of palm in the x-y plane, (d) Position of palm in the y-z plane, and (e) Position of 
+palm in the x-z plane. 
+4.3.3. Angle of grasp 
+The angle of grasp is one of the common measurements used to describe grasping. 
+Grasp angle describes the angle of the hand in grasping in relation to the Wrist position. 
+The grasp angle influences comfort and easiness in grasping. A little bend in the wrist 
+helps to maintain the grasp angle suitable for comfortable grasping of objects to get a 
+proper grip. This will aid controlling the gripped objects easier. Figure 6 shows the vari-
+ous analysis plots of grasp angle tracked during the experimental scenarios. 
+
+(a)Positionofpalm
+200
+(b)Trajectoryandscatterplotofpalm
+Position.x
+Position.y
+Trajectory
+150
+Position.z
+200
+Position
+(ww)
+100
+100
+axis
+Position
+50
+N
+0
+-100
+0
+200
+150
+50
+60
+-50
+100
+30
+40
+0
+200
+400
+600
+800
+1000
+1200
+50
+10
+20
+Time (cs)
+Yaxis
+Xaxis
+(c)Positionofpalm inX-Yplane
+(d)PositionofpalminY-Zplane
+(e)Positionofpalm inX-Zplane
+180
+150
+150
+160
+100
+100
+axis
+80
+8
+50
+50
+120
+N
+N
+O
+05
+O
+O
+0
+0
+100
+00
+O
+80
+-50
+-50
+10
+20
+30
+40
+50
+60
+80
+100
+120
+140
+160
+180
+10
+20
+30
+40
+50
+60
+Xaxis
+Yaxis
+XaxisSensors 2022, 22, x FOR PEER REVIEW 
+11 of 20 
+ 
+ 
+ 
+Figure 6. Plots for the angle of grasp: (a) Angle of grasp, (b) Box plot of grasp angle, (c) Grasp an-
+gle with a normal distribution, (d) Probability plot for normal distribution, and (e) Quantile-
+Quantile plot of grasp angle. 
+The angle of grasp is spanning from minimum 0°to maximum of 3.12° with an av-
+erage value of 1.33° and SD of 1.10 as shown in Figure 6(a). The grasp angle data are 
+charted as a box and whisker plot as shown in Figure 6(b). This will help to show the 
+shape of the distribution, its central value, and its variability. The first quartile of the 
+grasp angle values lies between 0° to 0.41°, second lies between 0.41° to 1.03°, third lies 
+between 1.03° to 2.44° and the final lies between 2.44° to 3.14°. 75% of the grasp angle is 
+below 2.44°. So, most of the grasp types can perform comfortably with a maximum angle 
+of grasp 2.44°. The ideal maximum angle of grasp for the proposed gripper module 
+should be between 2.5° to 4°. 
+Figure 6(c) plots a histogram of grasp angle in data using the number of bins equal 
+to the square root of the number of elements in data and fits a normal density function. 
+The bell curve fits the normal distribution with an SD of 1.10. 68% of the data falls with-
+in one SD of the mean 1.33°. The standard deviation controls the spread of the distribu-
+tion. Here the larger standard deviation indicates that the data is spread out around the 
+mean and the normal distribution is flat and wide. Figure 6(d) draws a normal probabil-
+ity plot, comparing the distribution of the grasp angle data to the normal distribution. 
+The plot includes a reference line helped to judge whether the data follow a normal dis-
+tribution. The plot shows that the normal line fit the data except the tails because of the 
+outliers. Figure 6(e) displays a quantile-quantile plot of the sample quantiles of grasp 
+angles versus theoretical quantiles from a normal distribution. The plot is close to linear 
+in the IQR, so the distribution of grasp angle is normal during the grasping. In the dura-
+tion of not grasping the plot shows the distribution is not normal. 
+4.3.4.Sphere of grasp 
+The spherical grip is the most used grasp in everyday life [64]. It is important to an-
+alyze the sphere of grasp in common grasping scenarios. Here the authors tracked the 
+center and radius of the sphere of grasp in all the grasping scenarios in the experimental 
+procedure. Figure 7 shows the various plots related to center and radius of the sphere of 
+grasp. Through the experiment, the position of the center of the sphere is spanning from 
+minimum (-147 mm, 48 mm, -48 mm) to maximum (181 mm, 259 mm, 85 mm) in x, y, 
+and z-axes. After filtering out the non-grasping samples, the span of sphere grasp is re-
+duced from minimum (-19 mm, 42 mm, -3 mm) to maximum (68 mm, 150 mm, 78 mm). 
+The scattering and trajectory of the sphere of grasp shown in Figure 7(b). 
+
+(a) Angle of grasp
+(b)Boxplotofgraspangle
+(c)Graspanglewithnormaldistribution
+4
+200
+Grasp angle
+3
+Normal distribution
+150
+degr
+100
+Angle
+50
+0
+0
+0
+0
+200
+400
+600
+800
+1000
+1200
+-2
+0
+2
+4
+6
+Time (cs)
+Angle (degree)
+(d)Probabilityplotfornormaldistribution
+(e)Quantile-Quantileplotofgraspangle
+0.9999
+8
+6
+Data
+Quantiles of Input Sample
+Duration of not grasping
+0.995
+Normal
+6
+Inter Quartile Range
+%%
+十
+Data
+Probability
+4
+0.75
++
+0.5
+2
+0.25
+0.05
+0
+0.005
+-2
+.0005
+0.0001
+0
+1
+2
+3
+4
+-2
+0
+2
+4
+Angle (degree)
+StandardNormalQuantilesSensors 2022, 22, x FOR PEER REVIEW 
+12 of 20 
+ 
+ 
+ 
+Figure 7. Center and radius plots for sphere of grasp: (a) Position of center of sphere of grasp, (b) 
+Trajectory and scatter plot of center of sphere, (c) Radius of sphere of grasp, (d) Histogram of 
+sphere radius with normal distribution, and (e) Box plot of sphere radius. 
+The volume of grasp can be represented by analyzing the radius of the sphere of 
+grasp. Figure 7(c)-(e) shows the plots for the radius of the sphere of grasp. In the whole 
+data tracked during the experiment, the radius of the sphere of grasp spanning from 
+minimum 30 mm to 188 mm. In the box plot showing in Figure 7(e) 75% of the data is 
+less than 10 mm and 50% of data is 64 mm. Again, after filtering out the noises the radi-
+us of the sphere of grasp is spanning from 31 mm to 50 mm. So, the authors targeting to 
+model the grasping module with a radius of the sphere of grasp is 50 mm.  
+4.3.5.Distance of Pinch 
+ 
+Figure 8.  Plots for the distance of pinch: (a) Distance of pinch, (b) Box plot of pinch distance, (c) 
+Pinch distance with a normal distribution, (d) Probability plot for normal distribution, and (e) 
+Quantile-Quantile plot of pinch distance. 
+The pinch distance is the distance between two fingers in grasping. Pinch gestures 
+are common for touch screens. The experiment interface tracked the pinch distance 
+
+(a)Positionofcenterofsphereofgrasp
+(b)Trajectoryandscatterplotofcenterofsphere
+300
+Center.x
+100
+Trajectory
+Center.y
+Position
+200
+Center.z
+50
+(mm)
+axis
+100
+Position
+N
+0
+0
+-50
+400
+-100
+200
+-200
+50
+100
+150
+200
+0
+500
+1000
+1500
+Yaxis
+0
+-150
+-100
+-50
+0
+Time(cs)
+Xaxis
+(c)Radiusofsphereofgrasp
+(d)Histogramofsphereradiuswithnormaldistribution
+(e)Boxplotofsphereradius
+250
+150
+data
+bellcurve
+200
+200
+Radius (mm)
+100
+150
+Radius
+100
+100
+50
+50
+50
+0
+0
+0
+500
+1000
+1500
+-100
+0
+100
+200
+300
+Time (cs)
+Radius (mm)(a) Distance of pinch
+(b)Boxplotofpinchdistance
+(c)Pinchdistancewithnormaldistribution
+120
+80
+Pinch distance
+Frequency of occurrence
+100
+100
+Normal distribution
+(ww)
+60
+80
+Distance (mm)
+80
+Distance
+60
+60
+40
+40
+40 
+20
+20
+0
+0
+200
+400
+600
+800
+1000
+1200
+-100
+-50
+0
+50
+100
+150
+200
+time (cs)
+Distance (mm)
+(d)Probability plotfornormal distribution
+(e)Quantile-Quantileplotofpinchdistance
+0.9999
+300
+Data
+Duration of not grasping
+Quantiles of Input Sample
+X
+Normal
+Inter Quartile Range
+0.995
+200
+Data
+%%
+Probability
+0.75
+100
+0.5
+0.25
+0
+%
+9.005
+-100
+0.0005
+0.0001
+-200
+0
+50
+100
+150
+200
+-4
+-2
+0
+2
+4
+Pinch distance (mm)
+StandardNormalQuantilesSensors 2022, 22, x FOR PEER REVIEW 
+13 of 20 
+ 
+ 
+throughout the experiment and various plots are shown in Figure 8. The maximum 
+pinch distance traced in the experiment setup is 110 mm. 75% of data is less than 90 mm 
+and 50% is less than 70 mm as per the boxplot showed in Figure 8(b). The normal proba-
+bility plot is shown in Figure 8(d) compared the distribution of the pinch distance to the 
+normal distribution. The reference normal line fits the data in a range of 20 mm – 100 
+mm. Figure 8(e) displays a quantile-quantile plot of the sample quantiles of grasp angles 
+versus theoretical quantiles from a normal distribution. This quantile-quantile plot is 
+close to linear in the IQR, so the distribution of pinch distance is normal during the 
+grasping. In the duration of not grasping the plot shows the distribution is not normal.  
+So, the ideal values for the radius of the sphere of grasp for our proposed model are 
+from 20 mm to 100 mm. 
+4.3.6.Finger motion parameters 
+When a person lifts any object, his fingers align in a particular way. This must be re-
+flected by the device so that the user is at ease when he uses the device. The experiment 
+setup traced the length and width of fingers and angular values between fingers in 
+grasping objects differ in size and dimensions. These angular values measured between 
+fingers must be replicated by the device. Unless the user feels comfortable when using 
+the device, its intended purpose cannot be met. A base point was marked at the center of 
+the outside of the palm. A line was drawn from the base point to the base of the middle 
+finger. This was the baseline at 0 degrees as seen in Figure 9(a). Lines were also drawn to 
+the base of the index and the thumb. The angles between the thumb and the index finger 
+and the angles between the middle and the index fingers were measured during the 
+grasping scenarios.  
+ 
+Figure 9.  Length, width, and angle of fingers. 
+The average length and width of fingers of the subjects took part in the experiment 
+shown in Figure 9(b). The average length and width of Thumb are 50 mm and 18 mm, 
+the index finger is 57 mm and 17 mm, the middle finger is 67 mm and 17 mm, ring finger 
+is 61 mm and 16 mm, and little finger is 48 mm and 14 mm. This will help the authors to 
+model the wearable multi-finger grasping interface to fit the user’s fingers. The meas-
+ured angles were tabulated and plotted as shown in Figure 9(c), proves that angular 
+measurements between fingers in grasping and lifting of objects does not depend on 
+gender. The angle between thumb and index finger is in range of 50-60degree, index and 
+middle finger is in range of 20-30 degree and thumb and middle finger is in range of 70-
+
+(a)Length,widthandangleoffingers
+(b)AverageLengthandwidthoffingersofsubjects
+ODegrees
+Length
+Width
+FingerLength
+70
+Length & Width (mm)
+60
+30Degrees
+FingerWidth
+30
+20
+10
+Thumb
+Thumb
+Index
+Middle
+Ring
+Little
+Range
+(c)Anglesbetweenfingers
+100
+90
+80
+70
+60
+90Degrees
+50
+40
+30
+20
+10
+0
+Average
+Maximun
+Minimum
+Average
+Maximum
+Minimum
+Angle
+Angle
+Men
+Women
+middle andindex
+IndexandThumb
+mMiddleandThumbSensors 2022, 22, x FOR PEER REVIEW 
+14 of 20 
+ 
+ 
+90 degree. This provided the data on how far apart the finger holders must be placed 
+when designing the prototype for comfortable grasping and lifting of objects. 
+ 
+Figure 10. Position of movement of fingers 
+The motion parameters of fingers in grasping are so important to characterize and 
+model the multi-finger grasping module. Here The authors traced all the five-finger’s 
+movement and force distributions. The movement of the tip of five fingers of the hand in 
+all the grasping scenarios during the experiment was plotted in Figure 10. The move-
+ment of the fingertip is minimal in both x and z-axes is around -50 mm to +50 mm. The 
+movement in x and z-axes are varied for fingers, especially it is very less in case of 
+thumb, ring finger and little finger in the range of -20 mm to +20 mm. For index and 
+middle finger, it is an almost same range of -50 mm to +50 mm. The movement in the y-
+axis is more compared to the other axes is around 50 mm to 200 mm for all fingers.  
+ 
+Figure 11. (a). Span of the trajectory of fingers and (b) Workspace triangle of three Virtual Fingers. 
+The span of the trajectory of all five-finger movement in space during the grasping 
+experiment is plotted together in Figure 11(a). This plot gave a clear idea about the ac-
+tive participation of all the fingers during grasping. Thumb is more active in grasping 
+and gradually decreasing towards little finger. The thump, index and middle fingers 
+more actively participate in the grasping scenarios than the ring and little fingers. Also, 
+
+(a)Position of thumb
+(b)Position of indexfinger
+(c) Position of middle finger
+300
+200
+200
+Position (mm)
+200
+Positions (mm)
+Position (mm)
+100
+100
+100
+-100
+-100
+-100
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+Samples
+Samples
+Samples
+(d) Position of ring finger
+(e) Position of little finger
+150
+150
+Tip position.x
+Tip position.y
+Position (mm)
+100
+Position (mm)
+100
+Tip position.z
+50
+50
+0
+-50
+-50
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+Samples
+Samples(a)Spanoftrajectoryoffingermovement
+(b)WorkspacetriangleofthreeVirtualFingers
+Thumb
+Thumb
+100
+Index finger
+100
+Index finger
+Middle finger
+Middle finger
+Ring finger
+80
+Littlefinger
+axis
+0
+60
+N
+-100
+40
+220
+axis
+20
+200
+N
+0.
+180
+-20
+160
+-40
+axis
+140
+120
+-60
+250
+100
+200
+150
+80
+100
+100
+60~
+60
+80
+50
+50
+20
+40
+Yaxis
+0
+40
+-20
+0
+0
+-50
+Xaxis
+-60
+-40
+XaxisSensors 2022, 22, x FOR PEER REVIEW 
+15 of 20 
+ 
+ 
+the workspace of the ring and little finger aligns with the middle finger as shown in Fig-
+ure 11(a).  So, authors proposed to group middle, ring, and little fingers to one Virtual 
+Finger (VF) for multi-finger grasping interfaces. Thumb and index fingers can act as oth-
+er two Virtual Fingers. The workspace triangle for three VF is shown in Figure 11(b).  
+4.3.7 Grasping forces on Fingers 
+The force values obtained through FSR sensors during the grasping exercises were 
+tabulated and the average force was calculated for each person. As shown in Figure 12, 
+forces are not more than 10N was experienced by the user. It can also be seen that the 
+thumb experienced the maximum force, whereas the middle and the index finger expe-
+rienced somewhat similar forces.  
+ 
+Figure 12. Average grasping force values on each finger. 
+5. Computational Model for Tripod Haptic Grasp 
+The multi-finger perception studies confirmed the hypothesis that the minimal con-
+figuration that allows most of the grasps is the three-finger tripod grasp. Based on the 
+human grasp analysis, a conceptual, computational model is presented here for the 
+three-finger tripod haptic grasping interface. Previous researchers worked on force 
+models [65] and virtual linkages [66] for multi-grasp manipulations. 
+The concept of the virtual finger [15] has been postulated as an abstract representa-
+tion through which the human brain plans are grasping tasks [60]. The virtual finger is a 
+functional unit of several fingers work together comprised of at least one real physical 
+finger (which may include the palm). This effectively reduces the many degrees of the 
+human hand to those that are deemed necessary to perform the grasping task. This con-
+cept replaces the analysis of the mechanical degrees of freedom of individual fingers by 
+the analysis of the functional roles of forces being applied in a grasp. Here the concept of 
+the virtual finger was implemented to reduce the realistic five-finger grasping to virtual 
+three-finger grasping. The characterization study revised the existing virtual fingers al-
+location and replaced with new tripod virtual fingers allocation.  
+
+Average Force (N) on Each Finger
+9
+8
+6
+5
+4
+3
+2
+1
+0
+7
+2
+3
+4
+5
+6
+7
+8
+10
+Thumb
+indlex
+MiddleSensors 2022, 22, x FOR PEER REVIEW 
+16 of 20 
+ 
+ 
+ 
+Figure 13. Haptic computation model for three-finger tripod haptic grasping interface. 
+The haptic computation model is shown in Figure 13 was implemented to create the 
+suitable force feedback for the tripod haptic grasping interface. Each finger holder is 
+connected to each slider and these sliders are responsible for creating forces to each fin-
+ger attachment point on the grasping interface through the closed-loop belt system. 
+Three proxies and Haptic Interactive Points (HIP) are assigned for three fingers. Based 
+on the position information received from each slider, the position of proxy and HIP are 
+updated. Collision detection algorithms detected the collision of each fingertip with the 
+virtual object, these collision points act as simple virtual walls. Then applying the God 
+object algorithms to these three proxies and calculated the resultant forces as shown in 
+Figure 13.  Each actuator in the slider generates forces and these forces are transferred to 
+the fingertip through the attached finger holder. In this proposed model for multi-finger 
+tripod haptic grasping, the thumb is assigned as 𝑉𝐹1, Index finger as 𝑉𝐹2 and other three 
+fingers as a single virtual finger 𝑉𝐹3. The proposed model including object and virtual 
+fingers in the virtual interface measures collision and progress of interactions in terms of 
+applying forces and movements of the fingers and computes force feedback to be pro-
+vided by the haptic interfaces. It covers the dependence of the force feedback and the ef-
+fect of finger motions on the tripod grasp. A basic grasping touch in virtual reality is de-
+fined as a touch that provides vertical force feedbacks on the contact surface coincident 
+with the reverse direction of the fingers in the space. 
+Forces and moments from the individual virtual fingers are considered for the ren-
+dering of resultant forces for the model. Three proxies and Haptic Interactive Points 
+(HIP) are assigned for three virtual fingers. Based on the position information received 
+from each 𝑉𝐹𝑖, the position of proxy and HIP are updated. When collision detection al-
+gorithms detected the collision of each 𝑉𝐹𝑖 with the virtual object, these collision points 
+act as simple virtual walls. The authors assume that only forces act at the grasp points. 
+Let 𝑢𝑖𝑗 is the unit vector along the virtual finger i to j, 𝑣𝑖 be the vector from the object ref-
+erence point o to the virtual finger grasp point. Let 𝐹𝑖 be the force exerted on objects 
+through each virtual finger 𝑉𝐹𝑖 and F be the vector. Based on the object shape, size, and 
+stiffness the applying forces on objects by the virtual fingers are different. 
+𝐹𝑖 = 𝑘 𝑋𝑖                                    
+(1) 
+where i = 1,2,3.                                       
+ 
+                      𝐹 = [𝐹1
+𝐹2
+𝐹3]𝑇 
+ 
+(2) 
+
+Slider3
+Slider2
+R2=-k.X2
+R3=-k.X3
+X3
+X2
+VirtualFinger3
+F3
+VirtualFinger2
+F2
+HIP
+Proxy
+F1
+X1
+VirtualFinger1
+R1=-k.X1
+Slider1Sensors 2022, 22, x FOR PEER REVIEW 
+17 of 20 
+ 
+ 
+Grasp perception rate depends on the force applied by the fingers 𝐹𝑖, the area of 
+contact between the fingers and the object A, the distance between the proxy and HIP X 
+and the change in position of the HIP ΔX. Mathematically, the perception rate is 
+ 𝑃 =
+𝑘 𝛥𝑋 𝐹𝑖
+𝐴
+  
+ 
+ 
+               (3) 
+where k is a constant that depends on the material of the object. Let 𝑅𝑖 be the result-
+ant force exerted on each virtual finger 𝑉𝐹𝑖 and R be the vector. Then applying the God 
+object algorithms [67] to three proxies and calculated the resultant forces 𝑅𝑖.   
+ 𝑅 = [𝑅1
+𝑅2
+𝑅3]𝑇 
+ 
+               (4) 
+The relationship between the applied forces F and resultant forces R would be giv-
+en by 
+ 𝑅 = 𝑢𝐹 
+ 
+ 
+ 
+ 
+ 
+ 
+(5) 
+were 
+ 
+ 𝑢 = [
+𝑢11
+𝑢12
+𝑢13
+𝑢21
+𝑢22
+𝑢23
+𝑢31
+𝑢32
+𝑢33
+]  
+               (6) 
+The relationships between the resultant force R, the resultant moment m, and the 
+applied forces F are given by 
+ 𝐹 = 𝑣 [𝑅
+𝑚]𝑇  
+ 
+               (7) 
+ 
+were  
+ 𝑣 = [𝑣1
+𝑣2
+𝑣3]𝑇 
+ 
+               (8) 
+and  
+ 𝑚 = [𝑚1
+𝑚2
+𝑚3]𝑇  
+               (9) 
+Force feedback on the grasping scenarios along virtual fingers has two components; 
+the force feedback resisting the grasping motion and a frictional force (𝐹𝑟), represented 
+by 
+ 𝑅 = 𝑢𝐹 + 𝐹𝑟 
+ 
+ 
+              (10) 
+ 
+An experiment was done to evaluate the rendered force to the fingers through the 
+virtual fingers using the proposed tripod haptic grasp model. The results plotted as 
+shown in Figure 14. Initially, the grasping forces in the real grasping cases were tracked 
+using the FSR and plotted as shown in Figure 14(a). In case of the real grasping scenari-
+os, the forces exerted by the thumb is in the range of 6-8N which is greater than the force 
+exerted by the index and middle finger. The force exerted by the index and middle fin-
+gers in in the range of 2-5N and 3-5N respectively. The same experiment was carried out 
+to measure the rendered forces in haptic grasping interfaces during virtual grasping.  
+Force is measured using the FSR sensor. The FSR was placed inside the finger holder of 
+the gripper at the location of the fingertip contact. The sensing part of FSR facing the flat 
+area of the holder and the user keeps a finger on top of the FSR. The motors were actuat-
+ed by the haptic rendering model. The tracked rendered forces were plotted in Figure 
+14(b). This graph shows that the range of rendered forces in virtual grasping is almost in 
+the same range of forces in real grasping scenarios. The range of force rendered at the 
+Thumb, index and middle finger holders is 5.7-7.5N, 2.6-4.4N and 3-4.5N respectively. 
+This demonstrates that the three-finger haptic grasping interfaces able to provide realis-
+tic grasping haptic feedbacks to the users. Also, the experiment setup was traced the 
+peak force that can provide the gripper device. As plotted in Figure 14(c), the device at-
+tained a peak force of 10.4N, 10.1N, and 10.2N at Thumb, index, and middle finger hold-
+ers respectively. It was found that the interface can give approximately 10N force which 
+is greater than the force when lifting objects as found out in Section 3.1. Finally, the Fig-
+
+Sensors 2022, 22, x FOR PEER REVIEW 
+18 of 20 
+ 
+ 
+ure 14(d) shows the performance index of three-finger haptic grasping interface in haptic 
+feedback and finger manipulability.  
+ 
+Figure 14. Evaluation results of the haptic grasping interface: (a) Exerted force tracked in the real 
+grasping scenarios, (b) Force rendered in haptic grasping interface during virtual grasping, (c) 
+Peak force generated in the haptic grasping interface, and (d) Performance index of the haptic 
+grasping interface. 
+6. Conclusions 
+The aim of this research was to come up with an analysis, model, and design of a 
+three-finger haptic interface. A detailed literature review was carried out regards the 
+anatomy, kinematics, and dynamics of hand. Also, a focused survey on the different 
+human grasps, prehension patterns and grasp taxonomy. As part of this work, authors 
+carried out characterization studies in most of the major aspects of human grasping. The 
+position, orientation, and forces of hand, wrist, palm, and fingers were analyzed, and the 
+results were discussed. The characterization studies confirmed the hypothesis that the 
+minimal configuration that allows most of the grasps in grasp taxonomy is the three fin-
+gers grasping. This detailed characterization study leads to the design of a three-finger 
+haptic grasping interface as an extension.  
+As a future work, I am planning to work on the multi-Finger grasping interface for 
+bimanual scenarios which provides the users a complete immersed grasping manipula-
+tion in the virtual and remote environment.  
+ 
+Funding: This research received no external funding.  
+Institutional Review Board Statement: Not applicable. 
+Informed Consent Statement: Not applicable. 
+Data Availability Statement: Not applicable. 
+Acknowledgments: I wholeheartedly thank the wonderful team at AMMACHI Labs for their 
+support, encouragement and for providing constructive criticism and valuable inputs during the 
+various stages of this work. 
+Conflicts of Interest: The author declares no conflicts of interest. The funders had no role in the 
+design of the study; in the collection, analyses, or interpretation of data; in the writing of the man-
+uscript; or in the decision to publish the results. 
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+Holographic inflation in f(R, T) gravity
+S. Taghavi, Kh.
+Saaidi,∗ and Z. Ossoulian
+Department of Physics, Faculty of Science, University of Kurdistan, Sanandaj, Iran.
+(Dated: January 9, 2023)
+We study the possibility of having HDE as the source of inflation in the frame of f(R, T) gravity
+theory. The length scale of the energy density is taken as GO cutoff which is a combination of the
+Hubble parameter and its time derivative. It is found that for specific ranges of the free parameters
+of the model, the scalar spectral index and the tensor-to-scalar ratio stand in good agreement with
+data. Also, by reconstructing the potential in terms of the scalar field the validity of the swampland
+criteria is considered. The results indicate that to have scalar field below the Planck mass, the
+parameter λ should be at least of the order of O(102). The same order of λ is also required to keep
+the gradient of the potential greater than one and satisfy the second conjecture of the swampland
+criteria.
+I.
+INTRODUCTION
+The modified versions of Einstein’s gravity have received tremendous attention and have been used for studying
+different cosmological and astrophysical phenomena. One of these modified gravity theories is f(R, T) gravity which
+was introduced in [1], where R is the Ricci scalar and T is the trace of the energy-momentum tensor. The theory
+replaces the Ricci scalar in Einstein-Hilbert action with an arbitrary function of R and T, indicating a non-minimal
+coupling between the geometric and matter parts. Different topics including dark energy[2], dark matter [3], worm-
+holes [4], gravitational waves [5], inflation [6–8], have been studied in the frame of f(R, T) gravity.
+One of the challenges in cosmological studies is understanding the true nature of dark energy. It is assumed to be
+the main reason for having an accelerated expansion phase for the current universe which is supported by much
+data. [9–14]. Different candidates for dark energy have been introduced such as cosmological constant, quintessence,
+tachyon, and so on (refer to [15] for a review of different models of dark energy). Another candidate is holographic
+dark energy (HDE) which recently becomes one of the most favorite ones [16–18]. The idea of HDE stands on the
+holographic principle [19] which states that the number of degree of freedom of a physical system is scaled with its
+bounding area rather than its volume [19–22]. It was first put forth by Cohen and colleagues [23] that in quantum
+field theory, due to the limit set by the formation of black holes, a short distant cutoff is related to a long distant
+cutoff. It suggests that the total energy in a region of size L should not exceed the mass of a black hole of the same
+size. Taking this assumption the HDE density was introduced as ρ = 3c2M 2
+p/L2 [18], where Mp = 8πG is the reduced
+Planck mass and c2 is a dimensionless positive constant. The infrared cutoff length L, which is our main concern in
+HDE, is usually selected as Hubble length, particle horizon, future event horizon, Ricci scalar curvature, and GO1
+cutoff [24, 25]. The HDE mostly has been used for studying the late time evolution of the universe as a possible
+reason for the current accelerated expansion [18, 26–29]. However, recently, the application of HDE as the origin of
+the early expansion of the universe, i.e. as a source of inflation, is rising [30–32].
+The universe is assumed to go through another phase of accelerated expansion at very early times, known as the
+inflationary phase. The scenario has received a tremendous amount of interest and has been modified on different
+levels [33–66]. In a common procedure, it is driven by a scalar field that stands on top of its potential and slowly rolls
+down toward the minimum of the potential [67–74]. Due to the slowly varying, the scalar field gets a kinetic energy
+density which is ignorable compared to the potential energy density. The scenario has been investigated for different
+types of the potential, mostly inspired from particle physics, in different gravity frames including f(R, T) gravity. A
+different approach, which has received interest recently, is to assumed that the dark energy which drives inflation is
+HDE [30–32]. Different type of the entropies with different length scales were studied in this matter however, up to
+our knowledge, no such formalism has been considered in f(R, T) gravity theory. Here, we are going to consider the
+HDE inflation in the frame of f(R, T) gravity theory. In this regard, the energy density is assumed to be originated
+from Tsallis entropy [75–78] addressed as Tsallis HDE2 (THDE). Since, during the slow-roll inflation, the total energy
+density is approximated by the potential term, we take the potential as the THDE with GO cutoff as the length
+∗ [email protected]
+1 stand for Granda and Oliveros
+2 It was shown by Tsallis and Cirto [79] that the entropy of a black hole should be generalized to S = γAδ where A stands for the area of
+the black hole, δ is the Tsallis parameter (or nonextensive parameter), and γ is an unknown parameter. Besides, the quantum gravity
+suggests a power-law function of the entropy inspired by Tsallis entropy [80, 81]
+arXiv:2301.02631v1  [gr-qc]  5 Jan 2023
+
+2
+scale. Then, the potential is determined in terms of the Hubble parameter instead of the usual approach where it
+is specified in terms of the scalar field. Applying the observational data and Python coding, the ranges of the free
+parameters of the model are determined. Using the result, the potential in terms of the scalar field is reconstructed
+and we find our way to examine the validity of the swampland criteria3.
+The paper is organized as follows: the f(R, T) gravity and its main dynamical equation will be presented in Sec.II in
+brief. In Sec.III, the scenario of the slow-roll inflation is introduced, and by applying the slow-roll approximations, the
+dynamical equations get simplified. Then, in Sec.IV, we take the potential equal to a THDE density with GO cutoff
+as the length scale. The perturbation parameters are estimated and by comparing with data, the free parameters of
+the model are determined. Then, in Sec.V, the potential is reconstructed versus the scalar field and the validity of
+the swampland criteria is studied. Finally, the results are summarized in Sec.VI.
+II.
+f(R, T) GRAVITY THEORY
+The f(R, T) gravity is a modified theory of gravity where the Ricci scalar, R, in the Einstein-Hilbert action is
+replaced by an arbitrary function of R and T, where it is the trace of the energy-momentum tensor. The general form
+of the action is given by [1]
+S =
+1
+2κ2
+�
+d4x√−g
+�
+f(R, T) + Lm
+�
+(1)
+where f(R, T) is an arbitrary function of R and T, the determinant of the metric gµν is given by g, the Lagrangian
+of the matter fields is indicated by Lm, and κ is related to the Newtonian gravitational constant as κ2 = 8πG.
+The field equation of the theory is obtained by taking variation of the above action with respect to the metric, which
+leads to [1]
+�
+gµν□ − ∇µ∇ν
+�
+f,R(R, T) + f,RRµν − 1
+2gµνf(R, T) = κ2Tµν − f,T (R, T)
+�
+Tµν + Θµν
+�
+,
+(2)
+where the tensor Θµν is defined as
+Θµν = gαβ δTαβ
+δgµν .
+(3)
+In the case of a perfect fluid with energy density ρ, pressure p, and four velocity uµ, the energy-momentum tensor is
+read as
+Tµν = (ρ + p)uµuν − pgµν
+(4)
+and the tensor Θµν is obtained from the definition (3) as
+Θ = −Tµν − pgµν.
+(5)
+Assuming a spatially flat FLRW metric and by taking f(R, T) = R + κ2λT, where λ is a constant, the Friedmann
+equations are acquired as
+H2 = κ2
+3
+��3
+2λ + 1
+�
+ρ − λ
+2 p
+�
+,
+(6)
+−3H2 − 2 ˙H = κ2
+�
+−λ
+2 ρ +
+�3
+2λ + 1
+�
+p
+�
+,
+(7)
+Substituting Eq.(6) in (7), one arrives at
+− 2 ˙H = κ2 (1 + λ) (ρ + p).
+(8)
+The conservation equation is achieved by taking the time derivative of Eq.(6) and using Eq.(8) as
+˙ρ + 3H(ρ + p) = −3λ
+2
+˙ρ + −λ
+2 ˙p − 3Hλ(ρ + p)
+(9)
+It is seen that the conservation equation is modified due to the term λT that implies a non-minimal coupling between
+matter and curvature. However, it comes back to the standard conservation equation by putting λ = 0.
+3 The swampland criteria, proposed in [82–84], is originated from string theory and it is a mechanism to separate the consistent low-energy
+effective field theory (EFT) from the inconsistent ones. Since inflation occurs at a low-energy scale, then, it is expected to be described
+by a consistent low-energy EFT.
+
+3
+III.
+SLOW-ROLL INFLATION
+Inflation is usually provided by an scalar field, known as inflaton, which is the dominant component of the universe
+at the time. The Lagrangian of the scalar field is read as
+Lφ = 1
+2∂µφ∂µφ − V (φ),
+(10)
+where V (φ) is the potential of the scalar field. The energy density and pressure of the scalar field is given by
+ρφ = 1
+2
+˙φ2 + V (φ)
+(11)
+pφ = 1
+2
+˙φ2 − V (φ).
+(12)
+By substituting above energy density and pressure in the Friedmann equations (6) and (8), one has
+H2 = κ2
+3
+�1
+2 (1 + λ) ˙φ2 + (1 + 2λ) V (φ)
+�
+(13)
+−2 ˙H = κ2(1 + λ) ˙φ2.
+(14)
+Also, the equation of motion of the scalar field takes the following form
+(1 + λ) ¨φ + 3H(1 + λ) ˙φ + (1 + 2λ)V ′(φ) = 0,
+(15)
+which is obtained by inserting Eq.(11) in Eq.(9).
+The conditions required for the slow-roll inflation are
+| ˙H| ≪ H2,
+˙φ2 ≪ V (φ),
+|¨φ| ≪ |H ˙φ|
+which is known as the slow-roll conditions. Applying the conditions on the Friedmann equations above, leads one to
+H2 = κ2
+3 (1 + 2λ) V (φ),
+(16)
+−2 ˙H = κ2(1 + λ) ˙φ2,
+(17)
+3H ˙φ = 1 + 2λ
+1 + λ V ′(φ).
+(18)
+The slow-roll conditions are encoded in the slow-roll parameters. The first slow-roll parameter states that the rate
+of the Hubble parameter during a Hubble time will be small, i.e. ϵ = − ˙H/H2. This condition provides a quasi-
+de Sitter expansion. A common approach to introduce other slow-roll parameters are through a hierarchy, namely
+ϵn+1 = ˙ϵn/Hϵ for n ≥ 1.
+To proceed further, it is required to determine the potential in terms of the scalar field. There are different types
+of the potential used to study the scenario. Chaotic potential, Hilltop potential, natural potential could be addressed
+as some of these potential.
+In the next section, we are going to follow a different path and propose a suitable function in terms of the Hubble
+parameter as the potential.
+IV.
+THDE AS THE POTENTIAL
+The early accelerated expansion of the universe is assumed to be provided by a dynamical scalar field, which is
+also a dark energy candidate. The scalar field is taken as the dominant component of the universe which governs the
+evolution. On the other hand, the kinetic energy of the scalar field could be ignored compared to the potential part
+and the energy density of the scalar field is approximated by the potential part, i.e. ρφ ≈ V (φ). Different types of
+the potential in terms of the scalar field have been introduced and studied in the literature. Another competent dark
+energy candidate is HDE. After prosperous application of HDE for the late time evolution of the universe, recently
+there has been a raising interest for taking it as the source of inflation. Here, we are going to change our prospective
+a little and take it as the potential of the scalar field, which is actually the main part of the energy density during
+
+4
+the inflationary times.
+The shape of the HDE depends on the entropy of the system which based on the holographic principle is scaled
+with the area rather than the volume[19–22]. Including the quantum correction, the standard entropy is modified
+in different ways. One of the modification is achived by following this argument that the thermodynamics of the
+gravitational and cosmological systems must be modified to the non-additive entropy. Based on this, Tsallis and
+Citro obtained that the entropy of the black hole should be altered to S = γAδ. The resulted HDE density from the
+entropy is ρ = Bc2L2δ−4 where B ≡ γ(4π)γ and L is the infrared cutoff. One of the candidate for the cutoff is GO
+cutoff which contains the time derivative of the Hubble parameter in addition to the square of the Hubble parameter,
+i.e. L−2 = αH2 + β ˙H.
+Following the above approach and taking the HDE density as the potential of the scalar field, the Friedmann equation
+(16) is rewritten as
+H2 = κ2
+3 (1 + 2λ) Bc2�
+αH2 + β ˙H
+�2−δ.
+(19)
+Working with the equation, one finds the term ˙H/H2, which is also the definition of the first slow-roll parameter.
+Then, we have
+ϵ1 = − ˙H
+H2 = 1
+β
+�
+α − AHξ�
+(20)
+where the defined constants A and ξ respectively are given as
+ξ ≡ 2δ − 2
+2 − δ ,
+A ≡
+�
+3M 2
+p
+Bc2(1 + 2λ)
+�
+1
+2−δ
+.
+To introduce the second slow-roll parameter, the hierarchy procedure is used. Then, the parameter is given by
+ϵ2 =
+˙ϵ1
+Hϵ1
+= ξ
+β AHξ.
+(21)
+Solving the problems of the hot big bang theory requires having enough amount of inflation. which is measured by
+the number of e-folds, defined as
+N =
+� te
+t⋆
+H dt =
+� He
+H⋆
+H
+˙H
+dt,
+(22)
+where the subscribes ”e” and ”⋆” respectively indicate that the parameters are estimated at the end of inflation and
+the horizon crossing time. Solving the integral and after some manipulation, the Hubble parameter at the time of
+horizon crossing is obtained as
+AHξ
+⋆ =
+α(α − β) e
+αξN
+β
+β + (α − β) e
+αξN
+β
+.
+(23)
+Utilizing the H⋆, the parameters could us estimated at the time of horizon crossing and it will allow us to compare
+the model with data. In this regard, we first need to acquired the perturbation parameters.
+Following [6–8], the scalar spectral index and the tensor-to-scalar ratio are read as
+ns = 1 −
+�
+2ϵ1 + ϵ2
+�
+= 1 − 2α
+β − ξ − 2
+β
+AHξ,
+(24)
+r =
+16
+1 + λ ϵ1 =
+16
+β(1 + λ)
+�
+α − AHξ�
+,
+(25)
+which are related to the slow-roll parameters. Applying Eq.(23), the scalar spectral index and the tensor-to-scalar
+ratio are computed at the time of horizon crossing.
+Fig.1 illustrates a three dimensional plot of the scalar spectral index versus α and β. It is realized that when the
+parameter β decreases there is wider range of the parameter α which takes ns into the data range. The plot gives
+some idea about the approximate values of the parameters α and β which we will use later for comparing the model
+
+5
+FIG. 1. The scalar spectral index is plotted versus the parameters α and β.
+(a)
+(b)
+with data.
+By getting some instict from above figure, the tensor-to-scalar ratio is plotted versus the scalar spectral index for
+different values of β, in Fig.IV, and δ, as in Fig.IV. The variable parameter in both plot is α which the arrow in the
+plots shows the increase direction of α.
+To enhance our understanding about the free parameters, Fig.2 portrays a parameteric space of (α, β) so that
+for every point in the space, the result about the scalar spectral index and the tensor-to-scalar ratio stand in good
+agreement with data. Table.I displays the results of the model for some selected (α, β) points from Fig.2.
+
+0.9
+0.9
+0.8
+0.8
+0.6
+0.7
+0.5
+0.6
+2.5
+5.0
+7.5
+2.0
+0.4
+-12.3
+0.2
+0.0
+15.0
+0.2
+-17.5
+0.4
+20.0β= - 10
+0.6
+β= - 15
+β= -20
+0.5
+0.4
+0.3
+0.2
+0.1 
+0.0
+0.92
+0.93
+0.94
+0.95
+0.960.40
+6= -1.5
+6= -0.5
+0.35
+6= +0.1
+0.30
+0.25
+0.20
+0.15
+0.10
+0.05
+0.940
+0.945
+0.950
+0.955
+0.960
+0.9656
+FIG. 2. The parametric space for the (α, β)) when δ = −1.5, λ = 5.5 and the number of e-folds is N = 65.
+α
+β
+ns
+r
+0.2500
+−15.0
+0.9608
+0.0595
+0.1500
+−10.0
+0.9619
+0.0662
+0.0800
+−5.0
+0.9613
+0.0621
+0.0220
+−1.5
+0.9622
+0.0676
+0.0078
+−0.5
+0.9616
+0.0637
+0.0055
+−0.3
+0.9596
+0.0533
+0.0016
+−0.1
+0.9613
+0.0621
+TABLE I. The result for the scalar spectal index and tensor-to-scalar ratio for different values of α and β taken from Fig.2 and
+number of e-folds N = 65.
+V.
+SCALAR FIELD, POTENTIAL AND SWAMPLAND CRITERIA
+The THDE was introduced as the potential of the scalar field where it is given in terms of the Hubble parameter.
+Following the equations, the kinetic terms also could be expressed versus the Hubble parameter, so that
+˙φ2 = −2M 2
+p
+(1 + λ)
+˙H.
+(26)
+Fig.3 describes the behavior of the potential-kinetic ratio during the inflationary times for different values of the
+parameter λ. The ratio decreases by the enhancement of the parameter λ. It is realized that the potential is the
+dominant part and it is greater than the kinetic term by the order of O(102).
+Taking integral of the above equation, the scalar field is obtained. Then, by combining the definition of the potential
+and using parametric plot, the potential is illustrated in terms of the scalar field. The result is depicted in Fig.4 for
+different values of the parameter λ. At the onset, the scalar field stands on the top of the potential, and by passing
+the time, it increases and rolls down. The potential magnitude reduces by enhancement of the parameter λ.
+Besides the observational constraints which we used in the previous section to determine the free parameters of the
+model, there are some theoretical constraint which is our interest to be satisfied by an inflationary model. One of
+these constraints is the swampland criteria proposed by [82–84]. It contains two conjectures, originated from string
+theory, to distinguish the consistent low-energy EFT from the inconsistent ones. The conjectures are
+• Distant conjecture: The scalar field excursion in the field space should satisfy the following upper bound
+∆φ
+Mp
+≤ c1,
+(27)
+where c1 is a constant of the order of one [82–84].
+• de Sitter conjecture: The conjecure is targeting the gradient of the potential and states that it should comply
+the following condition [82, 84]
+1
+Mp
+|V ′|
+V
+≥ c2
+(28)
+
+5
+6= -2.5
+0 
+-5
+-10 
+-15 
+-20
+-0.4
+-0.2
+0.0
+0.2
+0.47
+FIG. 3. The ratio of the potential over the kinetic term, i.e. V (φ)/ ˙φ2 for different values of λ.
+FIG. 4. The potential is plotted versus the scalar field for different values of the parameter λ.
+where c2 is a constant of the order of one. However, some consideration imply that it could be smaller of the
+order of O(0.1) [84, 85]. In the refined version of the conjecture, one of the following conditions should be
+satisfies [83, 84]
+1
+Mp
+|V ′|
+V
+≥ c2,
+or
+1
+Mp
+|V ′′|
+V
+≥ −c′
+2
+(29)
+The first conjecture could be checked out from the Fig.4 which describes the behavior of the potential versus the
+scalar field during the inflationary times. It is realized that field excursion decreases by increasing the parameter λ
+so that for λ ≳ 102 it will be smaller than one. Then, the model could satisfy the first conjecture.
+To consider the validity of the second conjecture, the behavior of the term V ′/V is plotted versus the number of e-folds
+in Fig.5 for different values of the parameter λ. At the beginning, the term could be bigger than one and it also gets
+larger for higher values of λ. Besides, the term in general increases by approaching to the end of inflation. Therefore,
+the second criterion is also satisfied by the model. For both conjectures, the parameter λ plays an important role and
+
+800
+入= 0.1
+入= 1.0
+入=300.0
+600
+400
+200
+0
+0
+10
+20
+30
+40
+50
+60
+701e-12
+3.0
+入= 400
+入=500
+2.5
+入=600
+2.0
+1.5
+1.0
+0.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.08
+FIG. 5. The behavior of the term V ′/V in the inflationary times is plotted for different values λ.
+λ
+α
+β
+ES
+|∆φ|
+V ′/V
+90
+0.0220
+−1.5
+1.84 × 10−3
+1.507
+0.877
+90
+0.0078
+−0.5
+1.81 × 10−3
+1.493
+0.851
+90
+0.0055
+−0.3
+1.73 × 10−3
+1.454
+0.779
+300
+0.0220
+−1.5
+1.36 × 10−3
+0.828
+1.595
+300
+0.0078
+−0.5
+1.34 × 10−3
+0.821
+1.548
+300
+0.0055
+−0.3
+1.28 × 10−3
+0.799
+1.416
+500
+0.0220
+−1.5
+1.20 × 10−3
+0.642
+2.058
+500
+0.0078
+−0.5
+1.18 × 10−3
+0.636
+1.997
+500
+0.0055
+−0.3
+1.13 × 10−3
+0.619
+1.827
+TABLE II. The table presents some results about the energy scale of inflation, and two conjecture of the swampland criteria
+for different values of α and β, selected from Fig.2, and λ. The other constant are picked as δ = −2.5 and N = 65.
+due to this parameter the model could satisfy swampland criteria.
+The results concerning the energy scale (ES) of
+inflation and two conjectures of the swampland criteria are presented in Table.II for different values of α and β picked
+from Fig.2. It is found that the energy scale of inflation is of the order of 10−3Mp stating that inflation occurs at
+energy scale below the Planck scale. Moreover, the role of the parameter λ is clarified when we are considering the
+validity of the swampland criteria. It is realized that for values of λ ≲ 300 the field distant becomes larger than one
+and the potential gradient less than one. Then, to guarantee the validity of swampland criteria, the parameter λ is
+required to approximately be λ ≳ 300.
+VI.
+CONCLUSION
+f(R, T) gravity theory, known as an alternative theory of gravity where matter and curvature have a non-minimal
+coupling, was utilized to study the scenario of slow-roll inflation. The common procedure in studying inflation is to
+assume that it is driven by a scalar field that stands first at the top of its potential, and then slowly rolls down. Due
+to the slow rolling, the kinetic energy of the scalar field is ignorable compared to the potential part, and a quasi-de
+Sitter expansion is provided. The potential plays an important role in inflationary times, however, there is no specific
+choice for it. Then, different choices for the potential have been considered and some of them provide an agreement
+between the model and data.
+Instead of introducing a function of the scalar field for the potential, here, the potential was assumed to be described
+by HDE built from Tsallis entropy. The IR cutoff was taken as the GO cutoff which in addition to the square of the
+Hubble parameter contains the time derivative of the parameter as well. Following this proposal, the perturbation
+parameters were obtained at the time of the horizon crossing. Comparing the model with data, the free parameters
+
+35
+入= 10
+入=100
+30
+入=600
+25
+20
+15
+10
+5
+0
+0
+10
+20
+30
+40
+50
+60
+N9
+of the model were determined and we could illustrate a range for the free parameter so that for every point within
+the range, the model completely agrees with the data.
+Then, we tried to reconstruct the potential in terms of the scalar field. It was found that the field range and the
+magnitude of the potential during inflation depend on the value of λ so that for smaller λ there is a wider field
+range and higher values for the potential. Finally, the validity of the swampland criteria was investigated. The result
+clarified that to satisfy both conjectures the parameter λ could at least be of the order of O(102).
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+Light-curves and nucleosynthesis of CNO-rp driven general relativistic
+instability supernovae in metal enriched supermassive protostars.
+Chris Nagele,1★ Hideyuki Umeda,1 Koh Takahashi, 2
+1Department of Astronomy, Graduate School of Science, the University of Tokyo, Tokyo, 113-0033, Japan
+2Astronomical Institute, Graduate School of Science, Tohoku University, Sendai, 980-8578, Japan
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+The assembly of supermassive black holes poses a challenge primarily because of observed quasars at high redshift, but
+additionally because of the current lack of observations of intermediate mass black holes. One plausible scenario for creating
+supermassive black holes is direct collapse triggered by the merger of two gas rich galaxies. This scenario allows the creation
+of supermassive stars with up to solar metallicity, where the enhanced metallicity is enabled by extremely rapid accretion. We
+investigate the behavior of metal enriched supermassive protostars which collapse due to the general relativistic radial instability.
+These stars are rich in both hydrogen and metals and thus may explode due to the CNO cycle (carbon-nitrogen-oxygen) and the
+rp process (rapid proton capture). We perform a suite of 1D general relativistic hydrodynamical simulations coupled to a 153
+isotope nuclear network with the effects of neutrino cooling. We determine the mass and metallicity ranges for an explosion. We
+then post process using a 514 isotope network which captures the full rp process. We present nucleosynthesis and lightcurves for
+selected models. These events are characterized by enhanced nitrogen, suppressed light elements (8 ≥ A ≥ 14), and low mass p
+nuclides and they are visible to JWST and other near infrared surveys as decades-long transients. Finally, we provide an estimate
+for the number of currently ongoing explosions in the Universe.
+Key words: gravitation — (stars:) supernovae: general — nuclear reactions, nucleosynthesis, abundances
+1 INTRODUCTION
+The study of supermassive stars has arisen from two peculiarities of
+the black hole population in the Universe. The first is that supermas-
+sive black holes (SMBHs) exist soon after the big bang (Mortlock
+et al. 2011; Wu et al. 2015; Bañados et al. 2018; Matsuoka et al. 2019;
+Wang et al. 2021). Our current understanding of cosmology requires
+that the SMBHs did not exist at the time of the big bang, implying
+that they were created in the intervening period. The second pecu-
+liarity is that the black hole mass function seems to be bimodal, with
+a noticeable lack of intermediate mass black holes (having masses
+in between solar mass black holes and SMBHs), although this may
+be due to observational bias (Wrobel et al. 2016; Baumgardt 2017;
+Kızıltan et al. 2017). If the bimodality is not due to observational bias,
+it stands in opposition to the distributions of other self gravitating
+objects (stars and galaxies), which have smooth mass functions.
+The direct collapse black hole (DCBH) scenario was proposed to
+resolve the first of these peculiarities (Bromm & Loeb 2003), and
+is sometimes invoked to explain the second (e.g. Banik et al. 2019).
+The scenario involves a gas cloud forming a single supermassive star
+instead of many individual stars. This can occur in the presence of
+local Lyman Werner radiation (Dijkstra et al. 2008; Agarwal et al.
+2012; Latif et al. 2014b) or baryon dark matter supersonic streaming
+(Latif et al. 2014a; Schauer et al. 2017; Hirano et al. 2017). The
+★ E-mail: [email protected]
+resultant supermassive star may be detectable directly (Surace et al.
+2018, 2019; Vikaeus et al. 2022), via a general relativistic instability
+supernova (GRSN, Chen et al. 2014; Whalen et al. 2013c; Nagele
+et al. 2020; Moriya et al. 2021; Nagele et al. 2022b,a), by the obser-
+vation of gravitational waves (Shibata et al. 2016; Li et al. 2018), or
+as an ultra long gamma ray burst (Sun et al. 2017).
+In this paper, however, we consider a slightly different scenario
+where the supermassive star formation is triggered by a merger of
+two gas rich galaxies (for a review, see Mayer & Bonoli 2019).
+The phenomenon of nuclear gaseous disks forming via multi-scale
+inflows was first investigated in the context of the M-sigma relation
+as a means of providing a source of dynamical friction for a SMBH
+binary in order to assist with its eventual merger (Kazantzidis et al.
+2005; Mayer et al. 2007). Since then, it has been shown that not only
+can this disk influence the behavior of existing SMBHs, but it can
+also collapse under its own gravity to form a new black hole (Mayer
+et al. 2010). Recently, Zwick et al. (2022) calculated observables
+from DCBHs resulting from galaxy mergers. The crucial element of
+this scenario as it pertains to the current study is that the scenario
+is agnostic to the metallicity of the interstellar medium (ISM). This
+means that supermassive stars will form out of metal enriched gas
+(Mayer et al. 2015).
+But what does it mean to have a metal enriched supermassive star?
+These objects radiate at very nearly their Eddington limit and mass
+loss rates from our local Universe (Vink et al. 2011) suggest such
+objects would not survive for long. The situation is further compli-
+© 2022 The Authors
+arXiv:2301.01941v1  [astro-ph.HE]  5 Jan 2023
+
+2
+C. Nagele et al.
+cated by the fact that these metal enriched stars may be accreting
+matter to replenish that lost to line driven winds. In this paper, we
+sidestep this difficulty by considering only metal enriched supermas-
+sive protostars which are massive enough to collapse via the general
+relativistic (GR) radial instability (Chandrasekhar 1964) before they
+reach the main sequence, thus avoiding any line driven mass loss.
+The behavior of such metal enriched supermassive protostars has
+been investigated previously (Fuller et al. 1986; Montero et al. 2012).
+In particular, it was shown that if the protostars collapse due to the GR
+radial instability, then this collapse can cause an explosion powered
+by the CNO cycle (which we will term a proton rich or pr-GRSN, to
+differentiate it from an 𝛼 process driven GRSN). pr-GRSNe were first
+investigated in Fuller et al. (1986). They used a 1D post Newtonian
+(PN) code with a 10 isotope nuclear reaction network and found
+several exploding models spanning the mass range 5 × 105 − 106
+M⊙. The metallicity floor for the lowest mass model was 5 × 10−3.
+Subsequently, Montero et al. (2012) used a 2D BSSN code with
+parameterized heating rates to investigate models with similar mass,
+and they were able to include the effects of rotation. For their non
+rotating models, they found explosions with the same masses as
+Fuller et al. (1986), but with slightly higher metallicity.
+We use a GR 1D hydrodynamics code coupled to a 153 isotope
+network. The large network allows us to more accurately follow the
+dynamics of the explosion at higher temperatures, and we thus find
+a lower metallicity floor than in previous works. After running our
+simulations, we post process the hydrodynamical trajectories with a
+514 isotope network designed to fully follow the rp-process on the
+proton rich side. Contrary to the conclusions of previous works, we
+find that the rp-process can play a critical role in the explosion.
+In Sec. 2 we outline our numerical procedures for stellar evolu-
+tion, hydrodynamics, post processing, and lightcurves. In Sec. 3.1,
+we present the results of the stellar evolution simulations. In Sec.
+3.2, we present the results of our hydrodynamical simulations and
+post processing, for a fiducial model, as well as for varying mass and
+metallicity. In Sec. 3.3 we present the results of our lightcurve calcu-
+lations and an estimate of pr-GRSN density. In Sec. 3.4, we discuss
+various feedback induced by a pr-GRSN. Finally, we conclude with
+a discussion in Sec. 4.
+2 METHODS
+In this section, we first describe our initial models and stellar evo-
+lution code, then provide details of our GR hydrodynamical code,
+after which we detail the open source code SNEC, which is used to
+calculate lightcurves.
+2.1 Stellar evolution
+The HOSHI code (Takahashi et al. 2016, 2018, 2019; Yoshida et al.
+2019) is a 1D stellar evolution code which solves the stellar structure
+and hydrodynamical equations using a Henyey type implicit method.
+Nagele et al. (2020) introduced the first order PN correction to the
+hydrostatic terms. The PN approximation is extremely accurate for
+SMSs in hydrostatic equilibrium because the effects of GR are mi-
+nor. These minor effects must be included, however, because SMSs
+are radiation dominated and therefore close to instability. Once the
+evolution of the star becomes dynamical, HOSHI’s lack of a shock
+capture scheme and the PN dynamical corrections neccesitate the
+use of another code. HOSHI includes a nuclear reaction network (52
+isotopes), neutrino cooling, mass loss, and rotation. The equation
+of state includes contributions from photons, averaged nuclei, elec-
+trons, and positrons. HOSHI uses the Rosseland mean opacity of the
+OPAL project (Iglesias & Rogers 1996) and solves the Saha equation
+to determine the ionization of hydrogen, helium, carbon, nitrogen,
+and oxygen.
+In this paper, 𝑀 is the total mass, 𝑅 the radius, 𝑇 the temperature,
+and 𝜌𝑏 the baryonic density where quantities with 𝑐 subscripts show-
+ing the central values. 𝑠𝑟 is the entropy due to radiation at a given
+mass (Shapiro & Teukolsky 1983)
+𝑠𝑟 = 0.942
+� 𝑀
+M⊙
+�1/2
+.
+(1)
+Finally, X is the mass fraction of a specified element.
+To assist with analysis, we define various global energy quantities.
+The internal energy is
+𝐸int =
+∫ 𝑀
+0
+𝜖 𝑑𝑚𝑟,
+(2)
+where 𝑚𝑟 is the mass coordinate and 𝜖 is the specific energy. The
+gravitational energy is
+𝐸grav = −
+∫ 𝑀
+0
+𝑔effective 𝑟 𝑑𝑚𝑟,
+(3)
+where 𝑔effective is the local gravity with the 1st order PN correction
+to the static terms (Nagele et al. 2022b). The accuracy of this ap-
+proximation degrades with increasing density and velocity, neither
+of which are particularly concerning for our purposes. The kinetic
+energy is
+𝐸kin =
+∫ 𝑀
+0
+𝑣2
+2 𝑑𝑚𝑟,
+(4)
+where 𝑣 is the radial velocity. The binding energy of the star is
+the negative of the thermal and gravitational energies (so that a
+more tightly bound star has higher 𝐸bind), while the total energy
+additionally includes kinetic energy:
+𝐸bind = −(𝐸int + 𝐸grav)
+(5)
+𝐸tot = 𝐸int + 𝐸grav + 𝐸kin.
+(6)
+As in our previous works, we define the explosion energy as the
+total energy at shock breakout. For HYDnuc, we also report the
+integration over energy generation due to the nuclear network and
+neutrino cooling (dots indicate time derivatives):
+𝐸nuc(𝑡) =
+∫ 𝑡
+0
+∫ 𝑀
+0
+�
+𝜖nuc 𝑑𝑚𝑟 𝑑𝑡
+(7)
+𝐸𝜈(𝑡) =
+∫ 𝑡
+0
+∫ 𝑀
+0
+�𝜖𝜈 𝑑𝑚𝑟 𝑑𝑡.
+(8)
+A summary of the results of the HOSHI simulations can be found in
+Table 2.
+We initiate the HOSHI code in a high entropy state, which imme-
+diately relaxes towards a constant entropy radiation dominated state,
+though the protostar does not reach all the way to the radiation value
+due to the small contribution of gas pressure. This configuration
+could represent one of two physically realizable scenarios. Either,
+it could be a supermassive protostar which has finished accreting
+due to depletion of accretion material, which then contracts to the
+radiation dominated constant entropy pre-ZAMS (zero age main se-
+quence) state, or it could represent the convective core of a currently
+MNRAS 000, 1–16 (2022)
+
+CNO-rp driven GR instability supernovae.
+3
+Table 1. Summary table for the nuclear networks. Entries show the range in A for the specified element.
+Element
+52
+153
+216
+514
+Element (ctd.)
+52
+153
+216
+514
+N
+1
+1
+1
+1
+V
+47
+45-51
+45-51
+42-53
+P
+1-3
+1-3
+1-3
+1-3
+Cr
+48
+47-54
+47-54
+44-55
+He
+3-4
+3-4
+3-4
+2-4
+Mn
+51
+49-55
+49-55
+46-57
+Li
+6-7
+6-7
+6-7
+4-7
+Fe
+52-56
+51-58
+51-58
+48-60
+Be
+7-9
+7-9
+7-9
+5-9
+Co
+55-56
+53-59
+53-59
+50-61
+B
+8-11
+8-11
+8-11
+6-11
+Ni
+56
+55-62
+55-62
+52-66
+C
+12-13
+12-13
+12-13
+8-14
+Cu
+—
+57-63
+57-63
+54-68
+N
+13-15
+13-15
+13-15
+10-16
+Zn
+—
+60-64
+60-64
+56-71
+O
+14-18
+14-18
+14-18
+12-18
+Ga
+—
+—
+63-69
+58-73
+F
+17-19
+17-19
+16-19
+14-20
+Ge
+—
+—
+65-70
+60-75
+Ne
+18-20
+18-22
+18-22
+16-22
+As
+—
+—
+68-75
+62-76
+Na
+23
+21-23
+20-23
+18-24
+Se
+—
+—
+72-76
+64-81
+Mg
+24
+22-26
+22-26
+20-26
+Br
+—
+—
+74-79
+66-82
+Al
+27
+25-27
+24-27
+22-28
+Kr
+—
+—
+76-80
+68-86
+Si
+28
+26-32
+26-30
+24-30
+Rb
+—
+—
+78-85
+70-87
+P
+31
+29-33
+28-31
+26-32
+Sr
+—
+—
+82-86
+72-89
+S
+32
+30-36
+30-36
+28-37
+Y
+—
+—
+84-89
+74-91
+Cl
+35
+33-37
+32-37
+30-39
+Zr
+—
+—
+86-90
+76-95
+Ar
+36
+34-40
+34-40
+32-43
+Nb
+—
+—
+—
+78-96
+K
+39
+37-41
+36-41
+34-45
+Mo
+—
+—
+—
+80-98
+Ca
+40
+38-43
+38-43
+36-48
+Tc
+—
+—
+—
+82-98
+Sc
+43
+41-45
+41-45
+38-49
+Ru
+—
+—
+—
+84-99
+Ti
+44
+43-48
+43-48
+40-51
+accreting supermassive star. In the latter case, the accretion envelope
+could in theory effect the stability of the system, but such effects are
+thought to be small (Haemmerlé 2020). We expect that the envelope
+will have no effect on the dynamical behavior of the protostar once
+the GR instability is reached besides increasing the overall gravity
+(see e.g. Fig. 14 of Nagele et al. 2022b).
+We determine the stability of the protostar in HOSHI by solving
+the pulsation equation for a hydrostatic, spherically symmetric object
+in general relativity (Chandrasekhar 1964):
+𝑒−2𝑎−𝑏 d
+d𝑟
+�
+𝑒3𝑎+𝑏Γ1
+𝑃
+𝑟2
+d
+d𝑟 (𝑒−𝑎𝑟2𝜉)
+�
+−4
+𝑟
+d𝑃
+d𝑟 𝜉+𝑒−2𝑎+2𝑏𝜔2(𝑃+𝜌𝑐2)𝜉
+−8𝜋𝐺
+𝑐4 𝑒2𝑏𝑃(𝑃 + 𝜌𝑐2)𝜉 −
+1
+𝑃 + 𝜌𝑐2
+� d𝑃
+d𝑟
+�2
+𝜉 = 0,
+(9)
+where 𝑎, 𝑏 are the metric coefficients as defined in Haemmerlé
+(2021a), 𝑟 is the radius, 𝑃 the pressure, Γ1 the local adiabatic in-
+dex at constant entropy (𝑠), 𝜌 = 𝜌𝑏(1 + 𝜖) the relativistic density,
+and 𝜖 the specific internal energy (we absorb rest mass due to mass
+excess of nuclei into this energy).
+The star, or in this case, protostar, is unstable if there exists a trial
+function 𝜉(𝑟) ∝ 𝑒𝑖𝜔𝑡 with 𝜔2 < 0, representing a perturbation which
+will grow exponentially. There are two main approaches to solving
+this equation, either by assuming a nearly linear trial function 𝜉 ∝ 𝑟𝑒𝑎
+(Haemmerlé 2021a), or by iteratively solving for the fundamental
+mode of the normal mode decomposition of perturbations to Eq. 9
+(Nagele et al. 2022b). Here we adopt the latter approach, as in our
+previous paper, but this choice should have a minimal bearing on the
+results.
+Even though we will eventually simulate the dynamics of metal
+enriched supermassive protostars, we only consider metal free proto-
+stars in HOSHI. This is because we are only interested in protostars
+which become unstable before the onset of nuclear burning, so metal-
+licity, will not effect the protostellar structure except as caused by
+Figure 1. Radial (upper) and velocity (lower) snapshots of the fiducial model
+at three timesteps, the initial time, the time when the central temperature is
+largest, and shock breakout.
+changes to the opacity. We consider the following range of masses:
+7 − 10 × 104 M⊙ in steps of 104 M⊙ and 1 − 3 × 105 M⊙ in steps of
+5 × 104 M⊙.
+2.2 Hydrodynamics
+HYDnuc is a 1D Lagrangian GR hydrodynamics code which uses
+a Roe-type approximate linearized Riemann solver (Yamada 1997;
+Takahashi et al. 2016; Nagele et al. 2020). It includes all of the
+physics from HOSHI except for convection and ionization. In this
+paper, we use a 153 isotope network (Table 1). The 153 isotope
+MNRAS 000, 1–16 (2022)
+
+le13
+initial
+1.5
+max T.
+shock breakout
+r [cm]
+1.0
+0.5
+0.0
+6
+4
+2
+0
+-2
+0
+20000
+40000
+60000
+80000
+100000
+mr[M。]4
+C. Nagele et al.
+Figure 2. Nucleosynthesis in the post processed 514 isotope network for the fiducial model, 𝑀 = 105 M⊙, 𝑍 = 𝑍⊙. Upper panels — isotope mass fractions for
+five snapshots: the initial time, when the temperature rises to 3/4 of the eventual maximum, the maximum temperature, the final time step, and 1012 seconds
+later. Lower panel — total energy compared to 𝐸nuc in both the 153 and 514 isotope calculations and compared to 𝐸nuc + 𝐸𝜈 for the 153 isotope calculation.
+network (Takahashi et al. 2018) covers the proton rich side (p side)
+at a depth of 3-8 isotopes up to zinc. Reaction rates for all networks
+are taken from JINA REACLIB (Cyburt et al. 2010).
+We use the same scheme to transport our models from HOSHI to
+HYDnuc as in Nagele et al. (2022b) which is based on the frequency
+function defined in Takahashi et al. (2019). We set the chemical
+composition to be constant throughout the star, as a fraction of the
+solar composition (Asplund et al. 2009). Due to the exploratory
+nature of this study and the requirement of a larger nuclear network,
+we use slightly less optimal numerical parameters than in Nagele
+et al. (2022b), specifically 255 mesh points and V = 10−4 (maximum
+allowed fractional variation of independent variables per timestep).
+The effect of these changes is to underestimate the energy generated
+by nuclear burning (see Fig. 6 of Nagele et al. 2022b). As in our
+previous works, we terminate the simulations when convergence
+issues arise due to large radius (𝑟 ∼ 1015 cm).
+2.3 Post-processing
+After performing the HYDnuc simulations, we post process the hy-
+drodynamical trajectories using a 514 isotope network (Table 1)
+designed to follow the rp process up to ruthenium. All isotopes on
+the p side are covered up to the line with slope 1 and y intercept 5.
+Even though this post processing is less computationally expensive,
+the network is too large to solve the composition at every timestep of
+HYDnuc. We choose to solve the composition with a frequency of
+100−1 timesteps−1, and have checked that a) the convergence of 𝐸nuc
+as the frequency increases and b) that 𝐸nuc with frequency 100−1
+agrees with 𝐸nuc with frequency 10−1 to within 0.1%. At the end
+of the HYDnuc simulation, the post-processed composition contains
+many radioactive isotopes. We then fix the temperature and density
+and continue to post-process for an additional 1012 seconds while
+logarithmically increasing the timestep. 1012 seconds is enough for
+most, but not all (e.g. 26Al) radioactive isotopes to decay.
+2.4 Lightcurves
+The SuperNova Explosion Code (SNEC) is an open source, 1D La-
+grangian, radiation hydrodynamics code designed to compute su-
+pernova lightcurves (Morozova et al. 2015). It includes artificial
+Figure 3. Nucleosynthetic yields (𝑡 = 𝑡final + 1012 s) from the 514 isotope
+network for the 𝑀 = 105 M⊙ for the indicated metallicities.
+viscosity, an equation of state with a Saha solver for ionization of
+hydrogen and helium, and equilibrium flux-limited photon diffusion
+using OPAL. As in (Nagele et al. 2022a), we port our HYDnuc mod-
+els to SNEC slightly before shock breakout and for the outer layer
+of the SNEC model, we use the HOSHI progenitor which enables
+increased surface resolution necessary for properly following the
+lightcurve. We terminate the SNEC simulations after 109 seconds at
+which point any plateau phase in the luminosity has finished.
+We then use the effective temperature to construct blackbody spec-
+tral energy distributions and assume a standard ΛCDM cosmology.
+With these components, we calculate apparent magnitudes for the
+passbands of various telescopes at a given redshift. In this study, we
+do not take extinction into account.
+MNRAS 000, 1–16 (2022)
+
+t= tinitial
+Tc= gTc,max
+Te= Tc,max
+t = tfinal
+t= trinal + 1012 s
+0
+20
+-2
+pnumber
+-4
+15
+Log
+-6
+10
+-8
+10
+15
+20
+25
+10
+15
+20
+25
+10
+15
+20
+25
+10
+15
+20
+25
+10
+15
+20
+25
+-10
+nnumber
+1e55
+2
+Etot (153 isotopes)
+Etot(t = 0) + Enuc (153)
+Etot(t = 0) + Enuc + Ev (153)
+Etot(t = 0) + Enuc (514 isotopes)
+0
+0
+20000
+40000
+60000
+80000
+100000
+t [s]Z/Z。=0.12254
+4
+Z/Z。=0.1226
+Z/Z。=0.123
+Z/Z。= 0.13
+Sc
+3
+Z/Z。= 0.2
+Z/Z。 = 1.0
+2
+[X/Fe]
+N
+Mo
+Ne
+0
+-1
+C
+Ar
+Co
+-2
+Mg
+5
+10
+15
+20
+25
+30
+35
+40
+45
+Z (proton number)CNO-rp driven GR instability supernovae.
+5
+Table 2. Summary of the state of the stellar evolution simulations at the GR instability. The columns are, total mass, radius, central temperature, baryonic density
+and entropy, central entropy as a fraction of radiative entropy (for a star of that mass), hydrogen mass fraction and binding energy.
+M [105 M⊙]
+R [1013 cm]
+𝑇𝑐 [108 K]
+𝜌𝑏,𝑐 [g/cc]
+𝑠𝑐 [𝑘𝑏/baryon]
+𝑠𝑐/𝑠𝑟 − 1
+X(1H)
+𝐸bind [1054 ergs]
+0.7
+264.8
+1.554
+1.881
+265.3
+0.06454
+0.4979
+1.694
+0.9
+32.24
+1.652
+1.968
+306.2
+0.08347
+0.7406
+2.702
+1.0
+2.061
+1.28
+0.8681
+322.4
+0.08224
+0.7599
+3.466
+1.5
+2.94
+0.7642
+0.1508
+390.4
+0.07005
+0.7599
+3.321
+2.0
+3.272
+0.7884
+0.1429
+447.6
+0.06238
+0.7599
+4.144
+2.5
+4.17
+0.6803
+0.08181
+499
+0.05935
+0.7599
+4.487
+3.0
+5.562
+0.5569
+0.04093
+544.7
+0.05577
+0.7599
+4.765
+Figure 4. Monotonic dependence of maximum temperature on metallicity
+for the 𝑀 = 105 M⊙ model.
+Figure 5. Central temperature evolution of the 𝑀 = 105 M⊙ for the indicated
+metallicities. These trajectories were used for the post processing.
+3 RESULTS
+In this section, we first describe the results of the HOSHI code, after
+which we provide details of the nucleosynthesis before moving on to
+a discussion of the lightcurves.
+3.1 Stellar Evolution
+The lowest mass model which we consider in this study (7×104 M⊙)
+becomes unstable only after entering the ZAMS phase, with the
+instability occurring after roughly half a million years. This model has
+a large radius and low proton mass fraction at the onset of instability
+(Table 2). Because of its relatively long lifetime, enhanced metallicity
+would likely have caused significant mass loss, thus prolonging the
+period before the GR instability occurred (less mass means less
+gravity). Then, there is the effect of accretion to consider, which, if
+present, may be able to restore some of the lost mass. We do not
+posses the proper tools to model these physical processes and so we
+do not perform any hydrodynamical simulations with this model.
+On the other hand, models with mass of 105 M⊙ or greater (9 ×
+104 M⊙ is a marginal case, which we include), collapse due to the
+GR radial instability before any nuclear burning has taken place. For
+increasing mass, these models have higher entropy, and are getting
+more and more radiation dominated (Table 2, 6th column). This
+corresponds to a decrease in the central temperature and density as
+well as an increase in the binding energy.
+The upper limit for stability in our study (7 × 104 M⊙) is slightly
+lower than that found in Fuller et al. (1986) (105 M⊙), which is
+likely due to small deviation from the GR pressure gradient in the
+PN code which they employed. Indeed, as was shown in Nagele et al.
+(2022b), the use of the baryonic density instead of the relativistic
+density in the PN correction changes the stellar structure. Our results
+are more massive than the cores of the accreting SMSs in Haemmerlé
+(2020) (Fig. 8), which is to be expected because we do not consider
+the gravity of the envelope. It should be noted that this is not a
+completely faithful comparison because the accreting SMSs will
+begin nuclear burning before they collapse due to GR (Hosokawa
+et al. 2013; Umeda et al. 2016; Haemmerlé et al. 2018). For the high
+accretion rates implied by the galaxy merger formation scenario,
+however, there is little chance that the hydrogen reservoir will be
+depleted before the GR instability, in which case the pr-GRSN would
+still occur. As we will show, the less massive the progenitor, the easier
+it is for the star to explode (Sec. 3.2).
+MNRAS 000, 1–16 (2022)
+
+1e8
+7
+6
+max T. [K]
+5
+4
+3
+10~5
+10-4
+10-3
+10-2
+10-1
+100
+Z/Zo-0.122531e8
+8
+Z/Z。=0.12254
+7
+Z/Z。=0.1226
+Z/Z。=0.123
+Z/Z。=0.13
+6
+Z/Z。=0.2
+Z/Z。=1.0
+5
+3
+2
+1
+0
+20000
+40000
+60000
+80000
+100000
+time [s]6
+C. Nagele et al.
+Figure 6. Same as Fig. 2 but for 𝑀 = 105 M⊙, 𝑍 = 0.12254 𝑍⊙, which is the lowest metallicity explosion for 𝑀 = 105 M⊙.
+Figure 7. Upper panel — same as upper panel Fig. 6 but showing the final composition at different meshes. Lower panel — mean molecular weight 𝜇 as a
+function of mass coordinate.
+3.2 Nucleosynthesis
+3.2.1 Fiducial model
+We take the M=105 M⊙, 𝑍 = Z⊙ as our fiducial model. This mass
+is near the lower end of our mass range and the metallicity is well
+above the threshold required for an explosion. In the hydrodynamical
+simulation, the star begins in a high entropy, but relatively compact
+state, due to the star not yet having settled into the main sequence.
+At the start of the simulation, the CNO cycle immediately turns on,
+but is not strong enough at those temperatures (𝑇𝑐 ≈ 108 K, Fig
+5) to arrest the collapse. The star continues to collapse for 50000
+seconds before reaching a temperature of 𝑇𝑐 ≈ 2.3 × 108 K, at which
+point the CNO cycle produces enough energy to reverse the motion
+of the star. The CNO cycle continues to produce energy, even as the
+temperature drops below 𝑇𝑐 ≈ 108 K, but the bulk of the energy
+production occurs with the central temperature 𝑇𝑐 > 1.5 × 108 K
+(Fig. 2). The star is completely unbound and explodes with energy
+𝐸exp = 1.81 × 1055 [ergs]. It’s final radial and velocity profiles as a
+function of mass coordinate can be seen in Fig. 1.
+After finishing the HYDnuc simulation, we post process the sim-
+ulation results as described in Sec. 2.3 with a 514 isotope network
+designed to follow the rp process. Fig. 2 shows the isotope mass frac-
+tion of the entire star in the 514 network at several time snapshots. In
+the fiducial model, the energy generation comes almost entirely from
+the CNO cycle, but proton captures on light elements do occur near
+the maximum temperature (Fig. 2) and this can be seen in the nucle-
+osynthetic yields (Fig. 3). The yields for the fiducial model (Z=Z⊙)
+are characterized by two features. The first is enhanced nitrogen and
+suppressed carbon and oxygen due to the CNO cycle, which produces
+roughly equal amounts of its eponymous elements. The second is a
+broad exchange of light elements (flourine, sodium, magnesium) for
+slightly heavier elements (aluminium through chlorine) due to mul-
+tiple proton captures which then decay back to stability at higher
+mass number than they originated (Fig. 2). The only exception to this
+trend appears to be neon which experiences proton captures, but is
+replenished from below by oxygen exiting the CNO cycle.
+3.2.2 Metallicity dependence
+Next we consider the behavior of the M=105 M⊙ model as we vary
+the metallicity. For each mass, there exists a threshold metallicity
+value below which there are not enough seed metals to fuel an ex-
+plosion. As one approaches this threshold, that is as one decreases
+the metallicity, the model needs to reach higher temperatures before
+the collapse can be reversed (Fig. 4). In addition, the time at which
+the model reaches the maximum temperature is pushed further back
+with decreasing metal content (Fig. 5).
+Fig. 6 shows the isotope mass fraction for several time snapshots,
+as in Fig. 2, but for Z= 0.12254 Z⊙ which is the metallicity closest
+to the explosion threshold (we will refer to this as the metal-poor
+MNRAS 000, 1–16 (2022)
+
+t= tinitial
+Te =gTc,max
+Te = Tc,max
+t= trinal
+t = tinal + 1012 s
+40
+-4
+-6
+Log
+-8
+X
+-10
+10
+12
+10
+20
+30
+40
+50
+20
+30
+40
+50
+10
+20
+30
+40
+50
+10
+20
+30
+40
+50
+10
+20
+30
+40
+50
+nnumber
+le55
+5.0
+Etot (153isotopes)
+Etot(t = 0) + Enuc (153)
+Etot(t = 0) + Enuc + Ev (153)
+Etot(t = 0) + Enuc (514 isotopes)
+0.0
+40000
+50000
+60000
+70000
+80000
+90000
+100000
+t [s]-2
+40
+-4
+number
+30
+-6
+Log
+20
+-8
++
+-10
+10
+-12
+10
+20
+30
+40
+50
+10
+20
+30
+40
+10
+20
+30
+40
+50
+10
+20
+30
+40
+50
+10
+20
+30
+40
+50
+nnumber
+0.63
+0.62
+0
+20000
+40000
+60000
+80000
+100000
+mr[M。]CNO-rp driven GR instability supernovae.
+7
+model). We emphasize that this fine tuning is not done in order to
+find the precise value of the threshold, but rather to demonstrate
+that near the threshold, temperatures high enough to trigger the rp
+process can be reached. Indeed, panels 2 and 3 of Fig. 6 show extreme
+levels of proton captures extending to technetium. Note that in the
+HYDnuc simulations with the 153 isotope network (as opposed to
+the post processing with 514), the composition cannot go higher
+than zinc (Table 1). In the post processing, the heavy p-side elements
+then decay back to stability at much higher mass number than they
+originated. Fig. 7 shows the final isotopic mass fraction distribution
+at different meshes throughout the star. Unlike in the 𝛼 process driven
+GRSN, most of the star undergoes nuclear burning and this is one of
+the reasons that these explosions can be so energetic.
+Fig. 3 shows the final abundances (relative to iron, relative to solar
+values Asplund et al. 2009) from the 514 isotope network for selected
+metallicities. As was the case in the fiducial model (Z⊙), there are
+two main characteristics, with the first being enhanced nitrogen.
+The second characteristic is a bulk transport of light elements to
+heavy elements, with lower metallicities experiencing a larger and
+wider transport. Note that oxygen is part of this transport, whereas
+carbon and nitrogen do not vary significantly with metallicity. For
+Z= 0.2 Z⊙, the transport extends up to scandium. For Z= 0.13 Z⊙,
+iron peak elements are produced, terminating around gallium. For the
+three smallest metallicities, the transport extends well past iron, up
+to molybdenum in the extreme case. For the lower metallicity cases,
+significant contributions to selected element are in the form of low
+mass p-nuclides, which reside away from the line of stability (Fig. 6,
+5th panel). Below Z= 1.0 Z⊙, all models show a peak at scandium,
+with Z= 0.123 Z⊙ peaking again at proton number 31-33 and the two
+lowest metallicity models peaking at proton number 31-36. Another
+signature of all models is high Cl/Ar and low Ar/K, both resulting
+from the bulk transport not necessarily preferring even elements. In
+the high temperature models, cobalt is suppressed because its lightest
+stable isotope (59Co) cannot be reached from the p side.
+3.2.3 Mass dependence
+Next, we vary the mass at solar metallicity. As the mass increases,
+the binding energy of the star increases, and more nuclear energy is
+required to unbind it. This means that the higher mass models will
+reach higher temperatures until eventually (M = 2.5 × 105) they can
+no longer explode at solar metallicity. Higher temperatures, in turn,
+mean that the yields from the 514 isotope post processing should
+show more heavier elements, and this is demonstrated in Fig. 8. We
+have already discussed 105 M⊙, and 9 × 104 M⊙ is nearly identical.
+The 1.5 × 105 M⊙ model shows the same bulk transport described
+in the previous section, but this time extending up to titanium, while
+the 2 × 105 M⊙ model (we will refer to this as the massive model)
+peaks around titanium, but reaches as high as gallium.
+Fig. 9 shows the explosion energy in HYDnuc (153 isotopes) as
+a function of mass and metallicity. The explosion energy depends
+only on mass, as it is a fixed fraction of the star’s binding energy,
+unless the model is sufficiently close to the explosion threshold, as
+can be seen with M= 2 × 105 M⊙, Z= 0.9 Z⊙. On the other hand,
+the metallicity threshold increases with mass for the reasons stated
+above.
+We have attempted to run additional simulations using a 216 iso-
+tope network (Table 1). However, there is a substantial increase in
+computational cost (3469 reactions compared with 2463 for the 153
+isotope network) and the network is still not large enough to follow
+the rp process to higher mass (Appendix A). For a rough estimate of
+Figure 8. Same as Fig. 3, but for different masses at solar metallicity.
+Figure 9. Dependence of explosion energy (denoted by color) on mass and
+metallicity. The black crosses are models which failed to explode. The gray
+region roughly corresponds to non-explosion.
+the energy generation rate of the rp process beyond what we investi-
+gate in this paper, see Appendix B.
+3.3 Lightcurves
+Fig. 10 shows the time evolution of photosphere radius, luminosity
+and effective temperature for each of the three named models. In
+addition, we have rerun the massive model with a much greater en-
+velope resolution in order to more accurately characterize the shock
+breakout (Fig. 11), though this increased resolution presents compu-
+tational challenges during the plateau phase. The shock breakout is
+accompanied by an extremely luminous (1049 ergs/s) burst with high
+effective temperature, which is potentially observable as an X-ray
+outburst. To date, the one observed X-ray outburst associated with
+shock breakout (Soderberg et al. 2008) had an X-ray luminosity six
+orders of magnitude lower (1043 ergs/s), implying that shock break-
+out of a pr-GRSN would be visible, even at high redshift. However,
+the short duration of the event and low rate of pr-GRSNe presents
+serious challenges to realizing an observation of this shock breakout.
+Besides the high luminosity and temperature at shock breakout,
+MNRAS 000, 1–16 (2022)
+
+Sc
+Z/Z。= 9e4
+3
+V
+Z/Z=1e5
+Z/Z。=1.5e5
+Z/Z。=2e5
+2
+[X/Fe]
+1
+N
+Ne
+0
+C
+-1
+Mg
+5
+10
+15
+20
+25
+30
+35
+40
+45
+Z (proton number)AA
+1e56
+1.0
+1.2
+0.8
+X
+1.0
+X
+0.85
+X
+[ergs
+N
+X
+0.6
+0.4
+X
+0.4
+x
+0.2
+0.2
+x
+0.0
+100000
+150000
+200000
+250000
+M[M。]8
+C. Nagele et al.
+Figure 10. Results of the SNEC simulations for the fiducial model, metal-poor
+model, and the massive model. Upper panel — photosphere radius. Middle
+panel — bolometric luminosity. Lower panel — effective temperature. The
+horizontal axis has been normalized so that shock breakout occurs at 10−3
+days.
+there is another difference between the lightcurves of the pr-GRSNe
+and the lightcurves of standard GRSNe (𝛼 process, Moriya et al.
+2021; Nagele et al. 2022a). This is the longer duration of the plateau
+phase which follows hydrogen recombination, nearly an order of
+magnitude longer than our previous GRSN models. This longer du-
+ration may be due to the increased mass or the increased energy in
+comparison with previous GRSN models. During this plateau phase,
+the hierarchy of the luminosities follows that of the associated ex-
+plosion energies. Throughout this phase, the effective temperature
+steadily falls and the photosphere radius steadily rises, the latter of
+which is slightly different to the stalled photosphere found in standard
+GRSNe.
+This effect can be seen in Fig. 12 which shows the time evolution
+of JWST NIRCAM wideband filters for the three named models
+at redshift five, which is consistent with both ZISM = 0.1 Z⊙ and
+ZISM = 1.0 Z⊙ (e.g. Pallottini et al. 2014). The prompt emission is
+visible in the bluer filters because of the higher effective temperature,
+but as this quantity drops, they rapidly fall away and only the four
+reddest filters remain during the plateau phase.
+3.3.1 Energy input from radioactive decays
+When calculating lightcurves in SNEC for the metal-poor model,
+there is an additional consideration, namely that the star is undergo-
+ing a significant amount of radioactive decays after the end of the
+HYDnuc calculation (see panels 4 and 5 of Fig. 6). In Fig. 13, we
+Figure 11. Same as Fig. 10, but showing shock breakout for the massive
+model. The simulation in this figure has higher surface resolution than those
+in Fig. 10. The horizontal axis has been normalized so that the peak luminosity
+occurs at 0 s.
+show the rate of change in the nuclear energy as a function of time.
+The vertical lines separate regions which are powered by different
+decays, and these regions are labeled by the mother nuclei. Swells in
+the heating rate can be seen for the decay of 56Ni and 57Co. Despite
+the staggering amount of energy produced by these radioactive de-
+cays (compare to Fig. 10), we do not think that they will affect the
+lightcurve for two reasons. The first is that these decays are occurring
+deep within the star and are well inside the photosphere. Later on,
+it is conceivable that they could contribute, but we have tested run-
+ning SNEC with 56Ni decays turned on and there is no difference in
+the lightcurve. The second reason is that the total amount of energy
+produced by these decays (∼ 1053 ergs) is order one percent of the
+explosion energy.
+3.3.2 Multiband lightcurves at different redshifts
+In this section, we show the observer time evolution of the four reddest
+JWST NIRCAM widebands at a variety of redshifts. Because the
+galaxy merger scenario does not depend strongly on redshift (there
+is a decrease below redshift two, but only by an order of magnitude,
+see Fig. 4 of Bonoli et al. 2014), we show redshifts down to one. Fig.
+14 shows this for the fiducial model up to redshift ten, above which
+we suspect solar metallicity conditions would be challenging (though
+not impossible, see e.g. Fig. 3 of Hartwig et al. 2022) to realize. Fig.
+15 is a similar figure, but for the metal poor model, which we show
+up to redshift 19, though it would not be detectable by JWST much
+above redshift 10. Appendix C shows multiband lightcurves of the
+MNRAS 000, 1–16 (2022)
+
+Photo-sphere radius [cm]
+1018
+fiducialmodel
+metal-poor model
+1017
+massivemodel
+1016
+1015
+1014
+1047
+I [ergs/s]
+1046
+1045
+Lbol
+1044
+1043
+1042
+106
+105
+104
+103
+102
+10-4
+10~3
+10-2
+10-1
+100
+101
+102
+103
+t [days]cm]
+2.795
+2.790
+2.785
+2.780
+2.775
+2.770
+1.5
+1.0
+0.5
+2:9
+2.0
+got]
+1.5
+1.0
+0.5
+0.0
+-2
+0
+2
+4
+6
+8
+10
+t [s]CNO-rp driven GR instability supernovae.
+9
+Figure 12. Rest frame time dependence of JWST NIRCAM magnitudes at
+𝑧 = 5 for the fiducial model (upper panel), metal-poor model (middle panel)
+and massive model (lower panel). The horizontal dotted line shows a typical
+limiting magnitude for JWST (29).
+Figure 13. Rate of energy produced by radioactive decays in the metal-poor
+model after the end of the HYDnuc simulation. Vertical dotted lines separate
+regions where the energy production is dominated by different decays. The
+mother nuclei of those decays label each region.
+fiducial, metal poor, and massive models for JWST, Euclid, Roman,
+and Rubin at various redshifts.
+Comparing Figs. 14,15, it is clear that from the point of view of
+observation, more metal enrichment is better. Not only is the mass
+range wider at higher metallicity (Fig. 9), but also the GRSN would
+have occurred at lower redshift, making it easier to observe.
+3.3.3 With hylotropic envelope
+Up to this point, we have not considered the effect that an accretion
+envelope (Hosokawa et al. 2013; Umeda et al. 2016; Haemmerlé
+et al. 2019) would have on the lightcurve. While the core of an
+accreting supermassive protostar has constant entropy, the envelope
+is thought to have entropy increasing as a power law Begelman
+(2010); Haemmerlé et al. (2019); Haemmerlé (2020, 2021b) and
+this structure has been termed a hylotrope (Gk. hyle, ‘matter’ +
+tropos, ‘turn’) in Begelman (2010). Specifically, hylotropes obey the
+equation of state
+𝑃 = 𝐴𝜌4/3𝑀 𝛼
+(10)
+where 𝛼 is often taken to be 2/3 as derived from the homology
+scalings. In this scenario, we instead take 𝛼 as a parameter and
+enforce the mass radius relation of rapidly accreting supermassive
+protostars (Eq. 1 of Hosokawa et al. 2013). This choice is motivated
+by the sensitivity of lightcurves to the stellar radius.
+We construct the hylotropic envelope in a similar manner to the
+integration of an 𝑛 = 3 polytrope, as described in Haemmerlé (2020).
+We find that, taking the fiducial model as the core, 𝛼 = 0.90481
+satisfies the mass radius relation. This result is not sensitive to the
+matching radius. This hylotropic model has 𝑀𝐻 = 4.58 × 105 M⊙ ,
+𝑅𝐻 = 1.23 × 1016 [cm], so that 𝑀𝐻 /𝑀 = 4.58 and 𝑅𝐻 /𝑅 = 653.
+This result resembles numerical models of accreting supermassive
+protostars, although self consistent simulations are needed to verify
+the results of this section.
+We run SNEC by attaching the final timestep of the HYDnuc sim-
+ulation to the hylotropic envelope which was constructed using the
+initial profile of the HYDnuc simulation. This results in a discon-
+tinuity in density, but such a feature is not unexpected around this
+radius (see Fig. 14 of Nagele et al. 2022b). We use the thermal bomb
+mode with the final explosion energy of the hylotropic model being
+the explosion energy of the fiducial model, as the added gravitational
+energy of the envelope could not be overcome otherwise. This ap-
+proach is not self consistent as it requires the injection of an additional
+∼ 1055 ergs, but larger explosion energies (1056 ergs) are achieved
+in the massive model, so our strategy is not unreasonable. The pho-
+tosphere remains at the surface of the hylotrope for ten days as the
+shock propagates through the envelope, after which the photosphere
+expands and the effective temperature drops. From a peak value of
+𝐿bol = 6.86 × 1046 [ergs/s], the luminosity then falls nearly mono-
+tonically (except for a slight increase at hydrogen recombination) in
+behavior reminiscent of the pulsations of large radius supermassive
+stars in Nagele et al. (2022a). The luminosity of the hylotropic model
+falls at a slower rate than for the fiducial model, passing 𝐿bol = 1043
+[ergs/s] only after 9.95 years in the rest frame. Fig. 16 shows the
+magnitudes of the hylotropic model at redshift 1 for JWST, Euclid,
+Roman, and Rubin.
+3.3.4 Rate estimate
+We now turn to the question of how frequently these pr-GRSN occur.
+Bonoli et al. (2014) calculated the number density of massive galaxy
+MNRAS 000, 1–16 (2022)
+
+AB mag at z = 5
+24
+F070W
+F150W
+F356W
+F090W
+F200W
+F444W
+fiducial model
+26
+F115W
+F277W
+28
+30
+32
+24
+metal-poor model
+26
+28
+30
+32
+24
+model
+26
+massive
+28
+30
+32
+10~3
+10~2
+10-1
+100
+101
+102
+103
+t [days]52Mn
+...........
+...........
+.................................
+1047
+725e
+.....
+.......
+:
+...
+1046
+oGe
+...............
+........
+......
+.....
+1045
+....
+.......
+............
+1044
+56Ni
+.....
+1043
+.......
+........
+1042
+.....
+.......
+......
+......
+1041
+.............
+..............
+.......
+..............
+'s5Fe
+81kr
+1040
+.......
+.........
+1039
+106
+107
+108
+109
+1010
+1011
+1012
+t [s]10
+C. Nagele et al.
+Figure 14. Redshift dependence of the four reddest JWST NIRCAM wide bands for the fiducial model. The horizontal axis shows the observer time.
+Figure 15. Same as Fig. 14 but for the metal-poor model. The horizontal axis shows the observer time.
+MNRAS 000, 1–16 (2022)
+
+20
+z= 1
+20
+/magnitude
+22
+/magnitude
+22
+24
+24
+5
+F200W
+26
+77W
+26
+28
+2
+28
+10
+30
+30
+20
+20
+magnitude
+22
+F444Wmagnitude
+22
+24
+24
+F356W
+26
+26
+28
+28
+30
+30
+0
+5000
+10000
+15000
+20000
+25000
+0
+5000
+10000
+15000
+20000
+25000
+observertime[days]
+observertime[days]20
+z= 1
+20
+/magnitude
+z = 3
+22
+magnitude
+22
+24
+Z =
+24
+26
+11
+F200W
+13
+77W
+26
+28
+15
+2
+28
+30
+30
+20
+20
+magnitude
+22
+F444Wmagnitude
+22
+24
+24
+356W
+26
+26
+28
+28
+30
+30
+0
+2000
+400060008000100001200014000
+0
+2000400060008000100001200014000
+observertime[days]
+observertime[days]CNO-rp driven GR instability supernovae.
+11
+Figure 16. Magnitudes of the hylotropic model in JWST (first panel), Euclid
+(second panel), Roman (third panel), and Rubin (fourth panel) at redshift 1.
+The horizontal axis is observer time.
+mergers (𝑀halo > 1011 M⊙) which fulfilled the criteria for merger
+induced direct collapse (Fig. 4). For a simple order of magnitude
+estimation, we will assume a merger rate of 𝜙 = 10−4 [cMpc−3
+Gyr−1] which is fairly conservative (note that Fig. 4 of Bonoli et al.
+(2014) has units of [cMpc−3 0.1 Gyr−1]). As discussed in Bonoli
+et al. (2014), relaxing the mass asymmetry condition further would
+increase the merger rate by an order of magnitude. Another increase
+could be had by relaxing the mass constraint, although the minimum
+mass at which the merger scenario creates a direct collapse object
+is unknown. For a more recent, empirical, estimate of the galaxy
+merger rate, see Fig. 3 of O’Leary et al. (2021).
+To determine the number of supermassive protostars produced per
+unit time by the galaxy merger scenario, we simply multiply the rate
+𝜙 by the volume between redshifts 0.1 and 10 (𝑉𝑧∈(0.1,10)).
+𝑁protostars = 𝜙𝑉𝑧∈(0.1,10) ≈ 0.3 yr−1
+(11)
+The lower bound of this range was chosen because very nearby GRSN
+will be visible to many instruments besides the four discussed in this
+paper, while the upper bound is nearing the observing threshold for
+JWST (Fig. 14).
+Next we introduce 𝑓 , the fraction of protostars which explode as
+pr-GRSN. We have no ability to estimate the initial mass function
+of these objects and thus no ability to estimate 𝑓 . However, it is
+reasonable to assume that the mass function decreases for increasing
+mass, as does an estimate of the mass function of supermassive stars
+(Toyouchi et al. 2022). If this is the case, then 𝑓 will depend heavily
+on the behavior of the protostars less massive than those considered
+in this paper (Fig. 9). As we have mentioned previously, however, it is
+feasible that many of these less massive protostars will also explode,
+so an 𝑓 of order unity is feasible.
+𝑁pr−GRSN = 𝑓 𝜙𝑉𝑧∈(0.1,10) yr−1
+(12)
+From the rate of pr-GRSN, we would then like to obtain a number
+of currently ongoing pr-GRSN in the sky. To do this, we need to
+know the expected observer duration for the pr-GRSN (⟨𝑡obs⟩). To
+do this we perform a Monte Carlo simulation, randomly selecting
+lookback times between redshifts 0.1 and 10 and computing the
+observer time during which the GRSN is visible to the JWST F444W
+band. Averaging the Monte Carlo draws results in
+⟨𝑡obs⟩ = 29.5 yr
+(13)
+so that the expected number of currently observable pr-GRSN is
+𝑁observable = 𝑓 𝜙𝑉𝑧∈(0.1,10) ⟨𝑡obs⟩ ≈ 9f
+(14)
+and we plot this quantity as a function of 𝑓 in Fig. 17, indicating that
+it is reasonable to expect a few GRSNe to be observable right now.
+On the one hand, this is a small number because JWST only covers
+a tiny section of the sky. In addition, the true value of 𝑓 may be small.
+However, we point out that the majority of the Monte Carlo draws
+occur in the low redshift Universe where gas should be sufficiently
+enriched to trigger an explosion. Thus, the pertinent question be-
+comes what is the mass function of the supermassive protostars? On
+the other hand, Fig. 17 is showing a large number. Bonoli et al. (2014)
+and Mayer et al. (2015) are investigating the formation of the most
+massive SMBHs in the Universe. This investigation is well founded
+upon observations of high redshift quasars with inferred masses in
+excess of 108 M⊙, but it does not consider the vast majority of the
+black hole population, which have smaller masses (e.g. Inayoshi et al.
+2020). For this reason, the mass and mass ratio constraints which are
+applied in Fig. 4 of Bonoli et al. (2014) could be far too stringent,
+in which case the population of direct collapse objects, and GRSNe,
+would be much greater.
+In summary, massive uncertainties exists regarding the frequency
+of this scenario, but we have shown that it is at least plausible to
+expect pr-GRSNe to be observable in the low redshift Universe, if
+the galaxy merger scenario is the dominant source of supermassive
+black holes.
+3.3.5 Observing strategy
+Because of the long duration of the pr-GRSNe, identifying them as
+transients may not be straightforward, although detection by multi-
+ple instruments would ameliorate this difficulty; as well as JWST,
+Appendix C shows the magnitudes of Euclid, Roman, and Rubin at
+various redshifts. In the rest of this section, however, we will consider
+only the four reddest JWST NIRCAM widebands (see Fig. 12).
+We divide the lightcurve up into phases, a rising phase, a falling
+phase, and a plateau phase in between the rising and falling phases.
+We define the rising phase as lasting until peak magnitude is reached,
+which occurs at different times in each band (Fig. 12) and the falling
+phase as beginning when the photosphere radius jumps (Fig. 10);
+the falling phase will not be visible in all bands, especially at high
+redshift. Observations of the rising or falling phases will be easily
+identifiable as transients (Fig. 14), but the plateau phase, while not
+having constant magnitude, will dim at a rate of the order of a few
+magnitudes or less per decade (Fig. 18). This rate of dimming is
+roughly constant as the duration of the plateau phase increases (for
+MNRAS 000, 1–16 (2022)
+
+16
+F070W
+18
+F090W
+F115W
+JWST
+20
+F150W
+F200W
+22
+F277W
+F356W
+F444W
+24
+16
+VIS
+18
+Y
+Euclid
+20
+H
+22
+24
+16
+F062
+18
+F087
+F106
+Roman
+20
+F129
+F158
+22
+F184
+F146
+24
+F213
+16
+Z
+18
+Rubin
+20
+y
+u
+22
+6
+24
+2500
+5000
+7500
+10000
+12500
+15000
+17500
+20000
+observertime[days]12
+C. Nagele et al.
+Table 3. Various quantities related to chemical, mechanical, and radiative feedback for the post-processed models. The columns are mass, metallicity, the change
+in metal content, the change in iron, the change in mean molecular weight, the kinetic energy (HYDnuc), the fade radius (at which the shock becomes subsonic),
+the radiated energy (SNEC), the number of ionizing photons, and the number of Lyman Werner photons, where the last two assume blackbody emission.
+M [105 M⊙]
+𝑍/𝑍⊙
+𝑀𝑍 [M⊙]
+𝑀Fe [M⊙]
+Δ𝜇
+𝐸kin [1055 ergs]
+𝑅fade [Mpc]
+𝐸rad [1052 ergs]
+𝑁𝛾,ion [1054]
+𝑁𝛾,LW [1054]
+1.0
+0.12254
+201.1
+24.73
+0.01721
+2.684
+1.423
+2.257
+55.9
+9.65
+1.0
+0.1226
+158.5
+26.25
+0.01663
+2.409
+1.278
+2.285
+25.28
+7.807
+1.0
+0.123
+105.8
+26.91
+0.01614
+2.108
+1.118
+2.32
+91.29
+9.491
+1.0
+0.13
+26.74
+2.669
+0.01591
+1.637
+0.8683
+2.43
+28.47
+5.851
+1.0
+0.2
+9.599
+-0.0008733
+0.01698
+1.45
+0.7692
+2.548
+8.964
+3.215
+1.0
+1.0
+4.933
+-0.002745
+0.02357
+1.593
+0.8446
+2.46
+7.814
+2.287
+0.9
+1.0
+4.509
+-0.002353
+0.02311
+1.278
+0.6777
+2.245
+380.7
+267.2
+1.5
+1.0
+48.43
+-0.004265
+0.05847
+8.055
+4.272
+3.914
+173.7
+22.5
+2.0
+1.0
+239.7
+-1.237
+0.0823
+17.26
+9.15
+5.485
+209.6
+41.23
+Figure 17. Expected number of concurrent GRSNe observable by JWST as
+a function of the fraction ( 𝑓 ) of supermassive protostars which explode as
+pr-GRSN.
+higher redshifts), until the photosphere radius jump is no longer
+visible (above 29 magnitude). For these fainter, high redshift, sources,
+the plateau phase is shorter and the magnitude changes more quickly.
+Fig. 19 shows color-color diagrams for the four reddest JWST
+NIRCAM filters. All of the data points, including different models
+and redshifts, fall neatly along a single line. The results of a linear fit
+to this line are shown in each plot. Based on these features, we propose
+that pr-GRSNe candidates are identified as NIRCAM sources falling
+within at least the three reddest bands which are consistent with Fig.
+19. These candidates may then be confirmed or ruled out with long
+cadence observations. GRSNe which have already dropped out of
+F270W and F2777W will not be identifiable by their color, and can
+only be identified with repeated observation.
+On the one hand, the long durations and shallow slopes of these
+lightcurves pose a challenge to observing these events as transients.
+On the other hand, if they are identified as such, their further clas-
+sification as GRSNe will be relatively unambiguous, as there exist
+no other events one hundred thousand times more energetic than a
+supernova.
+Figure 18. Rate of dimming during the plateau phase as a function of plateau
+phase duration for the four reddest NIRCAM bands (different panels). The
+colors denote redshifts and are shown in steps of 0.25, while the symbols
+show different models.
+3.4 Feedback
+The pr-GRSN likely would have been extremely disruptive to its
+host halo. Previously, several studies were conducted on the effect
+a standard GRSN would have on the halo, specifically on metal
+enrichment, gas evacuation, and star formation (Whalen et al. 2013a;
+Johnson et al. 2013; Whalen et al. 2013b). The pr-GRSN is more than
+an order of magnitude more energetic than in the standard case, and
+MNRAS 000, 1–16 (2022)
+
+8
+S
+4
+2
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ffiducial
+Z= 1
+0.4
+massive
+Z=2
+metalpoor
+Z=3
+0.3
+Z = 4
+F200W
+z = 5
+0.2
+Z=6
+Z = 7
+0.1
+z=9
+z = 10
+Plateauphase
+0.0
+0.4
+0.3
+F277W
+0.2
+0.1
+0.0
+0.4
+0.3
+F356W
+0.2
+0.1
+0.0
+0.4
+0.3
+0.2
+0.1
+0.0
+0
+10
+20
+30
+40
+50
+Plateauphaseduration[years]CNO-rp driven GR instability supernovae.
+13
+Figure 19. Two color-color diagrams for the four reddest NIRCAM wide-
+bands. As in Fig. 18, colors denote redshift and symbols denote models. Also
+shown is a linear fit (dotted line) as well as its slope and intercept.
+has a completely different chemical signature. Whereas the standard
+GRSN is characterized by excess silicon and magnesium, the pr-
+GRSN will produce nitrogen from the CNO cycle, plus elements
+from wherever the bulk transport lands on the line of stability (Figs.
+3,8).
+But this chemical enrichment will not be distributed evenly
+throughout the halo (at least not at first). Heavier elements are pref-
+erentially produced in the center of the protostar where higher tem-
+peratures are reached. In the resulting outflow, these regions have
+lower velocity, as the ejecta has nearly homologous structure (𝑣 ∝ 𝑟).
+Thus, portions of the ejecta with different chemical compositions
+will end up in different parts of the halo. We have attempted to vi-
+sualize this effect in Fig. 20, which shows the yields for different
+mass coordinates on the left, and the velocities of those correspond-
+ing coordinates on the right. As an example, in the middle panel
+(metal-poor model), consider the ratios of [Fe/Ni] and [C/Fe]. The
+former will be constant throughout the halo [Fe/Ni] = -3 (or 0 in the
+outermost regions), while the latter will vary [C/Fe] ∈ (-2,0).
+Table 3 summarizes various feedback effects for selected models.
+The third and fourth columns show the total amounts of metals and
+iron synthesized during the explosion. Note that models which do not
+reach sufficiently high temperature produce metals that do not extend
+up to iron which experiences a net loss. The change in mean molecular
+weight is shown in the next column. Next are the enormous effects
+of mechanical feedback, specifically the total kinetic energy and the
+radius at which the shock speed will become subsonic (𝑅fade, Magg
+et al. 2020), though this value does not take the effect of the halo’s
+gravity into account. Whalen et al. (2013a) found that the GRSN
+ejecta reached a radius of 1 kpc before falling back and igniting a
+violent starburst, although that was for a less energetic explosion.
+Finally, the radiated energy as well as the number of ionizing (E
+> 13.6 eV) and Lyman Werner (13.6 > E > 11.2 eV) photons. The
+number of high energy photons is fairly small because the effective
+temperature is low after shock breakout.
+4 DISCUSSION
+As the saying goes, hydrogen is flammable. We have shown that
+if supermassive protostars have sufficient seed metals when they
+collapse, then they will explode through a combination of the CNO
+cycle and rp process. The consequences of this are as follows.
+Our results present a challenge to the galaxy merger scenario
+in two senses of the word. The first is that if the galaxy mergers
+do not happen early enough in the Universe, then when they do
+occur, the gas may be sufficiently enriched to trigger a pr-GRSN,
+thereby preventing the formation of a SMBH seed. If this scenario
+is avoided, then SMBH seeds may form from galaxy mergers and
+this may explain the presence of the first quasars. However, as the
+Universe continues to enrich itself, galaxy mergers continue to occur,
+and if these later mergers produce a supermassive protostar, it will
+almost certainly contain the requisite metallicity for an explosion. In
+addition, these low redshift pr-GRSN will be easily observable. Thus,
+if current and future NIR surveys do not see these objects, then the
+galaxy merger scenario must posses a way to suppress massive object
+formation at lower redshift. Such a mechanism is naturally built into
+other scenarios such as atomic cooling halos by the stipulation that
+the gas must be nearly metal free (Z/Z⊙ < 10−3, Chon & Omukai
+2020; Hirano et al. 2022).
+As we have touched upon, the intermediate mass black hole popu-
+lation, or relative lack thereof, also presents a challenge to the study
+of black holes. If SMBHs originate primarily from the galaxy merger
+scenario, then our results present a natural explanation for the lack
+of intermediate mass black holes. This is because metal enriched
+supermassive stars and protostars which are not massive enough to
+collapse to supermassive black holes (say, 106 M⊙) do not then
+collapse to intermediate black holes, but instead either explode in a
+pr-GRSN or lose most of their mass due to line driven winds while
+on the main sequence. In the latter case, the mass of the final remnant
+will be determined roughly by how much helium can be produced
+before the envelope is completely lost, although mass loss from the
+helium core may also be possible in some situations.
+Although observation of the lightcurves from a pr-GRSN carries
+our best chance at observation, other pathways exist. We have pro-
+duced detailed nucleosynthetic signatures of the pr-GRSN and there
+is some possibility that we may be able to find traces of these signa-
+tures. A signature common to all of our models is enhanced nitrogen
+production. A large population of nitrogen-enhanced mildly metal
+poor stars has recently been observed (Fernández-Trincado et al.
+2020), but any explanation involving a pr-GRSN seems far-fetched
+due to the large amount of ejecta produced by a single event. Another
+signature found in many of our models is a dearth of light alpha el-
+ements, such as magnesium, which serve as the fuel for the proton
+captures. Yoshii et al. (2022) reported a quasar with [Mg/Fe] =-1.11
+± 0.12 at redshift 7.54. They discuss how it is challenging to produce
+so much iron so early in the Universe, but the pr-GRSN would nat-
+urally explain this phenomenon by producing iron and consuming
+MNRAS 000, 1–16 (2022)
+
+fiducial
+2.0
+massive
+metalpoor
+1.5
+F356W
+1.0
+z =1
+F277W-
+z = 2
+Z =3
+Z=4
+0.5
+z = 5
+Z=6
+z = 7
+..... linear fit
+8=Z
+0.0
+m= 1.768
+6=Z
+b=0.303
+z = 10
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+F200W-F277W
+1.75
+fiducial
+massive
+1.50
++metalpoor
+1.25
+P
+F444W
+1.00
+F356W-
+0.75
+Z = 1
+Z = 2
+z = 3
+0.50
+Z = 4
+Z = 5
+0.25
+Z = 6
+Z = 7
+0.00
+.....linear fit
+Z=8
+m= 1.158
+Z=9
+b = 0.266
+Z = 10
+-0.25
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+F277W-F356W14
+C. Nagele et al.
+Figure 20. Elemental yields at selected mass coordinates and velocity profiles for the three indicated models. Left column — elemental abundance for mass
+coordinates (not the average between mass coordinates) at multiples of 0.2 times the total mass. Right column — velocity profiles at a comparable timestep
+near the end of the HYDnuc simulation for each of the three models. The velocity profiles are nearly homologous. Colored circles indicate the mass coordinate
+shown in the left column.
+magnesium simultaneously. At this redshift, the ISM metallicity can
+be greater than the explosion threshold (Pallottini et al. 2014).
+We will now summarize some assumptions and shortcoming of the
+current study. First is our decision to ignore mass loss and accretion
+during the stellar evolution calculation, and thus limit ourselves to
+protostars greater than ∼ 105 M⊙. Because only a small mass fraction
+of protons is required to trigger the pr-GRSN, if a metal enriched
+supermassive star were to collapse via the GR radial instability at
+any point during the hydrogen burning phase (besides the very end),
+then it would likely explode. Similarly, if the GR instability triggers
+during the helium burning phase, a metal enriched version of the 𝛼
+process GRSN is plausible. A logical next step would be to apply our
+methods to realistic models of metal enriched accreting supermassive
+stars (e.g. Haemmerlé et al. 2019) which would also allow us to
+correctly account for the accretion envelope. The lightcurve of the
+pr-GRSNe is likely sensitive to the size of the accretion envelope
+(Fig. 16).
+The second major shortcoming of this study is that we do not
+have the resources to couple the 514 isotope network to our GR
+hydrodynamics code. Thus, all of the nucleosynthesis that we present
+is the result of post-processing, which may not be entirely accurate.
+In addition, since the energy generation by the 153 isotope network is
+smaller than that of 514 (Sec. 2), we likely underestimate the region
+MNRAS 000, 1–16 (2022)
+
+1.75
+8
+6
+1.50
+[X/Fe]
+4
+1.25
+P
+S
+cm/
+1.00
+model
+0
+0.75
+fiducial
+2
+Mg
+m, = 0.2 M
+-4
+0.50
+m,=0.4M
+-6
+m,=0.6M
+0.25
+m,=0.8M
+-8
+m,=1.0M
+0.00
+8
+1.75
+6
+1.50
+[X/Fe]
+Sc
+4
+1.25
+2
+N
+metal-poormodel
+cm/
+1.00
+0
+0.75
+m,=0.0M
+-4
+Co
+0.50
+Mg
+m.= 0.4 M
+-6
+m,=0.6M
+0.25
+m, = 0.8 M
+-8
+m.=10M
+0.00
+8
+1.75
+6
+1.50
+[X/Fe]
+1.25
+S
+cm/
+1.00
+model
+0
+60t]
+0.75
+massive
+-2
+m,=0.0M
+>
+Mq
+m, = 0.2 M
+0.50
+m,=0.4M
+-6
+m, = 0.6 M
+0.25
+m,=0.8M
+-8
+m,=1.0M
+0.00
+5
+10
+15
+20
+25
+30
+35
+40
+45
+0
+1
+2
+3
+4
+5
+6
+7
+protonnumber
+r [1014 cm]CNO-rp driven GR instability supernovae.
+15
+covered by the pr-GRSN (Fig. 9). Other caveats such as effects of
+rotation and initial conditions are salient, but likely less important.
+We have shown that supermassive protostars which encounter the
+GR radial instability before hydrogen burning will explode in a pr-
+GRSN if the metallicity is high enough. These events will be clearly
+visible to NIR surveys at lower redshifts, and marginally visible at
+higher ones. They will also leave distinct chemical imprints on their
+host halos, as well as massive amounts of radiative and mechanical
+feedback. It is likely that current and future surveys with unprece-
+dented breadth and depth will be able to constrain the population of
+merger induce DCBHs based on the observation or non observation
+of pr-GRSNe.
+DATA AVAILABILITY
+The data underlying this article will be shared on reasonable request
+to the corresponding author.
+ACKNOWLEDGEMENTS
+This study was supported in part by the Grant-in-Aid for the Sci-
+entific Research of Japan Society for the Promotion of Science
+(JSPS, Nos. JP19K03837, JP20H01905, JP20H00158, JP21H01123,
+JP22K20377) and by Grant-in-Aid for Scientific Research on Inno-
+vative areas (JP17H06357, JP17H06365) from the Ministry of Edu-
+cation, Culture, Sports, Science and Technology (MEXT), Japan.
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+MNRAS 000, 1–16 (2022)
+
+16
+C. Nagele et al.
+5 APPENDIX A
+This appendix shows the results of the 216 isotope simulation for M
+= 105 M⊙, Z = 0.1 Z⊙. The 216 isotope network is based on the 153
+isotope network, but contains additional heavy elements up to Zr.
+We had created this network because we noticed that our collapsing
+models were saturating at Zn, the heaviest element in the 153 network.
+Fig. 21 shows the isotopic mass fractions from the central mesh of
+this simulation as well as the proton mass fraction (bottom panel).
+As is apparent, proton captures stall around a central temperature of
+Log 𝑇𝑐 = 8.85, as the network runs out of room for additional proton
+captures. In addition, the decay times of the isotopes on the boundary
+of the network are longer than the dynamical timescale (∼ 500 s).
+This causes a buildup of isotopes at certain positions on the edge
+of the network. In an extreme case, 66Ge accumulates nearly half a
+percent (0.004) of the mass. This isotope has a half life of roughly
+8000 s, largely preventing progress past it.
+We offer this demonstration as a cautionary tale, that future net-
+works likely need to be larger than the ones we have used here. Such
+endeavours are beyond our current computational setup.
+6 APPENDIX B
+If we did have the resources to couple to a larger network, to what
+extent would that alter our results?
+To provide a simple estimate of the energy generation of the rp
+process, we calculate the energy density produced by 25% of the
+star undergoing the rp process up to an isotope, 𝐼𝑚. We assume that,
+in this region, every element with atomic number greater than 15 is
+converted to the most abundant solar isotope with proton number 𝑚
+(𝐼𝑚). Then, the chemical distribution of the inner 25% will change
+as follows:
+𝑋′
+𝐼𝑗 =
+�
+𝑋𝐼𝑗
+𝐴(𝐼 𝑗) < 16 or A(Ij) > A(Im)
+0
+15 < 𝐴(𝐼 𝑗) < 𝐴(𝐼𝑚)
+(15)
+𝑋′
+𝑝 = 𝑋𝑝 −
+∑︁
+𝐼𝑗
+(𝑋𝐼𝑗 − 𝑋′
+𝐼𝑗 ) 𝐴(𝐼𝑚) − 𝐴(𝐼 𝑗)
+𝐴(𝐼 𝑗)
+(16)
+𝑋′
+𝐼𝑚 = 𝑋𝐼𝑚 + 𝑋′
+𝑝 +
+∑︁
+𝐼𝑗
+(𝑋𝐼𝑗 − 𝑋′
+𝐼𝑗 )
+(17)
+Fig. 22 shows the resulting 𝐸nuc (in units of ergs/g) as a function of
+metallicity and m. Also shown are 𝐸nuc from the exploding models
+in Fig. 9. We have selected 25% so that the border between explosion
+and collapse will occur at the edge of our network, roughly m = 30.
+Note that this is a somewhat ad hoc approach, as we are ignoring
+neutrino cooling and inwards ram pressure, and we thus intend for
+this figure to be interpreted as an order of magnitude estimate. As
+such, it shows that a larger network would allow access to at least
+twice as much energy generated by the rp process if it extended up
+to technetium, a situation which we have shown to result from post
+processing (Fig. 6). Indeed, the post processed 𝐸nuc is roughly twice
+that produced by the 153 isotope network. Thus we can say that
+our mass and metallicity range is too conservative, but probably the
+correct order of magnitude.
+7 APPENDIX C
+This appendix shows lightcurves of JWST, Euclid, Roman and Rubin
+for the fiducial model, at redshifts 0.1, 0.3, 0.5, 1.0, 3.0, 5.0. Rubin
+can observe the pr-GRSN below redshift 1, while Euclid and Roman
+extend this up to redshift 3. Observation of more distant events will
+only be possible with JWST.
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–16 (2022)
+
+CNO-rp driven GR instability supernovae.
+17
+Figure 21. Upper panel — same as Fig. 6, but for only the central mesh of the M = 105 M⊙, Z = 0.1 Z⊙ with 216 isotopes. Lower panel — time evolution of
+proton mass fraction.
+Figure 22. 𝐸nuc (in units of ergs/g) generated by the rp process up to an
+isotope with mass proton number m for a given metallicity. The red triangles
+correspond to values of 𝐸nuc from the 153 isotope simulations (Fig. 9). The
+vertical dashed line shows the edge of the 153 isotope network.
+Figure 23. Same as Fig. 16, but for the fiducial model at redshift 0.1.
+MNRAS 000, 1–16 (2022)
+
+Log T。= 8.31
+Log T, = 8.65
+Log Te = 8.84
+Log Te = 8.90
+Log Te= 8.98
+40
+10
+-2
+-4
+Log
+-6
+20
+-8
+-10
+10E
+-12
+10
+20
+30
+40
+50
+10
+20
+30
+40
+5010
+20
+30
+40
+50
+10
+20
+30
+40
+50
+10
+20
+30
+40
+50
+nnumber
+0.75
+...
+47500
+50000
+52500
+55000
+57500
+60000
+62500
+65000
+t [s] 1e18
+1.0
+1.17
+1.04
+0.8
+0.91
+0.78
+[ergs/g]
+0.6
+0.65
+N
+0.52
+0.4
+0.39
+0.26
+0.2
+0.13
+0.00
+10
+15
+20
+25
+30
+35
+40
+protonnumberatrpterminus(m)15.0
+F070W
+F090W
+17.5
+F115W
+5
+F150W
+20.0
+M
+F200W
+22.5
+F277W
+F356W
+25.0
+F444W
+15.0
+VIS
+Y
+17.5
+Euclid
+H
+20.0
+22.5
+25.0
+15.0
+F062
+F087
+17.5
+F106
+Roman
+F129
+20.0
+F158
+22.5
+F184
+F146
+25.0
+F213
+15.0
+Z
+17.5
+Rubin
+20.0
+y
+u
+22.5
+b
+25.0
+0
+1000
+2000
+3000
+4000
+5000
+observertime[days]18
+C. Nagele et al.
+Figure 24. Same as Fig. 16, but for the fiducial model at redshift 0.3.
+Figure 25. Same as Fig. 16, but for the fiducial model at redshift 0.5.
+Figure 26. Same as Fig. 16, but for the fiducial model at redshift 1.
+Figure 27. Same as Fig. 16, but for the fiducial model at redshift 3.
+MNRAS 000, 1–16 (2022)
+
+17.5
+F070W
+F090W
+20.0
+F115W
+JWST
+22.5
+F150W
+F200W
+25.0
+F27XW
+F356W
+27.5
+F444W
+30.0
+17.5
+VS
+Y
+20.0
+Euclid
+22.5
+H
+25.0
+27.5
+30.0
+17.5
+F062
+F087
+20.0
+F106
+ue
+22.5
+F129
+Rom
+F158
+25.0
+F184
+F146
+27.5
+F213
+30.0
+17.5
+20.0
+Rubin
+22.5
+y
+u
+25.0
+6
+27.5
+30.0
+0
+1000
+2000
+3000
+4000
+5000
+observertime[days]17.5
+F070W
+F090W
+20.0
+F115W
+JWST
+F150W
+22.5
+F200W
+25.0
+F277W
+F356W
+27.5
+F444W
+30.0
+17.5
+VS
+Y
+20.0
+Euclid
+22.5
+H
+25.0
+27.5
+30.0
+17.5
+F062
+F087
+20.0
+F106
+ue
+22.5
+F129
+Rom
+F158
+25.0
+F184
+N146
+27.5
+P213
+30.0
+17.5
+Z
+r
+20.0
+Rubin
+22.5
+y
+u
+25.0
+b
+27.5
+30.0
+0
+1000
+2000
+3000
+4000
+5000
+observertime[days]20.0
+F070W
+22.5
+F090W
+F115W
+JWST
+25.0
+F15QW
+F200W
+27.5
+F277W
+30.0
+F356W
+F444W
+32.5
+20.0
+VS
+22.5
+Y
+Euclid
+25.0
+H
+27.5
+30.0
+32.5
+20.0
+F062
+22.5
+F087
+F106
+Roman
+25.0
+F129
+F158
+27.5
+F184
+30.0
+F146
+F213
+32.5
+20.0
+22.5
+Rubin
+25.0
+u
+27.5
+6
+30.0
+32.5
+0
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+observertime[days]22.5
+F070W
+F090W
+25.0
+F115W
+JWST
+F150W
+27.5
+F200W
+30.0
+F277WL
+F356W
+32.5
+F444W
+35.0
+22.5
+VIS
+Y
+25.0
+Euclid
+27.5
+H
+30.0
+32.5
+35.0
+22.5
+F062
+F087
+25.0
+F106
+Roman
+27.5
+F129
+F158
+30.0
+F184
+F146
+32.5
+F213
+35.0
+22.5
+Z
+25.0
+r
+i
+Rubin
+27.5
+u
+30.0
+6
+32.5
+35.0
+0
+2000
+4000
+6000
+8000
+10000
+observertime[days]CNO-rp driven GR instability supernovae.
+19
+Figure 28. Same as Fig. 16, but for the fiducial model at redshift 5.
+MNRAS 000, 1–16 (2022)
+
+22.5
+F070W
+F090W
+25.0
+F115W
+JWST
+27.5
+F150W
+F200W
+30.0
+F277W
+F356W
+32.5
+F444W
+35.0
+22.5
+VIS
+Y
+25.0
+Euclid
+27.5
+H
+30.0
+32.5
+35.0
+22.5
+F062
+F087
+25.0
+F106
+Roman
+27.5
+F129
+F158
+30.0
+F184
+F146
+32.5
+F213
+35.0
+22.5
+Z
+r
+25.0
+Rubin
+27.5
+u
+30.0
+6
+32.5
+35.0
+0
+2000
+4000
+6000
+8000
+10000
+observertime[days]

AdAyT4oBgHgl3EQfd_gY/content/tmp_files/2301.00311v1.pdf.txt

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+Non-linear corrections of overlap reduction functions for pulsar timing arrays
+Qing-Hua Zhu∗
+CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
+Chinese Academy of Sciences, Beijing 100190, China and
+School of Physical Sciences, University of Chinese Academy of Sciences,
+No.
+19A Yuquan Road, Beijing 100049, China
+(Dated: January 3, 2023)
+The signals from international pulsar timing arrays have presented a hint of gravitational
+stochastic background in nHz band frequency. Further confirmation will be based on whether
+the signals follow the angular correlation curves formulated by the overlap reduction func-
+tions, known as Hellings-Downs curves. This paper investigates the non-linear corrections of
+overlap reduction functions in the present of non-Gaussianity, in which the self-interaction
+of gravity is first taken into considerations. Based on perturbed Einstein field equations
+for the second order metric perturbations, and perturbed geodesic equations to the second
+order, we obtain non-linear corrections for the timing residuals of pulsar timing, and theo-
+retically study corresponding overlap reduction functions for pulsar timing arrays. There is
+order-one correction for the overlap reduction functions from the three-point correlations of
+gravitational waves, and thus the shapes of the overlap reduction functions with non-linear
+corrections can be distinguished from the Hellings-Downs curves.
+I.
+INTRODUCTION
+The stochastic gravitational wave background (SGWB) might contain lots of information of
+the Universe, since it can be originated from inflationary GWs [1–5], produced from early-time
+phase transitions [6–8], sourced by cosmic string [9–12], or formed by superpositions of unresolved
+individual GW sources such as binary systems [13–17], core-collapse supernovae [18–21], and de-
+formed rotating neutron stars [19, 22]. In the 10Hz–1kHz frequency band, the ground-based GW
+detectors LIGO/Virgo/KAGRA have presented an upper limits of SGWBs [23, 24]. In nHz fre-
+quency band, the international timing pulsar array projects (IPTA) [25–27] found and confirmed
+a common spectrum process from the pulsar-timing data sets, and suggested that further evidence
+for SGWBs might rely on its angular correlation signature [28–30].
+The angular correlations of output of a pair of GW detectors can give characteristic signature of
+∗ [email protected]
+arXiv:2301.00311v1  [gr-qc]  1 Jan 2023
+
+2
+GWs, which is formulated by overlap reduction functions (ORFs). For GW detector networks made
+by pulsar timing arrays (PTAs), the ORFs of GWs are known as Hellings-Downs curve for a pair
+of pulsars [31]. Motivated by the observation from IPTA on the angular correlations of SGWBs,
+it is necessary to clarify physical causes of deviations of the Hellings-Downs curve. It might come
+from the SGWBs beyond isotropy approximation [32, 33], polarized SGWBs [34–36], non-tensor
+modes from modified gravity [37–42], or simply a careful calculation on the pulsar terms [43–45].
+Recently, there was also study on the non-linear corrections for PTAs from higher order effect of
+gravity [46]. This correction for ORFs was shown to be order one in the present of non-Gaussianity
+of the GWs. In this paper, we will extend the study on the non-linear corrections of the ORFs, in
+which the self-interaction of gravity is taken into considerations.
+Due to the self-interaction of gravity, the linear order of GWs can generate the non-linear one
+even in vacuum, which can affect the propagation of light, thereby changing the response of GW
+detectors. There was shown to be order-one corrections for the ORFs in the present of the non-
+Gaussianity of GWs [46]. To obtain a solid derivation for GW detectors to the non-linear regime,
+we utilize perturbed Einstein field equations for evolution of metric perturbations, and perturbed
+geodesic equations for evaluating timing residuals of pulsar timing. We compute the ORFs with
+the non-linear corrections, and study its shapes.
+The rest of the paper is organized as follows. In Sec. II, the evolutions of metric perturbations
+to the second order are presented. In Sec. III, based on propagation of light formulated by the
+perturbed geodesic equations to the second order, we obtain timing residuals of pulsar timing with
+non-linear corrections. In Sec. IV, we compute ORFs with different shape of non-Gaussianity, and
+show its deviation from the Hellings-Downs curves. In Sec. V, conclusions and discussions are
+summarized.
+II.
+PROPAGATION OF SECONDARY GRAVITATIONAL WAVE WITHIN PTAS
+From the theory of perturbations in general relativity, the first order GWs, the transverse-
+traceless modes of metric perturbations, should give rise to secondary effects from the higher order
+metric perturbations. Because all the space-time fluctuations can affect propagation of light, there
+are inevitably corrections of response of GW detectors in non-linear regime. In this section, to
+study the secondary effects of GWs on PTAs, we will firstly show how the non-linear order metric
+perturbations freely propagate in vacuum.
+
+3
+The perturbed metric in Minkowski background is given by [47]
+ds2 = −dt2 +
+�
+δij(1 − ψ(2)) + ∂i∂jE(2) + 1
+2∂iC(2)
+j
++ 1
+2∂jC(2)
+i
++ h(1)
+ij + 1
+2h(2)
+ij
+�
+dxidxj ,
+(1)
+where δij is Kronecker symbol, h(1)
+ij is the first order transverse-traceless metric perturbation known
+as GWs, and h(2)
+ij , C(2)
+j , and ψ(2) are the second order tensor, vector, and scalar perturbations,
+respectively. Because GW detectors are reckon in the freely-falling frame, we have adopted Syn-
+chronous gauge for the perturbed metric.
+From Einstein field equations for the first order GWs, the motion of equations reduce to wave
+equations,
+h(1)
+ij
+′′ − ∆h(1)
+ij = 0 .
+(2)
+Thus, the solutions can be given by
+h(1)
+ij =
+�
+d3k
+(2π)3 ¯h(1)
+ij,ke−i(kt−k·x) ,
+(3)
+where the h(1)
+ij,k is the Fourier mode of ¯h(1)
+ij , which contains the physical information before the
+GWs reaching the detectors.
+Since the GW detectors can response to tensor, vector, and scalar modes of the metric pertur-
+bations, we should further consider all the second order metric perturbations. Using the metric in
+Eq. (1), the perturbed Einstein field equations in the second order take the form of [48, 49]
+h(2)
+ij
+′′ − ∆h(2)
+ij = −Λab
+ij Sab ,
+(4a)
+C(2)
+j
+′′ = −Vab
+j Sab ,
+(4b)
+−2ψ(2)′′ + ∆
+�
+E(2)′′ + ψ(2)�
+= −S(Ψ),abSab ,
+(4c)
+E(2)′′ + ψ(2) = −S(E),abSab ,
+(4d)
+where Λab
+ij , Vab
+j , S(Ψ),ab and S(E),ab are helicity decomposition operators [47], and the source on
+the right hand side of Eq. (4) is given by
+Sab = −2δcd∂0h(1)
+ac ∂0h(1)
+bd + 2h(1),cd∂c∂dh(1)
+ab − 2h(1),cd∂c∂ah(1)
+db
+−2h(1),cd∂c∂bh(1)
+da − 2∂dh(1)
+ac ∂ch(1)
+bd + 2δcd∂jh(1)
+ad ∂jh(1)
+bc + ∂ah(1),cd∂bh(1)
+cd + 2h(1),cd∂a∂bh(1)
+cd
++δab
+�3
+2∂0h(1)
+cd ∂0h(1),cd + 2h(1)
+cd ∂2
+0h(1),cd − 2h(1)
+cd ∆h(1),cd + ∂jh(1)
+cd ∂dh(1),cj − 3
+2∂jh(1)
+cd ∂jh(1),cd
+�
+.
+(5)
+
+4
+Differed from Eq. (2), the second metric perturbations are sourced the by the first order GWs. By
+making use of Eq. (3) and (4), we obtain the solutions in the form of
+ψ(2) (t, x) =
+�
+d3k
+(2π)3
+��
+d3p
+(2π)3
+�
+ˆF ab
+ψ ¯Sab (k, p) e−i(|k−p|+p)t�
+eik·x
+�
+,
+(6a)
+E(2) (t, x) =
+�
+d3k
+(2π)3
+��
+d3p
+(2π)3
+�
+ˆF ab
+E ¯Sab (k, p) e−i(|k−p|+p)t�
+eik·x
+�
+,
+(6b)
+C(2)
+j
+(t, x) =
+�
+d3k
+(2π)3
+��
+d3p
+(2π)3
+�
+ˆF ab
+C,j ¯Sab (k, p) e−i(|k−p|+p)t�
+eik·x
+�
+,
+(6c)
+h(2)
+ij (t, x) =
+�
+d3k
+(2π)3
+��
+d3p
+(2π)3
+�
+ˆF ab
+h,ij ¯Sab (k, p) e−i(|k−p|+p)t�
+eik·x
+�
+,
+(6d)
+where
+ˆF ab
+h,ij (k, p) ≡ −
+Λab
+ij (k)
+k2 − (|k − p| + p)2 ,
+(7a)
+ˆF ab
+C,j (k, p) ≡
+Vab
+j (k)
+(|k − p| + p)2 ,
+(7b)
+ˆF ab
+ψ (k, p) ≡ −S(Ψ),ab (k) + k2S(E),ab (k)
+2 (|k − p| + p)2
+,
+(7c)
+ˆF ab
+E (k, p) ≡
+S(E),ab (k)
+(|k − p| + p)2 − S(Ψ),ab (k) + k2S(E),ab (k)
+2 (|k − p| + p)4
+.
+(7d)
+The ¯Sab (k, p) in Eq. (6) is defined with
+¯Sij (k, p) = fcdab
+ij
+(k, p) ¯h(1)
+cd,k−p¯h(1)
+ab,p ,
+(8)
+where
+fbclm
+ij
+(k, p) = δblδcm (−3δij (p |k − p| − p · (k − p)) + pi(kj − 2pj) + ki(pj − 2kj))
++2
+�
+pb(−δl
+iδm
+j pc + δcm(δl
+jpi + δl
+ipj + δij(pl − kl))
++δb
+j
+�
+δcm �
+δl
+i (p |k − p| − p · (k − p)) + ki(kl − pl) + pi(pl − kl)
+�
++ δl
+ipc(km − pm)
+�
++δb
+i
+�
+δcm �
+δl
+j (p |k − p| − p · (k − p)) + kj(kl − pl) + pj(pl − kl)
+�
++ δl
+jpc(km − pm)
+�
+−δb
+i δc
+j(klkm − 2klpm + plpm)
+�
+.
+(9)
+The ¯Sab (k, p) is derived from the source in Eq. (5) via
+Sab (t, x) =
+�
+d3k
+(2π)3 Sab (t, k, p) eik·x =
+�
+d3k
+(2π)3 ¯Sab (k, p) e−i(|k−p|+p)t+ik·x .
+(10)
+Note that the Sab (t, k, p) is the Fourier mode of the source, and the ¯Sab (k, p) is not. Here, as
+shown in Eqs. (6), the explicit solutions of perturbations ψ(2), E(2), C(2)
+j
+and h(2)
+ij can be obtained
+based on the known h(1)
+ij in Eq. (3).
+
+5
+We consider the non-linear corrections for GW detectors originated from the gravity self-
+interaction. In this case, all the second order perturbations are generated by the first order one.
+In fact, there could be second order perturbations that are independent of the h(1)
+ij . Because the
+response of the second order perturbations of this type to the GWs detector has nothing different
+from the response for the first order GWs, we did not consider this situation in the present study.
+III.
+PERTURBED GEODESIC EQUATIONS
+The space-time fluctuations can affect time of arrivals of radio beams from a pulsar. Thus, via
+monitoring pulsar timing, the PTA observations can reflect the space-time fluctuations over the
+Universe in principle. To formulate it, and extend it to the non-linear regime, we will calculate the
+perturbed geodesic equations to the second order.
+Based on geodesic equations in Minkowski background, namely,
+P µ∂µP ν = 0 ,
+(11)
+one can obtain the 4-momentum of a light ray,
+P µ = P 0(1, −ˆnj) ,
+(12)
+where ˆnj is a constant unit vector, and can be used to locate a pulsar. By making use of above
+4-momentums, one can obtain trajectories of light rays from a pulsar to the detectors. The pulsar
+emits a radio beam at the event (t − L, Lˆnj), and the beam is detected on the earth at the event
+of (t, 0), where the L is distance between the pulsar and the detectors.
+From the first order perturbed geodesic equations,
+0 = δP µ∂µP ν + P µ∂µδP ν + gνρ
+�
+∂µh(1)
+λρ − 1
+2∂ρh(1)
+µλ
+�
+P µP λ ,
+(13)
+we can evaluate it by using Eq. (12), namely,
+(∂0 − ˆn · ∂)
+�δP 0
+P 0
+�
+= −1
+2 ˆnaˆnb∂0h(1)
+ab ,
+(14a)
+(∂0 − ˆn · ∂)
+�δP j
+P 0
+�
+= δjbˆna
+�
+∂0h(1)
+ab − ˆnc
+�
+∂ch(1)
+ab − 1
+2∂bh(1)
+ac
+��
+,
+(14b)
+where δP µ is the first order perturbed 4-momentum. The temporal and spatial components of
+the above perturbed geodesic equations take different forms, because the time components of h(1)
+µν
+in Eq. (13) vanish in the Synchronous gauge. From Eq. (14), the solutions in Fourier space take
+
+6
+the form of
+δP 0
+k
+P 0 ≡ K0,ab (k, ˆn) ¯h(1)
+ab,ke−ikt = −
+1
+2(1 + ˆn · ˆk)
+ˆnaˆnb¯h(1)
+ab,ke−ikt ,
+(15a)
+δP j
+k
+P 0 ≡ Kj,ab (k, ˆn) ¯h(1)
+ab,ke−ikt =
+�
+δjbˆna −
+ˆkjˆnaˆnb
+2(1 + ˆn · ˆk)
+�
+¯h(1)
+ab,ke−ikt ,
+(15b)
+where the k is wave number, and the ˆk describes propagation direction of the first order GWs.
+The results shown in Eqs. (15) are the basis of GW detectors, and astrometric detection with Gaia
+[31, 50].
+In the non-linear regime, we extend the calculations to the second order perturbed geodesic
+equations, namely,
+0 = δ2P µ∂µP ν + 2δP µ∂µδP ν + P µ∂µδ2P ν + 2gνρP µδP λ(∂µh(1)
+λρ − ∂ρh(1)
+µλ + ∂λh(1)
+µρ )
++P µP λ
+�
+gνρ
+�
+∂λδg(2)
+µρ − 1
+2∂ρδg(2)
+µλ
+�
++ gνκgρωh(1)
+ωκ(∂ρh(1)
+µλ − 2∂λh(1)
+µρ )
+�
+,
+(16)
+where where δ2P µ is the second order perturbed 4-momentum, and the second order metric per-
+turbations in Synchronous gauge is
+δg(2)
+00 = δg(2)
+i0 = 0 ,
+(17a)
+δg(2)
+ij = −2δijψ(2) + 2∂i∂jE(2) + ∂iC(2)
+j
++ ∂jC(2)
+i
++ h(2)
+ij .
+(17b)
+To obtain the timing residuals to the second order, we calculate temporal components of Eq. (16)
+by making use of Eqs. (12) and (17), namely,
+(∂0 − ˆn · ∂)
+�δ2P 0
+P 0
+�
+= −1
+2 ˆnaˆnb∂0δg(2)
+ab − 2
+�δP µ
+P 0
+�
+∂µ
+�δP 0
+P 0
+�
++ 2ˆna
+�δP b
+P 0
+�
+∂0h(1)
+ab .
+(18)
+Differed from the linearized equations in Eq. (14a), the second and the third terms on the right hand
+side of above equation should be attributed to the non-linear contributions from perturbed geodesic
+equations. It was partly considered in previous study [46], in which the non-linear corrections were
+obtained by expanding the proper time to the second order. By making use of Eqs. (15), the
+solutions of Eq. (18) can be obtained,
+δ2P 0
+k
+P 0
+=
+�
+d3q
+(2π)3
+�
+Fcd,ab (k, q) ¯h(1)
+cd,k−q¯h(1)
+ab,qe−i(|k−q|+q)t�
+,
+(19)
+where
+Fcd,ab (k, q, ˆn) ≡ −
+1
+|k − q| + q + ˆn · k
+�
+�Fcd,ab
+L
+(k, q) + |k − q| + q
+2
+�
+∗=ψ,E,C,h
+Fcd,ab
+∗
+(k, q)
+�
+� ,
+(20)
+
+7
+and
+Fcd,ab
+L
+(k, q) ≡ (|k − q| + q) K0,cd
+k−qK0,ab
+q
+− qjKj,cd
+k−qK0,ab
+q
+,
+−(kj − qj)K0,cd
+k−qKj,ab
+q
+− qˆnaKb,cd
+k−q − |k − q| ˆncKd,ab
+q
+,
+(21a)
+Fcd,ab
+ψ
+(k, q) ≡ −2 ˆF ij
+ψ (k, q) fcdab
+ij
+(k, q) ,
+(21b)
+Fcd,ab
+E
+(k, q) ≡ −2(ˆn · k)2 ˆF ij
+E (k, q) fcdab
+ij
+(k, q) ,
+(21c)
+Fcd,ab
+C
+(k, q) ≡ iˆnj(ˆn · k) ˆF ml
+C,j (k, q) fcdab
+ml
+(k, q) ,
+(21d)
+Fcd,ab
+h
+(k, q) ≡ ˆniˆnj ˆF ml
+h,ij (k, q) fcdab
+ml
+(k, q) ,
+(21e)
+where Kµ,ab
+k
+has been defined in Eq. (15) and ˆF ij
+ψ (k, q), ˆF ij
+E (k, q), ˆF ij
+C (k, q) and ˆF ij
+h (k, q) have
+been given in Eq. (7). The Fcd,ab
+L
+(k, q) is derived from the second and the third terms of Eq. (18),
+and the Fcd,ab
+ψ
+(k, q), Fcd,ab
+E
+(k, q), Fcd,ab
+C
+(k, q), and Fcd,ab
+h
+(k, q) are obtained by the solving the
+motion of equations of second order scalar, vector, and tensor perturbations, respectively.
+The difference of the time of arrivals can be quantified by the redshift caused by GWs fluctua-
+tions, namely,
+z = uµ ˜P µ|obs
+uµ ˜P µ|src
+,
+(22)
+where ˜P µ(≡ P µ + δP µ + 1
+2δ2P µ + O(3)) is the total 4-momentum of a radio beam from a pulsar.
+For comoving observers in Synchronous gauge uµ = (1, 0, 0, 0), we obtain the redshift and its
+fluctuations as follows,
+z(0) = 0 ,
+(23a)
+z(1) = δP 0
+obs
+P 0
+obs
+− δP 0
+src
+δP 0src
+,
+(23b)
+z(2) = δ2P 0
+obs
+P 0
+obs
+− δ2P 0
+src
+P 0src
+− 2
+�δP 0
+src
+P 0src
+� �δP 0
+obs
+P 0
+obs
+− δP 0
+src
+δP 0src
+�
+,
+(23c)
+where the perturbed 4-momentums can be given by δ(n)P 0/P 0 =
+�
+d3k
+(2π)3
+�
+(δ(n)P 0
+k/P 0)eik·x�
+, and
+the expressions of δ(n)P 0
+k/P 0 have been given in Eqs. (15) and (19). From Eq. (12), we have known
+the events of a pulsar emitting a radio beam tsrc = t − L and xj
+src = Lˆnj, and the events of the
+beam reaching the earth tobs = t and xj
+obs = 0. Therefore, the redshifts and its fluctuations in
+
+8
+Eqs. (23) can be rewritten in the form of
+z(1) =
+�
+d3k
+(2π)3
+�
+K0,ab (k, ˆn) ¯h(1)
+ab,ke−ikt(1 − eikL(1+ˆn·ˆk))
+�
+,
+(24a)
+z(2) =
+� d3kd3q
+(2π)6
+�
+Fcd,ab (k, q, ˆn) ¯h(1)
+cd,k−q¯h(1)
+ab,qe−i(|k−q|+q)t �
+1 − eiL(|k−q|+q+ˆn·k)�
++K0,cd (k, ˆn) K0,ab (q, ˆn) ¯h(1)
+cd,k¯h(1)
+ab,qe−i(k+q)t
+×1
+2
+�
+eiqL(1+ˆn·ˆq)(1 − eikL(1+ˆn·ˆk)) + eikL(1+ˆn·ˆk)(1 − eiqL(1+ˆn·ˆq))
+��
+.
+(24b)
+In practical, the observables are the timing residuals of pulsar timing. It can be obtained by
+integration of the redshifts in Eqs. (24) over observation duration t, namely, R(n)(t) =
+� t
+0 z(n)(¯t)d¯t.
+Thus, one can obtain expressions of the timing residuals in the form of
+R(1) =
+�
+d3k
+(2π)3
+� 1
+ikK0,ab (k, ˆn) ¯h(1)
+ab,k(1 − e−ikt)(1 − eikL(1+ˆn·ˆk))
+�
+,
+(25)
+R(2) =
+� d3kd3q
+(2π)6
+�
+1
+i (|k − q| + q)Fcd,ab (k, q, ˆn) ¯h(1)
+cd,k−q¯h(1)
+ab,q(1 − e−i(|k−q|+q)t)
+�
+1 − eiL(|k−q|+q+ˆn·k)�
++
+1
+i(k + q)K0,cd (k, ˆn) K0,ab (q, ˆn) ¯h(1)
+cd,k¯h(1)
+ab,q(1 − e−i(k+q)t)
+×1
+2
+�
+eiqL(1+ˆn·ˆq)(1 − eikL(1+ˆn·ˆk)) + eikL(1+ˆn·ˆk)(1 − eiqL(1+ˆn·ˆq))
+��
+.
+(26)
+Due to kt ≪ 1 for the nHz band PTAs, the timing residuals can be expanded as kt → 0. In
+this approximation, the leading-order timing residuals reduce to R(n) = tz(n)|t=0. It indicates that
+the corrections of the outputs (timing residuals) of PTAs can be obtained via the correlations of
+redshifts, namely, ⟨R(x)R(x′)⟩ = t2⟨z(x)z(x′)⟩.
+IV.
+SPATIAL CORRELATIONS AND OVERLAP REDUCTION FUNCTIONS
+A.
+Non-linear correction of the correlations and non-Gaussianity
+The timing residuals can reflect physical information of SGWBs in the space. Thus, it also
+should be studied statistically, due to stochastic nature of the SGWBs. The spatial correlations of
+the timing residuals, as mentioned above, are proportional to spatial correlations of the redshifts
+⟨z(x)z(x′)⟩, where the x and x′ can represent the locations of pulsar pairs. For illustration, we let
+zα ≡ z(x) and zβ ≡ z(x′) in the rest of the paper.
+To calculate the correlations order by order, we expand correlations for the redshifts zα in the
+form of
+⟨zαzβ⟩ = ⟨z(1)
+α z(1)
+β ⟩ + 1
+2(⟨z(1)
+α z(2)
+β ⟩ + ⟨z(2)
+α z(1)
+β ⟩) + O(4) .
+(27)
+
+9
+For PTAs, the angular correlation derived from ⟨z(1)
+α z(1)
+β ⟩ is known as Hellings-Downs curve [31].
+In this section, we will extend the spatial correlations to the non-linear regime with ⟨z(1)
+α z(2)
+β ⟩ and
+⟨z(2)
+α z(1)
+β ⟩.
+The purpose of PTAs is extracting the physical information of h(1)
+ij based on observation of the
+timing residuals. Since z(1) ∝ h(1) and z(2) ∝ (h(1))2 shown in Eq. (24), the ⟨z(1)
+α z(1)
+β ⟩ and ⟨z(1)
+α z(2)
+β ⟩
+could encode two-point and three-point correlations of h(1)
+ij , respectively. Here, we strict our study
+to the isotropic and unpolarized GWs. In this case, the two-point correlations of hλ
+k in Fourier
+space can be given by
+�
+h(1),λ1
+k1
+h(1),λ2
+k2
+�
+= (2π)3δ (k1 + k2) δλ1λ2P(k2) ,
+(28)
+where the P(k) is power spectrum, the λ∗(= +, ×) is polarization index, and the Kronecker symbol
+δλ1λ2 indicates that h(1),λ
+k
+is unpolarized. In the non-linear regime, the three-point correlations in
+Fourier space can be given by
+�
+h(1),λ1
+k1
+h(1),λ2
+k2
+h(1),λ3
+k3
+�
+= (2π)6δ(3) (k1 + k2 + k3) Bλ1λ2λ3(k1, k2, k3) ,
+(29)
+where the Bλ1λ2λ3(k1, k2, k3) is bi-spectrum.
+It does not vanish due to the non-Gaussianity of
+SGWBs. For the unpolarized GWs, i) different polarizations of h(1),λ have no correlations, namely,
+Bλ1λ2λ3(k1, k2, k3) does not vanish, only if λ1 = λ2 = λ3, and ii) different components of the bi-
+spectrum are equally weighted, namely, the B+++(k1, k2, k3) = B×××(k1, k2, k3). To quantifying
+shapes of the non-Gaussianity, we follow the parameterization scenario used in Ref. [46]. With all
+the above assumptions, the bi-spectrum in Eq. (29) can be given in the form of
+Bλ1λ2λ3(k1, k2, k3) = Hλ1λ2λ3(k3)P(k3)δ(k1 − χk3)δ(k2 − ζk3) ,
+(30)
+where we have H+++(k3) = H×××(k3) ≡ κ(k3) due to the unpolarized GWs, the P(k) is the power
+spectrum defined in Eq. (28), and the ζ and χ are the dimensionless quantities formulating the
+shape of bi-spectrum. In principle, the shape of bi-spectrum quantified by κ, ζ and χ should be
+given based on generation mechanism of GWs. Here, we did not involve any physical models for
+the generation mechanism, but utilize the non-Gaussianity based on a phenomenological param-
+eterization. Because of k1 + k2 + k3 = 0 in Eq. (29), the parameters ζ and χ should satisfy the
+relations,
+χ + ζ ⩾ 1 , and |χ − ζ| ⩽ 1 .
+(31)
+In Fig. 1, we show scheme diagram for the shape of the bi-spectrum, in which the ζ and χ are
+defined with ζ ≡ BC/AB and χ ≡ CA/AB.
+In Fig. 2, we show the parameter space (ζ, χ),
+
+10
+k3
+k1
+k2
+A
+B
+C
+Figure 1: The scheme diagram of the parameterized bi-spectrum defined in Eq. (30).
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+ζ
+χ
+ζ  0.60
+χ  0.60
+ζ  0.87
+χ  0.87
+ζ  1.1
+χ  1.1
+ζ  1.4
+χ  1.4
+ζ  1.7
+χ  1.7
+ζ  1.9
+χ  1.9
+ζ  1.0
+χ  1.0
+ζ  1.3
+χ  0.87
+ζ  1.4
+χ  0.87
+ζ  1.9
+χ  1.1
+Figure 2: Left panel: parameter space of (ζ, χ) for the parameterized non-Gaussianity of hλ
+ij,k. The
+points locate the available domain of the parameters. The dashed curve is formulated by ζ = χ, and the
+dotted curve is formulated by 1
+2ζχ
+�
+1 −
+�
+ζ2+χ2−1
+2ζχ
+�2
+=
+√
+3
+4 . Right panel: The shape of non-Gaussianity for
+selected parameters ζ and χ on the dashed and dotted curves.
+where the points locate the domain of available parameters in Eqs. (31). Here, every single point
+represents a distinguishable shape of non-Gaussianity. As shown in the right panel of Fig. 2, for
+example, the orange points on the dashed line represent isosceles triangles, and the green points
+on the dotted curve represent triangles with the same height.
+B.
+Overlap reduction functions of PTAs in second order
+By making use of Eqs. (24a) and (28), the linear-order correlations of the redshifts for a pulsar
+pair can be given by
+⟨z(1)
+α z(1)
+β ⟩ =
+� k2dk
+2π2 P(k)
+� dΩ
+4π
+�
+K0,ab (k, ˆnα) K0,cd (k, ˆnβ) eλ
+ab(ˆk)eλ
+cd(ˆk)
+×(1 − eikLα(1+ˆnα·ˆk))(1 − e−ikLβ(1+ˆnβ·ˆk)) ,
+�
+(32)
+
+11
+where the eλ
+cd(ˆk) is polarization tensor for h(1)
+ij,k, the θαβ ≡ cos−1(ˆnα · ˆnβ), and the P(k) is the
+power spectrum defined in Eq. (28). The ORFs describe angular correlations of outputs of the GW
+detectors, and can be obtained via surface integrals over the unit sphere ˆk. Rewriting Eq. (32) in
+the form of
+⟨z(1)
+α z(1)
+β ⟩ ≡
+� k2dk
+2π2 P(k)Γ(2)(k, θαβ) ,
+(33)
+one can read the ORFs,
+Γ(2)(k, θab) =
+� dΩ
+4π
+�
+K0,ab (k, ˆnα) K0,cd (k, ˆnβ) eλ
+ab(ˆk)eλ
+cd(ˆk)
+×(1 − eikLα(1+ˆnα·ˆk))(1 − e−ikLβ(1+ˆnβ·ˆk))
+�
+.
+(34)
+Because of the approximation kL ≫ 1 for the known frequency band and arms length of PTAs,
+above ORFs would reduce to Hellings-Downs curve [31], namely,
+ΓHD(θab) ≡ Γ(2)(k, θab)|kLα≫1,kLβ≫1 =
+� dΩ
+4π
+�
+K0,ab (k, ˆnα) K0,cd (k, ˆnβ) eλ
+ab(ˆk)eλ
+cd(ˆk)
+�
+. (35)
+Here, the oscillation parts in Eq. (34) is suppressed by the factor (kL)−1, and thus can be neglected
+for PTAs. Besides, the ORFs without the approximation were also studied [43–45].
+To the second order, we further compute non-linear corrections for the correlations of redshift
+in Eq. (24), namely,
+⟨z(1)
+α z(2)
+β ⟩ =
+� d3kd3k′d3q′
+(2π)12
+��
+h(1)
+k,mlh(1)
+k′−q′,cd
+∗h(1)
+q,ab
+∗�
+K0,ml (k, ˆnα) Fcd,ab �
+k′, q′, ˆnβ
+�
+×e−ikt(1 − eikLα(1+ˆnα·ˆk))eit(|k′−q′|+q′) �
+1 − e−iLβ(|k′−q′|+q′+ˆnβ·k′)�
++1
+2
+�
+h(1)
+k,mlh(1)
+k′,cd
+∗h(1)
+q,ab
+∗�
+K0,ml (k, ˆnα) K0,cd �
+k′, ˆnβ
+�
+K0,ab �
+q′, ˆnβ
+�
+e−ikt(1 − eikLα(1+ˆnα·ˆk))ei(k′+q′)t
+×
+�
+e−iq′Lβ(1+ˆnβ·ˆq′)(1 − e−ik′Lβ(1+ˆnβ·ˆk′)) + e−iq′Lβ(1+ˆnβ·ˆq′)(1 − e−ik′Lβ(1+ˆnβ·ˆk′))
+��
+,
+(36)
+where above three-point correlations of h(1)
+ij,k can be written in terms of polarization components
+h(1),λ
+k
+,
+�
+h(1)
+k,cdh(1)
+ab,ph(1)
+q,ml
+�
+= eλ1
+cd(ˆk)eλ2
+ab(ˆp)eλ3
+ml(ˆq)
+�
+h(1),λ1
+k
+h(1),λ2
+p
+h(1),λ3
+q
+�
+.
+(37)
+
+12
+By making use of Eqs. (29) and (30), the Eq. (34) is evaluated to be
+⟨z(1)
+α z(2)
+β ⟩ + ⟨z(2)
+α z(1)
+β ⟩ =
+�
+d3k
+(2π)3
+� dφq
+2π
+�
+eλ1
+ml(ˆk)eλ2
+cd( �
+k − q)eλ3
+ab(ˆq)Hλ1λ2λ3(k)P(k)
+×
+��
+K0,ml (k, ˆnα) Fcd,ab (k, q, ˆnβ) (1 − eikLα(1+ˆnα·ˆk))(1 − e−ikLβ(χ+ζ+ˆnβ·ˆk))
++1
+2K0,ml (k, ˆnα) K0,cd (k − q, ˆnβ) K0,ab (q, ˆnβ)
+×e−iζkLβ(1+ˆnβ·ˆq)(1 − eikLα(1+ˆnα·ˆk))(1 − e−iLβ(χk+ˆnβ·(k−q)))
++1
+2K0,ml (k, ˆnα) K0,cd (q, ˆnβ) K0,ab (k − q, ˆnβ)
+×e−iLβ(χk+ˆnβ·(k−q))(1 − eikLα(1+ˆnα·ˆk))(1 − e−iζkLβ(1+ˆnβ·ˆq)))
+�
+e−ikt(1−χ−ζ)
++
+�
+Fcd,ab (k, q, ˆnα) K0,ml (k, ˆnβ) (1 − eikLα(ζ+χ+ˆnα·ˆk))(1 − e−ikLβ(1+ˆnβ·ˆk))
++1
+2Kml (k, ˆnβ) K0,cd (k − q, ˆnα) K0,ab (q, ˆnα)
+×eiζkLα(1+ˆnα·ˆq)(1 − eiLα(χk+ˆnα·(k−q)))(1 − e−ikLβ(1+ˆnβ·ˆk))
++1
+2Kml (k, ˆnβ) K0,cd (q, ˆnα) K0,ab (k − q, ˆnα)
+×eiLα(χk+ˆnα·(k−q))(1 − eiζkLα(1+ˆnα·ˆq))(1 − e−ikLβ(1+ˆnβ·ˆk))
+�
+eikt(1−χ−ζ)��
+,
+(38)
+where the momentums are give by
+q =
+�1 + ζ2 − χ2
+2
+�
+k +
+�
+((ζ + χ)2 − 1)(1 − (ζ − χ)2)
+2
+k (cos φqu + sin φqυ) ,
+(39a)
+ˆq = 1 + ζ2 − χ2
+2ζ
+ˆk +
+�
+((ζ + χ)2 − 1)(1 − (ζ − χ)2)
+2ζ
+(cos φqu + sin φqυ) ,
+(39b)
+�
+k − q = χ2 − ζ2 + 1
+2χ
+ˆk −
+�
+((ζ + χ)2 − 1)(1 − (ζ − χ)2)
+2χ
+(cos φqu + sin φqυ) .
+(39c)
+Here, the u and υ represent polarization vectors with respect to the ˆk. Using Eq. (39a), one can
+verify the relations of q = ζk and |k − q| = χk shown in Fig. 1. From Eq. (38), it is found that the
+three-point correlations are proportional to e±ikt(1−χ−ζ). It indicates that the values of correlations
+would oscillate with time around zero. In practical, due to kt ≪ 1 for nHz band PTAs, we here
+can let e±ikt(1−χ−ζ) ≃ 1.
+Similarly, the correlations in Eq. (38) can be rewritten in the form of
+⟨z(1)
+α z(2)
+β ⟩ + ⟨z(2)
+α z(1)
+β ⟩ =
+� k2dk
+2π2 P(k)Γ(3)(k, θab) ,
+(40)
+
+13
+where the ORFs in the non-linear order are given by
+Γ(3)(k, θαβ) =
+� dΩ
+4π
+� dφq
+2π
+�
+eλ1
+ml(ˆk)eλ2
+cd( �
+k − q)eλ3
+ab(ˆq)Hλ1λ2λ3(k)P(k)
+×
+�
+K0,ml (k, ˆnα) Fcd,ab (k, q, ˆnβ) (1 − eikLα(1+ˆnα·ˆk))(1 − e−ikLβ(χ+ζ+ˆnβ·ˆk))
++Fcd,ab (k, q, ˆnα) K0,ml (k, ˆnβ) (1 − eikLα(ζ+χ+ˆnα·ˆk))(1 − e−ikLβ(1+ˆnβ·ˆk))
++1
+2K0,ml (k, ˆnα) K0,cd (k − q, ˆnβ) K0,ab (q, ˆnβ)
+×e−iζkLβ(1+ˆnβ·ˆq)(1 − eikLα(1+ˆnα·ˆk))(1 − e−iLβ(χk+ˆnβ·(k−q)))
++1
+2K0,ml (k, ˆnα) K0,cd (q, ˆnβ) K0,ab (k − q, ˆnβ)
+×e−iLβ(χk+ˆnβ·(k−q))(1 − eikLα(1+ˆnα·ˆk))(1 − e−iζkLβ(1+ˆnβ·ˆq)))
++1
+2Kml (k, ˆnβ) K0,cd (k − q, ˆnα) K0,ab (q, ˆnα)
+×eiζkLα(1+ˆnα·ˆq)(1 − eiLα(χk+ˆnα·(k−q)))(1 − e−ikLβ(1+ˆnβ·ˆk))
++1
+2Kml (k, ˆnβ) K0,cd (q, ˆnα) K0,ab (k − q, ˆnα)
+×eiLα(χk+ˆnα·(k−q))(1 − eiζkLα(1+ˆnα·ˆq))(1 − e−ikLβ(1+ˆnβ·ˆk))
+��
+.
+(41)
+The expression of Hλ1λ2λ3(k) has been given in the Eq. (30). Since the oscillation parts in the
+integration is suppressed by the factor (kL)−1 for PTAs, we also adopt kLα ≫ 1 and kLβ ≫ 1 for
+evaluating Eq. (41). Namely, the ORFs can be simplified in the form of
+Γnl(k, θαβ) ≡ Γ(3)(k, θab)|kLα≫1,kLβ≫1
+= κ
+� dΩ
+4π
+� dφq
+2π
+��
+e+
+ml(ˆk)e+
+cd( �
+k − q)e+
+ab(ˆq) + e×
+ml(ˆk)e×
+cd( �
+k − q)e×
+ab(ˆq)
+�
+×
+�
+K0,ml (k, ˆnα) Fcd,ab (k, q, ˆnβ) + Fcd,ab (k, q, ˆnα) K0,ml (k, ˆnβ)
+��
+.
+(42)
+In the following, we would show the results of Γnl(k, θab) with selected parameters ζ and χ. In
+Fig. 3, we show the ORFs over the κ as function of parameter (ζ, χ) for given angle θαβ. It is
+found that the values of Γln/κ tend to be zero for large ζ and χ, and to be larger in the case of
+ζ + χ = 1. In Fig. 4, we show the ORFs for select parameters given in right panel of Fig. 2. It
+is confirmed that there is a larger value of ORFs for the parameters ζ, χ → 0. Differed from the
+ORFs in the linear order, there are three extreme points in the curves of the non-linear ORFs. In
+order to clarify the extreme cases, such as ζ − χ = 1 or ζ + χ = 1, we show the ORFs as function
+of θαβ for ζ − χ → 1 and ζ + χ → 1 in Figs. 7 and 6, respectively. From Fig. 7, the values of ORFs
+tend to be zero as ζ − χ → 1, and these ORFs have the same zero of the Γ(k, θαβ) with respect
+to θαβ. From Fig. 6, the values of ORFs tend to be larger, and the numbers of extreme points are
+reduced as ζ + χ → 1, which is different from the results shown in the left panel of Fig. 4 for a
+larger ζ + χ.
+In the case of ζ + χ = 1, we can further simplify the expression of ORFs as
+
+14
+Figure 3: Non-linear ORFs over the κ as function of parameter (ζ, χ) for θαβ = 0, π/4, π/2, π.
+Γnl(k, θαβ) = κ
+� dΩ
+4π
+��
+e+
+ml(ˆk)e+
+cd(�k)e+
+ab(ˆk) + e×
+ml(ˆk)e×
+cd(�k)e×
+ab(ˆk)
+�
+×
+�
+K0,ml (k, ˆnα) Fcd,ab (k, ζk, ˆnβ) + Fcd,ab (k, ζk, ˆnα) K0,ml (k, ˆnβ)
+��
+.
+(43)
+Here, the integration over the angle φq simply gives 2π. In Fig. 7, we show the ORFs in the case
+of ζ + χ = 1 for different ζ. It is found that the values of ORFs get smaller as ζ → 0.
+In principle, the parameter κ in Eq. (42) depends on the wave number k, and the shape of
+non-Gaussianity quantified by ζ and χ. There seems no reason that the non-linear corrections
+come from the non-Gaussianity with one of the available parameters (ζ, χ). For a general case, the
+total ORFs to the non-linear order should be the sum of all the shape of non-Gaussianity weighted
+by parameter κ, namely,
+Γ(k, θαβ) = ΓHD(θαβ) + 1
+2
+�
+ζ,χ
+Γnl(k, θαβ)∆σ ,
+(44)
+
+15
+ζ  χ  0.60
+ζ  χ  0.87
+ζ  χ  1.1
+ζ  χ  1.4
+ζ  χ  1.7
+ζ  χ  1.9
+0.0 0.5 1.0 1.5 2.0 2.5 3.0
+-0.05
+0.00
+0.05
+0.10
+0.15
+0.20
+θαβ
+Γnlθαβ
+ζ  χ  1.0
+ζ  1.3 , χ  0.87
+ζ  1.4 , χ  0.87
+ζ  1.9 , χ  1.1
+0.0 0.5 1.0 1.5 2.0 2.5 3.0
+-0.04
+-0.02
+0.00
+0.02
+0.04
+0.06
+0.08
+θαβ
+Γnlθαβ
+Figure 4: Non-linear ORFs for selected parameters ζ and χ shown in right panel of Fig. 2, and κ = 1.
+Left panel: the non-Gaussianity in the shape of isosceles triangles. Right panel: the non-Gaussianity in the
+shape of the triangles with the same height.
+ζ  0.500 , χ  1.40
+ζ  0.500 , χ  1.47
+ζ  0.500 , χ  1.49
+0.0 0.5 1.0 1.5 2.0 2.5 3.0
+-0.002
+-0.001
+0.000
+0.001
+0.002
+0.003
+0.004
+0.005
+θαβ
+Γnlθαβ
+ζ  1.00 , χ  1.90
+ζ  1.00 , χ  1.97
+ζ  1.00 , χ  1.99
+0.0 0.5 1.0 1.5 2.0 2.5 3.0
+-0.0005
+0.0000
+0.0005
+0.0010
+0.0015
+0.0020
+θαβ
+Γnlθαβ
+ζ  0.500
+χ  1.40
+ζ  0.500
+χ  1.47
+ζ  0.500
+χ  1.49
+ζ  1.00
+χ  1.90
+ζ  1.00
+χ  1.97
+ζ  1.00
+χ  1.99
+Figure 5: Non-linear ORFs for selected parameters ζ − χ → 1, and κ = 1.
+
+16
+ζ  χ  0.600
+ζ  χ  0.533
+ζ  χ  0.510
+ζ  χ  0.501
+ζ  χ  0.500
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+θαβ
+Γnlθαβ
+ζ  0.600
+χ  0.600
+ζ  0.533
+χ  0.533
+ζ  0.510
+χ  0.510
+ζ  0.501
+χ  0.501
+ζ  0.500
+χ  0.500
+Figure 6: Non-linear ORFs for selected parameters ζ + χ → 1, and κ = 1.
+ζ  χ  0.50
+ζ  0.75 , χ  0.25
+ζ  0.90 , χ  0.10
+ζ  0.99 , χ  0.010
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+θαβ
+Γnlθαβ
+ζ  0.500
+χ  0.500
+ζ  0.750
+χ  0.250
+ζ  0.900
+χ  0.100
+ζ  0.990
+χ  0.0100
+Figure 7: Non-linear ORFs for selected parameters ζ + χ = 1, ζ → 1, and κ = 1.
+where ∆σ is the size of grids of points in the parameter space (ζ, χ).
+For example, we have
+∆σ = 0.018 for the grids in the left panel of Fig. 2. Here, the Γnl(k, θαβ) is proportional to the
+parameter κ(k; ζ, χ) shown in Eq. (42). The dependence of Γ(k, θαβ) on the parameters ζ and
+
+17
+ΓHD(θαβ)
+Γtot(θαβ)
+ζ,χ<2,κ=1
+Γtot(θαβ)
+ζ,χ<10,κ=1
+Γtot(θαβ)
+ζ,χ<2,κ=-1
+Γtot(θαβ)
+ζ,χ<10,κ=-1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+-0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+θαβ
+Γθαβ
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+ζ
+χ
+0
+2
+4
+6
+8
+10
+0
+2
+4
+6
+8
+10
+ζ
+χ
+Figure 8: Left panel: total ORFs with non-linear corrections. The non-linear ORFs are sum of the
+selected parameters κ = ±1 and (ζ, χ) shown with the points in the right panel. Right panel: The sets of
+selected parameters in the plots of (ζ, χ). The ζ, χ ∈ (0, 2), and ζ, χ ∈ (0, 10) for the top-right panel, and
+bottom-right panel, respectively.
+χ should be based on specific physical models, which are not involved in our study. Here, we
+phenomenologically show the ORFs in Eq. (44) by letting |κ| ≡ 1 on the left panel of Fig. 8.
+Because of the parameter space ζ, χ ∈ (0, ∞) in Eq. (31), it is not practical to consider to all the
+shape of non-Gaussianity with |κ| = 1. In principle, if the deviation from Hellings-Downs curve is
+completely ascribed to the non-linear corrections of the ORFs, one can fit the κ(k; ζ, χ) with real
+data [28].
+V.
+CONCLUSIONS AND DISCUSSIONS
+In this paper, we extended the study on the non-linear corrections of the ORFs in the present
+non-Gaussianity, in which self-interaction of gravity is first taken into considerations. Due to the
+self-interaction of gravity, the linear order GWs can generate the non-linear one, which will change
+the response of GW detectors. Based on the perturbed Einstein field equations for the second order
+metric perturbations, and perturbed geodesic equations to the second order, we obtained non-linear
+order timing residuals of pulsar timing, and compute the ORFs with non-linear corrections in the
+
+18
+PTA frequency band.
+We considered the self-interaction of gravity through evaluating Einstein field equations in
+vacuum for the second order metric perturbations. Namely, the space-time fluctuations are freely
+propagating within the GW detectors described in Einstein’s gravity.
+It is suggested that the
+influence from secondary effect of GWs on the detectors could be different in the alternative theory
+of gravity, or in the present of (dark) matter.
+This paper showed that the leading order non-linear corrections for the ORFs come from the
+three-point correlations of h(1)
+k,ij. It is different from pioneers’ study that the correlations are from
+the four-points functions [46]. It is because the contributions from three-point correlations in our
+study are all derived from the self-interaction of gravity shown in Eqs. (21a)–(21e), which was not
+considered in pioneers’ study.
+Acknowledgments.
+The author thanks Prof. Qing-Guo Huang and Prof. Sai Wang for useful
+discussions.
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+1
+An RTL Implementation of the Data Encryption
+Standard (DES)
+Ruby Kumari, Student Member, IEEE, Jai Gopal Pandey, Senior Member,IEEE, and Abhijit Karmakar,
+Abstract—Data Encryption Standard (DES) is based on the
+Feistel block cipher, developed in 1971 by IBM cryptography
+researcher Horst Feistel. DES uses 16 rounds of the Feistel
+structure. But with the changes in recent years, the internet is
+starting to be used more to connect devices to each other. These
+devices can range from powerful computing devices, such as
+desktop computers and tablets, to resource constrained devices,
+When it comes to these constrained devices, using a different key
+for each round cryptography algorithms fail to provide necessary
+security and performance.
+Index Terms—Keywords: Cryptography, DES , SDES, Feistel
+block Cipher.
+I. INTRODUCTION
+T
+HIS Security is a prevalent concern in information and
+data systems of all types. Historically, military and na-
+tional security issues drove the need for secure communica-
+tions. Recently, security issues have pervaded the business and
+private sectors. E-commerce has driven the need for secure
+internet communications. Many businesses have fire- walls to
+protect internal corporate information from competitors. In the
+private sector, personal privacy is a growing concern. Products
+are available to scramble both e-mail and telephone communi-
+cations. One means of providing security in communications
+is through encryption. By encryption, data is transformed in a
+way that it is rendered unrecognizable. Only by decryption can
+this data be recovered. Ostensibly, the process of decryption
+can only be performed correctly by the intended recipients.
+The validity of this assertion determines the “strength” or
+“security” of the encryption scheme. Many communications
+products incorporate encryption as a feature to provide se-
+curity. This application report studies the implementation of
+one of the most historically famous and widely implemented
+encryption algorithms, the Data Encryption Standard (DES).
+The Data Encryption Standard is a symmetric-key block
+cipher published by the National Institute of Standards and
+Technology (NIST) for the encryption of digital data. DES
+is probably one of the best-known cryptographic algorithms
+and has been widely used since its introduction in 1976.
+Although its short key length of 56 bits makes it too inse-
+1 cure for applications, it has been highly influential in the
+advancement of cryptography. The DES must be stronger than
+the other cryptosystems in security. The goal of this project is
+to develop a python code for SDES and DES. Before building
+our design, we need an overview of cryptography, followed
+by a description of the DES algorithm.
+Manuscript received December 19, 2022
+1) Overview of Cryptography: Cryptography is a type of
+rule or technique by which private or sensitive information is
+secured from the public or other members. It plays a vital role
+in preserving data integrity, confidentiality and user privacy.
+An encryption algorithm can convert imported essential data
+to encrypted data (plaintext into cipher- text). This data
+would be of no use to a person that does not possess the
+encryption key. The use of Cryptography in passwords is a
+very famous example. Cryptography is based on mathematical
+theory and some Computer Science principles. There are many
+terminologies related to cryptography. Some terms are defined
+below.
+• Ciphertext: Conversion of plain text into intelligible text
+is called ciphertext.
+• Cipher: It is a technique of encryption and decryption.
+Critical and algorithms play vital role in this technique.
+• Symmetric: It is a kind of cryptosystem. It uses same key
+for encryption and decryption. It is faster than asymmetric.
+• Asymmetric: It is also a kind of cryptosystem. It uses
+a public key for the encryption and a private key for the
+decryption of any message.
+• Cryptanalysis: It studies cracking the encryption of the
+algorithms.
+A. Symmetric Ciphers Model
+Symmetric-key (or private-key) encryption can be simply
+illustrated with the schematic shown in Figure 1.
+Fig. 1: Symmetric Cryptosystem model.
+A symmetric encryption scheme has five main parts, that is,
+• Encryption algorithm: The encryption algorithm per-
+forms various substitutions and transformations on plain-
+text.
+• Secret key: The secret key is also input to the encryption
+algorithm. The key is a value independent of the plaintext
+and the algorithm. The algorithm will produce a different
+arXiv:2301.05530v1  [cs.CR]  13 Jan 2023
+
+Cryptanalyst
+Message
+Encryption
+Decryption
+Sources
+Destination
+Algorithm
+Algorithm
+Secure Channel
+Key
+Sources2
+output depending on the specific key. The exact substi-
+tutions and transformations performed by the algorithm
+depend on the key.
+• Ciphertext: This is the scrambled message produced as
+output. It depends on the plaintext and the secret key.
+For a given message, two different keys will produce two
+different ciphertexts. The ciphertext is an random stream
+of data.
+• Decryption algorithm: This is essentially the encryption
+algorithm run in reverse. It takes the ciphertext and the
+secret key and produces the original plaintext.
+Alice and Bob want to communicate over an un-secure
+channel, but Oscar is trying to read the message. So Alice and
+Bob must use a cryptosystem to prevent Oscar from reading
+the message. Let us take a closer look at the essential elements
+of a symmetric encryption scheme using Figure 1. A source
+produces a message in plaintext, X = [X1, X2, . . . , XM].
+The M elements of X are letters in some finite alphabet.
+Traditionally, the alphabet usually consisted of t6 capital
+letters. Nowadays, the binary alphabet 0, 1 is typically used.
+For encryption, a key of the form K = [K1, K2, . . . . . .., KJ]
+is generated. If the key is generated at the message source,
+then it must also be provided to the destination using some
+secure channel. Alternatively, a third party could generate key
+and securely deliver it to both source and destination. The
+encryption algorithm forms the ciphertext as given in 1.
+Y = [Y 1, Y 2, ...., Y N]
+(1)
+with the message X and the encryption key K as it. We can
+write this as given in 2.
+Y = E(K, X)
+(2)
+This notation indicates that Y is produced by using en-
+cryption algorithm E as a function of the plaintext X, with
+the specific process determined by the value of the key K.
+The intended receiver, in possession of the key, can invert the
+transformation: X = D(K, Y ). An opponent, observing Y but
+not having access to K or X, may attempt to recover X or K
+or both X and K. It is assumed that the opponent knows the
+encryption (E) and decryption (D) algorithms. If the opponent
+is interested in only this particular message, then the focus of
+the effort is to recover X by generating a plaintext estimate
+X. Often, however, the opponent is interested in being able
+to read future messages as well, in which case an attempt is
+made to recover K by generating an estimate K.
+B. Simplified Data Encryption Standard
+The S-DES encryption algorithm takes an 8-bit block of
+plaintext and a 10-bit key as input and produces an 8-bit block
+of ciphertext as output. The S-DES decryption algorithm takes
+an 8- bit block of ciphertext and the same 10-bit key used to
+produce that ciphertext as input and produces the original 8-bit
+block of plaintext. Simplified DES (SDES) was designed for
+educational purposes only, to help students learn about modern
+cryptanalytic techniques [1]. SDES has similar properties and
+structure as DES but has been simplified to make it much
+easier to perform encryption and decryption by hand with
+Fig. 2: Simplified DES (SDES)
+Pencil and paper. Some people feel that learning SDES gives
+insight into DES and other block ciphers, and insight into
+various cryptanalytic attacks against them.
+An adversary trying to interrupt two communicating parties
+may have one of the four main goals:
+1) Read the secret message.
+2) Find the secret key, so that they can read all messages
+encrypted with that key.
+3) Modify the message sent by Alice and go unnoticed by
+both parties.
+4) Act like Alice and send a message to Bob, to make Bob
+think he is communicating with Alice when in reality
+he is communicating with the adversary.
+In order to prevent an adversary from reaching his
+goals, some security measures are Applied to cryptosystems,
+
+10-bit Key
+4
+P10
+Encryption
+Decryption
+8-bit Plaintext
+8-bit Plaintext
+Shift
+IP
+IP-1
+P8
+K1
+Ki
+Fk
+Fk
+Shift
+SW
+SW
+P8
+K2
+K2
+Fk
+Fk
+IP
+IP-1
+8-bit Ciphertext
+8-bit Ciphertext3
+namely confidentiality, data integrity, authentication, and non-
+repudiation.
+1) Confidentiality means the transmitted message or infor-
+mation is kept secret, and only the authorized parties
+have the means to decipher the information.
+2) Data integrity makes sure that the messages are not being
+modified. This stops the adversary from reaching their
+third goal.
+3) Authentication helps Bob to correctly identify the sender
+as Alice, thus stopping the adversary from posing as
+Alice.
+4) Non-repudiation prevents Alice from denying she sent
+the message.
+Cryptographic algorithms are gathered under two main
+branches; symmetric algorithms and asymmetric algorithms. In
+symmetric algorithms both Alice and Bob have the same key.
+Since the communication channel is insecure, this key must
+be previously decided on through secure ways. The encryption
+and decryption keys are either the same, or very similar that
+the decryption key can easily be derived from the encryption
+key. But sometimes Alice and Bob cannot agree on a key
+beforehand. They could be very far away from each other
+and cannot get together to determine a secret key, and there
+may not be a secure way for Alice to send Bob the secret
+key. She cannot just send Bob a secret key through any open
+channel, because an adversary can interrupt the channel and
+get their hands on the key. Thus making the key useless. To
+get around this problem asymmetric algorithms, usually called
+public key algorithms, are used. In public key algorithms each
+party has their key pairs, one public and one private key. As
+can be understood from their names, private keys are kept
+secret, and public keys can be known by everyone. The public
+key is computed from the private key in a way that finding the
+private key from the public key is infeasible. Alice encrypts
+the message she wants to send using Bob’s public key. The
+message can only be decrypted with the corresponding private
+key, which only Bob has. Therefore Alice can send a secret
+message even though they are far away and cannot decide on
+a common key together [2].
+Further, the details of the DES cipher is given in the next
+chpater [3].
+II. DATA ENCRYPTION STANDARD
+Developed in 1974 by IBM in cooperation with the National
+Securities Agency (NSA), DES has been the worldwide en-
+cryption standard for more than 20 years. For these 20 years,
+it has held up against cryptanalysis remarkably well and is still
+secure against all but possibly the most powerful adversaries.
+Because of its prevalence throughout the encryption market,
+DES [4] is an excellent interoperability standard between
+different encryption equipment. The predominant weakness of
+DES is its 56-bit key which, more than sufficient for the time
+period which it was developed [5], has become insufficient to
+protect against brute-force attacks modern computers [6]. As
+a result of the need for a greater encryption strength, DES
+evolved into triple-DES [7].
+Fig. 3: Encryption and Decryption
+III. DES ENCRYPTION
+The Data Encryption Standard is a Feistel cipher. In which
+round function consists of an expansion, a bitwise XOR-
+operation XOR operation round key, an S-box layer and a
+permutation [3]. In encryption n scheme, there are two inputs
+to the encryption function [8]. the plaintext to be encrypted
+and the key. In this case, the plaintext must be 64 bits in length
+and the key is 56 bits in length [9].
+On the left-hand side of the figure, we can see that the
+plaintext processing proceeds in three phases. First, the 64-
+bit plaintext passes through an initial permutation (IP) that
+rearranges the Bits to produce the permuted input [10]. This
+is followed by a phase consisting of sixteen rounds of the same
+function, which involves both permutation and substitution
+functions. The output of the last (sixteenth) round consists of
+64 bits that are a function of the input plaintext and the key.
+The left and right halves of the output are swapped to produce
+the pre-output. Finally, the pre-output is passed through a per-
+mutation [IP −1], inverse of the initial permutation function,
+to produce the make ciphertext. With the exception of initial
+and final permutations.
+On the right-hand portion of figure 6-5,6-a-bitkey is used.
+Initially, the key is passed through a permutation function.
+Then, for each of the sixteen rounds, a subkey (Ki) is produced
+by the combination f.t Initially, the key is passed through a
+permutation function. Then, for each of the sixteen rounds, a
+subkey (Ki) is produced by the combination of a left.
+A. Initial Permutation and Final Permutation
+Each of these permutations takes a 64-bit input and per-
+mutes them according to a predefined rule. These permutations
+are keyless straight permutations that are the inverse of each
+other. For example, in the initial permutation [IP], the 58th bit
+in the input becomes the first bit in the output. Similarly, in the
+final permutation [IP −1], the first bit in the input becomes the
+58th bit in the output. In other words, if the rounds between
+these two permutations do not exist, the 58th bit entering the
+initial permutation is the same as the 58th bit leaving the final
+permutation. The initial permutation is given in TABLE I
+The final permutation is given in TABLE II.
+
+DES Reverse
+DES Cipher
+Cipher4
+Fig. 4: Structure of DES
+Fig. 5: Initial and final permutation step in DES
+IV. ROUNDS
+DES uses 16 rounds. Each round of DES is a Feistel cipher.
+Fig. 6 shows the internal structure of a single round. Again,
+begin by focusing on the left-hand side of the diagram. The left
+and right halves of each 64-bit intermediate value are treated
+as separate 32-bit quantities, labeled L (left) and R (right). As
+in the Feistel cipher, the overall processing at each round can
+be summarized in the following formulas:
+Li = Ri − 1 Ri = Li − 1 XOR FE(Ri − 1, Ki)
+The round takes Li − 1 and Ri − 1 from the previous game
+(or the initial permutation box) and creates Li and Ri, which
+go to the next round (or final permutation box).
+A. Initial Permutation
+A single initial permutation is needed at the beginning of
+the encryption process. IP is necessary on each block of 64
+bits in DES once the entire plaintext has been divided into
+TABLE I: Initial Permutation
+58
+50
+42
+34
+26
+18
+10
+2
+60
+52
+44
+36
+28
+20
+12
+4
+62
+54
+46
+38
+30
+22
+14
+6
+64
+56
+48
+40
+32
+24
+16
+8
+57
+49
+41
+33
+25
+17
+9
+1
+59
+51
+43
+35
+27
+19
+11
+3
+61
+53
+45
+37
+29
+21
+13
+5
+63
+55
+47
+39
+31
+23
+15
+7
+TABLE II: Final Permutation
+40
+8
+48
+16
+56
+24
+64
+32
+39
+7
+47
+15
+55
+23
+63
+31
+38
+6
+46
+14
+54
+22
+62
+30
+37
+5
+45
+13
+53
+21
+61
+29
+36
+4
+44
+12
+52
+20
+60
+28
+35
+3
+53
+11
+51
+19
+59
+27
+34
+2
+42
+10
+50
+18
+58
+26
+33
+1
+41
+9
+49
+17
+57
+25
+such blocks. The transposition process goes through this initial
+permutation. Only once, just before the first round, does the
+first permutation appear. As seen in the Table I, it provides
+decisions for how the IP transposition process has to go. It is
+possible to claim for example, that the IP replaced the first
+bit of the original plain-text block with the 58th bit of the
+original plain-text block, the second bit with the 50th bit of
+the original plain-text block, etc. This is nothing more than
+bit shuffling with respect to the original plaintext block.
+B. Expansion D-Box
+Since Ri − 1 is a 32-bit input and KI is a 48-bit key, we
+first need to expand Ri − 1 to 48 bits. Ri − 1 is divided into
+8 4-bit sections. Each 4-bit section is then expanded to 6 bits.
+For each section, input bits 1, 2, 3, and 4 are copied to output
+bits 2, 3, 4, and 5, respectively. Output bit ‘1’ comes from bit
+4 of the previous section; output bit 6 comes from bit 1 of the
+next section. If sections 1 and 8 can be considered adjacent
+sections, the same rule applies to bits 1 and 32.
+The main part of DES is the DES function. The DES
+function applies a 48-bit key to the rightmost 32 bits (Ri −1)
+to produce a 32-bit output. This function is made up of four
+sections: an expansion D-box, a whitener (that adds key), a
+group of S-boxes, and a straight D-box, as shown in Fig 6.
+C. Whitener (XOR)
+After the expansion permutation, DES uses the XOR op-
+eration on the expanded right section and the round key. It
+XORed expansion permutation and key input and gives 48-bit
+TABLE III: Expansion Permutation
+32
+1
+2
+3
+4
+5
+64
+32
+4
+5
+6
+7
+8
+9
+63
+31
+8
+9
+10
+11
+12
+13
+62
+30
+12
+13
+14
+15
+16
+17
+61
+29
+16
+17
+18
+19
+20
+21
+60
+28
+20
+21
+22
+23
+24
+25
+59
+27
+24
+25
+26
+27
+28
+29
+58
+26
+28
+29
+30
+31
+32
+1
+57
+25
+
+Initial Permutation
+Permuted Choice 1
+Round 1
+Permuted Choice 2
+Left circular Shift
+Round 2
+Permuted Choice 2
+Left Circular Shift
+Round 16
+Permuted Choice 2
+Left circular shift
+32 bit Swap
+Inverse Initial Permutation1
+2
+25
+40
+50
+58
+60
+Initial
+Permutation
+1
+2
+...-8
+25
+40
+50
+58
+60
+16 Rounds
+1
+2...8
+25
+40
+50
+58
+60
+Final
+Permulalion
+★
+25
+40
+1
+50
+58
+605
+Fig. 6: Single round of the DES Algorithm [11].
+Fig. 7: DES function
+input to s-boxes. Note that both the right section and the key
+are 48-bits in length [12].
+D. S-Boxes
+The S-boxes do the real mixing (confusion). DES uses 8
+S-boxes, each with a 6-bit input and a 4-bit output. The 48-
+bit data from the second operation is divided into eight 6-bit
+chunks, and each chunk is fed into a box [13]. The result
+of each box is a 4-bit chunk; when these are combined the
+result is a 32-bit text. The substitution in each box follows a
+pre-determined rule based on a 4-row by 16- column table.
+Fig. 8: S-box
+TABLE IV: Straight Permutation Table
+16
+07
+20
+21
+29
+12
+28
+17
+01
+15
+23
+26
+05
+18
+31
+10
+02
+08
+24
+14
+32
+27
+03
+09
+19
+13
+30
+06
+22
+11
+04
+25
+16
+17
+18
+19
+20
+21
+60
+28
+20
+21
+22
+23
+24
+25
+59
+27
+24
+25
+26
+27
+28
+29
+58
+26
+28
+29
+30
+31
+32
+1
+57
+25
+E. Final Permutation
+The last operation in the DES function is a permutation
+with a 32-bit input and a 32-bit output. The input/output
+relationship for this operation is shown in Table II.
+V. EXAMPLES OF DES
+Let M be the plain text message M = 0123456789ABCDEF
+where M is in hexadecimal (base 16) format. Rewriting M in
+binary format, we get the 64-bit block of text:
+M = 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001
+1010 1011 1100 1101 1110 1111,
+R = 1000 1001 1010 1011 1100 1101 1110 1111,
+L = 0000 0001 0010 0011 0100 0101 0110 0111.
+The first bit of M is ‘0’. The last bit is ‘1’. We read from
+left to right. DES operates on the 64-bit blocks using key sizes
+of 56- bits. The keys are actually stored as being 64 bits long,
+but every 8th bit in the key is not used (i.e., its numbered 8,
+16, 24, 32, 40, 48, 56, and 64). However, we will nevertheless
+number the bits from 1 to 64, going left to right, in the
+following calculations. But, as you will see, the eight bits just
+mentioned get eliminated when we create subkeys. Let K be
+the hexadecimal key K = 133457799BBCDFF1. This gives
+us binary key (setting 1 = 0001, 3 = 0011, etc., and grouping
+together every eight bits, of which the last one in each group
+will be unused) [14]. K = 00010011 00110100 01010111
+01111001 10011011 10111100 11011111 11110001 .
+VI. KEY GENERATION
+The 64-bit key is permuted according to the following table,
+PC−1. Since the first entry in the table is “57”, this means
+that the 57th bit of the original key K becomes the first bit
+of the permuted key K+. The 49th bit of the original key
+becomes the second bit of the permuted key. The 4th bit of
+the original key is the last bit of the permuted key. Note only
+56 bits of the original key appear in the permuted key.
+From the original 64-bit key
+K = 0001001100110100010101110111100110011011
+
+4
+32 bits
++
+32 bits
++
+4
+28 bits
++
+28 bits
+Li1
+Ri.1
+C(i-1
+Expansion/Perimutation (E Table)
+Left Shit(s)
+Right Shit(s)
+48
+F
+Permutation/Contraction
+XOR
+48 1
+(Perimuted Choice 2)
+48
+Substitution/Choice (S-Box)
++
+32
+Permutation (P)
+1
+32
+XOR
+7
+Li
+R;
+Ci
+D,Expansion/Permutation (E Table
+48
+F
+XOR
+48/
+48
+Substitution/Choice (S-Box)
+32
+Permutation (P)
+3248-bit input
+Array of S-Boxes
+S-Box
+S-Box
+S-Box
+S-Box
+S-Box
+S-Box
+S-Box
+S-Box
+32-bit output6
+TABLE V: Permuted Choice-1
+57
+49
+41
+33
+25
+17
+9
+17
+1
+58
+50
+42
+34
+26
+18
+10
+10
+2
+59
+51
+43
+35
+27
+09
+19
+11
+3
+60
+52
+44
+36
+25
+63
+55
+47
+39
+31
+23
+15
+28
+7
+62
+54
+46
+38
+30
+22
+27
+14
+6
+61
+53
+45
+37
+29
+26
+21
+13
+5
+28
+20
+12
+4
+25
+101111001101111111110001 we get the 56-bit permutation
+K+ = 11110000110011001010101011110101010
+101100110011110001111
+Now, From the permuted key K+, we get
+C0 = 1111000011001100101010101111
+C1 = 1110000110011001010101011111
+C2 = 1100001100110010101010111111
+C3 = 0000110011001010101011111111
+C4= 0011001100101010101111111100
+C5 = 1100110010101010111111110000
+D0 = 0101010101100110011110001111
+D1 = 1010101011001100111100011110
+D2 = 0101010110011001111000111101
+D3 = 0101011001100111100011110101
+D4 = 0101100110011110001111010101
+D5 = 0110011001111000111101010101
+C6 = 0011001010101011111111000011
+D6 = 1001100111100011110101010101
+C7 = 1100101010101111111100001100
+D7 = 0110011110001111010101010110
+C8 = 0010101010111111110000110011
+D8 = 1001111000111101010101011001
+C9 = 0101010101111111100001100110
+D9 = 0011110001111010101010110011
+C10 = 0101010111111110000110011001
+D10 = 1111000111101010101011001100
+C11 = 0101011111111000011001100101
+D11 = 1100011110101010101100110011
+C12 = 0101111111100001100110010101
+D12 = 0001111010101010110011001111
+C13 = 0111111110000110011001010101
+D13 = 0111101010101011001100111100
+C14 = 1111111000011001100101010101
+D14 = 1110101010101100110011110001
+C15 = 1111100001100110010101010111
+D15 = 1010101010110011001111000111
+C16 = 1111000011001100101010101111
+D16 = 0101010101100110011110001111
+TABLE VI: Permuted choice-2
+14
+17
+11
+24
+1
+5
+9
+17
+3
+28
+15
+6
+21
+10
+18
+10
+23
+19
+12
+4
+26
+8
+27
+09
+16
+7
+27
+20
+13
+2
+36
+25
+41
+52
+31
+37
+47
+55
+15
+28
+30
+40
+51
+45
+33
+48
+22
+27
+44
+49
+39
+56
+34
+53
+29
+26
+46
+42
+50
+36
+29
+32
+4
+25
+After we apply the permutation PC-2, it becomes
+K1 = 000110 110000 001011 101111 111111 000111 000001
+110010
+K2 = 011110 011010 111011 011001 110110 111100 100111
+100101
+K3 = 010101 011111 110010 001010 010000 101100 111110
+011001
+K4 = 011100 101010 110111 010110 110110 110011 010100
+011101
+K5 = 011111 001110 110000 000111 111010 110101 001110
+101000
+K6 = 011000 111010 010100 111110 010100 000111 101100
+101111
+K7 = 111011 001000 010010 110111 111101 100001 100010
+111100
+K8 = 111101 111000 101000 111010 110000 010011 101111
+111011
+K9 = 111000 001101 101111 101011 111011 011110 011110
+000001
+K10 = 101100 011111 001101 000111 101110 100100 011001
+001111
+K11 = 001000 010101 111111 010011 110111 101101 001110
+000110
+K12 = 011101 010111 000111 110101 100101 000110 011111
+101001
+K13 = 100101 111100 010111 010001 111110 101011 101001
+000001
+K14 = 010111 110100 001110 110111 111100 101110 011100
+111010
+K15 = 101111 111001 000110 001101 001111 010011 111100
+001010
+K16 = 110010 110011 110110 001011 000011 100001 011111
+110101
+Encode each 64-bit block of data Applying the initial
+permutation to the block of text M, given previously, we get
+M = 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001
+1010 1011 1100 1101 1110 1111
+IP = 1100 1100 0000 0000 1100 1100 1111 1111 1111 0000
+1010 1010 1111 0000 1010 1010
+Here the 58th bit of M is ‘1’, which becomes the first bit
+of IP. The 50th bit of M is ‘1’, which becomes the second
+bit of IP. The 7th bit of M is ‘0’, which becomes the last bit
+of IP.
+Next, divide the permuted block IP into a left half L0 of
+32 bit, and a right half R0 of 32 bits.
+From IP, we get L0 and R0
+L0 = 1100 1100 0000 0000 1100 1100 1111 1111
+R0 = 1111 0000 1010 1010 1111 0000 1010 1010
+We now proceed through 16 iterations, for 1≤ n ≤ 16, using
+a function f which operates on two blocks–a data block of 32
+bits and a key Kn of 48 bits–to produce a block of 32 bits. Let
++ denote XOR addition, (bit-by-bit addition modulo 2). Then
+for n going from 1 to 16 we calculate (3) and (4).
+Ln = Rn − 1
+(3)
+Rn = Ln−1 + f(Rn−1, Kn)
+(4)
+
+7
+This results in a final block, for n = 16, of L16R16. That
+is, in each iteration, we take the right 32 bits of the previous
+result and make them the left 32 bits of the current step. For
+the right 32 bits in the current step, we XOR the left 32 bits
+of the previous step with the calculation f.
+For n = 1, we have
+K1 = 000110 110000 001011 101111 111111 000111 000001
+110010
+L1 = R0 = 1111 0000 1010 1010 1111 0000 1010 1010
+R1 = L0 + f(R0, K1)
+It remains to explain how the function f works. To calculate
+f, first expand each block Rn −1 from 32 bits to 48 bits. This
+is done by using a selection table that repeats some of the
+bits in Rn − 1. We’ll call the use of this selection table the
+function E. Thus E(Rn − 1) has a 32 bit input block, and a
+48 bit output block. After this, We calculate E(R0) from R0
+as follows:
+R0 = 1111 0000 1010 1010 1111 0000 1010 1010
+E(R0) = 011110 100001 010101 010101 011110 100001
+010101 010101 Next in the f calculation, we XOR the output
+E(Rn − 1) with the key Kn : Kn + E(Rn − 1). For K1,
+E(R0), we have
+K1 = 000110 110000 001011 101111 111111 000111 000001
+110010
+E(R0) = 011110 100001 010101 010101 011110 100001
+010101 010101
+K1+E(R0) = 011000 010001 011110 111010 100001 100110
+010100 100111
+To this point we have expanded Rn-1 from 32 bits to 48 bits,
+using the selection table, and XORed the result with the key
+Kn. We now have 48 bits, or eight groups of six bits. We now
+do something strange with each group of six bits: we use them
+as addresses in tables called ”S boxes”. Each group of six bits
+will give us an address in a different S box. Located at that
+address will be a 4-bit number. This 4-bit number will replace
+the original 6 bits. The net result is that the eight groups of
+6 bits are transformed into eight groups of 4 bits (the 4-bit
+outputs from the S boxes) for 32 bits total.
+Write the previous result, which is 48 bits, in the form:
+Kn + E(Rn − 1) = B1B2B3B4B5B6B7B8
+where each Bi is a group of six bits. We now calculate it as
+S1(B1)S2(B2)S3(B3)S4(B4)S5(B5)S6(B6)S7(B7)S8(B8)
+where Si(Bi) refers to the output of the ith S box.
+To repeat, each of the functions S1, S2, ..., S8, takes a 6-bit
+block as input and yields a 4-bit block as output. For the first
+round, we obtain as the output of the eight S boxes:
+K1 + E(R0) = 011000 010001 011110 111010 100001
+100110 010100 100111
+S1(B1)S2(B2)S3(B3)S4(B4)S5(B5)S6(B6)S7(B7)S8(B8)
+= 0101 1100 1000 0010 1011 0101 1001 0111
+The final stage in the calculation of f is to do a permutation
+P of the S-box output to obtain the final value of f:
+f = P(S1(B1)S2(B2)...S8(B8))
+(5)
+The permutation P is defined in the following table. P yields
+a 32-bit output from a 32-bit input by permuting the bits of
+the input block
+TABLE VII: Permutation
+16
+7
+20
+21
+1
+5
+9
+17
+29
+12
+28
+17
+21
+10
+18
+10
+1
+15
+23
+26
+26
+8
+27
+09
+5
+18
+31
+10
+13
+2
+36
+25
+2
+8
+24
+14
+47
+55
+15
+28
+32
+27
+3
+9
+33
+48
+22
+27
+19
+23
+30
+6
+34
+53
+29
+26
+22
+11
+4
+25
+29
+32
+4
+25
+From the output of the eight S boxes:
+S1(B1)S2(B2)S3(B3)S4(B4)S5(B5)S6(B6)S7(B7)S8(B8)
+= 0101 1100 1000 0010 1011 0101 1001 0111
+we get,
+f = 0010 0011 0100 1010 1010 1001 1011 1011
+R1 = L0 + f(R0, K1)
+R1 = 1100 1100 0000 0000 1100 1100 1111 1111 + 0010
+0011 0100 1010 1010 1001 1011 1011 = 1110 1111 0100
+1010 0110 0101 0100 0100
+In the next round, we will have L2 = R1, which is the
+block we just calculated, and then we must calculate R2 =
+L1 + f(R1, K2), and so on for 16 rounds. At the end of the
+sixteenth round we have the blocks L16 and R16. We then
+reverse the order of the two blocks into the 64-bit block as
+shown in equation (6).
+R16L16
+(6)
+Now, apply a final permutation IP−1 and the output of the
+algorithm has bit 40 of the preoutput block as its first bit, bit
+8 as its second bit, and so on, until bit 25 of the preoutput
+block is the last bit of the output.
+If we process all 16 blocks using the method defined
+previously, we get, on the 16th round,
+L16 = 0100 0011 0100 0010 0011 0010 0011 0100
+R16 = 0000 1010 0100 1100 1101 1001 1001 0101
+We reverse the order of these two blocks and apply the
+final permutation to
+R16L16
+=
+00001010
+01001100
+11011001
+10010101
+01000011 01000010 00110010 00110100
+IP −1 = 10000101 11101000 00010011 01010100 00001111
+00001010 10110100 00000101 which in hexadecimal format
+is 85E813540F0AB405.
+This is the encrypted form of M = 0123456789ABCDEF:
+namely, C = 85E813540F0AB405. Decryption is simply the
+inverse of encryption, following the same steps as above, but
+reversing the order in which the subkeys are applied.
+VII. SYMMETRIC CIPHERS
+If we examine the symmetric ciphers in detail, we can see
+that symmetric ciphers can be divided into two categories;
+stream ciphers and block ciphers [15].
+Stream ciphers use a key-stream, obtained from the original
+key, and encrypts the plain-text bit by bit. Encryption is usually
+done by combining the plain-text bits with the corresponding
+key-stream bits with an XOR operation. In some cases stream
+
+8
+ciphers have some advantages over block ciphers, because
+there is no error propagation. It means that an error made
+in one bit of cipher-text during transmission only affects the
+decryption of that bit and doesn’t affect other bits [16].
+Block ciphers, on the other hand, take the plain-text bits
+in blocks. Each block is encrypted with the same encryption
+function and the cipher-text blocks are produced. When the
+length of the plain-text is not a multiple of the block size
+some padding is applied to the plain-text [17]. This padding
+is usually done by adding a ‘1’ bit followed by necessary
+amount of ‘0’ bits. Because the encryption function does not
+change from one block to another, same blocks of plain-
+text are encrypted to same blocks of cipher-text. When an
+adversary captures the cipher-text, they can accurately guess
+some information about the plaintext by using this property.
+In order to stop any information leakage, some modes of
+operation are used.
+VIII. ASYMMETRIC CIPHERS
+While modern symmetric ciphers such as AES are very
+secure, they have some drawbacks in practicality, namely key
+distribution problem, and the number of keys [18].
+The key distribution problem occurs when Alice and Bob
+want to determine a secret key. This would be easy if they
+can come together and decide, but if they have no means
+to decide on a key in person, they have to decide on the
+key through a secure channel [19]. Since the communication
+channel is always assumed to be insecure, because it can be
+easily hacked, this poses a problem. Even if they can somehow
+solve this problem, they would be facing another problem, the
+number of keys [20]. If there are n users in a network, and
+all of the users want to communicate with each other secretly,
+the number of encryption keys needed would be n∗(n−1)
+2
+,and
+each user would have n − 1 key pairs they need to know
+and keep secret. This becomes exponentially infeasible as the
+number of people increase. The usage of asymmetric ciphers
+eliminate these problems. Since every user has a pair of keys,
+and anything encrypted with a specific public key can only be
+decrypted with the corresponding private key, Alice and Bob
+doesn’t need to agree on a secret key together beforehand. In
+addition, nobody would need to store n − 1 key pairs, they
+only need to store their own private and public keys, and the
+number of key pairs needed in the network would be reduced
+to n [11]. Cryptographic protocols can be considered as a third
+main branch of cryptography, and one of the most important
+primitives they use is called a hash function. Therefore it
+would be useful to go over the definition of hash functions.
+In order to understand how public key algorithms work
+we can imagine a box [21]. For Alice to send Bob a secret
+message, first Bob sends Alice a box with an open padlock,
+for which he has the key. Alice then can put her message in the
+box, and lock it with the padlock. When Bob receives the box
+he simply unlocks the padlock and reads the message [11].
+Of course there are still some security concerns, for example
+an adversary can intercept the box and replace the padlock
+with their own lock, or put their own message in the box and
+act like Alice. To achieve authentication and to prevent these
+problems, cryptographers have developed some procedures.
+A. Modes of Operation
+There are several modes of operation that can be used when
+encrypting a plaintext with a block cipher. NIST recommends
+the usage of 5 modes of operation. [11]:
+• Electronic Codebook (ECB)
+• Cipher Block Chaining (CBC)
+• Cipher Feedback (CFB)
+• Output Feedback (OFB)
+• Counter (CTR)
+In ECB mode, each block is encrypted and decrypted in-
+dependently from each other. Because the encryption function
+does not change, identical blocks of plaintext are encypted to
+identical blocks of ciphertext [22].
+In CBC mode, the ciphertext of one block is XORed with
+the plaintext of the next block before the encryption [23].
+For the first plaintext block an Initialization Vector IV is
+used. In CFB mode, ciphertext blocks are encrypted with the
+encryption function instead of the plaintext blocks. Plaintext
+blocks are XORed with the results of encryption function to
+obtain the ciphertext blocks. For the first block an IV is used
+[24].
+In OFB mode [25], IV is repeatedly encrypted with the
+encryption function and the results are XORed with the
+plaintext blocks to obtain ciphertext blocks [26].
+In CTR mode, a nonce and counter is encrypted and the
+result is XORed with the plaintext block [27]. The counter is
+increased each time.
+All of these modes while having different advantages also
+have some disadvantages. For example, some of them have
+parallelizable encryption and decryption but others don’t. The
+decision of which modes of operation is to be used should
+be made based on the desired security and performance levels
+[28].
+IX. RESULTS
+Implementation of DES has been performed using VHDL
+and the results is shown in TABLE IX and TABLE VIII.
+TABLE VIII: Performance Matrix of DES on Virtex-7 FPGA
+Device.
+Operating
+Frequency (MHz)
+Datapath Dalay
+(nS)
+Maximum
+Frequency (MHz)
+Dynamic
+Power (mW)
+100
+1.829
+246
+8
+TABLE IX: Resource Utilization of DES on Virtex-7 FPGA
+Device.
+Slices
+LUTs
+Flip-Flops
+69
+244
+139
+X. CONCLUSION
+Architecture Exploration of Simplified Data Encryption
+Standard (SDES) and Data Encryption Standard (DES) has
+been done. Simplified DES (SDES) was designed for educa-
+tional purposes only, to help learn about modern cryptanalytic
+techniques. SDES has similar properties and structure as DES
+but has been simplified to make it much easier to perform
+
+9
+encryption and decryption by hand with pencil and paper.
+Some people feel that learning SDES gives insight into DES
+and other block ciphers, and insight into various cryptanalytic
+attacks against them [29]. In DES, 64-bit input is encrypted
+and decrypted using 56-bit key. At the encryption site, DES
+takes a 64-bit plaintext and creates a 64-bit ciphertext; at the
+decryption site, DES takes a 64- bit ciphertext and creates a
+64-bit block of plaintext. The same 56-bit cipher key is used
+for both encryption and decryption. Implementation of SDES
+and DES has been performed using Python 3.7 version and
+VHDL. During this project I have learned thoroughly about
+various cryptography techniques and ciphers.
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+ing (ICECCE).
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+ecb mode,” in 2006 International Symposium on Computer Networks.
+IEEE, 2006, pp. 134–139.
+[23] C. Tan, X. Deng, and L. Zhang, “Identification of block ciphers under
+cbc mode,” Procedia Computer Science, vol. 131, pp. 65–71, 2018.
+[24] S. Mister and R. Zuccherato, “An attack on cfb mode encryption as
+used by openpgp,” in International Workshop on Selected Areas in
+Cryptography.
+Springer, 2006, pp. 82–94.
+[25] Y.-L. Huang, F.-Y. Leu, J.-C. Liu, J.-H. Yang, C.-W. Yu, C.-C. Chu, and
+C.-T. Yang, “Building a block cipher mode of operation with feedback
+keys,” in 2013 IEEE International Symposium on Industrial Electronics.
+IEEE, 2013, pp. 1–4.
+[26] T. Iwata, K. Minematsu, J. Guo, S. Morioka, and E. Kobayashi, “Silc:
+simple lightweight cfb,” CAESAR submission, 2014.
+[27] H. Lipmaa, P. Rogaway, and D. Wagner, “Ctr-mode encryption,” in First
+NIST Workshop on Modes of Operation, vol. 39.
+Citeseer. MD, 2000.
+[28] S. Almuhammadi and I. Al-Hejri, “A comparative analysis of aes
+common modes of operation,” in 2017 IEEE 30th Canadian Conference
+on Electrical and Computer Engineering (CCECE), 2017, pp. 1–4.
+[29] R. C. Merkle and M. E. Hellman, “On the security of multiple encryp-
+tion,” Communications of the ACM, vol. 24, no. 7, pp. 465–467, 1981.
+R uby Kumari, Integrated Dual Degree Ph.D (IDDP)
+scholar at Academy of Scientific Innovative Re-
+search (AcSIR), working area Integrated Circuits
+and System Group, CSIR-CEERI. Completed B.tech
+in Electronics and Communication Engineering from
+Maulana Abul Kalam Azad University of Technol-
+ogy. Her research interest includes Cryptography,
+Lightweight Ciphers, Digital Logic Design, VLSI
+Architecture and RTL Design.
+Jai Gopal Pandey is a Principal Scientist and work-
+ing in the CSIR-CEERI, Pilani, India since 2005. He
+is an M.Tech. (Electronics Design and Technology)
+from U. P. Technical University, Lucknow, in 2003
+and a Ph.D. in Electronics Engineering from Birla
+Institute of Technology and Science (BITS), Pilani,
+India in 2015.
+His research interests include High-performance
+Architecture, System-on-chips (SoCs), Embedded
+Systems, Cryptography, FPGAs, and ASIC designs.
+Dr. Pandey is a Senior Member of IEEE and an IETE
+Fellow.
+
+10
+Abhijit Karmakar received the B.E. degree in
+Electronics and Telecommunication Engineering in
+1993 from Jadavpur University, India, and M.Tech.
+degree in Electrical Engineering from Indian Insti-
+tute of Technology (IIT), Madras, India, in 1995. He
+recieved the Ph.D. degree in Electrical Engineering
+from IIT, Delhi, India, in 2007. Since 1995, he has
+been working with the CSIR - Central Electronics
+Engineering Research Institute (CEERI), Pilani, In-
+dia. His research interest span the area of VLSI
+Design, Signal Processing and related areas.
+

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+Strain-mediated ion-ion interaction in
+rare-earth-doped solids
+A. Louchet-Chauvet
+E-mail: [email protected]
+ESPCI Paris, Universit´e PSL, CNRS, Institut Langevin, 75005 Paris, France
+T. Chaneli`ere
+Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France
+Abstract.
+It was recently shown that the optical excitation of rare-earth ions produces a
+local change of the host matrix shape, attributed to a change of the rare-earth ion’s
+electronic orbital geometry. In this work we investigate the consequences of this piezo-
+orbital backaction and show from a macroscopic model how it yields a disregarded
+ion-ion interaction mediated by mechanical strain.
+This interaction scales as 1/r3,
+similarly to the other archetypal ion-ion interactions, namely electric and magnetic
+dipole-dipole interactions. We quantitatively assess and compare the magnitude of these
+three interactions from the angle of the instantaneous spectral diffusion mechanism, and
+reexamine the scientific literature in a range of rare-earth doped systems in the light of
+this generally underestimated contribution.
+arXiv:2301.05531v1  [cond-mat.mtrl-sci]  13 Jan 2023
+
+Strain-mediated ion-ion interaction in rare-earth-doped solids
+2
+1. Introduction
+Atomic ensembles have attracted a lot of attention over more than 20 years due to their
+inherent capacity to efficiently interact with light [1]. They are at the center of a number of
+quantum storage protocols, in the form of gases [2, 3] or solid state [4, 5]. Among the most
+interesting solid state candidates, rare-earth ion-doped crystals (REIC) are particularly
+attractive due to their long optical coherence lifetimes at cryogenic temperatures [6] and
+are at the center of a number of actively developed quantum memory protocols [7, 8, 9].
+In these light-matter interfaces, the optical depth of the medium is an important figure of
+merit since it enables high storage and retrieval efficiency [10, 11, 5]. However, reaching
+large optical depths usually comes with working with large atomic concentrations, leading
+to reinforced ion-ion interactions and thereby enhanced decoherence [12, 13, 14, 15, 16, 17].
+Interestingly, although ion-ion interactions are potentially detrimental to the
+performance of quantum devices, they have also emerged as the foundational mechanism
+for quantum computing architectures because they allow multiqubit gate operations [18].
+Quantum computing was initially developed in physical systems with strong readout
+capacity ranging from dilute systems (trapped atoms) to condensed matter (quantum
+dots or superconducting qubits).
+Rare-earth ion-doped crystals were only recently
+considered as relevant for such applications [19] thanks to the elaboration of efficient
+single ion readout schemes that compensate for the optical transition’s weak oscillator
+strength [20, 21].
+Either way, proper understanding and quantifying of ion-ion interactions in REIC
+are crucial.
+In most systems, the order of magnitude of the measured interaction
+strength is compatible with magnetic and/or electric dipole-dipole interaction. In both
+mechanisms, the excitation of some ions induces a change in their electric or magnetic
+dipole moment, modifying the local field accordingly in their vicinity.
+This modified
+field affects the surrounding ions in proportion to the Stark or Zeeman sensitivity of
+their energy levels.
+In a limited number of REIC, however, the actual magnitude of
+this interaction is larger than expected by several orders of magnitude [22, 23]. In this
+paper we consider a strain-mediated ion-ion interaction that has not been investigated
+so far and that may explain this discrepancy. The interaction we consider stems from
+the apparition of an excitation-induced stress field, affecting the surrounding ions via
+their piezospectroscopic sensivity. This fundamental effect has been pointed out early
+in the context of paramagnetic resonance under RF excitation, named virtual phonon
+exchange interaction initially considering transition metals ions [24, 25, 26]. At the time,
+the description exploited the equivalent operators formalism for the spin-lattice coupled
+system, but the different modeling parameters are difficult to infer from experimental
+measurements. Despite a series of studies including rare-earth salts [27, and references],
+phonon-mediated interactions seem to have been overlooked for years, before reappearing
+in the context of quantum technologies with original proposals of phononic engineering
+[28, 29].
+
+Strain-mediated ion-ion interaction in rare-earth-doped solids
+3
+This paper is organized as follows. In Sec. 2, the physical origin of this interaction
+is presented. A scaling law is given and an estimation of its strength is provided. In
+Sec. 3, we quantitatively compare the magnitude of this strain-mediated interaction with
+respect to the electromagnetic dipole-dipole interactions in a number of host-dopant
+combinations and confront our predictions with the experimental data found in the
+scientific literature. The strength of the instantaneous spectral diffusion mechanism is
+used as a comparison criterion. Finally, in Sec. 4, we discuss the implications of this work
+in quantum technology-related applications.
+2. Strain-mediated ion-ion interaction
+2.1. General description
+When an atomic particle is promoted to a different electronic level, its outer shape and
+size are bound to change due to the modification of its electronic wavefunction. If the
+particle is embedded in a solid matrix, this piezo-orbital backaction effect will additionally
+give rise to a stress field around the excited particle, making it detectable via a change
+of shape of the solid itself. This was recently evidenced in a bulk rare-earth ion-doped
+crystal in which only a finite volume of the crystal was illuminated, leading to a distortion
+of the nearby crystal surface [30].
+Conversely, it has been known for several decades that the optical lines of rare-earth
+ions in solids are sensitive to stress via the piezospectroscopic effect [31, 32]. Indeed,
+in the elastic regime, a compressive or tensile stress modifies the interatomic distances.
+This affects the crystal field and in turn shifts the rare-earth ion’s energy levels. This
+sensitivity to stress was recently proposed as a tool to sense mechanical vibrations in a
+cryogenic environment [33, 34].
+The piezo-orbital backaction and the piezospectroscopic effect are in fact two facets
+of the same coupling mechanism. Their combination leads to a shift of the transition
+frequency in the ions surrounding a given excited rare-earth ion. In the following we coin
+this as the strain-mediated ion-ion interaction.
+2.2. Magnitude of the strain-mediated interaction
+For simplicity we assume that the ion is spherical with a radius r1 and that the piezo-
+orbital backaction acts as a simple ionic radius change (see Fig. 1). This impacts the
+surrounding matrix by creating a radial stress field σ(r) around the ion. With a spherically
+symmetric continuum mechanics model detailed in appendix Appendix A, we calculate
+the elastic strain energy that is necessary to establish this stress field. Due to energy
+conservation, this elastic energy corresponds to the energy shift of the electronic levels
+due to the internal stress within the ionic volume (see appendix Appendix B). This allows
+
+Strain-mediated ion-ion interaction in rare-earth-doped solids
+4
+Figure 1. Simplified view of the piezo-orbital backaction around a spherical rare-earth
+ion. The stress field is symbolized by a radial color gradient around the ion.
+us to relate the ionic radius variation ∆r to the piezospectroscopic sensitivity κ:
+∆r = hκ
+4πr2
+1
+(1)
+where h is the Planck constant. With this we derive the radial stress field σ(r) and write
+the strain-mediated ion-ion interaction energy between two ions separated by a distance
+r:
+Estr(r) = hκσ(r) =
+E
+1 + ν
+(hκ)2
+2πr3
+(2)
+where E is the Young modulus and ν is the Poisson’s ratio of the crystal. We emphasize
+the relatively strong hypotheses underlying this result: the piezo-orbital backaction is
+assumed to occur in the form of a mere ionic radius change of the spherical excited ion,
+and the piezospectroscopic sensitivity is assumed to be scalar. The anisotropic nature of
+the crystalline matrix may naturally translate into a non-spherical strain field. Because
+of the large distance between dopants (larger than the crystal cell parameter at low
+concentration levels), we may expect the strain anisotropy to be weak. On the contrary,
+the piezospectroscopic sensitivity is usually described by a tensor [35], so its scalar nature
+appears as a crude assumption in order to derive an order of magnitude.
+Using Eq. 1 we can estimate the corresponding relative ionic radius change due
+to the piezo-orbital backaction in rare-earth-doped crystals.
+Taking typical values
+κ = 100 Hz/Pa [33] and r1 = 1 ˚A [36], we obtain ∆r/r1 ≃ 5·10−3. This value remarkably
+agrees with the relative ionic radius change of a free ion among its 4f states that can be
+derived from a Hartree-Fock calculation (see [37] for Ce3+ and [38] for Eu3+) even if the
+exact rearrangement of the outer shell defining the ionic radius surrounding the 4f-shell
+deserves further analysis.
+3. Comparing the three ion-ion interactions
+In this section we propose to study how this interaction compares with the other usual
+ion-ion interactions generally considered in rare-earth doped systems, i.e. electric and
+magnetic dipole-dipole interactions.
+We also confront these estimations to published
+measurements of ion-ion interactions in rare-earth doped crystals.
+
+(a) lon at rest
+(b) Excited ion
+ri ＋△r
+r1Strain-mediated ion-ion interaction in rare-earth-doped solids
+5
+Interestingly, the strain-mediated interaction scales as 1/r3, exactly like the electric
+and magnetic dipole-dipole interactions, although originating from a radically different
+physical mechanism. All three interactions can be characterized by a coefficient Ai such
+that their energies read as:
+Ei(r) = hAi
+2πr3
+(3)
+where the {Ai} are defined by the following:
+Ael
+= π¯h
+ϵ0ϵr
+∆µ2
+el
+(4)
+Amag = µ0¯h
+4π ∆µ2
+mag
+(5)
+Astr =
+E
+1 + ν hκ2
+(6)
+∆µel and ∆µmag are the electric and magnetic dipole moment variation when the ion
+is promoted to the excited state.
+They are expressed in Hz m V−1 and in Hz T−1,
+respectively. We note that in all three expressions, the sensitivity of the optical transition
+to electric field, magnetic field or strain (respectively) appears as a square law, in
+agreement with what is expected in an ion-ion interaction.
+This scaling has been early derived for paramagnetic impurities under RF
+excitation [24, 25], in the so-called zero-retardation limit of the virtual phonon exchange
+(VPE) interaction [26]. At the time, McMahon et al. also showed that VPE and magnetic
+interactions have the same order of magnitude for transition metal paramagnetic dopants
+[25].
+With these expressions one can anticipate the variability of the three considered
+interactions amongst REIC. For example, the strain-mediated interaction should be
+quite constant across different ions or crystals, since the mechanical properties of typical
+crystalline rare-earth doped oxides and the piezospectroscopic sensitivity of their optical
+lines are rather similar [33]. On the other hand, the magnetic dipole-dipole interactions
+vary by several orders of magnitude between Kramers and non-Kramers ions because some
+exhibit an electronic spin while some only have a nuclear spin behaviour. A variability
+of 6 orders of magnitude is expected, since (µB/µN)2 ≃ 3 · 106, where µB and µN are
+the Bohr magneton and the nuclear magneton, respectively. A large variability is also
+expected for the electric dipole-dipole interaction because the appearance of an electric
+dipole is only permitted by the crystal field that weakly perturbs the free ion electronic
+structure. Indeed different hosts, depending on the crystal site symmetry, allow or inhibit
+permanent electric dipole moment for the rare earth ion dopant.
+The overall ion-ion interactions taking place in an ensemble of rare-earth ions can be
+probed experimentally by the observation of instantaneous spectral diffusion (ISD) [39],
+ie the effect of a substantial population transfer on the spectral width of a narrow subset
+of ions within a given volume. When it builds upon interactions scaling as 1/r3, ISD
+
+Strain-mediated ion-ion interaction in rare-earth-doped solids
+6
+manifests as an additional term in the homogeneous linewidth Γh that is proportional to
+the volumic density of excited particles ne:
+Γeff = Γh + β
+2 ne
+(7)
+where β
+=
+�
+i βi is the sum of individual contributions due to each considered
+interaction [40, 14]:
+βi = 8π
+9
+√
+3106Ai,
+(8)
+the 106 factor stemming from the choice of units (m3 s−1 for Ai and Hz cm3 for βi).
+ISD is generally observed via high-resolution spectroscopy experiments such as
+photon echoes [41, 42, 43]. It has been evidenced in a large number of rare-earth ion-
+doped materials, but a quantitative value is only given in a handful of them. The published
+experimental values for β in ten different rare-earth ion-doped crystals are given in Table 1.
+These materials represent a rather complete sampling of the REIC diversity, including
+non-Kramers and Kramers ions, different crystal and site symmetries, and crystals with
+and without charge compensation. The measured values of β are contained within a 2
+order-of-magnitude range (between 10−13 and 10−11 Hz cm3).
+Cristal
+βexp (Hz cm3)
+Tm:YAG
+2.3 · 10−12 [43]
+Tm:YGG
+2.6 · 10−13 [43]
+Tm:LiNbO3
+1.0 · 10−11 [43]
+Er:YSO
+1.3 · 10−12 [44]
+Er:LiNbO3
+1.0 · 10−13 [45]
+Eu:YSO
+9.0 · 10−13 [46]
+Eu:Y2O3
+1.3 · 10−13 [47]
+EuCl3·6D2O
+4.6 · 10−13 [22]
+Pr:YSO
+1.2 · 10−11 [48]
+Pr:La2(WO4)3
+7.0 · 10−12 [49]
+Table 1. Measured values for the ISD coefficient βexp in a selection of rare-earth ion-
+doped crystals.
+The three ion-ion interactions considered in this work (see equations 4, 5 and 6)
+should contribute to ISD. Focusing our attention to the selection of REICs for which
+quantitative ISD measurements are available, we calculate the βi parameters for each ion-
+ion interactions using Eq. 8 and display the results graphically in Figure 2. The material
+parameters relevant to this calculation and resulting numerical estimations are given in
+appendix Appendix C. Again, we point out that the values are mostly indicative since
+they rely on a very simplified modelization of the interactions.
+We verify that both magnetic and electric dipole-dipole interactions vary among
+the materials within a 6 order-of-magnitude span depending on the existence of an
+
+Strain-mediated ion-ion interaction in rare-earth-doped solids
+7
+Figure 2.
+Calculated (bars) and measured (triangles) values for βi and Ai (where
+i = {mag, el, str, exp}) for the 10 rare-earth ion-doped crystals for which quantitative
+values for βexp are available (see Table 1). Note the logarithmic vertical scale spanning
+10 orders of magnitude.
+electronic spin and the presence of a centrosymmetry in the doping site (βmag is comprised
+between 7 · 10−20 and 2 · 10−13 Hz cm3and βel between 8 · 10−19 and 8 · 10−12 Hz cm3,
+respectively). Conversely, the strain-mediated interaction is comprised within a much
+smaller interval (βstr between 6 · 10−13 and 2 · 10−12 Hz cm3) since we use a typical value
+for the piezospectroscopic sensitivity κ for all and because of the very similar values of
+the Young moduli between crystals.
+It is interesting to note that the strain-mediated interaction is of the order of
+the largest of the electric or magnetic dipole-dipole contributions found among all
+materials.
+This means that when one or both of the dipole-dipole interactions are
+strong, they should coexist with the strain-mediated interaction. On the other hand,
+when both electromagnetic contributions are in the low range (weak Stark effect and
+no electronic spin for the non-Kramers ions, e.g. Tm-doped garnets or in a lesser part
+stoechiometric europium chloride), the strain-mediated interaction dominates by several
+orders of magnitude.
+In all considered REICs, the sum of the three predicted effects
+satisfactorily accounts for the measured values of βexp, finally providing an explanation
+to the previously large discrepancy between theoretical estimations and measurements of
+the ISD mechanism in some media.
+
+10-10
+(Hz cm²)
+） 10-12
+S
+10-20
+A
+10-14
+ISD coeficient β
+Magnetic
+Electric
+Strain
+10-22
+Exp
+10
+16
+Interaction
+10-
+24
+10-18
+10-26
+10-20
+YSO
+Tm:YAG
+Tm:YGG
+Eu:YSO
+EuCl'Strain-mediated ion-ion interaction in rare-earth-doped solids
+8
+4. Discussion
+Besides providing a better understanding of the physical origin and strength of ISD in
+REIC, the strain-mediated interaction in REIC could have interesting applications in
+the field of quantum computing.
+Indeed, quantum computing schemes rest upon the
+existence of a long-range atom-atom interaction enabling multiqubit operations [18]. This
+interaction is often spontaneously assimilated to dipole-dipole interaction, and particularly
+in REIC [50, 19].
+We argue that the strain-mediated interaction could play this role
+in quantum-computing to replace the dipole-dipole coupling. Such a scheme could be
+performed in almost any rare-earth ion-doped crystal since the strength of the interaction
+is rather similar over a broad variety of ion and host combinations [33]. In particular,
+the possession of a permanent electric dipole moment would not be an exclusive criteria
+for quantum computing compatibility. On the contrary, in some crystals (namely Tm-
+doped garnets, although there could be other candidates) this strain-mediated interaction
+dominates its electromagnetic counterparts by at least 4 orders of magnitude. This means
+that not only is the ion-ion interaction particularly pure in such media, but these strain-
+coupled qubits would be more robust against other types of decoherence process, such
+as magnetic field fluctuations due to spin flips in the host matrix [51, 13], or electric
+field noise due to charge fluctuations that may occur at the surface of rare-earth-doped
+nanoparticles [52].
+One may argue that since the strain-mediated interaction is intrinsically slow since it
+relies on the propagation of stress at the speed of sound (between 3000 and 8000 m/s in
+most considered crystals). We estimate its propagation delay by considering the average
+ion-ion distance dion−ion, given by
+3√nRE (where nRE is the volumic density of rare-earth
+ions). dion−ion ranges from a few nm in highly concentrated materials (e.g. ∼ 2 nm in
+1% doped YAG) up to hundreds of nm in low concentration materials (e.g. ∼ 650 nm in
+200ppm doped YSO). This leads to a strain propagation time shorter than the nanosecond,
+to be compared with typical µs-scale light-matter interactions occuring in such media.
+Therefore, the strain-mediated interaction can still be considered instantaneous within
+rare-earth-doped crystals with typical concentrations.
+5. Conclusion
+Based on recent evidence of a conservative optomechanical backaction mechanism in rare-
+earth ion-doped crystals, we have unveiled a disregarded ion-ion interaction based on a
+physical mechanism fundamentally different from generally considered electromagnetic
+dipole-dipole interactions: the sensitivity of rare-earth ions to piezo-orbitally induced
+stress. With a simple mechanical model, we have estimated the strength of this strain-
+mediated interaction and shown that it is largely dominant in some rare-earth ion-doped
+crystals, opening interesting perspectives for strain-based quantum computing in solids.
+
+Strain-mediated ion-ion interaction in rare-earth-doped solids
+9
+6. Acknowledgments
+The authors are grateful to Lars Rippe and Xiaoping Jia for helpful discussions.
+The authors acknowledge support from the French National Research Agency (ANR)
+through the projects ATRAP (ANR-19-CE24-0008), MIRESPIN (ANR-19-CE47-0011)
+and MARS (ANR-20-CE92-0041). This work has received support under the program
+“Investissements d’Avenir” launched by the French Government.
+Appendix A. Continuum mechanics in a REIC
+Appendix A.1. Spherical defect in an isotropic elastic medium
+Let us consider a sphere with radius r1, embedded in an infinite, isotropic, continuous
+elastic medium with a Young modulus E and a Poisson’s ratio ν. When a homogeneous
+radial stress σ0 is applied in the sphere, the displacement field at a distance r outside the
+sphere is radial and obeys [53]:
+u(r) = σ0
+1 + ν
+2E
+r3
+1
+r2,
+(A.1)
+while the stress field around the sphere is also radial and reads as:
+σ(r) = σ0
+r3
+1
+r3 for r > r1
+(A.2)
+Defining ∆r = u(r1) as the radius change, we obtain a simple relationship between
+σ0 et ∆r:
+σ0 = ∆r
+r1
+2E
+1 + ν
+(A.3)
+Figure A1.
+Radial stress σ(r) corresponding to the dilation of a sphere within an
+infinite isotropic medium. The stress is homogeneous within the sphere and decays with
+a 1/r3 law outside.
+The radial stress can then be written:
+σ(r) = −∆r
+r1
+2E
+1 + ν
+r3
+1
+r3
+(A.4)
+
+/0
+()
+1
+1 stress (
+0.8
+Normalized radial
+0.6
+0.4
+0.2
+0
+0
+1
+2
+3
+4
+5
+Normalized distance to ion r/riStrain-mediated ion-ion interaction in rare-earth-doped solids
+10
+Appendix A.2. Strain energy
+We now want to assess the elastic strain energy contained in the medium when the stress
+σ0 is applied.
+We successively calculate the energy contained inside and outside the
+sphere [54].
+Inside the sphere:
+The strain energy inside the sphere reads as:
+Uint = 1
+2σ0ε0V
+(A.5)
+where ε0 = ∆V/V = 3∆r/r1 is the volumic change of the sphere. We finally obtain:
+Uint = 2π∆rσ0r2
+1
+(A.6)
+Outside the sphere:
+The sphere being included in an infinite medium, we must also
+consider the strain energy that was necessary to establish the whole stress field around
+the sphere. The volumic energy outside the sphere at a distance r reads as:
+uV (r) = 1
+2σ(r)ε(r)
+(A.7)
+The volumic strain ε(r) is related to the local stress via the volumic Hooke’s law
+σ(r) = Kε(r) (where K =
+E
+1−2ν is the bulk modulus). Using Eq. A.2 we get ε(r) = ε0r3
+1/r3,
+which finally leads to:
+uV (r) = 3∆r
+2 σ0
+r5
+1
+r6
+(A.8)
+We integrate this volumic energy over the infinite volume of the medium:
+Uext = 4π
+� ∞
+r=r1
+uV (r)r2dr = 6π∆rσ0r5
+1
+� ∞
+r=r1
+1
+r4dr
+(A.9)
+and obtain:
+Uext = 2π∆rσ0r2
+1
+(A.10)
+Total strain energy:
+Based on Eqs. A.6 and A.10, we derive the total strain energy that
+is necessary to distort the sphere and generate the associated stress field around it:
+Ustrain = Uint + Uext = 4π∆rσ0r2
+1
+(A.11)
+Appendix B. Relating the strain energy with the atomic energy
+Due to piezo-orbital backaction [30], the size of a rare-earth ion is expected to change
+under optical excitation. Assuming this change of shape is merely a change of radius on
+a spherical ion, we can apply the calculation presented in appendix Appendix A. Due
+to energy conservation, this elastic energy is taken from the ion’s energy levels. We can
+
+Strain-mediated ion-ion interaction in rare-earth-doped solids
+11
+therefore write the following, considering that the shift in the ion’s energy levels ∆E is
+related to the stress within the ion σ0.
+Ustrain = ∆E = hκσ0
+(B.1)
+where κ is the piezospectroscopic sensitivity, assumed scalar. Injecting the expression of
+Ustrain given by Eq. A.11, we get:
+∆r = hκ
+4πr2
+1
+(B.2)
+This allows us to obtain a simple relationship between the radial stress σ(r) and the ionic
+radius change ∆r, using Eq. A.4:
+σ(r) =
+2E
+1 + ν
+hκ
+4πr3
+(B.3)
+Appendix C. Instantaneous Spectral Diffusion
+In Table C1 we list the physical parameters that are needed for the estimation of ISD
+strengths for an ensemble of 10 REIC. Note that some values had to be estimated using
+data measured in similar materials. The value for the piezospectroscopic sensitivity κ
+being basically unknown for most REICs, we choose to take κ = 100 Hz/Pa for all. This
+choice is supported by the observed similarity of the value of κ in a broad variety of hosts,
+dopants and transitions [33].
+Crystal
+E
+ν
+∆µmag
+ϵr
+∆µel
+Tm:YAG
+270 GPa [55]
+0.256 [55]
+320 MHz/T [56]
+10.6 [57]
+65 Hz cm/V [58]
+Tm:YGG
+224 GPa [59]
+0.28 [59]
+44 MHz/T [60]
+12 [57]
+⋆65 Hz cm/V
+Tm:LiNbO3
+170 GPa [61]
+0.25 [61]
+1.3 GHz/T [62]
+†65 [63]
+18 kHz cm/V [62]
+Er:YSO
+150 GPa [64]
+0.26 [64]
+26 GHz/T [65]
+10 [66]
+50 kHz cm/V [67]
+Er:LiNbO3
+170 GPa [61]
+0.25 [61]
+16 GHz/T [68]
+†65 [63]
+25 kHz cm/V [69]
+Eu:YSO
+150 GPa [64]
+0.26 [64]
+10 MHz/T [70]
+10 [66]
+35 kHz cm/V [71, 72]
+Eu:Y2O3
+⋆120 GPa
+⋆0.25
+⋆10 MHz/T
+15 [73]
+⋆35 kHz cm/V
+EuCl3·6D2O
+⋆120 GPa
+⋆0.25
+8 MHz/T [74]
+3.6 [22]
+1.57 kHz cm/V [22]
+Pr:YSO
+150 GPa [64]
+0.26 [64]
+150 MHz/T [75]
+10 [66]
+111 kHz cm/V [76]
+Pr:La2(WO4)3
+90 GPa [77]
+0.3 [77]
+⋆150 MHz/T
+20 [78]
+⋆100 kHz cm/V
+Table C1. Material parameters used to compute the interaction strengths. E is the
+mechanical Young modulus, ν the Poisson’s ratio, and ϵr is the dielectric constant of
+the host matrix.
+∆µmag and ∆µel are the difference in the rare earth magnetic or
+electric moments between ground and excited states. When no values could be found
+in the literature, we chose a value among similar materials and indicated this with the
+symbol ”⋆”. †: The lithium niobate (LNO) crystal is anisotropic and exhibits different
+values of ϵr depending on the crystallographic direction [63]. For simplicity we choose
+an intermediate value.
+
+Strain-mediated ion-ion interaction in rare-earth-doped solids
+12
+In Table C2 we present the different ISD coefficients βi calculated for the three
+possible ion-ion interactions (magnetic dipole-dipole interaction, electric dipole-dipole
+interaction, and strain-mediated interaction), using the parameters listed in Table C1
+and Eqs. 4, 5 and 6.
+Crystal
+βmag
+βel
+βstr
+Tm:YAG
+6.9 · 10−17
+2.0 · 10−18
+2.3 · 10−12
+Tm:YGG
+1.3 · 10−18
+8.3 · 10−19
+1.9 · 10−12
+Tm:LiNbO3
+1.1 · 10−15
+3.0 · 10−14
+1.5 · 10−12
+Er:YSO
+4.5 · 10−13
+3.8 · 10−13
+1.3 · 10−12
+Er:LiNbO3
+1.7 · 10−13
+5.8 · 10−14
+1.5 · 10−12
+Eu:YSO
+6.7 · 10−20
+7.4 · 10−13
+1.3 · 10−12
+Eu:Y2O3
+6.7 · 10−20
+4.9 · 10−13
+1.2 · 10−12
+EuCl3·6D2O
+4.3 · 10−20
+4.1 · 10−15
+8.2 · 10−13
+Pr:YSO
+1.5 · 10−17
+7.5 · 10−12
+1.3 · 10−12
+Pr:La2(WO4)3
+1.5 · 10−17
+3.0 · 10−12
+6.3 · 10−13
+Table C2. Estimated values for the ISD coefficients βi in Hz cm3.
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+

AtE5T4oBgHgl3EQfSw9y/content/tmp_files/load_file.txt

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+page_content='Strain-mediated ion-ion interaction in  rare-earth-doped solids  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Louchet-Chauvet  E-mail: anne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='louchet-chauvet@espci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='fr  ESPCI Paris, Universit´e PSL, CNRS, Institut Langevin, 75005 Paris, France  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Chaneli`ere  Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  It was recently shown that the optical excitation of rare-earth ions produces a  local change of the host matrix shape, attributed to a change of the rare-earth ion’s  electronic orbital geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' In this work we investigate the consequences of this piezo-  orbital backaction and show from a macroscopic model how it yields a disregarded  ion-ion interaction mediated by mechanical strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  This interaction scales as 1/r3,  similarly to the other archetypal ion-ion interactions, namely electric and magnetic  dipole-dipole interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' We quantitatively assess and compare the magnitude of these  three interactions from the angle of the instantaneous spectral diffusion mechanism, and  reexamine the scientific literature in a range of rare-earth doped systems in the light of  this generally underestimated contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='05531v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='mtrl-sci]  13 Jan 2023  Strain-mediated ion-ion interaction in rare-earth-doped solids  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Introduction  Atomic ensembles have attracted a lot of attention over more than 20 years due to their  inherent capacity to efficiently interact with light [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' They are at the center of a number of  quantum storage protocols, in the form of gases [2, 3] or solid state [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Among the most  interesting solid state candidates, rare-earth ion-doped crystals (REIC) are particularly  attractive due to their long optical coherence lifetimes at cryogenic temperatures [6] and  are at the center of a number of actively developed quantum memory protocols [7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  In these light-matter interfaces, the optical depth of the medium is an important figure of  merit since it enables high storage and retrieval efficiency [10, 11, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' However, reaching  large optical depths usually comes with working with large atomic concentrations, leading  to reinforced ion-ion interactions and thereby enhanced decoherence [12, 13, 14, 15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Interestingly, although ion-ion interactions are potentially detrimental to the  performance of quantum devices, they have also emerged as the foundational mechanism  for quantum computing architectures because they allow multiqubit gate operations [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Quantum computing was initially developed in physical systems with strong readout  capacity ranging from dilute systems (trapped atoms) to condensed matter (quantum  dots or superconducting qubits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Rare-earth ion-doped crystals were only recently  considered as relevant for such applications [19] thanks to the elaboration of efficient  single ion readout schemes that compensate for the optical transition’s weak oscillator  strength [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Either way, proper understanding and quantifying of ion-ion interactions in REIC  are crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  In most systems, the order of magnitude of the measured interaction  strength is compatible with magnetic and/or electric dipole-dipole interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' In both  mechanisms, the excitation of some ions induces a change in their electric or magnetic  dipole moment, modifying the local field accordingly in their vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  This modified  field affects the surrounding ions in proportion to the Stark or Zeeman sensitivity of  their energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  In a limited number of REIC, however, the actual magnitude of  this interaction is larger than expected by several orders of magnitude [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' In this  paper we consider a strain-mediated ion-ion interaction that has not been investigated  so far and that may explain this discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' The interaction we consider stems from  the apparition of an excitation-induced stress field, affecting the surrounding ions via  their piezospectroscopic sensivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' This fundamental effect has been pointed out early  in the context of paramagnetic resonance under RF excitation, named virtual phonon  exchange interaction initially considering transition metals ions [24, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' At the time,  the description exploited the equivalent operators formalism for the spin-lattice coupled  system, but the different modeling parameters are difficult to infer from experimental  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Despite a series of studies including rare-earth salts [27, and references],  phonon-mediated interactions seem to have been overlooked for years, before reappearing  in the context of quantum technologies with original proposals of phononic engineering  [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Strain-mediated ion-ion interaction in rare-earth-doped solids  3  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' 2, the physical origin of this interaction  is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' A scaling law is given and an estimation of its strength is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' In  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' 3, we quantitatively compare the magnitude of this strain-mediated interaction with  respect to the electromagnetic dipole-dipole interactions in a number of host-dopant  combinations and confront our predictions with the experimental data found in the  scientific literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' The strength of the instantaneous spectral diffusion mechanism is  used as a comparison criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' 4, we discuss the implications of this work  in quantum technology-related applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Strain-mediated ion-ion interaction  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' General description  When an atomic particle is promoted to a different electronic level, its outer shape and  size are bound to change due to the modification of its electronic wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' If the  particle is embedded in a solid matrix, this piezo-orbital backaction effect will additionally  give rise to a stress field around the excited particle, making it detectable via a change  of shape of the solid itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' This was recently evidenced in a bulk rare-earth ion-doped  crystal in which only a finite volume of the crystal was illuminated, leading to a distortion  of the nearby crystal surface [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Conversely, it has been known for several decades that the optical lines of rare-earth  ions in solids are sensitive to stress via the piezospectroscopic effect [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Indeed,  in the elastic regime, a compressive or tensile stress modifies the interatomic distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  This affects the crystal field and in turn shifts the rare-earth ion’s energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' This  sensitivity to stress was recently proposed as a tool to sense mechanical vibrations in a  cryogenic environment [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  The piezo-orbital backaction and the piezospectroscopic effect are in fact two facets  of the same coupling mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Their combination leads to a shift of the transition  frequency in the ions surrounding a given excited rare-earth ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' In the following we coin  this as the strain-mediated ion-ion interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Magnitude of the strain-mediated interaction  For simplicity we assume that the ion is spherical with a radius r1 and that the piezo-  orbital backaction acts as a simple ionic radius change (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' This impacts the  surrounding matrix by creating a radial stress field σ(r) around the ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' With a spherically  symmetric continuum mechanics model detailed in appendix Appendix A, we calculate  the elastic strain energy that is necessary to establish this stress field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Due to energy  conservation, this elastic energy corresponds to the energy shift of the electronic levels  due to the internal stress within the ionic volume (see appendix Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' This allows  Strain-mediated ion-ion interaction in rare-earth-doped solids  4  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Simplified view of the piezo-orbital backaction around a spherical rare-earth  ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' The stress field is symbolized by a radial color gradient around the ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  us to relate the ionic radius variation ∆r to the piezospectroscopic sensitivity κ:  ∆r = hκ  4πr2  1  (1)  where h is the Planck constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' With this we derive the radial stress field σ(r) and write  the strain-mediated ion-ion interaction energy between two ions separated by a distance  r:  Estr(r) = hκσ(r) =  E  1 + ν  (hκ)2  2πr3  (2)  where E is the Young modulus and ν is the Poisson’s ratio of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' We emphasize  the relatively strong hypotheses underlying this result: the piezo-orbital backaction is  assumed to occur in the form of a mere ionic radius change of the spherical excited ion,  and the piezospectroscopic sensitivity is assumed to be scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' The anisotropic nature of  the crystalline matrix may naturally translate into a non-spherical strain field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Because  of the large distance between dopants (larger than the crystal cell parameter at low  concentration levels), we may expect the strain anisotropy to be weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' On the contrary,  the piezospectroscopic sensitivity is usually described by a tensor [35], so its scalar nature  appears as a crude assumption in order to derive an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' 1 we can estimate the corresponding relative ionic radius change due  to the piezo-orbital backaction in rare-earth-doped crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Taking typical values  κ = 100 Hz/Pa [33] and r1 = 1 ˚A [36], we obtain ∆r/r1 ≃ 5·10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' This value remarkably  agrees with the relative ionic radius change of a free ion among its 4f states that can be  derived from a Hartree-Fock calculation (see [37] for Ce3+ and [38] for Eu3+) even if the  exact rearrangement of the outer shell defining the ionic radius surrounding the 4f-shell  deserves further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Comparing the three ion-ion interactions  In this section we propose to study how this interaction compares with the other usual  ion-ion interactions generally considered in rare-earth doped systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' electric and  magnetic dipole-dipole interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  We also confront these estimations to published  measurements of ion-ion interactions in rare-earth doped crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  (a) lon at rest  (b) Excited ion  ri ＋△r  r1Strain-mediated ion-ion interaction in rare-earth-doped solids  5  Interestingly, the strain-mediated interaction scales as 1/r3, exactly like the electric  and magnetic dipole-dipole interactions, although originating from a radically different  physical mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' All three interactions can be characterized by a coefficient Ai such  that their energies read as:  Ei(r) = hAi  2πr3  (3)  where the {Ai} are defined by the following:  Ael  = π¯h  ϵ0ϵr  ∆µ2  el  (4)  Amag = µ0¯h  4π ∆µ2  mag  (5)  Astr =  E  1 + ν hκ2  (6)  ∆µel and ∆µmag are the electric and magnetic dipole moment variation when the ion  is promoted to the excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  They are expressed in Hz m V−1 and in Hz T−1,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' We note that in all three expressions, the sensitivity of the optical transition  to electric field, magnetic field or strain (respectively) appears as a square law, in  agreement with what is expected in an ion-ion interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  This scaling has been early derived for paramagnetic impurities under RF  excitation [24, 25], in the so-called zero-retardation limit of the virtual phonon exchange  (VPE) interaction [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' At the time, McMahon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' also showed that VPE and magnetic  interactions have the same order of magnitude for transition metal paramagnetic dopants  [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  With these expressions one can anticipate the variability of the three considered  interactions amongst REIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' For example, the strain-mediated interaction should be  quite constant across different ions or crystals, since the mechanical properties of typical  crystalline rare-earth doped oxides and the piezospectroscopic sensitivity of their optical  lines are rather similar [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' On the other hand, the magnetic dipole-dipole interactions  vary by several orders of magnitude between Kramers and non-Kramers ions because some  exhibit an electronic spin while some only have a nuclear spin behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' A variability  of 6 orders of magnitude is expected, since (µB/µN)2 ≃ 3 · 106, where µB and µN are  the Bohr magneton and the nuclear magneton, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' A large variability is also  expected for the electric dipole-dipole interaction because the appearance of an electric  dipole is only permitted by the crystal field that weakly perturbs the free ion electronic  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Indeed different hosts, depending on the crystal site symmetry, allow or inhibit  permanent electric dipole moment for the rare earth ion dopant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  The overall ion-ion interactions taking place in an ensemble of rare-earth ions can be  probed experimentally by the observation of instantaneous spectral diffusion (ISD) [39],  ie the effect of a substantial population transfer on the spectral width of a narrow subset  of ions within a given volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' When it builds upon interactions scaling as 1/r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' ISD  Strain-mediated ion-ion interaction in rare-earth-doped solids  6  manifests as an additional term in the homogeneous linewidth Γh that is proportional to  the volumic density of excited particles ne:  Γeff = Γh + β  2 ne  (7)  where β  =  �  i βi is the sum of individual contributions due to each considered  interaction [40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' 14]:  βi = 8π  9  √  3106Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  (8)  the 106 factor stemming from the choice of units (m3 s−1 for Ai and Hz cm3 for βi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  ISD is generally observed via high-resolution spectroscopy experiments such as  photon echoes [41, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' It has been evidenced in a large number of rare-earth ion-  doped materials, but a quantitative value is only given in a handful of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' The published  experimental values for β in ten different rare-earth ion-doped crystals are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  These materials represent a rather complete sampling of the REIC diversity, including  non-Kramers and Kramers ions, different crystal and site symmetries, and crystals with  and without charge compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' The measured values of β are contained within a 2  order-of-magnitude range (between 10−13 and 10−11 Hz cm3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Cristal  βexp (Hz cm3)  Tm:YAG  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='3 · 10−12 [43]  Tm:YGG  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='6 · 10−13 [43]  Tm:LiNbO3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='0 · 10−11 [43]  Er:YSO  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='3 · 10−12 [44]  Er:LiNbO3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='0 · 10−13 [45]  Eu:YSO  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='0 · 10−13 [46]  Eu:Y2O3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='3 · 10−13 [47]  EuCl3·6D2O  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='6 · 10−13 [22]  Pr:YSO  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='2 · 10−11 [48]  Pr:La2(WO4)3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='0 · 10−12 [49]  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Measured values for the ISD coefficient βexp in a selection of rare-earth ion-  doped crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  The three ion-ion interactions considered in this work (see equations 4, 5 and 6)  should contribute to ISD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Focusing our attention to the selection of REICs for which  quantitative ISD measurements are available, we calculate the βi parameters for each ion-  ion interactions using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' 8 and display the results graphically in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' The material  parameters relevant to this calculation and resulting numerical estimations are given in  appendix Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Again, we point out that the values are mostly indicative since  they rely on a very simplified modelization of the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  We verify that both magnetic and electric dipole-dipole interactions vary among  the materials within a 6 order-of-magnitude span depending on the existence of an  Strain-mediated ion-ion interaction in rare-earth-doped solids  7  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Calculated (bars) and measured (triangles) values for βi and Ai (where  i = {mag, el, str, exp}) for the 10 rare-earth ion-doped crystals for which quantitative  values for βexp are available (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Note the logarithmic vertical scale spanning  10 orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  electronic spin and the presence of a centrosymmetry in the doping site (βmag is comprised  between 7 · 10−20 and 2 · 10−13 Hz cm3and βel between 8 · 10−19 and 8 · 10−12 Hz cm3,  respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Conversely, the strain-mediated interaction is comprised within a much  smaller interval (βstr between 6 · 10−13 and 2 · 10−12 Hz cm3) since we use a typical value  for the piezospectroscopic sensitivity κ for all and because of the very similar values of  the Young moduli between crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  It is interesting to note that the strain-mediated interaction is of the order of  the largest of the electric or magnetic dipole-dipole contributions found among all  materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  This means that when one or both of the dipole-dipole interactions are  strong, they should coexist with the strain-mediated interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' On the other hand,  when both electromagnetic contributions are in the low range (weak Stark effect and  no electronic spin for the non-Kramers ions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Tm-doped garnets or in a lesser part  stoechiometric europium chloride), the strain-mediated interaction dominates by several  orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  In all considered REICs, the sum of the three predicted effects  satisfactorily accounts for the measured values of βexp, finally providing an explanation  to the previously large discrepancy between theoretical estimations and measurements of  the ISD mechanism in some media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content="  10-10  (Hz cm²)  ） 10-12  S  10-20  A  10-14  ISD coeficient β  Magnetic  Electric  Strain  10-22  Exp  10  16  Interaction  10-  24  10-18  10-26  10-20  YSO  Tm:YAG  Tm:YGG  Eu:YSO  EuCl'Strain-mediated ion-ion interaction in rare-earth-doped solids  8  4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Discussion  Besides providing a better understanding of the physical origin and strength of ISD in  REIC, the strain-mediated interaction in REIC could have interesting applications in  the field of quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Indeed, quantum computing schemes rest upon the  existence of a long-range atom-atom interaction enabling multiqubit operations [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' This  interaction is often spontaneously assimilated to dipole-dipole interaction, and particularly  in REIC [50, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  We argue that the strain-mediated interaction could play this role  in quantum-computing to replace the dipole-dipole coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Such a scheme could be  performed in almost any rare-earth ion-doped crystal since the strength of the interaction  is rather similar over a broad variety of ion and host combinations [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' In particular,  the possession of a permanent electric dipole moment would not be an exclusive criteria  for quantum computing compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' On the contrary, in some crystals (namely Tm-  doped garnets, although there could be other candidates) this strain-mediated interaction  dominates its electromagnetic counterparts by at least 4 orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' This means  that not only is the ion-ion interaction particularly pure in such media, but these strain-  coupled qubits would be more robust against other types of decoherence process, such  as magnetic field fluctuations due to spin flips in the host matrix [51, 13], or electric  field noise due to charge fluctuations that may occur at the surface of rare-earth-doped  nanoparticles [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  One may argue that since the strain-mediated interaction is intrinsically slow since it  relies on the propagation of stress at the speed of sound (between 3000 and 8000 m/s in  most considered crystals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' We estimate its propagation delay by considering the average  ion-ion distance dion−ion, given by  3√nRE (where nRE is the volumic density of rare-earth  ions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' dion−ion ranges from a few nm in highly concentrated materials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' ∼ 2 nm in  1% doped YAG) up to hundreds of nm in low concentration materials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' ∼ 650 nm in  200ppm doped YSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' This leads to a strain propagation time shorter than the nanosecond,  to be compared with typical µs-scale light-matter interactions occuring in such media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Therefore, the strain-mediated interaction can still be considered instantaneous within  rare-earth-doped crystals with typical concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Conclusion  Based on recent evidence of a conservative optomechanical backaction mechanism in rare-  earth ion-doped crystals, we have unveiled a disregarded ion-ion interaction based on a  physical mechanism fundamentally different from generally considered electromagnetic  dipole-dipole interactions: the sensitivity of rare-earth ions to piezo-orbitally induced  stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' With a simple mechanical model, we have estimated the strength of this strain-  mediated interaction and shown that it is largely dominant in some rare-earth ion-doped  crystals, opening interesting perspectives for strain-based quantum computing in solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Strain-mediated ion-ion interaction in rare-earth-doped solids  9  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Acknowledgments  The authors are grateful to Lars Rippe and Xiaoping Jia for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  The authors acknowledge support from the French National Research Agency (ANR)  through the projects ATRAP (ANR-19-CE24-0008), MIRESPIN (ANR-19-CE47-0011)  and MARS (ANR-20-CE92-0041).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' This work has received support under the program  “Investissements d’Avenir” launched by the French Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Continuum mechanics in a REIC  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Spherical defect in an isotropic elastic medium  Let us consider a sphere with radius r1, embedded in an infinite, isotropic, continuous  elastic medium with a Young modulus E and a Poisson’s ratio ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' When a homogeneous  radial stress σ0 is applied in the sphere, the displacement field at a distance r outside the  sphere is radial and obeys [53]:  u(r) = σ0  1 + ν  2E  r3  1  r2,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='1)  while the stress field around the sphere is also radial and reads as:  σ(r) = σ0  r3  1  r3 for r > r1  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='2)  Defining ∆r = u(r1) as the radius change, we obtain a simple relationship between  σ0 et ∆r:  σ0 = ∆r  r1  2E  1 + ν  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='3)  Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Radial stress σ(r) corresponding to the dilation of a sphere within an  infinite isotropic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' The stress is homogeneous within the sphere and decays with  a 1/r3 law outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  The radial stress can then be written:  σ(r) = −∆r  r1  2E  1 + ν  r3  1  r3  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='4)  /0  ()  1  1 stress (  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='8  Normalized radial  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='2  0  0  1  2  3  4  5  Normalized distance to ion r/riStrain-mediated ion-ion interaction in rare-earth-doped solids  10  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Strain energy  We now want to assess the elastic strain energy contained in the medium when the stress  σ0 is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  We successively calculate the energy contained inside and outside the  sphere [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Inside the sphere:  The strain energy inside the sphere reads as:  Uint = 1  2σ0ε0V  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='5)  where ε0 = ∆V/V = 3∆r/r1 is the volumic change of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' We finally obtain:  Uint = 2π∆rσ0r2  1  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='6)  Outside the sphere:  The sphere being included in an infinite medium, we must also  consider the strain energy that was necessary to establish the whole stress field around  the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' The volumic energy outside the sphere at a distance r reads as:  uV (r) = 1  2σ(r)ε(r)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='7)  The volumic strain ε(r) is related to the local stress via the volumic Hooke’s law  σ(r) = Kε(r) (where K =  E  1−2ν is the bulk modulus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='2 we get ε(r) = ε0r3  1/r3,  which finally leads to:  uV (r) = 3∆r  2 σ0  r5  1  r6  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='8)  We integrate this volumic energy over the infinite volume of the medium:  Uext = 4π  � ∞  r=r1  uV (r)r2dr = 6π∆rσ0r5  1  � ∞  r=r1  1  r4dr  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='9)  and obtain:  Uext = 2π∆rσ0r2  1  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='10)  Total strain energy:  Based on Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='6 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='10, we derive the total strain energy that  is necessary to distort the sphere and generate the associated stress field around it:  Ustrain = Uint + Uext = 4π∆rσ0r2  1  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='11)  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Relating the strain energy with the atomic energy  Due to piezo-orbital backaction [30], the size of a rare-earth ion is expected to change  under optical excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Assuming this change of shape is merely a change of radius on  a spherical ion, we can apply the calculation presented in appendix Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Due  to energy conservation, this elastic energy is taken from the ion’s energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' We can  Strain-mediated ion-ion interaction in rare-earth-doped solids  11  therefore write the following, considering that the shift in the ion’s energy levels ∆E is  related to the stress within the ion σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Ustrain = ∆E = hκσ0  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='1)  where κ is the piezospectroscopic sensitivity, assumed scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Injecting the expression of  Ustrain given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='11, we get:  ∆r = hκ  4πr2  1  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='2)  This allows us to obtain a simple relationship between the radial stress σ(r) and the ionic  radius change ∆r, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='4:  σ(r) =  2E  1 + ν  hκ  4πr3  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='3)  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Instantaneous Spectral Diffusion  In Table C1 we list the physical parameters that are needed for the estimation of ISD  strengths for an ensemble of 10 REIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Note that some values had to be estimated using  data measured in similar materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' The value for the piezospectroscopic sensitivity κ  being basically unknown for most REICs, we choose to take κ = 100 Hz/Pa for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' This  choice is supported by the observed similarity of the value of κ in a broad variety of hosts,  dopants and transitions [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Crystal  E  ν  ∆µmag  ϵr  ∆µel  Tm:YAG  270 GPa [55]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='256 [55]  320 MHz/T [56]  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='6 [57]  65 Hz cm/V [58]  Tm:YGG  224 GPa [59]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='28 [59]  44 MHz/T [60]  12 [57]  ⋆65 Hz cm/V  Tm:LiNbO3  170 GPa [61]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='25 [61]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='3 GHz/T [62]  †65 [63]  18 kHz cm/V [62]  Er:YSO  150 GPa [64]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='26 [64]  26 GHz/T [65]  10 [66]  50 kHz cm/V [67]  Er:LiNbO3  170 GPa [61]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='25 [61]  16 GHz/T [68]  †65 [63]  25 kHz cm/V [69]  Eu:YSO  150 GPa [64]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='26 [64]  10 MHz/T [70]  10 [66]  35 kHz cm/V [71, 72]  Eu:Y2O3  ⋆120 GPa  ⋆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='25  ⋆10 MHz/T  15 [73]  ⋆35 kHz cm/V  EuCl3·6D2O  ⋆120 GPa  ⋆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='25  8 MHz/T [74]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='6 [22]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='57 kHz cm/V [22]  Pr:YSO  150 GPa [64]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='26 [64]  150 MHz/T [75]  10 [66]  111 kHz cm/V [76]  Pr:La2(WO4)3  90 GPa [77]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='3 [77]  ⋆150 MHz/T  20 [78]  ⋆100 kHz cm/V  Table C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Material parameters used to compute the interaction strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' E is the  mechanical Young modulus, ν the Poisson’s ratio, and ϵr is the dielectric constant of  the host matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  ∆µmag and ∆µel are the difference in the rare earth magnetic or  electric moments between ground and excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' When no values could be found  in the literature, we chose a value among similar materials and indicated this with the  symbol ”⋆”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' †: The lithium niobate (LNO) crystal is anisotropic and exhibits different  values of ϵr depending on the crystallographic direction [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' For simplicity we choose  an intermediate value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Strain-mediated ion-ion interaction in rare-earth-doped solids  12  In Table C2 we present the different ISD coefficients βi calculated for the three  possible ion-ion interactions (magnetic dipole-dipole interaction, electric dipole-dipole  interaction, and strain-mediated interaction), using the parameters listed in Table C1  and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' 4, 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='  Crystal  βmag  βel  βstr  Tm:YAG  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='9 · 10−17  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='0 · 10−18  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='3 · 10−12  Tm:YGG  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='3 · 10−18  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='3 · 10−19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='9 · 10−12  Tm:LiNbO3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='1 · 10−15  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='0 · 10−14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content='5 · 10−12  Er:YSO  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}
+page_content=' Data 2 313–410' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'}

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+arXiv:2301.13633v1  [hep-ph]  31 Jan 2023
+QCD equation of state at finite isospin density from the linear sigma model with
+quarks: The cold case
+Alejandro Ayala1,2,3, Aritra Bandyopadhyay3,4,5, Ricardo L. S. Farias3, L. A. Hern´andez6,2, Jos´e Luis Hern´andez7,8
+1Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, Apartado Postal 70-543, CdMx 04510, Mexico.
+2Centre for Theoretical and Mathematical Physics, and Department of Physics,
+University of Cape Town, Rondebosch 7700, South Africa.
+3Departamento de F´ısica, Universidade Federal de Santa Maria, Santa Maria, RS 97105-900, Brazil.
+4Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter,
+South China Normal University, Guangzhou 510006, China.
+5Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany.
+6Departamento de F´ısica, Universidad Aut´onoma Metropolitana-Iztapalapa,
+Av.
+San Rafael Atlixco 186, CdMx 09340, Mexico.
+7Instituto de Ciencias del Espacio (ICE, CSIC),
+c.Can Magrans s.n., 08193 Cerdanyola del Vall`es, Catalonia, Spain.
+8Facultat de F´ısica, Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028 Barcelona, Spain.
+We use the two-flavor linear sigma model with quarks to study the phase structure of isospin
+asymmetric matter at zero temperature.
+The meson degrees of freedom provide the mean field
+chiral- and isospin-condensates on top of which we compute the effective potential accounting for
+quark fluctuations at one-loop order.
+Using the renormalizability of the model, we absorb the
+ultraviolet divergences into suitable counter-terms that are added respecting the original structure
+of the theory. These counter-terms are determined from the stability conditions which require the
+effective potential to have minima in the condensates directions at the classical values, as well as
+the transition from the non-condensed to the condensed phase to be smooth as a function of the
+isospin chemical potential. We use the model to study the evolution of the condensates as well as
+the pressure, energy and isospin densities and the sound velocity as functions of the isospin chemical
+potential. The approach does a good average description up to isospin chemical potentials values
+not too large as compared to the vacuum pion mass.
+Keywords: Quantum Chromodynamics, Linear Sigma Model with Quarks, Isospin Asymmetry
+I.
+INTRODUCTION
+Multiple implications of the remarkably rich phase
+structure of Quantum Chromodynamics (QCD) have
+been extensively explored over the last years. QCD at
+finite density is usually characterized by the baryon µB
+and the isospin µI chemical potentials. Nature provides
+us with physical systems at finite baryon densities with
+non zero µI in the form of isospin asymmetric matter, for
+example, compact astrophysical objects such as neutron
+stars. Because of this, along with the imminent arrival
+of new generation relativistic heavy-ion collision experi-
+ments at the FAIR [1] and NICA [2] facilities, the study of
+the phase structure in the temperature T and the chem-
+ical potentials µB and µI has become an ideal subject of
+scrutiny within the heavy-ion and astroparticle physics
+communities [3, 4].
+A typical T − µB − µI phase diagram is anticipated to
+be full of rich phase structures [5]. However, from the
+theoretical perspective, systems with finite µB are not
+straightforwardly accessible to the first-principle meth-
+ods of Lattice QCD (LQCD), due to the well-known
+fermion determinant sign problem [6, 7]. Hence, studies
+on the µB − µI plane have been performed mainly using
+low energy effective models. These models have revealed
+the existence of an exciting phase structure that includes
+Gapless Pion Condensates (GPC), a Bose-Einstein Con-
+densed (BEC) phase with gaped single particle excita-
+tions, a BEC-BCS crossover, etc [8, 9].
+On the other hand, LQCD calculations for vanishing
+and even small µB do not suffer from the sign problem.
+These calculations have predicted the existence of a su-
+perfluid pion condensate phase for high enough µI [10–
+15]. At zero temperature, they show that a second order
+phase transition at a critical isospin chemical potential
+(corresponding to the vacuum pion mass), separates the
+hadron from the pion condensate phase [14].
+In addi-
+tion to LQCD, these phases are also found using chiral
+perturbation theory (χPT) [16–28], Hard Thermal Loop
+perturbation theory (HTLPt) [29], the Nambu-Jona-
+Lasinio (NJL) model [9, 30–45] and its Polyakov loop
+(PNJL) extended version [46, 47], the quark meson model
+(QMM) [48–51] and other low energy effective models ex-
+ploiting functional RG studies [52]. Calculations using a
+LQCD equation of state for finite µI have investigated
+the viability of the existence of pion stars, with a pion
+condensate as the dominant core constituent [24, 53].
+Since LQCD calculations with µI ̸= 0, µB = µs = T = 0
+can be carried out without being hindered by the sign
+problem, they can be used as a benchmark to test effec-
+tive model predictions. For example, recently, the NJL
+model has been used in this domain and it has been
+found that results agree exceptionally well with LQCD
+results [54, 55].
+In this work we study another effective QCD model,
+the Linear Sigma Model with quarks (LSMq), extended
+
+2
+to consider a finite µI to describe the properties of
+strongly interacting systems with an isospin imbalance.
+The LSMq is a renormalizable theory that explicitly im-
+plements the QCD chiral symmetry.
+It has been suc-
+cessfully employed to study the chiral phase transition
+at finite T and µB [56–59], as well as in the presence
+of a magnetic field [60–67].
+The Linear Sigma Model
+has been used at finite µI, albeit considering the meson
+degrees of freedom as an effective classical background,
+in the Hartree or Hartree Fock approximations within
+the Cornwall-Jackiw-Tomboulis (CJT) formalism [68]. In
+contrast, in the LSMq mesons are treated as dynamical
+fields able to contribute to quantum fluctuations. Part of
+the reason for other models to avoid considering mesons
+as dynamical fields, for example the QMM, is that when
+mesons become true quantum fields and chiral symmetry
+is only spontaneously broken, their masses are subject to
+change as a result of medium effects. During this change,
+the meson square masses can become zero or even neg-
+ative.
+At zero temperature, this drawback is avoided
+by considering an explicit symmetry breaking term that
+provides pions with a vacuum finite mass. At finite tem-
+perature, the plasma screening effects need to also be
+included.
+In this work we use the LSMq to describe the evolu-
+tion of the chiral and isospin (pion) condensates, as well
+as thermodynamical quantities such as pressure, isospin
+and energy densities and the sound velocity at zero tem-
+perature and finite µI. We restrict ourselves to consider-
+ing only the effects of fermion quantum fluctuations, re-
+serving for a future work the inclusion of meson quantum
+fluctuations effects. We make use of the renormalizability
+of the LSMq and describe in detail the renormalization
+procedure which is achieved by implementing the stabil-
+ity conditions. The results thus obtained are valid for
+the case where µ2
+I) is small compared to the sum of the
+squares of the chiral and isospin condensates multiplied
+by the square of the boson-fermion coupling constant g.
+The work is organized as follows: In Sec. II we write the
+LSMq Lagrangian using degrees of freedom appropriate
+to describe an isospin imbalanced system. We work with
+an explicit breaking of the chiral symmetry introducing a
+vacuum pion mass and expanding the charged pion fields
+around the values of their condensates. The effective po-
+tential is constructed by adding to the tree-level poten-
+tial the one-loop contribution from the fermion degrees
+of freedom. Renormalization is carried out by introduc-
+ing counter-terms to enforce that the tree-level structure
+of the effective potential is preserved by loop corrections.
+We first work out explicitly the treatment in the con-
+densed phase to then work out the non-condensed phase.
+In Sec. III we study the condensates evolution with µI as
+well as that of the pressure, isospin and energy density
+and the sound velocity, and compare to recent LQCD
+results. We finally summarize and conclude in Sec. IV.
+We reserve for a follow up work the computation of the
+meson quantum fluctuations as well as finite temperature
+effects. The appendix is devoted to the explicit computa-
+tion of the one-loop fermion contribution to the effective
+potential.
+II.
+LSMQ AT FINITE ISOSPIN CHEMICAL
+POTENTIAL
+The LSMq is an effective theory that captures the ap-
+proximate chiral symmetry of QCD. It describes the in-
+teractions among small-mass mesons and quarks.
+We
+start with a Lagrangian invariant under SU(2)L ×
+SU(2)R chiral transformations
+L = 1
+2(∂µσ)2 + 1
+2(∂µ⃗π)2 + a2
+2 (σ2 + ⃗π2) − λ
+4 (σ2 + ⃗π2)2
++ i ¯ψγµ∂µψ − ig ¯ψγ5⃗τ · ⃗πψ − g ¯ψψσ,
+(1)
+where ⃗τ = (τ1, τ2, τ3) are the Pauli matrices,
+ψL,R =
+�
+u
+d
+�
+L,R
+,
+(2)
+is a SU(2)L,R doublet, σ is a real scalar field and ⃗π =
+(π1, π2, π3) is a triplet of real scalar fields. π3 corresponds
+to the neutral pion whereas the charged ones are repre-
+sented by the combinations
+π− =
+1
+√
+2
+(π1 + iπ2),
+π+ =
+1
+√
+2
+(π1 − iπ2).
+(3)
+The parameters a2, λ and g are real and positive definite.
+Equation (1) can be written in terms of the charged and
+neutral-pion degrees of freedom as
+L = 1
+2[(∂µσ)2 + (∂µπ0)2] + ∂µπ−∂µπ+ + a2
+2 (σ2 + π2
+0)
++ a2π−π+ − λ
+4 (σ4 + 4σ2π−π+ + 2σ2π2
+0 + 4π2
+−π2
++
++ 4π−π+π2
+0 + π4
+0) + i ¯ψ/∂ψ − g ¯ψψσ − ig ¯ψγ5(τ+π+
++ τ−π− + τ3π0)ψ,
+(4)
+where we introduced the combination of Pauli matrices
+τ+ =
+1
+√
+2(τ1 + iτ2),
+τ− =
+1
+√
+2(τ1 − iτ2).
+(5)
+The Lagrangian in Eq. (4) possesses the following sym-
+metries: A SU(Nc) global color symmetry, a SU(2)L ×
+SU(2)R chiral symmetry and a U(1)B symmetry. The
+sub-index of the latter emphasizes that the conserved
+charge is the baryon number B.
+A conserved isospin
+charge can be added to the LSMq Hamiltonian, multi-
+plied by the isospin chemical potential µI. The result is
+that the Lagrangian gets modified such that the ordinary
+derivative becomes a covariant derivative [69]
+∂µ → Dµ = ∂µ + iµIδ0
+µ,
+∂µ → Dµ = ∂µ − iµIδµ
+0 , (6)
+
+3
+As a result, Eq. (4) is modified to read as
+L = 1
+2[(∂µσ)2 + (∂µπ0)2] + Dµπ−Dµπ+ + a2
+2 (σ2 + π2
+0)
++ a2π−π+ − λ
+4
+�
+σ4 + 4σ2π−π+ + 2σ2π2
+0 + 4π2
+−π2
++
++ 4π−π+π2
+0 + π4
+0
+�
++ i ¯ψ/∂ψ − g ¯ψψσ + ¯ψµIτ3γ0ψ
+− ig ¯ψγ5(τ+π+ + τ−π− + τ3π0)ψ.
+(7)
+Because of the spontaneous breaking of the chiral sym-
+metry in the Lagrangian given in Eq. (7), the σ field ac-
+quires a non-vanishing vacuum expectation value
+σ → σ + v.
+To make better contact with the meson vacuum proper-
+ties and to include a finite vacuum pion mass, m0, we
+can add an explicit symmetry breaking term in the La-
+grangian such that
+L → L′ = L + h(σ + v).
+(8)
+The constant h is fixed by requiring that the model ex-
+pression for the neutral vacuum pion mass squared in
+the non-condensed phase, Eq. (11a), corresponds to m2
+0.
+This yields
+h = m2
+0
+�
+a2 + m2
+0
+λ
+,
+≡ m2
+0fπ,
+(9)
+where fπ is the pion decay constant and have used its ex-
+plicit model expression. Equation (9) provides a relation
+for the model parameters a and λ in terms of fπ.
+Before diving into the formalism details, here we first
+pause to discuss the symmetry properties of the theory.
+Notice that the introduction of µI and h modifies the
+structure of the effective Lagrangian given in Eq. (8). In
+the presence of a finite µI, the U(1)B ×SU(2)L×SU(2)R
+symmetry is reduced to U(1)B × U(1)I3L × U(1)I3R for
+h = 0, and to U(1)B × U(1)I3 for h ̸= 0, thereby repre-
+senting the explicit breaking of the chiral symmetry [70].
+The notation also emphasizes that the third component
+of the isospin charge, I3, corresponds to the generator
+of the remaining symmetry U(1)I3. Since in the present
+work, we are interested in the dynamics of the pion fields,
+further simplifications in the pseudoscalar channels can
+be obtained using the ansatz ⟨ ¯ψiγ5τ3ψ⟩ = 0 combined
+with ⟨¯uiγ5d⟩ = ⟨ ¯diγ5u⟩∗ ̸= 0 [9]. This further breaks the
+residual U(1)I3 symmetry and corresponds to a Bose-
+Einstein condensation of the charged pions. Then, the
+charged pion fields can be referred from their conden-
+sates as
+π+ → π+ + ∆
+√
+2eiθ,
+π− → π− + ∆
+√
+2e−iθ,
+(10)
+where the phase factor θ indicates the direction of the
+U(1)I3 symmetry breaking. We take θ = π for defini-
+tiveness. The shift in the sigma field produces that the
+fermions and neutral bosons acquire masses given by
+mf = gv
+(11a)
+m2
+π0 = λv2 − a2 + λ∆2
+(11b)
+m2
+σ = 3λv2 − a2 + λ∆2.
+(11c)
+The charged pions also acquire masses.
+However, in
+the condensed phase (∆ ̸= 0) they need to be described
+in terms of the π1,2 fields [71]. Since for our purposes,
+pions are not treated as quantum fluctuations, hereby
+we just notice that, as a consequence of the breaking
+of the U(1)I3 symmetry, one of these fields becomes a
+Goldstone boson. In the absence of the explicit symmetry
+breaking term in the Lagrangian of Eq. (8), this mode’s
+mass would vanish.
+However, a finite h prevents this
+mode from being massless.
+A.
+Condensed phase
+In the condensed phase the tree-level potential, ex-
+tracted from Eqs. (7) and (8), can be written as
+Vtree = −a2
+2
+�
+v2 + ∆2�
++ λ
+4
+�
+v2 + ∆2�2 − 1
+2µ2
+I∆2 − hv.
+(12)
+The fermion contribution to the one-loop effective po-
+tential becomes
+�
+f=u,d
+V 1
+f = −2Nc
+�
+d3k
+(2π)3
+�
+Eu
+∆ + Ed
+∆
+�
+,
+(13)
+with (see Appendix A)
+Eu
+∆ =
+���
+k2 + m2
+f + µI
+�2
++ g2∆2
+�1/2
+,
+(14a)
+Ed
+∆ =
+���
+k2 + m2
+f − µI
+�2
++ g2∆2
+�1/2
+,
+(14b)
+where we chose that
+µd = µI
+µu = −µI.
+(15)
+Equation (13) is ultraviolet divergent. Ultraviolet diver-
+gences are a common feature of loop vacuum contribu-
+tions. However, since Eq. (13) depends on µI, this di-
+vergence needs to be carefully treated given that mat-
+ter contributions cannot contain ultraviolet divergences.
+To identify the divergent terms, we work in the approx-
+imation whereby the fermion energies, Eqs. (14), are ex-
+panded in powers of µ2
+I/[g2(v2 +∆2)]. Considering terms
+up to O(µ4
+I), we obtain
+�
+f=u,d
+Ef
+∆ ≃ 2
+�
+k2 + m2
+f + g2∆2 +
+µ2
+Ig2∆2
+(k2 + m2
+f + g2∆2)3/2
++
+µ4
+I
+�
+4(k2 + m2
+f)g2∆2 − g4∆4�
+4
+�
+k2 + m2
+f + g2∆2
+�7/2
++ O(µ6
+I).
+(16)
+
+4
+Notice that the ultraviolet divergent part corresponds
+only to the first and second terms on the right-hand side
+of Eq. (16). In this approximation, and up to terms of
+order µ2
+I, the expression for the leading fermion contri-
+bution to the one-loop effective potential is given by
+�
+f=u,d
+V 1
+f = −2Nc
+�
+d3k
+(2π)3
+�
+2
+�
+k2 + m2
+f + g2∆2
++
+µ2
+Ig2∆2
+(k2 + m2
+f + g2∆2)3/2
+�
+(17)
+This expression can be readily computed using dimen-
+sional regularization in the MS scheme, with the result
+(see Appendix A)
+�
+f=u,d
+V 1
+f = 2Nc
+g4 �
+v2 + ∆2�2
+(4π)2
+�1
+ǫ + 3
+2 + ln
+� Λ2/g2
+v2 + ∆2
+��
+− 2Nc
+g2µ2
+I∆2
+(4π)2
+�1
+ǫ + ln
+� Λ2/g2
+v2 + ∆2
+��
+,
+(18)
+where Nc = 3 is the number of colors, Λ is the dimen-
+sional regularization ultraviolet scale and the limit ǫ → 0
+is to be understood. The explicit computation of Eq. (18)
+is described also in Appendix A. Notice that Eq. (18)
+contains an ultraviolet divergence proportional to µ2
+I∆2.
+Since a term with this same structure is already present
+in the tree-level potential, Eq. (12), it is not surpris-
+ing that this ultraviolet divergence can be handled by
+the renormalization procedure with the introduction of a
+counter-term with the same structure, as we proceed to
+show.
+To carry out the renormalization of the effective po-
+tential up to one-loop order, we introduce counter-terms
+that respect the structure of the tree-level potential and
+determine them by accounting for the stability condi-
+tions. The latter are a set of conditions satisfied by the
+tree-level potential and that must be preserved when con-
+sidering loop corrections. These conditions require that
+the position of the minimum in the v- and ∆-directions
+remain the same as the tree-level potential ones.
+The tree-level minimum in the v, ∆ plane is found from
+∂Vtree
+∂v
+=
+�
+λv3 − (a2 − λ∆2)v − h
+�����
+v0, ∆0
+= 0
+(19a)
+∂Vtree
+∂∆
+=
+�
+λ∆2 − (µ2
+I − λv2 + a2)
+�����
+v0, ∆0
+= 0. (19b)
+Notice that the second of Eqs. (19) admits a real, non-
+vanishing solution, only when
+µ2
+I > λv2 − a2 = m2
+0,
+(20)
+which means that a non-zero isospin condensate is devel-
+oped only when, for positive values of the isospin chem-
+ical potential, the latter is larger than the vacuum pion
+mass. This is what we identify as the condensed phase.
+The simultaneous solutions of Eqs. (19) are
+v0 = h
+µ2
+I
+,
+(21a)
+∆0 =
+�
+µ2
+I
+λ − h2
+µ4
+I
++ a2
+λ .
+(21b)
+Hereafter, we refer to the expressions in Eq. (21) as the
+classical solution.
+The effective potential, up to one-loop order in the
+fermion fluctuations, including the counter-terms, can be
+written as
+Veff = Vtree +
+�
+f=u,d
+V 1
+f − δλ
+4 (v2 + ∆2)2
++ δa
+2 (v2 + ∆2) + δ
+2∆2µ2
+I.
+(22)
+The counter-terms δλ and δ are determined from the gap
+equations
+∂Veff
+∂v
+����
+v0, ∆0
+= 0,
+(23a)
+∂Veff
+∂∆
+����
+v0, ∆0
+= 0.
+(23b)
+These conditions suffice to absorb the infinities of
+Eq. (18). The counter-term δa is determined by requiring
+that the slope of Veff vanishes at µI = m0,
+∂Veff
+∂µI
+����
+µI=m0
+= 0,
+(24)
+or in other words, that the transition from the non-
+condensed to the condensed phase be smooth. The ef-
+fective potential thus obtained is ultraviolet finite as well
+as Λ-independent.
+B.
+Non-condensed phase
+In the non-condensed phase, 0 ≤ µI ≤ m0, the only
+allowed solution for the second of Eqs. (19) is ∆ = 0. For
+this case, the first of Eqs. (19) becomes a cubic equation
+in v. The only real solution is
+˜v0 = (
+√
+3
+√
+27h2λ4 − 4a6λ3 + 9hλ2)1/3
+(18)2/3λ
++
+(2/3)1/3a2
+(
+√
+3
+√
+27h2λ4 − 4a6λ3 + 9hλ2)1/3 .
+(25)
+In the limit when h is taken as small one gets
+˜v0 ≃
+a
+√
+λ
++ h
+2a2 ,
+(26)
+an approximation that some times is considered. How-
+ever, hereafter we work instead with the full expression
+given by Eq. (25).
+
+5
+Δ [GeV]
+v [GeV]
+0.0
+0.5
+1.0
+1.5
+2.0
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+μI/m0
+Figure 1: v- and ∆-condensates as functions of the
+scaled variable µI/m0. For µI ≥ m0, the v-condensate
+decreases while the ∆-condensate increases.
+The effective potential V noncond
+eff
+up to one-loop order
+can be obtained from the corresponding one in the con-
+densed phase, by setting ∆ = 0. Therefore, we can write
+V noncond
+eff
+= λ
+4 v4 − a2
+2 v2 − hv −
+˜δ1
+4 v4 +
+˜δ2
+2 v2
++ 2Nc
+g4v4
+(4π)2
+�1
+ǫ + 3
+2 + ln
+� Λ2
+g2v2
+��
+. (27)
+In this case, only two conditions are needed to stabilize
+the vacuum. We take these as the requirement that the
+position and curvature of V noncond
+eff
+remain at its classical
+value when evaluated at ˜v0, namely,
+∂V noncond
+eff
+∂v
+����
+˜v0
+= 0
+(28a)
+∂2V noncond
+eff
+∂v2
+����
+˜v0
+= 3λ˜v2
+0 − a2,
+(28b)
+from where the counter-terms ˜δ1, ˜δ2 can be determined.
+Therefore, in the non-condensed phase, in addition to
+∆ = 0, the v-condensate is simply given by the constant
+˜v0 given in Eq. (25). As for the case of the condensed
+phase, in the non-condensed phase the effective potential
+is ultraviolet finite as well as Λ-independent.
+III.
+THERMODYNAMICS OF THE
+CONDENSED PHASE
+Armed with the expressions for the effective potential,
+we can now proceed to study the dependence of the con-
+densates as well as of the thermodynamical quantities as
+functions of µI.
+Since the µI-dependence in the non-
+condensed phase is trivial, we concentrate in the descrip-
+tion of the behavior of these quantities in the condensed
+phase.
+LQCD 24×48
+LQCD 32×48
+Tree Level
+One loop
+SU(2) NJL
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+μI/m0
+PN/m0
+4
+Figure 2: Normalized pressure as a function of the
+scaled variable µI/m0. Shown are the tree-level and
+one-loop fermion improved pressures compared to the
+results from Ref. [54] together with the LQCD results
+from Ref. [72].
+The model requires fixing three independent parame-
+ters: the boson self-coupling λ, the boson-fermion cou-
+pling g and the mass parameter a. For a vacuum pion
+mass m0 = 135 MeV, these parameters are fixed by re-
+quiring that the pion vacuum decay constant is fπ = 93
+MeV, the light quark mass is mq = 235 MeV and the
+sigma mass is mσ = 400 MeV. The phase space for these
+parameters is limited since for certain combinations, the
+gap equation conditions in the v-∆ plane become saddle
+points rather than global minima.
+Figure 1 shows the v- and ∆-condensates as functions
+of the scaled variable µI/m0. The behavior is qualita-
+tively as expected: for µI ≥ m0, the v-condensate de-
+creases while the ∆-condensate increases.
+Figure 2 shows the normalized pressure, defined as the
+negative of the effective potential referred from its value
+at µI = m0, as a function of the scaled variable µI/m0
+and divided by m4
+0. Shown are the results obtained by
+using the tree-level and the fermion one-loop corrected
+effective potentials, compared to the results from Ref. [54]
+and the LQCD results from [72]. Notice that the one-loop
+improved calculation does a better description than the
+tree-level one and that deviations from the LQCD result
+appear for µI ≳ 1.5 m0.
+Figure 3 shows the normalized isospin density, nI =
+dP/dµI, divided by m3
+0 as a function of the scaled vari-
+able µI/m0 compared to results obtained using the tree-
+level potential as well as to the results from Ref. [54]
+together with the LQCD results from Ref. [72]. Notice
+that the one-loop improved calculation is close to the NJL
+one up to µI ∼ 1.5 m0 but the latter does a better job
+describing the LQCD results for µI ≳ 1.5 m0. However,
+it is fair to say that neither the current calculation nor
+the NJL result reproduce the change of curvature that
+
+6
+Tree Level
+One loop
+SU(2) NJL
+1.0
+1.2
+1.4
+1.6
+1.8
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+μI/m0
+nI/m0
+3
+Figure 3: Normalized isospin density as a function of
+the scaled variable µI/m0. Shown are the tree-level and
+one-loop fermion improved effective potentials
+compared to a recent SU(2) NJL calculation [54] and
+the LQCD results from Ref. [72].
+seems to be present in the LQCD result.
+Figure 4 shows the normalized energy density, ǫ/m4
+0,
+as a function of the scaled variable µI/m0, compared to
+the results from Ref. [54] together with the LQCD results
+from Ref. [72]. Although the change in curvature shown
+by the LQCD results is not described by the present cal-
+culation, it is fair to say that neither the NJL calculation
+captures such trend. The one-loop improved calculation
+does a better average description of the LQCD result al-
+though deviations appear for µI ≳ 1.5 m0.
+Figure 5 shows the equation of state, pressure vs. en-
+ergy density, compared to the results from Ref. [54] to-
+gether with the LQCD results from Ref. [72].
+Notice
+that for the latter, the vacuum pion mass is taken as
+m0 = 135 MeV. As can be seen, the initial increasing
+trend of LQCD results is properly described by the low-
+energy models considered. Given that the accuracy of
+our results is limited to the low µI domain the NJL cal-
+culation does a better description of the LQCD results.
+Figure 6 shows the square of the speed of sound, c2
+s, as
+a function of the scaled variable µI/m0. Shown are the
+one-loop results compared to the results from Ref. [54] to-
+gether with the LQCD results from Ref. [72]. The appar-
+ent peak in the LQCD results is not reproduced by any
+model. However, notice that for the range of shown µI
+values, the one-loop improved result is above, although
+closer to the conformal bound, shown as a horizontal line
+at c2
+s = 1/3.
+IV.
+SUMMARY AND CONCLUSIONS
+In this work we have used the LSMq, with two quark
+flavors, to study the phase structure of isospin asymmet-
+LQCD 24×48
+LQCD 32×48
+Tree Level
+One loop
+SU(2) NJL
+1.0
+1.2
+1.4
+1.6
+1.8
+0.0
+0.5
+1.0
+1.5
+μI/m0
+ϵ/m0
+4
+Figure 4: Normalized energy density as a function of
+the scaled variable µI/m0. Shown are the tree-level and
+one-loop fermion improved effective potentials
+compared to the results from Ref. [54] together with the
+LQCD results from Ref. [72].
+ric matter at zero temperature. The meson degrees of
+freedom are taken as providing the mean field on top of
+which we include quantum quark fluctuations at one-loop
+order. We have used the renormalization of the LSMq to
+absorb the ultraviolet divergences with the addition of
+counter-terms that respect the original structure of the
+theory. An interesting aspect of the method is that it al-
+lows the proper handling of the disturbing µI-dependent
+ultraviolet divergence. The one-loop quark contributions
+are treated in the approximation whereby µ2
+I is taken as
+small compared to g2(v2 +∆2) and working up to O(µ2
+I).
+After determining the model parameters, we have stud-
+ied the evolution of the chiral and isospin condensates
+as well as the pressure, energy and isospin densities and
+the sound velocity. We have compared the model results
+with a recent NJL calculation of the same quantities and
+with LQCD data. The model does a good description
+for µI ≲ 1.5 m0, except perhaps for the sound veloc-
+ity for which it does not reproduce the peak seemingly
+appearing in the LQCD calculations.
+The results are encouraging and set the stage to ex-
+plore whether the method can be used to incorporate the
+effect of meson fluctuations. The method also lends itself
+to include in the description higher powers of µ2
+I as well
+as finite temperature effects. We are currently exploring
+these avenues and will report on the findings elsewhere
+in the near future.
+ACKNOWLEDGMENTS
+The authors are grateful to G. Endr¨odi and B. B.
+Brandt for kindly sharing their LQCD data in tabular
+form.
+Support for this work was received in part by
+
+LQCD 24×48
+LQCD32×487
+LQCD 24×48
+LQCD 32×48
+Tree level
+One loop
+SU(2) NJL
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.0
+0.5
+1.0
+1.5
+PN/m0
+4
+ϵ/m0
+4
+Figure 5: Equation of state, pressure vs. energy
+density. Shown are the tree-level and one-loop fermion
+improved effective potentials compared to the results
+from Ref. [54] together with the LQCD results from
+Ref. [72]. For the latter, the vacuum pion mass is taken
+as m0 = 135 MeV.
+UNAM-PAPIIT IG100322 and by Consejo Nacional de
+Ciencia y Tecnolog´ıa grant number A1-S-7655. L. A. H.
+acknowledges support from a PAPIIT-DGAPA-UNAM
+fellowship. This work was partially supported by Con-
+selho Nacional de Desenvolvimento Cient´ıfico e Tecno-
+l´ogico (CNPq), Grant No.
+309598/2020-6 (R.L.S.F.);
+Funda¸c˜ao de Amparo `a Pesquisa do Estado do Rio
+Grande do Sul (FAPERGS), Grants Nos.
+19/2551-
+0000690-0 and 19/2551-0001948-3 (R.L.S.F.). A.B. ac-
+knowledges the support from the Alexander von Hum-
+boldt Foundation postdoctoral research fellowship in
+Germany.
+Appendix A: One-loop quark contribution to the
+effective potential
+The thermodynamic potential accounting for the quark
+contribution at one-loop order is given by
+V 1
+f = iV −1 ln
+�
+Z1
+f
+�
+,
+(A1)
+where
+ln (Z1
+f ) = ln
+�
+det
+��
+S−1
+mf
+���
+,
+(A2)
+and V is the space-time volume. Also here, S−1
+mf is the
+inverse propagator of the two light-quark species. There-
+fore, we are bound to compute the determinant of a ma-
+trix M of the form
+M =
+�
+A B
+C D
+�
+,
+(A3)
+where A, B, C, D can be thought of as p×p, p×q, q ×p,
+and q × q complex matrices, respectively. When A, and
+NJL SU(2)
+Conformal Bound
+LQCD 24×48
+LQCD
+1.0
+1.2
+1.4
+1.6
+1.8
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+μI/m0
+cs
+2
+Figure 6: Square of the speed of sound as a function of
+the scaled variable µI/m0. Shown are the tree-level and
+one-loop fermion improved effective potentials
+compared to a recent SU(2) NJL calculation [54] and
+the LQCD results from Ref. [72].
+D, are invertible, the determinant of M is given by
+det{(M)} = det{(A)} det
+�
+(D − CA−1B)
+�
+,
+(A4)
+det{(M)} = det{(D)} det
+�
+(A − BD−1C)
+�
+.
+(A5)
+Equation (A4) can be written as
+det{(M)} = det{(A)} det
+�
+(D − CA−1B)
+�
+= det{(A)} det
+�
+(C−1C)
+�
+det
+�
+(D − CA−1B)
+�
+= det
+�
+(−C2A−1BC−1A + CDC−1A)
+�
+,
+(A6)
+whereas Eq. (A5) as
+det{(M)} = det{(D)} det
+�
+(A − BD−1C)
+�
+= det{(D)} det
+�
+(C−1C)
+�
+det
+�
+(A − BD−1C)
+�
+= det
+�
+(−CB + CAC−1D)
+�
+.
+(A7)
+For our purposes, B = C = ig∆γ5.
+Thus, from
+Eqs. (A6) and (A7), we obtain
+det{(M)} = det
+�
+(−C2 + CDC−1A)
+�
+,
+(A8)
+det{(M)} = det
+�
+(−C2 + CAC−1D)
+�
+.
+(A9)
+We explicitly compute both expressions.
+Fist, we use
+that the standard spin projectors Λ± satisfy
+γ0Λ±γ0 = ˜Λ∓,
+(A10)
+and
+γ5Λ±γ5 = ˜Λ±,
+(A11)
+
+LQCD32X48
+Tree Level
+One Loop8
+with the projectors ˜Λ± defined as
+˜Λ± = 1
+2
+�
+1 ± γ0(⃗γ · ⃗k − gv)
+Ek
+�
+.
+(A12)
+Next, we notice that A = S−1
+u
+and D = S−1
+d . There-
+fore, working first in the absence of an isospin chemical
+potential, for which
+S−1
+u
+= S−1
+d
+= k0γ0 − ⃗γ · ⃗k − gv,
+(A13)
+D1 ≡ −C2 + CDC−1A
+= g2∆2 + (ig∆γ5)S−1
+d
+� 1
+ig∆γ5
+�
+S−1
+u
+= g2∆2 −
+�
+k2
+0 − (Eu
+k )2�
+Λ− −
+�
+k0 −
+�
+Ed
+k
+�2�
+Λ+,
+(A14)
+and
+D2 ≡ −C2 + CAC−1D
+= g2∆2 + γ5S−1
+u γ5S−1
+d
+= g2∆2 −
+�
+k2
+0 −
+�
+Ed
+k
+�2�
+Λ− −
+�
+k2
+0 − (Eu
+k )2�
+Λ+.
+(A15)
+Thus, using that Λ+ + Λ− = 11 and defining Eq
+∆ =
+�
+(Eq
+k)2 + g2∆2, we have
+D1 = −
+�
+k2
+0 − (Eu
+∆)2�
+Λ− −
+�
+k0 −
+�
+Ed
+∆
+�2�
+Λ+, (A16)
+D2 = −
+�
+k2
+0 −
+�
+Ed
+∆
+�2�
+Λ− −
+�
+k2
+0 − (Eu
+∆)2�
+Λ+, (A17)
+and
+det
+�
+(S−1
+mf )
+�
+= det{(D1)} = det{(D2)}.
+(A18)
+Note that
+ln (Z1
+f ) = ln
+�
+det
+��
+S−1
+mf
+���
+= 1
+2 ln
+�
+det
+��
+S−1
+mf
+�2��
+= 1
+2 ln (det{(D1D2)})
+= 1
+2Tr [ln (D1D2)] ,
+(A19)
+and since the product D1D2 is given by
+D1D2 =
+�
+k2
+0 − (Eu
+∆)2� �
+k2
+0 −
+�
+Ed
+∆
+�2�
+,
+(A20)
+we get
+ln (Z1
+f ) = 1
+2
+�
+q=u,d
+Tr
+�
+ln
+�
+k2
+0 − (Eq
+∆)2 ��
+,
+(A21)
+where the trace is taken in Dirac, color (factors of 4 and
+Nc, respectively), and in coordinate spaces, namely,
+ln (Z1
+f ) = 2Nc
+�
+q=u,d
+�
+d4x
+�
+x
+��� ln
+�
+k2
+0 − (Eq
+∆)2 ����x
+�
+= 2Nc
+�
+q=u,d
+�
+d4x
+�
+d4k
+(2π)4 ln
+�
+k2
+0 − (Eq
+∆)2 �
+.
+(A22)
+Therefore
+ln (Z1
+f) = 2V Nc
+�
+q=u,d
+�
+d4k
+(2π)4 ln
+�
+k2
+0 − (Eq
+∆)2 �
+.
+(A23)
+In order to obtain a more compact expression, we inte-
+grate and differentiate with respect to Eq
+∆ as follows
+ln (Z1
+f) = 2V Nc
+�
+q=u,d
+�
+d4k
+(2π)4
+�
+dEq
+∆
+Eq
+∆
+k2
+0 − (Eq
+∆)2 .
+(A24)
+Performing a Wick rotation k0 → ik4, we obtain
+ln (Z1
+f) = 4iV Nc
+�
+q=u,d
+�
+d4kE
+(2π)4
+�
+dEq
+∆
+Eq
+∆
+k2
+0 − (Eq
+∆)2 ,
+(A25)
+and integrating over k4 and Eq
+∆, in this order, we get
+ln (Z1
+f) = 2iV Nc
+�
+q=u,d
+�
+d3k
+(2π)3 Eq
+∆,
+(A26)
+with Re[(Eq
+∆)2] ≥ 0. Therefore, the quark contribution
+to the effective potential at one-loop order is given by
+V 1
+f = iV −1 ln (Z1
+f ).
+(A27)
+Thus,
+V 1
+f = −2Nc
+�
+q=u,d
+�
+d3k
+(2π)3 Eq
+∆.
+(A28)
+In the presence of an isospin chemical potential for which
+S−1
+u
+= (k0 + µI)γ0 − ⃗γ · ⃗k − gv,
+S−1
+d
+= (k0 − µI)γ0 − ⃗γ · ⃗k − gv,
+(A29)
+and repeating the steps starting from Eq. (A14), we ob-
+tain Eq. (A28), with the energies Eu
+∆ and Ed
+∆ given by
+Eqs. (14).
+We now proceed to the explicit computation of
+Eq. (13). In the limit where µ2
+I/[g2(v2 + ∆2)] is small,
+Eq. (A28) can be written as in Eq. (17). We use dimen-
+sional regularization. The first of the integrals on the
+right hand side of Eq. (17) is expressed as
+�
+d3k
+(2π)3
+�
+k2 + g2v2 + g2∆2 → Λ3−d Γ
+�
+− 1
+2 − d
+2
+�
+(4π)
+d
+2 Γ
+�
+− 1
+2
+�
+×
+�
+1
+g2v2 + g2∆2
+�− 1
+2 − d
+2
+.
+(A30)
+
+9
+Taking d → 3 − 2ǫ and working in the MS scheme
+Λ2 → Λ2eγE
+4π
+,
+(A31)
+where γE is the Euler-Mascheroni constant, we get
+�
+d3k
+(2π)3
+�
+k2 + g2v2 + g2∆2 → −(g2v2 + g2∆2)2
+2(4π)2
+�1
+ǫ + 3
+2 + ln
+�
+Λ2
+g2v2 + g2∆2
+��
+.
+(A32)
+The second of the integrals on the right hand side of Eq. (17) is expressed as
+�
+d3k
+(2π)3
+1
+(k2 + g2v2 + g2∆2)3/2 → Λ3−d Γ
+� 3
+2 − d
+2
+�
+(4π)
+d
+2 Γ
+� 3
+2
+�
+�
+1
+g2v2 + g2∆2
+� 3
+2 − d
+2
+.
+(A33)
+Taking d → 3 − 2ǫ and working in the MS scheme we get
+�
+d3k
+(2π)3
+1
+(k2 + g2v2 + g2∆2)3/2 →
+2
+(4π)2
+�1
+ǫ + ln
+�
+Λ2
+g2v2 + g2∆2
+��
+,
+(A34)
+from where the result of Eq. (18) follows.
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+Divergence-Conforming Isogeometric Collocation Methods for the
+Incompressible Navier-Stokes Equations
+Ryan M. Aronsona,∗, John A. Evansb
+aStanford University, 94305, Stanford, CA, USA
+bUniversity of Colorado Boulder, 80309, Boulder, CO, USA
+Abstract
+We develop two isogeometric divergence-conforming collocation schemes for incompress-
+ible flow. The first is based on the standard, velocity-pressure formulation of the Navier-
+Stokes equations, while the second is based on the rotational form and includes the vorticity
+as an unknown in addition to the velocity and pressure. We describe the process of discretiz-
+ing each unknown using B-splines that conform to a discrete de Rham complex and collocat-
+ing each governing equation at the Greville abcissae corresponding to each discrete space.
+Results on complex domains are obtained by mapping the equations back to a parametric do-
+main using structure-preserving transformations. Numerical results show the promise of the
+method, including accelerated convergence rates of the three field, vorticity-velocity-pressure
+scheme when compared to the two field, velocity-pressure scheme.
+Keywords:
+Isogeometric analysis, Collocation, Incompressible flow, Divergence-conforming
+discretizations, Velocity-pressure formulation, Vorticity-velocity-pressure formulation
+1. Introduction
+Isogeometric Analysis (IGA) is a technology [1, 2] which replaces the standard polynomial
+basis functions used in traditional Finite Element Analysis (FEA) with B-splines, Non-
+Uniform Rational B-splines (NURBS), and other classes of splines in an aim to reduce the
+gap between geometry and analysis. IGA has a distinct advantage over traditional FEA
+due to its ability to exactly represent the geometries commonly seen in Computer-Aided
+Design (CAD). Moreover, the basis functions used in IGA are globally more smooth than
+those of FEA and it has been shown that IGA can exhibit improved accuracy and robustness
+over FEA. For example, higher continuity splines are shown to have optimal approximating
+power in the sense of Kolmogorov n-widths [3] and spline discretizations have more favorable
+dissipation and dispersion properties than standard, high-order FEA discretizations [2].
+To improve on the complexity and implementation details of IGA, the feasibility of isoge-
+ometric collocation methods has been explored [4, 5]. In Galerkin IGA methods, the discrete
+system of equations is formed by integrating the PDE residual against the test function space.
+∗Corresponding author
+Email address: [email protected] (Ryan M. Aronson)
+Preprint submitted to Elsevier
+January 11, 2023
+arXiv:2301.03783v1  [math.NA]  10 Jan 2023
+
+This requires numerical integration, which renders system assembly quite expensive. Collo-
+cation, on the other hand, forms the discrete system by simply requiring that the residual
+of the governing equations vanish at a set of discrete locations in the domain.
+Many recent studies have shown the efficacy of isogeometric collocation methods in both
+static and dynamic solid mechanics problems [6, 7, 8], and detailed comparisons between iso-
+geometric Galerkin and collocation methods have been made [9]. In addition, recent studies
+have also investigated the use of mixed collocation methods for use in nearly incompressible
+elasticity [10, 11]. Isogeometric collocation has even been used for acoustic problems [12],
+computing Karhunen-Loeve expansions [13, 14], and introduced into physics-informed neural
+networks [15]. Finally, a method was recently introduced for IGA collocation in immersed
+domains by combining with the finite cell method near the boundaries [16]. These results
+all suggest that isogeometric collocation methods retain some of the improved qualities of
+standard IGA, while reducing the computational cost and improving sparsity of the discrete
+systems.
+Isogeometric collocation methods have not been as well explored in the context of fluid
+mechanics, though the idea of using B-spline collocation to solve incompressible fluid me-
+chanics problems has been investigated in the past [17, 18]. In addition, spline collocation
+has been employed in fundamental Direct Numerical Simulation (DNS) studies of turbulent
+flows [19]. However, the methods previously introduced are typically limited to simple ge-
+ometries and are not divergence-conforming, meaning that the discrete velocity field does
+not exactly satisfy the continuity equation in incompressible flow.
+In addition, a mixed
+isogeometric collocation method has recently been proposed for use in poromechanics [20],
+though the preliminary results were limited to one-dimensional problems.
+In the context of incompressible fluid mechanics, divergence-conforming Galerkin meth-
+ods based on B-spline basis functions have been developed for both the Stokes and the
+Navier-Stokes equations [21, 22, 23]. These methods are provably inf-sup stable and yield
+discrete velocity fields that are exactly pointwise divergence free, among other desirable qual-
+ities such as pressure robustness. An excellent summary of divergence-conforming methods
+is given by [24]. These discretizations have prospered in areas such as turbulent flow simula-
+tion [25, 26, 27] and fluid-structure interaction [28]. Moreover, efficient multigrid solvers have
+been developed based on these discretizations [29]. Divergence-conforming Galerkin meth-
+ods have also been developed for more advanced spline discretizations, such as hierarchical
+B-splines [30] and LR B-splines [31].
+In this paper we develop similar divergence-conforming methodologies for incompressible
+flow using collocation. In particular, we introduce two collocation methods, one based on
+the standard velocity-pressure form of the steady Navier-Stokes equations, and one based on
+a three field (velocity-vorticity-pressure) form of the steady Navier-Stokes equations. The
+latter form of the Navier-Stokes equations has recently been used to develop alternative
+structure-preserving finite element discretizations [32, 33, 34] and we find that collocation
+methods based on the resulting system of first order differential-algebraic equations returns
+improved convergence rates compared to the rates obtained using collocation in conjunction
+with the standard velocity-pressure form of the equations. In our collocation schemes, each
+unknown is discretized with compatible B-spline spaces that preserve the structure of the
+governing equations. Both collocation methods in this paper are shown to return velocity
+fields which are still exactly pointwise divergence free, similar to the Galerkin methods
+2
+
+mentioned above.
+We lay out this paper as follows: In Section 2 we describe the steady form of the Navier-
+Stokes equations using velocity and pressure unknowns as well as vorticity, velocity, and
+pressure unknowns. This is followed in Section 3 by a discussion of the de Rham complex and
+isogeometric discrete differential forms, the tools used to develop a divergence-conforming
+method. Section 4 describes the collocation schemes for square domains in two dimensions.
+Then results are presented in the two dimensional setting in Section 5 which detail the
+high-order convergence rates of the methods as well as agreement with standard benchmark
+problems.
+We then discuss the necessary changes to make the methods work for cubic
+domains in three dimensions and illustrate that the methods performs similarly in this setting
+in Sections 6 and 7. Finally, we return to 2D in Section 8 and consider the Stokes equations
+in more complicated domains. We show that by mapping the equations and unknowns via
+divergence and integral preserving transformations we can also obtain results for flow in
+complex geometries. Section 9 summarizes these results.
+2. Velocity-Pressure and Vorticity-Velocity-Pressure Forms of the Navier-Stokes
+Equations
+In this paper we consider the steady, incompressible Navier-Stokes equations on a Lips-
+chitz open set Ω of points in either R2 or R3 when subjected to Dirichlet boundary conditions.
+The standard form of this problem with d = 2, 3 is stated as follows:
+�
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+�
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+�
+�
+�
+�
+Given ν ∈ R+, f : Ω → Rd, and g : ∂Ω → Rd, find u : Ω → Rd and
+p : Ω → R such that:
+−ν∆u + u · ∇u + ∇p = f
+in
+Ω
+(1)
+∇ · u = 0
+in
+Ω
+(2)
+u = g
+on
+∂Ω.
+(3)
+Equations (1) and (2) represent the standard forms of the momentum and mass conservation
+equations, while Equation (3) sets the Dirichlet boundary values. Above, we denote the
+velocity field by u, the kinematic pressure field by p, the constant kinematic viscosity by ν,
+the applied forcing as f, and the prescribed Dirichlet boundary values as g.
+For the purposes of this paper we will not only work with this set of equations, but also
+introduce vorticity ω as a separate unknown variable and introduce ω − ∇ × u = 0 as a
+constitutive relation. Substituting the two vector calculus identities:
+∆u = ∇(∇ · u) − ∇ × (∇ × u) = −∇ × ω,
+(4)
+u · ∇u = (∇ × u) × u + 1
+2∇(u · u) = ω × u + 1
+2∇(u · u),
+(5)
+into Equation (1) when d = 3, we arrive at the vorticity-velocity-pressure formulation of the
+problem in 3D:
+3
+
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+�
+Given ν ∈ R+, f : Ω → R3, and g : ∂Ω → R3, find u : Ω → R3,
+P : Ω → R, and ω : Ω → R3 such that:
+ν∇ × ω + ω × u + ∇P = f
+in
+Ω
+(6)
+∇ · u = 0
+in
+Ω
+(7)
+ω − ∇ × u = 0
+in
+Ω
+(8)
+u = g
+on
+∂Ω.
+(9)
+Note that in the above formulation we have replaced the kinematic pressure p with the total
+pressure P, which are related via P = p + 1
+2u · u.
+For later sections in this paper it is useful to employ the component forms of the vec-
+tor equations above describing the vorticity-velocity-pressure formulation. When explicitly
+broken into its components, the momentum conservation equation, given by Equation (6),
+becomes:
+ν(∂ωz
+∂y − ∂ωy
+∂z ) + (ωyuz − ωzuy) + ∂P
+∂x = fx,
+(10)
+ν(∂ωx
+∂z − ∂ωz
+∂x ) + (ωzux − ωxuz) + ∂P
+∂y = fy,
+(11)
+ν(∂ωy
+∂x − ∂ωx
+∂y ) + (ωxuy − ωyux) + ∂P
+∂z = fz,
+(12)
+and the constitutive relation given by Equation (8) reads:
+ωx − (∂uz
+∂y − ∂uy
+∂z ) = 0,
+(13)
+ωy − (∂ux
+∂z − ∂uz
+∂x ) = 0,
+(14)
+ωz − (∂uy
+∂x − ∂ux
+∂y ) = 0.
+(15)
+The above component form of the equations is also useful for considering 2D problems in
+the three field formulation, as we do not need to redefine operations such as cross products
+when the vorticity reduces to a scalar unknown. Thus we can arrive at the problem statement
+for 2D domains by simply removing any terms involving ωx, ωy, or any derivatives in the z
+direction. In full, the 2D problem reads:
+4
+
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+�
+�
+Given ν ∈ R+, f : Ω → R2, and g : ∂Ω → R2, find u : Ω → R2,
+P : Ω → R, and ω : Ω → R such that:
+ν ∂ω
+∂y − ωuy + ∂P
+∂x = fx
+in
+Ω
+(16)
+−ν ∂ω
+∂x + ωux + ∂P
+∂y = fy
+in
+Ω
+(17)
+∇ · u = 0
+in
+Ω
+(18)
+ω − (∂uy
+∂x − ∂ux
+∂y ) = 0
+in
+Ω
+(19)
+u = g
+on
+∂Ω.
+(20)
+With the governing equations fully defined we can move to a more in-depth description
+of the collocation scheme, starting with the definition of discrete approximation spaces in
+the following section.
+3. The de Rham Complex and Isogeometric Discrete Differential Forms
+The first step in creating a collocation scheme is to define the sets of basis functions used
+to approximate the unknown variables. To construct the collocation methods presented in
+this paper, we leverage the de Rham complex which aids in the development of exactly
+divergence-conforming finite element spaces. After recalling the de Rham complex, we de-
+scribe the process of constructing B-spline basis functions and conclude with the definition
+of B-spline spaces that conform to the de Rham complex.
+3.1. The de Rham Complex
+The de Rham complex is a cochain complex that is often used as a starting point for
+developing mixed finite element methods which preserve topological properties of the con-
+tinuous problem and are typically more stable in practice [24]. In 3D, it is typically written
+as:
+R
+Φ
+Ψ
+V
+Q
+0,
+∇
+∇×
+∇·
+(21)
+where:
+Φ := H 1(Ω),
+(22)
+Ψ := H(curl, Ω),
+(23)
+V := H(div, Ω),
+(24)
+Q := L2(Ω).
+(25)
+In the context of fluid flow, these infinite dimensional spaces can be interpreted as the
+spaces of scalar potential fields (Φ), vector potential fields (Ψ), velocity fields (V), and
+5
+
+pressure fields (Q). This complex is exact for simply connected domains, meaning that the
+range of each map is the same as the null space of the following map.
+For completeness, we also state the rotated 2D de Rham complex:
+R
+Ψ
+V
+Q
+0,
+∇⊥
+∇·
+(26)
+where:
+Ψ := H1(Ω),
+(27)
+V := H(div, Ω),
+(28)
+Q := L2(Ω),
+(29)
+and the rotor operator ∇⊥ acting on a scalar function ω is defined as ∇⊥ω = ( ∂ω
+∂y , − ∂ω
+∂x).
+Not only is the topological structure of the incompressible Navier-Stokes equations em-
+bedded in the de Rham complex, but stability conditions result from the complex as well. By
+creating approximation spaces for the unknown variables that conform to discrete analogs of
+Equations (21) and (26) we can generate numerical methods which inherit these properties.
+More concretely, for 3D problems let the space Φh contain the discrete scalar potentials, Ψh
+contain the discrete vector potentials as well as the discrete vorticity, Vh contain the dis-
+crete velocity, and Qh contain the discrete pressure. Then if there exist projection operators
+ΠΦ : Φ → Φh, ΠΨ : Ψ → Ψh, ΠV : V → Vh, and ΠQ : Q → Qh such that the following
+commuting diagram holds
+R −−−→ Φ
+∇
+−−−→ Ψ
+∇×
+−−−→ V
+∇·
+−−−→ Q −−−→ 0
+���ΠΦ
+���ΠΨ
+���ΠV
+���ΠQ
+R −−−→ Φh
+∇
+−−−→ Ψh
+∇×
+−−−→ Vh
+∇·
+−−−→ Qh −−−→ 0,
+(30)
+a Galerkin finite element method employing Vh for the discrete velocity space and Qh for the
+discrete pressure space will be inf-sup stable and will yield discrete velocity approximations
+that are divergence free almost everywhere [35]. We shall prove later on that the divergence-
+conforming property is maintained if we utilize these spaces in our collocation scheme.
+The same holds in 2D, where we instead let Ψh define the discrete space to which the
+vorticity belongs (as well as the streamfunction), Vh define the discrete velocity space, and
+Qh define the discrete pressure space. The required commuting diagram in this case is
+R −−−→ Ψ
+∇⊥
+−−−→ V
+∇·
+−−−→ Q −−−→ 0
+���ΠΨ
+���ΠV
+���ΠQ
+R −−−→ Ψh
+∇⊥
+−−−→ Vh
+∇·
+−−−→ Qh −−−→ 0.
+(31)
+Of course, we have yet to define the specifics of how to construct discrete spaces such
+that these discrete complexes hold. For the purposes of this paper we will use compatible
+B-spline spaces, and the following section is devoted to introducing the basics of B-spline
+basis functions.
+6
+
+3.2. Univariate and Multivariate B-Splines
+The construction of a B-spline basis in one dimension requires two objects: the de-
+gree of the basis (denoted k) and a series of numbers called the knot vector (denoted
+Ξ = {ξ1, ...ξn+k+1}). The knots ξi are non-decreasing and denote the locations in paramet-
+ric space where the parametrization can change, similar to element boundaries in standard
+FEA. The number n in the previous relation represents the total number of functions in the
+basis. The basis functions themselves are defined through the Cox-de Boor recursion: The
+k = 0 basis functions are built as
+Ni,0(ξ) =
+�
+1
+ξi ≤ ξ ≤ ξi+1
+0
+otherwise,
+(32)
+and higher-order bases are defined through
+Ni,k(ξ) =
+ξ − ξi
+ξi+k − ξi
+Ni,k−1(ξ) +
+ξi+k+1 − ξ
+ξi+k+1 − ξi+1
+Ni+1,k−1(ξ).
+(33)
+Note that in the above relations, we must utilize the convention that any occurrence of zero
+divided by zero is equal to zero.
+In higher dimensions (two or three for the purposes of this paper), B-spline basis functions
+are constructed by simply taking the tensor product of one dimensional B-spline bases in
+each parametric direction. Note that different polynomial degrees and knot vectors can be
+used in each direction.
+B-spline basis functions possess a number of useful properties for numerical method
+development.
+In particular, the smoothness of the global basis at the knot locations is
+controlled by the repetition of the knot value in Ξ. The basis at these locations is Ck−r,
+where r is the multiplicity of the knot. Compared to a standard finite element basis, this
+basis has improved global continuity, which enables the use of collocation as more classical
+derivatives of the functions are well defined. Note that if the first and last entries in the knot
+vector are repeated k +1 times, the spline basis will become interpolatory at those locations.
+Such knot vectors are referred to as open knot vectors, and allow for easy specification of
+Dirichlet boundary conditions. For the results within this paper, all other entries in the
+knot vector have multiplicity one, meaning we are utilizing a B-spline basis with the highest
+possible order of continuity. Further, let us define the so-called regularity vector α for a
+basis. The size of this vector is equal to the number of distinct knots, with entries equal
+to the polynomial degree of the basis minus the multiplicity of the corresponding knot. In
+terms of the regularity vector, the global basis functions are Cαj-continuous across the αj
+unique knot. To simplify notation, the space of functions spanned by a 1D B-spline basis of
+degree k and a provided knot vector is denoted as:
+Sk
+α = span{Ni,k}n
+i=1.
+(34)
+We extend this notation to higher dimensions by adding extra sub- and superscripts, repre-
+senting the polynomial degrees and regularities in each spatial direction.
+7
+
+3.3. Isogeometric Discrete Differential Forms
+Using the basics of B-spine functions above allows us to develop discrete approximation
+spaces for the vorticity, velocity, and pressure. These results are built upon work in the
+area of isogeometric discrete differential forms [36, 37], which we will not fully develop here.
+The construction of these types of spaces in the context of Galerkin approximation for the
+Navier-Stokes equations can also be found in [22, 23]. For 3D problems, the B-spline spaces
+used to discretize our unknown fields are given by:
+Φh := {φh ∈ Sk1,k2,k3
+α1,α2,α3},
+(35)
+Ψh := {ψh ∈ Sk1−1,k2,k3
+α1−1,α2,α3 × Sk1,k2−1,k3
+α1,α2−1,α3 × Sk1,k2,k3−1
+α1,α2,α3−1},
+(36)
+Vh := {wh ∈ Sk1,k2−1,k3−1
+α1,α2−1,α3−1 × Sk1−1,k2,k3−1
+α1−1,α2,α3−1 × Sk1−1,k2−1,k3
+α1−1,α2−1,α3},
+(37)
+Qh := {qh ∈ Sk1−1,k2−1,k3−1
+α1−1,α2−1,α3−1}.
+(38)
+It can be shown that these spaces satisfy the discrete complex in Equation (30).
+In practice we usually define k1 = k2 = k3, and thus we can define the polynomial degree
+of the spline bases constructed in the above manner using a single number k′ = k1 − 1 =
+k2−1 = k3−1. This indicates that the pressure space Qh will have degree equal to k′ in each
+direction. Then, according to the above, each velocity component will have degree k′ + 1 in
+one direction and degree k′ in the other two. Similarly, the vorticity components will have
+degree k′ + 1 in two directions and degree k in the last.
+In 2D, we define the following spline spaces:
+Ψh := {ψh ∈ Sk1,k2
+α1,α2},
+(39)
+Vh := {wh ∈ Sk1,k2−1
+α1,α2−1 × Sk1−1,k2
+α1��1,α2},
+(40)
+Qh := {qh ∈ Sk1−1,k2−1
+α1−1,α2−1}.
+(41)
+Similar to the 3D setting, these spaces are related as in Equation (31).
+4. Collocation Methods on Square Domains
+Using the discrete spaces developed above, this section focuses on the construction of
+collocation methods for the Navier-Stokes equations using divergence-conforming bases. Here
+we develop methods based on the velocity-pressure form of the Navier-Stokes equations as
+well as the vorticity-velocity-pressure form. As the vorticity changes between a scalar in the
+2D case and a vector in the 3D case, we start by considering only square domains in 2D.
+This selection also lends itself to easier visualization of the methods. After briefly reviewing
+the form of a typical divergence-conforming isogeometric Galerkin method, we define the
+collocation grids for each unknown and then describe the imposition of boundary conditions.
+The section concludes by summarizing the form of the discrete system.
+8
+
+4.1. Review of Galerkin Methods
+We start by reviewing the form of the divergence-conforming isogeometric Galerkin meth-
+ods which inspired our collocation schemes. Let us consider a problem with Dirichlet bound-
+ary conditions on the velocity for concreteness. Then we define the discrete test and trial
+function spaces for velocity as Vh,0 and Vh,g, which are defined as the same Vh from Equa-
+tion (40) with either no penetration boundary conditions strongly enforced (for the test
+space Vh,0) or with the normal velocity prescribed as given by the boundary data g at spec-
+ified collocation points (for the trial space Vh,g). Similarly, define the test and trial space
+for pressure as Qh,0, where Qh is the same space as in Equation (41) but with the added
+condition that the pressure must have zero integral. Then the Galerkin formulation for the
+velocity-pressure form would read
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Given ν ∈ R+, f : Ω → R2, and g : ∂Ω → R2, find uh ∈ Vh,g and ph ∈ Qh,0 such that,
+∀(wh, qh) ∈ (Vh,0, Qh,0):
+�
+Ω
+(ν∇wh · ∇uh + wh · (uh · ∇uh) − ph∇ · wh)dΩ
+− ν
+�
+∂Ω
+((∇uh · n) · wh − Cpen
+h uh · wh)dΓ =
+�
+Ω
+wh · fdΩ + ν
+�
+∂Ω
+Cpen
+h g · whdΓ
+(42)
+�
+Ω
+qh(∇ · uh)dΩ = 0.
+(43)
+Note that in the above we have used a Nitsche approach to enforce the tangential bound-
+ary conditions in the momentum equations. This Galerkin formulation is valid if and only if
+the minimum entry in the regularity vectors satisfy min{α1 − 1} ≥ 0 and min{α2 − 1} ≥ 0,
+where α1 and α2 are the regularity vectors from Equations (39) - (41). We can write a
+similar Galerkin form of the vorticity-velocity-pressure form of the Navier-Stokes equations,
+which would yield
+9
+
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Given ν ∈ R+, f : Ω → R2, and g : ∂Ω → R2, find uh ∈ Vh,g, P h ∈ Qh,0, and ωh ∈ Ψh
+such that, ∀(wh, qh, ψh) ∈ (Vh,0, Qh,0, Ψh):
+�
+Ω
+ν ∂ωh
+∂y wh
+xdΩ −
+�
+Ω
+ωhuh
+ywh
+xdΩ −
+�
+Ω
+∂wh
+x
+∂x P hdΩ +
+�
+Γ
+P hwh
+xdy =
+�
+ω
+fxwh
+xdΩ
+(44)
+−
+�
+Ω
+ν ∂ωh
+∂x wh
+ydΩ +
+�
+Ω
+ωhuh
+xwh
+ydΩ −
+�
+Ω
+∂wh
+y
+∂y P hdΩ +
+�
+Γ
+P hwh
+ydx =
+�
+ω
+fywh
+ydΩ (45)
+�
+Ω
+(∇ · uh)qhdΩ = 0
+(46)
+�
+Ω
+ωhψhdΩ +
+�
+Ω
+(∂ψh
+∂x uh
+y − ∂ψh
+∂y uh
+x)dΩ =
+�
+∂Ω
+ψh(g · s)dΓ.
+(47)
+In this case the tangential velocity boundary conditions appear as natural boundary condi-
+tions in the weak form of the constitutive equation, with s representing the unit tangent on
+the boundary (oriented counter-clockwise). In contrast with the velocity-pressure Galerkin
+formulation, this three field Galerkin formulation is valid if and only if the minimum regu-
+larity of the discrete spaces satisfies min{α1 − 1} ≥ −1 and min{α2 − 1} ≥ −1 due to the
+reduced differential order of the strong form.
+However, we wish to pursue a collocation method inspired by the divergence-free mixed
+finite element form, which we describe below.
+Collocation imposes additional regularity
+requirements on the spaces of unknowns, as the unknown fields and their derivatives will be
+evaluated at points rather than integrated over the domain. The spaces developed above are
+not only divergence-conforming, meaning discrete velocity approximations will be pointwise
+divergence-free, but are also regular enough to use in collocation provided the polynomial
+degree is sufficiently large and the global basis is sufficiently regular.
+4.2. Collocation Grids
+Similarly to the Galerkin setting, each discrete unknown in the collocation schemes is
+assumed to lie in the corresponding space from above: The discrete velocity uh ∈ Vh,g, the
+discrete pressure ph ∈ Qh,0, and, when applicable, the discrete vorticity ωh ∈ Ψh. For now we
+ignore any boundary conditions; these will be discussed in the following section. To generate
+the system of equations needed to solve for the coefficients for each basis function, we define
+sets of collocation points of the form τ i for i = 1, ..., N for each of the governing equations.
+Note that the total number of collocation points should be equal to the total number of
+degrees of freedom in the discretization. The full discrete system is formed by requiring the
+strong form of the governing equations to hold at each of the collocation points.
+We choose the Greville abscissae of a B-spline space as collocation points. In one dimen-
+sion, the Greville abscissae are defined by
+ˆξi = ξi+1 + ... + ξi+p
+p
+,
+(48)
+10
+
+A First Collocation Attempt for Stokes:
+The Unit Square
+ 
+ˆξi = ξi+1 +…+ξi+p
+p
+Greville abscissae:
+=	First	Momentum	Collocation	Pt.
+=	Second	Momentum	Collocation	Pt.
+=	Continuity	Collocation	Pt.
+k = 2, 4 x 4 elements
+(a) Before strong enforcement of normal ve-
+locity boundary conditions
+A First Collocation Attempt for Stokes:
+The Unit Square
+k = 2, 4 x 4 elements
+Boundary	Conditions:
+We	enforce	no-penetration BCs strongly	and
+remove	the	corresponding	collocation	points.
+We	enforce	no-slip BCs weakly by	modifying
+the	momentum	equations at	the	boundary	
+with	a	suitable	penalty	term.
+ 
+−div
+! grad
+" ˆu
+( )
+(
+)+ grad
+" ˆp
+( )− ˆf
+(
+)
+             + Cpen
+2
+h2
+ˆu − ˆuBC
+(
+) = 0
+(b) After strong enforcement of normal ve-
+locity boundary conditions
+Figure 1: Example of collocation grid for k′ = 2, 4 x 4 elements for the velocity-pressure scheme. Horizontal
+triangles represent the collocation points for the first momentum equation, vertical triangles represent the
+points for the second momentum equation, and squares are collocation points for the continuity equation.
+and in higher dimensions we simply take the tensor product of the Greville abscissae in each
+direction. By construction there will be the same number of Greville points as there are basis
+functions in the considered space. There are choices for collocation points other than the
+Greville abscissae, such as the Cauchy-Galerkin points [38, 39] or the Demko abscissae [40].
+However, the Greville points are very easy to compute and have already been demonstrated
+to give satisfactory results in practical applications (see for example [6, 18]).
+As each of the discrete unknowns lie in a different B-spline space, each of the governing
+equations will be collocated at a different set of Greville points. In particular, for both
+the velocity-pressure formulation and the vorticity-velocity-pressure formulation we use the
+Greville abscissae associated with the basis of the x-velocity (Sk1,k2−1
+α1,α2−1) as collocation points
+for the x-momentum equation, the Greville abscissae associated with the basis of the y-
+velocity (Sk1−1,k2
+α1−1,α2) as collocation points for the y-momentum equation, and the Greville
+abscissae associated with the basis of the pressure (Sk1−1,k2−1
+α1−1,α2−1) as collocation points for the
+continuity equation. The constitutive relation within the three field formulation is collocated
+at the Greville abscissae for the vorticity basis (Sk1,k2
+α1,α2). The left of Figure 1 details an
+example of this construction for the velocity-pressure scheme, while the left of Figure 2
+shows example grids for the vorticity-velocity-pressure scheme.
+4.3. Boundary Condition Enforcement
+The last unspecified aspect of the method is enforcement of the Dirichlet boundary con-
+ditions. The enforcement of the normal boundary condition is done strongly and collocation
+points along a boundary for the velocity component orthogonal to that boundary are re-
+moved, as the boundary condition specifies the value of the solution at these points. This
+is shown on the right of Figure 1 and Figure 2, which depict the same scenarios as their
+counterparts but with normal boundary conditions enforced.
+11
+
+A Second Collocation Attempt for Stokes:
+The Unit Square
+ 
+ˆξi = ξi+1 +…+ξi+p
+p
+Greville abscissae:
+=	First	Momentum	Collocation	Pt.
+=	Second	Momentum	Collocation	Pt.
+=	Continuity	Collocation	Pt.
+k = 2, 4 x 4 elements
+=	Constitutive	 Collocation	Pt.
+(a) Before strong enforcement of normal ve-
+locity boundary conditions
+A Second Collocation Attempt for Stokes:
+The Unit Square
+k = 2, 4 x 4 elements
+Boundary	Conditions:
+We	enforce	no-penetration BCs strongly	and
+remove	the	corresponding	collocation	points.
+We	enforce	no-slip BCs weakly	by	modifying
+the	constitutive	equations	at	the	boundary	
+with	a	suitable	penalty	term.
+ 
+ˆω ˆx, ˆy
+(
+)− curl
+! ˆu ˆx, ˆy
+(
+)
+(
+)
+(
+)
+         + Cpen
+h
+ˆu⋅ ˆs − ˆuBC ⋅ ˆs
+(
+) = 0
+An	exact	expression	for	the	penalty	
+constant	is	obtained	by	appealing	to	
+isogeometric finite	volumes.
+(b) After strong enforcement of normal ve-
+locity boundary conditions
+Figure 2: Example of collocation grid for k′ = 2, 4 x 4 elements for the vorticity-velocity-pressure scheme.
+Horizontal triangles represent the collocation points for the first momentum equation, vertical triangles
+represent the points for the second momentum equation, squares are collocation points for the continuity
+equation, and circles are the points for the constitutive relation.
+Enforcement of the tangential boundary condition is slightly more subtle. Recall that
+in Equation (42) we utilized Nitsche’s method to enforce this boundary condition. This
+motivates the enforcement in the velocity-pressure collocation scheme. Indeed if we take this
+equation and undo the integration by parts, the consistency term vanishes by construction
+and we are left with just the penalty terms.
+If we approximate the integral of the test
+function as done in [41] the collocated momentum equations will be of the form
+− ν∆uh + uh · ∇uh + ∇ph + C2
+pen
+h2 (uh − g) = f,
+(49)
+where Cpen is a penalty constant and h is the Greville mesh size perpendicular to the bound-
+ary. Note that because this construction is used to only enforce the tangential boundary
+conditions, this penalty term only appears in the equation for the momentum balance along
+the tangential direction of each boundary.
+In the vorticity-velocity-pressure scheme we do not use the same Nitsche-based approach.
+The method utilized here is directly inspired by the Enhanced Collocation method for en-
+forcing Neumann boundary conditions in isogeometric collocation schemes [41].
+We start by considering the weak form of the constitutive relation given by Equation
+(47). The final term on the left hand side is the boundary term which would be used to
+enforce natural boundary conditions in a Galerkin method. In a similar vein to the Enhanced
+Collocation approach, we can undo the integration by parts to arrive at
+�
+Ω
+ψh(ωh − (∂uh
+y
+∂x − ∂uh
+x
+∂y ))dΩ +
+�
+∂Ω
+ψh(uh · s − g · s)ds = 0.
+(50)
+By approximating the integrals of the test functions as done in [41, 38] we arrive at a
+modified strong form statement of the constitutive relation which can be collocated along
+12
+
+the boundaries
+ωh − (∂uh
+y
+∂x − ∂uh
+x
+∂y ) + Cpen
+h (uh · s − g · s) = 0,
+(51)
+where again Cpen is a penalty constant and h is the Greville mesh size perpendicular to the
+boundary.
+4.4. Final Collocated Equations
+Finally, the results of the previous sections are collected and we present the final form of
+the discrete equations used to solve for the discrete unknowns. The velocity-pressure scheme
+is considered first. Let us define τ ux
+ℓ
+for ℓ = 1, ..., M ux to be the set of Greville points for
+Sk1,k2−1
+α1,α2−1 with the points corresponding to no-penetration boundaries removed as discussed
+previously. Define in a similar manner τ uy
+ℓ
+for ℓ = 1, ..., M uy, which are the Greville points
+of Sk1−1,k2
+α1−1,α2 with no-penetration boundary points removed. Lastly, τ p
+ℓ for k = 1, ..., N p are
+the Greville points of Qh. Then the discrete 2D problem reads:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Find uh ∈ Vh,g and P h ∈ Qh,0 such that:
+�
+−ν ∂2uh
+x
+∂x2 − ν ∂2uh
+x
+∂y2 + uh
+x
+∂uh
+x
+∂x + uh
+y
+∂uh
+x
+∂y + ∂ph
+∂x
+�
+(τ ux
+ℓ ) = fx(τ ux
+ℓ )
+∀τ ux
+ℓ
+∈ Ω
+(52)
+�
+−ν ∂2uh
+x
+∂x2 − ν ∂2uh
+x
+∂y2 + uh
+x
+∂uh
+x
+∂x + uh
+y
+∂uh
+x
+∂y + ∂ph
+∂x + C2
+pen
+h2 (uh
+x − gx)
+�
+(τ ux
+ℓ )
+= fx(τ ux
+ℓ )
+∀τ ux
+ℓ
+∈ ∂Ω
+(53)
+�
+−ν ∂2uh
+y
+∂x2 − ν ∂2uh
+y
+∂y2 + uh
+x
+∂uh
+y
+∂x + uh
+y
+∂uh
+y
+∂y + ∂ph
+∂y
+�
+(τ uy
+ℓ ) = fy(τ uy
+ℓ )
+∀τ uy
+ℓ
+∈ Ω
+(54)
+�
+−ν ∂2uh
+y
+∂x2 − ν ∂2uh
+y
+∂y2 + uh
+x
+∂uh
+y
+∂x + uh
+y
+∂uh
+y
+∂y + ∂ph
+∂y + C2
+pen
+h2 (uh
+y − gy)
+�
+(τ uy
+ℓ )
+= fy(τ uy
+ℓ )
+∀τ uy
+ℓ
+∈ ∂Ω
+(55)
+�
+∂uh
+x
+∂x + ∂uh
+y
+∂y
+�
+(τ p
+ℓ) = 0
+∀τ p
+ℓ ∈ Ω ∪ ∂Ω.
+(56)
+In the above we have split the momentum equations into expressions valid on the interior
+collocation points (Equations (52) and (54)) and expressions valid on the remaining boundary
+collocation points (Equations (53) and (55)).
+Similarly, for the vorticity-velocity-pressure scheme we also define τ ω
+ℓ for ℓ = 1, ..., N ω as
+the Greville points of Ψh. With this scheme the discrete 2D problem reads:
+13
+
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Find uh ∈ Vh,g, P h ∈ Qh,0, and ωh ∈ Ψh such that:
+�
+ν ∂ωh
+∂y − ωhuh
+y + ∂P h
+∂x
+�
+(τ ux
+ℓ ) = fx(τ ux
+ℓ )
+∀τ ux
+ℓ
+∈ Ω ∪ ∂Ω
+(57)
+�
+−ν ∂ωh
+∂x + ωhuh
+x + ∂P h
+∂y
+�
+(τ uy
+ℓ ) = fy(τ uy
+ℓ )
+∀τ uy
+ℓ
+∈ Ω ∪ ∂Ω
+(58)
+�
+∂uh
+x
+∂x + ∂uh
+y
+∂y
+�
+(τ p
+ℓ) = 0
+∀τ p
+ℓ ∈ Ω ∪ ∂Ω
+(59)
+�
+ωh − ∂uh
+y
+∂x − ∂uh
+x
+∂y
+�
+(τ ω
+ℓ ) = 0
+∀τ ω
+ℓ ∈ Ω
+(60)
+�
+ωh − ∂uh
+y
+∂x − ∂uh
+x
+∂y + Cpen
+h (uh · s − g · s)
+�
+(τ ω
+ℓ ) = 0
+∀τ ω
+ℓ ∈ ∂Ω.
+(61)
+In the three field formulation we split the constitutive law into an expression for the
+interior collocation points (Equation (60)) and another expression for boundary collocation
+points (Equation (61)).
+Resulting from these equations are nonlinear systems of equations which we can use to
+solve for the unknown coefficients of velocity, pressure, and vorticity using a Newton-Raphson
+method.
+4.5. Proof of Divergence Conforming Property
+From the commuting diagrams our spaces form, shown in Equations (30) and (31), it is
+simple to show that both of the resulting collocation methods return an exactly pointwise
+divergence free velocity field. The commuting diagrams reveal that the divergence of the
+discrete velocity lies within the discrete pressure space, ∇·uh ∈ Qh. We can thus equivalently
+write the divergence of the velocity as a linear combination of the pressure basis functions:
+∇ · uh =
+Np
+�
+i=0
+ciqi,
+(62)
+where qi ∈ Qh are the basis functions for the pressure and ci ∈ R are the coefficients. As
+part of the collocation scheme, we strongly enforce that the velocity field has zero divergence
+at a number of collocation points equal to the dimension of the discrete pressure space. This
+condition can be written as a system of linear equations
+Mc = 0,
+(63)
+where M is a square matrix whose entries are the pressure basis functions evaluated at each
+collocation point and c is the vector of coefficients.
+14
+
+If the choice of collocation points yields a set of linearly independent equations, that is
+to say M is invertible, then we know that the solution to Equation (63) is c = 0, and thus
+the velocity field is exactly divergence free pointwise.
+5. Numerical Results on Square Domains
+The developed schemes are now tested on multiple 2D problems on the unit square. First,
+a manufactured vortex problem is considered to experimentally compute the convergence
+rates of the error and test for pressure and Reynolds number robustness. Then, the classical
+lid-driven cavity problem is considered at a variety of Reynolds numbers.
+5.1. Two-Dimensional Manufactured Solution
+As a first numerical experiment, we consider a vortex problem on the unit square con-
+structed using the method of manufactured solutions. This solution was originally developed
+in [21] and employs the following velocity and pressure fields:
+˜u =
+�
+2ex(−1 + x)2x2(y2 − y)(−1 + 2y)
+(−ex(−1 + x)x(−2 + x(3 + x))(−1 + y)2y2)
+�
+,
+(64)
+˜p
+=
+(−424 + 156e + (y2 − y)(−456 + ex(456 + x2(228 − 5(y2 − y))+
+2x(−228 + (y2 − y)) + 2x3(−36 + (y2 − y)) + x4(12 + (y2 − y))))).
+(65)
+For the velocity-pressure scheme we define the forcing term to be
+f = −ν∆˜u + ˜u · ∇˜u + ∇˜p,
+(66)
+while for the vorticity-velocity-pressure formulation we define the forcing term in the mo-
+mentum equations to be:
+f = ν∇⊥˜ω + ˜ω × ˜u + ∇ ˜P,
+(67)
+with ˜ω = ∇ × ˜u and ˜P = ˜p + 1
+2(˜u · ˜u). Enforcing homogeneous boundary conditions and
+requiring that the integral of pressure is zero, it is clear to see that the velocity and kinematic
+pressure solutions should be equal to ˜u and ˜p.
+To understand the accuracy of this collocation method, we test the convergence rates on
+a variety of grids and with different degree B-spline bases. For this test we set the Reynolds
+number to Re = 1
+ν = 1. We measure error using the L2 norm as well as the H1 semi-norm.
+Figure 3 details the convergence rates of velocity and pressure when using the two field
+formulation. In this case the errors in both velocity and pressure converge at a rate of k′
+when k′ is even and k′ − 1 for odd k′. These are the standard, expected rates that have been
+seen in other studies of isogeometric collocation, and are one and two orders suboptimal in
+L2 for odd and even k′.
+Figure 4 details the convergence of velocity, kinematic pressure, and vorticity as we refine
+the meshes in the three field scheme. Using this scheme, all of the unknowns converge in the
+L2 norm at a rate of approximately k′ for even values of k′ and at a rate of k′ + 1 for odd
+values of k′. These rates match the rates achieved using even k′ in the two field formulation,
+15
+
+10-2
+10-1
+10-10
+10-5
+(a) Velocity L2 error
+10-2
+10-1
+10-10
+10-5
+(b) Velocity H1 error
+10-2
+10-1
+10-10
+10-5
+(c) Pressure L2 error
+10-2
+10-1
+10-10
+10-5
+100
+(d) Pressure H1 error
+Figure 3: Errors in 2D manufactured vortex solution for velocity-pressure formulation
+16
+
+and these rates are two orders faster for odd k′. In fact, this formulation recovers optimal
+convergence rates in the L2 norm for odd k′.
+In the H1 semi-norm, we see convergence rates of k′ for all polynomial degrees for the
+velocity and pressure. These rates are optimal in the H1 semi-norm for all degrees and again
+are as fast or better than the corresponding velocity-pressure scheme results. Interestingly,
+the H1 convergence of vorticity seems to be at a rate of k′ + 1 for odd k′ and a rate of k′ for
+even values.
+To further study our new collocation schemes, we can also directly compare the errors
+produced with divergence-conforming Galerkin schemes of the same orders. Figure 5 shows
+the L2 norm and H1 semi-norm errors in velocity as well as the L2 errors in pressure for both
+collocation schemes along with the Galerkin results for the same problem from [22]. This
+comparison highlights the severe suboptimality of the velocity-pressure results with odd k′.
+We also note that the H1 errors obtained with the three field formulation nearly match the
+Galerkin results.
+5.2. Pressure Robustness
+Next we perform some ancillary tests related to the manufactured solution to test some
+secondary robustness properties of the method. The first test relates to so-called pressure
+robustness [24].
+In particular, we take the kinematic pressure ˜p from the manufactured
+solution and multiply it by a scalar σ. Thus the pressure term in the forcing function f
+will also be multiplied by σ, and the exact solution to which our numerical solution should
+converge has the same velocity as before but with a scaled kinematic pressure field.
+For a pressure robust method this increase in the pressure magnitude, and thus the
+pressure approximation errors, will not affect the velocity approximation error. Conversely,
+a non-pressure robust method will see its velocity errors increase as the pressure is scaled
+larger. Figure 6 shows the convergence of the velocity errors for the two field scheme with
+k′ = 2 and increasing values of the scalar σ, while Figure 7 shows the same for the three field
+formulation. Clearly the velocity error increases in both schemes as σ increases, meaning
+the method is not pressure robust. This is interesting as the divergence-conforming Galerkin
+method upon which this work is based is pressure robust.
+5.3. Reynolds Robustness
+Similar to pressure robustness, we also want to test how the errors in the solution behave
+as the Reynolds number is increased. We increase the Reynolds number by decreasing the
+viscosity ν. This affects the viscous term in the forcing vector f, but the exact solution to
+the problem is identical to the original manufactured solution.
+Figures 8 and 9 detail the convergence of the velocity errors as the Reynolds number
+increases, again for k′ = 2, in the two and three field schemes. Once again, the error in the
+velocity field increases as we increase the Reynolds number, in contrast to the divergence-
+conforming Galerkin setting, where the velocity error is agnostic to increasing Reynolds
+number [22].
+5.4. Two-Dimensional Lid-Driven Cavity Flow
+The next 2D numerical test problem that we consider is the square lid-driven cavity flow.
+The left, right, and bottom walls of the cavity remain fixed while the top wall slides in the
+17
+
+10-2
+10-1
+10-10
+10-5
+(a) Velocity L2 error
+10-2
+10-1
+10-10
+10-5
+100
+(b) Velocity H1 error
+10-2
+10-1
+10-10
+10-5
+100
+(c) Pressure L2 error
+10-2
+10-1
+10-8
+10-6
+10-4
+10-2
+100
+(d) Pressure H1 error
+10-2
+10-1
+10-10
+10-5
+(e) Vorticity L2 error
+10-2
+10-1
+10-8
+10-6
+10-4
+10-2
+100
+(f) Vorticity H1 error
+Figure 4: Errors in 2D manufactured vortex solution for vorticity-velocity-pressure formulation
+18
+
+10-2
+10-1
+10-10
+10-9
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+(a) Velocity L2 error
+10-2
+10-1
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+(b) Velocity H1 error
+10-2
+10-1
+10-9
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+(c) Pressure L2 error
+Figure 5: Errors in 2D manufactured vortex solution comparison
+19
+
+10-2
+10-1
+10-6
+10-4
+10-2
+(a) Velocity L2 error
+10-2
+10-1
+10-5
+100
+(b) Velocity H1 error
+Figure 6: Errors in 2D manufactured vortex solution with varying pressure scaling for velocity-pressure
+formulation
+10-2
+10-1
+10-6
+10-4
+10-2
+(a) Velocity L2 error
+10-2
+10-1
+10-5
+100
+(b) Velocity H1 error
+Figure 7: Errors in 2D manufactured vortex solution with varying pressure scaling for vorticity-velocity-
+pressure formulation
+20
+
+10-2
+10-1
+10-6
+10-4
+10-2
+(a) Velocity L2 error
+10-2
+10-1
+10-5
+100
+(b) Velocity H1 error
+Figure 8: Errors in 2D manufactured vortex solution with varying Reynolds number for velocity-pressure
+formulation
+10-2
+10-1
+10-6
+10-4
+10-2
+(a) Velocity L2 error
+10-2
+10-1
+10-5
+100
+(b) Velocity H1 error
+Figure 9: Errors in 2D manufactured vortex solution with varying Reynolds number for vorticity-velocity-
+pressure formulation
+21
+
+positive x direction, causing vortices to develop within the domain. Due to the inconsistency
+in the boundary conditions, pressure singularities exist in the corners of the domain, making
+this a challenging test case for a numerical scheme to properly capture.
+For our simulations, we set both the speed of the top wall U = 1 and the wall lengths
+H = 1. The kinematic viscosity ν defines the Reynolds number through Re = UH
+ν
+= 1
+ν. In
+particular, we consider the flows produced with Re = 100, Re = 400, and Re = 1000. To
+validate our results, we compare the centerline velocity profiles at each Reynolds number
+with the results from Ghia et al [42].
+Figure 10 details the two field formulation results across the three considered Reynolds
+numbers and two mesh sizes: a 32 element stretched mesh and a 64 element stretched mesh.
+The stretched mesh is formed by setting the interior knots of the knot vectors defining the
+bases in each direction as
+ξi = 1
+2
+�
+1 + tanh(4ih − 2)
+tanh(2)
+�
+∀ξi ∈ Ξ,
+(68)
+where h is the mesh size in each direction. Figure 11 shows the same results for the three field
+formulation. The collocation results from both schemes agree very well with the reference
+data in all cases, and we see that the results are converging with increasing resolution. At a
+Reynolds number of 100, all of our results show that the maximum and minimum values of
+the vertical velocity are larger in magnitude than those of Ghia et al. This is similar to the
+behavior seen in the Galerkin method [22], and we note that there are some inaccuracies in
+the Ghia data for this low Reynolds number case [22, 43]. For Reynolds number 400, the
+two field formulation predicts extrema in velocity that are slightly smaller than the three
+field predictions, which match the corresponding Galerkin results very well. This trend is
+also valid at a Reynolds number of 1000. Moreover, while we have used stretched meshes
+here, the results with a non-stretched mesh are similar.
+As a more quantitative comparison, we compute the minimum horizontal velocity along
+the vertical centerline as well as the maximum and minimum vertical velocities along the
+horizontal centerline for each of simulations presented above. These results are shown for a
+Reynolds number of 100 in Table 1, along with the values from [42] and [17]. These results
+show the inadequacy of the Ghia results at this Reynolds number, and for the most part the
+k′ = 2 collocation results outperform the Ghia data when compared to the pseudospectral
+results. To highlight the potential possibilities of the collocation methods, we also compute
+results using an unstretched mesh of 8 elements in each direction and k′ = 20 for both the
+two and three field formulations. While this would be essentially infeasible with a Galerkin
+method, as the quadrature would be prohibitively expensive, it is handled with ease by the
+collocation schemes. We see that these results match the pseudospectral results, even on the
+utilized coarse meshes.
+6. Collocation Methods on Cubic Domains
+The previous two sections detailed the construction of the divergence-conforming colloca-
+tion methods in 2D and tested their behavior numerically. In the following, we will highlight
+the required modifications to the methods to solve problems in 3D cubic domains.
+22
+
+0
+0.2
+0.4
+0.6
+0.8
+1
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+-0.5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(a) Re = 100 velocities with h = 1/32
+0
+0.2
+0.4
+0.6
+0.8
+1
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+-0.5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(b) Re = 100 velocities with h = 1/64
+0
+0.2
+0.4
+0.6
+0.8
+1
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+-0.5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(c) Re = 400 velocities with h = 1/32
+0
+0.2
+0.4
+0.6
+0.8
+1
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+-0.5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(d) Re = 400 velocities with h = 1/64
+0
+0.2
+0.4
+0.6
+0.8
+1
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+-0.5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(e) Re = 1000 velocities with h = 1/32
+0
+0.2
+0.4
+0.6
+0.8
+1
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+-0.5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(f) Re = 1000 velocities with h = 1/64
+Figure 10: Centerline velocity profiles for 2D lid-driven cavity with velocity-pressure formulation, k′ = 2.
+Red curves and axes represent the vertical velocity along the horizontal centerline, while blue curves and
+axes represent the horizontal velocity along the vertical centerline.
+23
+
+0
+0.2
+0.4
+0.6
+0.8
+1
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+-0.5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(a) Re = 100 velocities with h = 1/32
+0
+0.2
+0.4
+0.6
+0.8
+1
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+-0.5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(b) Re = 100 velocities with h = 1/64
+0
+0.2
+0.4
+0.6
+0.8
+1
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+-0.5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(c) Re = 400 velocities with h = 1/32
+0
+0.2
+0.4
+0.6
+0.8
+1
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+-0.5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(d) Re = 400 velocities with h = 1/64
+0
+0.2
+0.4
+0.6
+0.8
+1
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+-0.5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(e) Re = 1000 velocities with h = 1/32
+0
+0.2
+0.4
+0.6
+0.8
+1
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+-0.5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(f) Re = 1000 velocities with h = 1/64
+Figure 11: Centerline velocity profiles for 2D lid-driven cavity with vorticity-velocity-pressure formulation,
+k′ = 2. Red curves and axes represent the vertical velocity along the horizontal centerline, while blue curves
+and axes represent the horizontal velocity along the vertical centerline.
+24
+
+Table 1: Velocity extrema for 2D lid-driven cavity at Re = 100
+Method
+ux,min
+uy,max
+uy,min
+Collocation, 2 field formulation, k′ = 2 and h = 1/32
+−0.21348
+0.17941
+−0.25307
+Collocation, 2 field formulation, k′ = 2 and h = 1/64
+−0.21389
+0.17953
+−0.25358
+Collocation, 2 field formulation, k′ = 20 and h = 1/8
+−0.21404
+0.17957
+−0.25380
+Collocation, 3 field formulation, k′ = 2 and h = 1/32
+−0.21800
+0.18392
+−0.25908
+Collocation, 3 field formulation, k′ = 2 and h = 1/64
+−0.21511
+0.18075
+−0.25521
+Collocation, 3 field formulation, k′ = 20 and h = 1/8
+−0.21404
+0.17957
+−0.25380
+Pseudospectral (Ref. [43])
+−0.21404
+0.17957
+−0.25380
+Ghia et al. (Ref. [42])
+−0.21090
+0.17527
+−0.24533
+6.1. Review of Galerkin Methods
+Similar to 2D, we start by reviewing the form of the divergence-conforming isogeometric
+Galerkin methods for 3D problems. Again assume the velocity is subject to Dirichlet bound-
+ary conditions along the entire boundary. We then define the discrete test and trial function
+spaces for velocity as Vh,0 and Vh,g, which are defined as the same Vh from Equation (37)
+with either no penetration boundary conditions strongly enforced (for the test space Vh,0)
+or with the normal velocity prescribed at collocation points as given by the boundary data g
+(for the trial space Vh,g). We also define the test and trial space for pressure as Qh,0, where
+Qh is the same space as in Equation (38) but with the added condition that the pressure
+must have zero integral. Then the Galerkin formulation for the velocity-pressure form would
+read
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Given ν ∈ R+, f : Ω → R3, and g : ∂Ω → R3, find uh ∈ Vh,g and ph ∈ Qh,0 such that,
+∀(wh, qh) ∈ (Vh,0, Qh,0):
+�
+Ω
+(ν∇wh · ∇uh + wh · (uh · ∇uh) − ph∇ · wh)dΩ
+− ν
+�
+∂Ω
+(∇uh · n) · wh − Cpen
+h uh · whdA =
+�
+Ω
+wh · fdΩ + ν
+�
+∂Ω
+Cpen
+h g · whdA
+(69)
+�
+Ω
+qh(∇ · uh)dΩ = 0.
+(70)
+This weak form is essentially unchanged from the 2D case, with the only major difference
+being that the velocity has 3 components. The vorticity-velocity-pressure Galerkin form,
+however, is more different. In this case the discrete problem reads
+25
+
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Given ν ∈ R+, f : Ω → R3, and g : ∂Ω → R3, find uh ∈ Vh,g, P h ∈ Qh,0, and
+ωh ∈ Ψh such that, ∀(wh, qh, ψh) ∈ (Vh,0, Qh,0, Ψh):
+�
+Ω
+(ν∇ × ωh) · vhdΩ +
+�
+Ω
+(ωh × uh) · vhdΩ −
+�
+Ω
+P h(∇ · vh)dΩ =
+�
+Ω
+f · vhdΩ
+(71)
+�
+Ω
+(∇ · uh)qhdΩ = 0
+(72)
+�
+Ω
+(ωh · ψh)dΩ +
+�
+Ω
+uh · (∇ × ψh)dΩ −
+�
+∂Ω
+(ψh × g) · ndA = 0.
+(73)
+Again the no-slip velocity boundary conditions appear as natural boundary conditions in
+the weak form of the constitutive equation.
+Within the collocation schemes, the unknowns are selected to reside in the same spaces as
+the corresponding Galerkin scheme, as in 2D. In the following we highlight the main changes
+to the method for 3D problems with regards to the choice of collocation grids and boundary
+condition enforcement before again summarizing the final form of the discrete equations.
+6.2. Collocation Grids
+Much like the two dimensional case, in 3D we choose to collocate at Greville abscissae
+and the grids are different for each of the governing equations. For both formulations the
+schemes for the momentum and pressure equations are essentially unchanged; each momen-
+tum equation component is collocated at the Greville abscissae of the corresponding discrete
+velocity component space, and the continuity equation is collocated at the Greville abscissae
+of the discrete pressure space. Thus the velocity-pressure formulation extends fairly trivially
+to 3D.
+The constitutive equation in the vorticity-velocity-pressure formulation, on the other
+hand, is now split into components much like how the momentum equations are treated.
+We choose to collocate the x component of the constitutive equation at the Greville ab-
+scissae associated with the discrete x vorticity space (Sk1−1,k2,k3
+α1−1,α2,α3), the y component of the
+constitutive equation at the Greville abscissae associated with the discrete y vorticity space
+(Sk1,k2−1,k3
+α1,α2−1,α3), and the z component of the constitutive equation at the Greville abscissae
+associated with the discrete z vorticity space (Sk1,k2,k3−1
+α1,α2,α3−1).
+6.3. Boundary Condition Enforcement
+The no-penetration boundary condition is enforced identically to 2D case: We strongly
+enforce the normal velocity on face collocation points corresponding to the normal velocity
+component and remove these points from the set used to collocate the momentum equations.
+The no-slip boundary condition in the velocity-pressure scheme is also essentially enforced
+identically to the 2D case and again leads to equations of the form
+26
+
+− ν∆uh + uh · ∇uh + ∇ph + C2
+pen
+h2 (uh − g) = f.
+(74)
+As the constitutive law relating velocity and vorticity is a vector relation in 3D, the weak
+enforcement of no-slip boundary conditions is slightly altered in the three field formulation.
+We again start by considering the weak form shown above, particularly Equation (73). The
+last term in this equation represents the boundary term which would be used to enforce
+boundary conditions by replacing terms with their prescribed values. Following the Enhanced
+Collocation method of [41], the equation can be integrated by parts once again, to arrive at
+a strong form representation given by:
+�
+Ω
+ψ · (ω − ∇ × u)dΩ +
+�
+∂Ω
+(ψ × u − ψ × g) · ndA = 0.
+Using the properties of the scalar triple product, we can re-write this as:
+�
+Ω
+ψ · (ω − ∇ × u)dΩ +
+�
+∂Ω
+(u × n − g × n) · ψdA = 0.
+By approximating these integrals as is done in [41, 38], we arrive at a strong form state-
+ment including boundary conditions suitable for collocation:
+ωh − ∇ × uh + Cpen
+h (uh × n − g × n) = 0.
+(75)
+6.4. Final Collocated Equations
+Once again the entire collocation scheme based on the velocity-pressure formulation is
+summarized first. Let us again define τ ux
+ℓ
+for ℓ = 1, ..., M ux to be the set of Greville points
+for the basis of the x velocity component (Sk1,k2−1,k3−1
+α1,α2−1,α3−1) with the points corresponding to
+no-penetration boundaries removed as discussed previously. Define in a similar manner τ uy
+ℓ
+for ℓ = 1, ..., M uy and τ uz
+ℓ
+for ℓ = 1, ..., M uz. The pressure Greville points are defined as τ p
+ℓ
+for ℓ = 1, ..., N p. For this formulation the discrete 3D problem reads:
+27
+
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
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+�
+�
+�
+�
+�
+�
+�
+�
+�
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+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
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+�
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+�
+�
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+�
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+�
+�
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+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Find uh ∈ Vh,g and P h ∈ Qh,0 such that:
+�
+−ν ∂2uh
+x
+∂x2 − ν ∂2uh
+x
+∂y2 − ν ∂2uh
+x
+∂z2 + uh
+x
+∂uh
+x
+∂x + uh
+y
+∂uh
+x
+∂y + uh
+z
+∂uh
+x
+∂z + ∂ph
+∂x
+�
+(τ ux
+ℓ )
+= fx(τ ux
+ℓ )
+∀τ ux
+ℓ
+∈ Ω
+(76)
+�
+−ν ∂2uh
+x
+∂x2 − ν ∂2uh
+x
+∂y2 − ν ∂2uh
+x
+∂z2 + uh
+x
+∂uh
+x
+∂x + uh
+y
+∂uh
+x
+∂y + uh
+z
+∂uh
+x
+∂z + ∂ph
+∂x +
+C2
+pen
+h2 (uh
+x − gx)
+�
+(τ ux
+ℓ ) = fx(τ ux
+ℓ )
+∀τ ux
+ℓ
+∈ ∂Ω
+(77)
+�
+−ν ∂2uh
+y
+∂x2 − ν ∂2uh
+y
+∂y2 − ν ∂2uh
+y
+∂z2 + uh
+x
+∂uh
+y
+∂x + uh
+y
+∂uh
+y
+∂y + uh
+z
+∂uh
+y
+∂z + ∂ph
+∂y
+�
+(τ uy
+ℓ )
+= fy(τ uy
+ℓ )
+∀τ uy
+ℓ
+∈ Ω
+(78)
+�
+−ν ∂2uh
+y
+∂x2 − ν ∂2uh
+y
+∂y2 − ν ∂2uh
+y
+∂z2 + uh
+x
+∂uh
+y
+∂x + uh
+y
+∂uh
+y
+∂y + uh
+z
+∂uh
+y
+∂z + ∂ph
+∂y +
+C2
+pen
+h2 (uh
+y − gy)
+�
+(τ uy
+ℓ ) = fy(τ uy
+ℓ )
+∀τ uy
+ℓ
+∈ ∂Ω
+(79)
+�
+−ν ∂2uh
+z
+∂x2 − ν ∂2uh
+z
+∂y2 − ν ∂2uh
+z
+∂z2 + uh
+x
+∂uh
+z
+∂x + uh
+y
+∂uh
+z
+∂y + uh
+z
+∂uh
+z
+∂z + ∂ph
+∂z
+�
+(τ uz
+ℓ )
+= fz(τ uz
+ℓ )
+∀τ uz
+ℓ ∈ Ω
+(80)
+�
+−ν ∂2uh
+z
+∂x2 − ν ∂2uh
+z
+∂y2 − ν ∂2uh
+z
+∂z2 + uh
+x
+∂uh
+z
+∂x + uh
+y
+∂uh
+z
+∂y + uh
+z
+∂uh
+z
+∂z + ∂ph
+∂z +
+C2
+pen
+h2 (uh
+z − gz)
+�
+(τ uz
+ℓ ) = fz(τ uz
+ℓ )
+∀τ uz
+ℓ ∈ ∂Ω
+(81)
+�
+∂uh
+x
+∂x + ∂uh
+y
+∂y + ∂uh
+z
+∂z
+�
+(τ p
+ℓ) = 0
+∀τ p
+ℓ ∈ Ω ∪ ∂Ω.
+(82)
+Similarly to the velocity, in the three field formulation we also define collocation points
+for the vorticity component-wise. In particular, let τ ωx
+ℓ
+for ℓ = 1, ..., N ωx be the Greville
+points for the x component of the vorticity, and define τ ωy
+ℓ
+for ℓ = 1, ..., N ωy and τ ωz
+ℓ
+for
+ℓ = 1, ..., N ωz similarly. The final, discrete 3D problem for the vorticity-velocity-pressure
+collocation scheme reads as:
+28
+
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Find uh ∈ Vh,g, P h ∈ Qh,0, and ωh ∈ Ψh such that:
+�
+ν(∂ωh
+z
+∂y − ∂ωh
+y
+∂z ) + ωh
+yuh
+z − ωh
+z uh
+y + ∂P h
+∂x
+�
+(τ ux
+ℓ ) = fx(τ ux
+ℓ )
+∀τ ux
+ℓ
+∈ Ω ∪ ∂Ω
+(83)
+�
+ν(∂ωh
+x
+∂z − ∂ωh
+z
+∂x ) + ωh
+z uh
+x − ωh
+xuh
+z + ∂P h
+∂y
+�
+(τ uy
+ℓ ) = fy(τ uy
+ℓ )
+∀τ uy
+ℓ
+∈ Ω ∪ ∂Ω
+(84)
+�
+ν(∂ωh
+y
+∂x − ∂ωh
+x
+∂y + ωh
+xuh
+y − ωh
+yuh
+x + ∂P h
+∂z
+�
+(τ uz
+ℓ ) = fz(τ uz
+ℓ )
+∀τ uz
+ℓ ∈ Ω ∪ ∂Ω
+(85)
+�
+∂uh
+x
+∂x + ∂uh
+y
+∂y + ∂uh
+z
+∂z
+�
+(τ p
+ℓ) = 0
+∀τ p
+ℓ ∈ Ω ∪ ∂Ω
+(86)
+�
+ωh
+x − (∂uh
+z
+∂y − ∂uh
+y
+∂z )
+�
+(τ ωx
+ℓ ) = 0
+∀τ ωx
+ℓ
+∈ Ω
+(87)
+�
+ωh
+x − (∂uh
+z
+∂y − ∂uh
+y
+∂z )+
+Cpen
+h ((uh
+y − gy)nz − (uh
+z − gz)ny)
+�
+(τ ωx
+ℓ ) = 0
+∀τ ωx
+ℓ
+∈ ∂Ω
+(88)
+�
+ωh
+y − (∂uh
+x
+∂z − ∂uh
+z
+∂x )
+�
+(τ ωy
+ℓ ) = 0
+∀τ ωy
+ℓ
+∈ Ω
+(89)
+�
+ωh
+y − (∂uh
+x
+∂z − ∂uh
+z
+∂x )+
+Cpen
+h ((uh
+z − gz)nx − (uh
+x − gx)nz)
+�
+(τ ωy
+ℓ ) = 0
+∀τ ωy
+ℓ
+∈ ∂Ω
+(90)
+�
+ωh
+z − (∂uh
+y
+∂x − ∂uh
+x
+∂y )
+�
+(τ ωz
+ℓ ) = 0
+∀τ ωz
+ℓ
+∈ Ω
+(91)
+�
+ωh
+z − (∂uh
+y
+∂x − ∂uh
+x
+∂y )+
+Cpen
+h ((uh
+x − gx)ny − (uh
+y − gy)nx)
+�
+(τ ωz
+ℓ ) = 0
+∀τ ωz
+ℓ
+∈ ∂Ω.
+(92)
+7. Numerical Results on Cubic Domains
+To verify that the schemes properly extend into 3D, two sample problems are considered.
+First, a manufactured solution gives even more insight into the convergence properties of
+29
+
+the methods. Then the three-dimensional lid-driven cavity problem is considered and the
+results are compared with established literature.
+7.1. Three-Dimensional Manufactured Solution
+In 3D, we also start our numerical studies by considering a manufactured solution. In
+this case, the exact solution represents the flow around a single vortex filament within the
+unit cube. We define a potential function as
+˜φ =
+�
+�
+x(x − 1)y2(y − 1)2z2(z − 1)2
+0
+x2(x − 1)2y2(y − 1)2z(z − 1)
+�
+� ,
+(93)
+through which we can define the velocity field as
+˜u = ∇ × ˜φ,
+(94)
+and the vorticity as
+˜ω = ∇ × ˜u.
+(95)
+Finally, we specify the pressure field as
+˜p = sin(πx) sin(πy) − 4
+π2.
+(96)
+For the velocity-pressure scheme we define the forcing term on the right hand sign of the
+momentum equations as
+f = −ν∆˜u + ˜u · ∇˜u + ∇˜p,
+(97)
+while for the vorticity-velocity-pressure scheme the forcing term is given by
+f = −ν∆˜u + ˜ω × ˜u + ∇ ˜P.
+(98)
+Once again we enforce homogeneous Dirichlet boundary conditions everywhere and require
+that the kinematic pressure field has zero average. With these conditions the discrete solution
+should again converge to the quantities above with mesh refinement.
+Similar to the 2D case, we set Re = 1
+ν = 1 and measure the errors produced on a variety of
+grids in the L2 norm and H1 semi-norm. Figure 12 shows the results for the velocity-pressure
+scheme while Figure 13 details the errors for the vorticity-velocity-pressure scheme.
+We start by noting that when k′ < 3 in both cases, everything behaves in the same
+manner as in the 2D setting. Once k′ ≥ 3 we start to see very fast convergence rates and
+some pre-asymptotic type behavior in the velocity errors produced by both schemes. This
+can be explained by talking a closer look at the exact velocity field for this problem. In fact,
+the exact velocity field is given by a quartic polynomial in each direction and this solution
+is actually contained within the discrete velocity approximation space for k′ ≥ 3. If we
+were using a pressure robust Galerkin method, the velocity error would be zero. Since the
+collocation scheme is not pressure robust, we obtain superconvergence rather than exactly
+zero error.
+30
+
+10-1
+10-10
+10-5
+(a) Velocity L2 error
+10-1
+10-10
+10-5
+(b) Velocity H1 error
+10-1
+10-8
+10-6
+10-4
+10-2
+(c) Pressure L2 error
+10-1
+10-6
+10-4
+10-2
+100
+(d) Pressure H1 error
+Figure 12: Errors in 3D manufactured vortex solution for velocity-pressure formulation
+31
+
+10-1
+10-10
+10-5
+(a) Velocity L2 error
+10-1
+10-10
+10-5
+100
+(b) Velocity H1 error
+10-1
+10-8
+10-6
+10-4
+10-2
+100
+(c) Pressure L2 error
+10-1
+10-6
+10-4
+10-2
+100
+(d) Pressure H1 error
+10-1
+10-10
+10-5
+100
+(e) Vorticity L2 error
+10-1
+10-10
+10-5
+100
+(f) Vorticity H1 error
+Figure 13: Errors in 3D manufactured vortex solution for vorticity-velocity-pressure formulation
+32
+
+0
+0.2
+0.4
+0.6
+0.8
+1
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+-0.5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(a) Velocity-pressure formulation
+0
+0.2
+0.4
+0.6
+0.8
+1
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+-0.5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(b) Vorticity-velocity-pressure formulation
+Figure 14: Centerline velocity profile for 3D lid-driven cavity using both formulations, k′ = 2. Red curves
+and axes represent the vertical velocity along the horizontal centerline, while blue curves and axes represent
+the horizontal velocity along the vertical centerline.
+The pressure convergence results also show some interesting behavior. While the vorticity-
+velocity-pressure scheme seems to behave in the same manner as in 2D, the velocity-pressure
+scheme seems to be recovering the faster rates seen in the three field scheme. We believe
+that this is a consequence of the superconvergence of velocity.
+7.2. Three-Dimensional Lid-Driven Cavity
+The next numerical study that we perform is on the 3D lid-driven cavity flow. Consider
+again the cavity setup describing the 2D flow, but now extend the square cavity by unit
+length in the out-of-page direction, thus making it a cube. The point singularities of the 2D
+case now extend along the top edges of the cube and we expect to see more influence of 3D
+boundary effects [44].
+In our tests we again set the wall speed U = 1, the side length H = 1, and consider
+Re = UH
+1
+= 100. We use an unstretched mesh with 32 elements per side and k′ = 2, and
+compare the x velocity along the vertical centerline and the y velocity along the horizontal
+centerline with the pseudospectral results from [44]. Figure 14 shows the results with each
+formulation. Once again the results match very well with the literature, and it seems as
+though the results from the three field formulation match with the reference results slightly
+better that the two field results.
+8. Collocation Methods on Mapped Domains
+As the last main component of this paper we shift our focus to problems posed on more
+complicated domains. We will present some theory for both 2D and 3D problems, but for
+simplicity we will focus the development of numerical schemes for the 2D, linear Stokes
+equations. However, the results would generalize to the nonlinear, 3D setting as well. We
+will also focus on the rotational form of the equations, as the first order nature enables easier
+mappings between domains.
+33
+
+The main idea of this section is the mapping back to a parametric reference domain,
+i.e. a square in 2D or a cube in 3D. The previous sections detail how to develop collocation
+schemes on these simple geometries, thus simply pulling the equations and unknowns back to
+the reference domain, collocating as before, and pushing the results forward to the physical
+domain gives our numerical solution.
+Let ˆΩ be the parametric domain (the unit square in 2D or the unit cube in 3D), and let
+Ω be the physical domain. We define the function F as mapping from ˆΩ to Ω. Let DF be
+the Jacobian of the parametric mapping, and define
+J = Det(DF),
+(99)
+C = (DF)T(DF).
+(100)
+Next we can define the pull-back operators in 3D as
+ιΦ(φ) = (φ ◦ F),
+(101)
+ιω(ψ) = (DF)T(ψ ◦ F),
+(102)
+ιu(v) = J(DF)−1(v ◦ F),
+(103)
+ιp(q) = J(q ◦ F).
+(104)
+We define the pulled-back unknowns on the reference domain via the ι maps, specifically
+ˆu = ιu(u), ˆp = ιp(p), and ˆω = ιω(ω). These are the unknowns for which we solve us-
+ing collocation, and the physical domain solution is then obtained via the corresponding
+push-forward. Importantly, the push-forward of velocity as defined above maps divergences
+to divergences and preserves nullity of normal components. Similarly the push-forward of
+pressure preserves the nullity of the integral operator. These facts imply that the following
+commuting diagram exists:
+R −−−→ Φ
+∇
+−−−→ Ψ
+∇×
+−−−→ V
+∇·
+−−−→ Q −−−→ 0
+���ιΦ
+���ιω
+���ιu
+���ιp
+R −−−→ Φ
+∇
+−−−→ ˆΨ
+∇×
+−−−→ ˆV
+∇·
+−−−→ ˆQ −−−→ 0,
+(105)
+where now the hat spaces correspond to the ones defined over the parametric domain, and
+are identical to the ones used in the previous sections of this paper. Moreover, by composing
+the ι maps with the projectors from the de Rham complex in the square domain setting, we
+arrive at a new commuting diagram between the physical domain continuous spaces and the
+discrete spaces in the physical domain defined by the push-forward of the discrete spaces
+chosen for the unit square.
+For completeness we also define the 2D pull-back operators
+ιω(ψ) = ψ ◦ F,
+(106)
+ιu(v) = J(DF)−1(v ◦ F),
+(107)
+ιp(q) = J(q ◦ F).
+(108)
+34
+
+In 2D a commuting diagram also exists:
+R −−−→ Ψ
+∇⊥
+−−−→ V
+∇·
+−−−→ Q −−−→ 0
+���ιω
+���ιu
+���ιp
+R −−−→ ˆΨ
+∇⊥
+−−−→ ˆV
+∇·
+−−−→ ˆQ −−−→ 0.
+(109)
+Next we begin the process of mapping the governing equations back to the reference
+domain. We start with Equations (6) - (8) for the rotational form of the 3D Navier-Stokes
+equations. The Stokes equations are recovered by simply removing the nonlinear term in the
+momentum equation, Equation (6), and noting now that the pressure becomes the standard
+kinematic pressure p. In the momentum equation, the viscous term is mapped back to the
+reference domain via
+(∇ × ω) ◦ F = J−1(DF)( ˆ∇ × ιω(ω)) = J−1(DF)( ˆ∇ × ˆω),
+(110)
+and the pressure term is mapped to
+(∇p) ◦ F = (DF)−T ˆ∇(ιΦ(p)) = (DF)−T ˆ∇(J−1ιp(p)) = (DF)−T ˆ∇(J−1ˆp).
+(111)
+Within the continuity equation, Equation (7), the divergence is mapped via
+(∇ · u) ◦ F = J−1 ˆ∇ · (ιu(u)) = J−1 ˆ∇ · ˆu.
+(112)
+Finally, in the constitutive law, Equation (8), the curl term is mapped similarly to the
+viscous momentum term
+(∇ × u) ◦ F = J−1(DF)( ˆ∇ × ιω(u)) = J−1(DF)( ˆ∇ × ((DF)Tu))
+= J−1(DF)( ˆ∇ × ((DF)T(J−1(DF)ˆu)))
+= J−1(DF)( ˆ∇ × (J−1Cˆu)).
+(113)
+Now we pull each equation back to the reference domain via the corresponding ι map,
+so the momentum equations are pulled back via ιu, the continuity equation is pulled back
+with ιp and the constitutive law is pulled back with ιω. For brevity, we will not state the
+full form of the mapped equations in 3D, but instead state just the 2D form. This arises in
+a similar way as the 2D rotational form of the Navier-Stokes equations was generated from
+the 3D equations. In particular we can simply write the equations out component-wise and
+note that z velocities as well as derivatives in the z direction are zero. This yields:
+35
+
+(a) Before strong enforcement of no pene-
+tration conditions
+(b) After strong enforcement of no penetra-
+tion conditions
+Figure 15: Example of collocation grid on a mapped domain for vorticity-velocity-pressure scheme
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Given ν ∈ R+, ˆf : ˆΩ → R2, and ˆg : ∂ ˆΩ → R2, find ˆu : ˆΩ → R2, ˆp : ˆΩ → R, and
+ˆω : ˆΩ → R such that:
+ν ∂ˆω
+∂ˆy + JC−1
+11
+∂(J−1ˆp)
+∂ˆx
++ JC−1
+12
+∂(J−1ˆp)
+∂ˆy
+= ˆf1
+in
+ˆΩ
+(114)
+− ν ∂ˆω
+∂ˆx + JC−1
+21
+∂(J−1ˆp)
+∂ˆx
++ JC−1
+22
+∂(J−1ˆp)
+∂ˆy
+= ˆf2
+in
+ˆΩ
+(115)
+ˆ∇ · ˆu = 0
+in
+ˆΩ
+(116)
+ˆω − J−1( ∂
+∂ˆx(J−1(C21ˆux + C22ˆuy)) − ∂
+∂ˆy(J−1(C11ˆux + C12ˆuy))) = 0
+in
+ˆΩ
+(117)
+ˆu = ˆg
+on
+∂ ˆΩ,
+(118)
+where ˆf = ιu(f) and ˆg = ιu(g).
+We collocate these equations in the same manner as in the previous sections to solve
+for the parametric domain variables ˆu, ˆp, and ˆω. The collocation points are chosen as the
+Greville abscissae in the parametric domain, and an example of the resulting points pushed
+forward into the physical domain is shown in Figure 15. No penetration boundary conditions
+are enforced strongly and no slip boundary conditions are enforced weakly with a suitable
+penalty term. For brevity we omit the full statement of the discrete problem and simply
+note that it leads to a linear system of equations (as we are focused in this section on Stokes
+flow).
+36
+
+9. Numerical Results on Mapped Domains
+In this penultimate section we verify the performance of the vorticity-velocity-pressure
+collocation scheme on non-square domains. We first consider linear Couette flow to confirm
+that the expected convergence rates are maintained and then move on to modified lid-driven
+cavity flows in non-square setups.
+9.1. Cylindrical Couette Flow
+The first problem posed on a mapped domain that we consider is Couette flow. This
+models the behavior of a fluid between 2 concentric cylinders, with the outer fixed and the
+inner rotating at a constant rate. We solve the problem over a quarter circle domain as shown
+in Figure 15, enforcing homogeneous Dirichlet boundary conditions on the outer cylindrical
+wall, zero normal and unit tangential velocity on the inner cylindrical wall, and zero pressure
+gradient on the horizontal and vertical boundaries. The last Neumann boundary condition
+is enforced using the Enhanced Collocation approach [41].
+The exact velocity field is given in polar coordinates as:
+¯u =
+� (Ar + B/r) sin θ
+(Ar + B/r) sin θ
+�
+,
+(119)
+with A = −Ωin
+δ2
+1−δ2, B = Ωin
+r2
+in
+1−δ2, Ωin =
+U
+rin, δ =
+rin
+rout, rin = 1 is the radius of the inner
+cylinder, rout = 2 is the radius of the outer cylinder, and the velocity of the inner cylinder
+has magnitude U = 1. The exact pressure field is zero everywhere, and the exact vorticity
+is a constant equal to 2A. We use a polar mapping to map between the parametric and
+physical domains:
+F(ξ1, ξ2) =
+� ((rout − rin)ξ2 + rin) sin(2πξ1)
+((rout − rin)ξ2 + rin) cos(2πξ1)
+�
+.
+(120)
+In solving this problem with this mapping one can show analytically that the collocation
+approximation to the exact solution ˆu is a function of ˆy only, the collocation approximation
+to ˆv is zero, the collocation approximation to ˆp is zero, and the collocation approximation
+to ˆω is a constant. However, we assemble and solve the full linear system without utilizing
+this structure.
+Figure 16 shows the errors in the solution as a function of resolution. For the L2 norm
+and H1 semi-norm errors of velocity we recover the same rates are in the square domain
+setting. The collocation scheme also captures the zero pressure up to finite precision on the
+coarsest mesh as both the L2 and H1 errors are essentially zero. As the mesh is refined we see
+this error increase, which we attribute to worsening matrix conditioning and roundoff error
+effects. We also see the same rates as in square domains for the L2 convergence of vorticity.
+Note that a constant vorticity is also recovered even on the coarsest mesh, as evidenced by
+the numerically zero H1 semi-norm error. Like the pressure errors the H1 error grows with
+mesh refinement, and we believe the explanation is the same.
+37
+
+10-2
+10-1
+10-10
+10-8
+10-6
+10-4
+10-2
+(a) Velocity L2 error
+10-2
+10-1
+10-10
+10-5
+100
+(b) Velocity H1 error
+10-2
+10-1
+4
+6
+8
+10
+12
+14
+16
+10-16
+(c) Pressure L2 error
+10-2
+10-1
+10-15
+10-14
+10-13
+(d) Pressure H1 error
+10-2
+10-1
+10-10
+10-5
+(e) Vorticity L2 error
+10-2
+10-1
+10-15
+10-14
+10-13
+(f) Vorticity H1 error
+Figure 16: Errors in Couette flow solution for vorticity-velocity-pressure formulation
+38
+
+x
+y
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+x
+y
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.1
+0.2
+x
+y
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.1
+0.2
+Lid-Driven Cavity Flow Over a Wavy Wall
+Figure 17: Mapped Stokes results for lid-driven cavity with varying numbers of bumps
+9.2. Lid-Driven Cavity Over Wavy Wall
+Our final numerical test case concerns the Stokes flow in a 2D lid-driven cavity, similar
+to the square domain examples, but now with a non-flat bottom surface of the cavity. In
+particular, the mapping from parametric to physical domain is given by
+F(ξ1, ξ2) =
+�
+ξ1
+A(B(1 − ξ2) sin(Cπξ1) + ξ2)
+�
+,
+(121)
+where A, B, and C are constants which control the shape of the domain. We use three
+combinations in this paper, in particular A = 1, B = 0.75, and C = 1 gives a domain with
+one bump, A = 0.25, B = 0.3, and C = 3 gives a domain with two bumps, and A = 0.25,
+B = 0.3, and C = 5 gives a domain with three bumps.
+Figure 17 shows the streamfunctions obtained with 64 elements and k′ = 2. Clearly we
+are able to recover symmetric fields in all cases which are appropriate for Stokes flow.
+39
+
+10. Conclusions
+In this paper, two divergence-conforming collocation methodologies have been presented
+for solution of the steady, incompressible Navier-Stokes equations using a velocity-pressure
+formulation and a vorticity-velocity-pressure formulation.
+By employing B-spline spaces
+that conform to the de Rham complex, these methods produce velocity fields which are
+exactly pointwise divergence free. Moreover, by the nature of collocation methods, these
+methods are much less computationally expensive than traditional Galerkin finite element
+formulations as no costly numerical integrations are required. By applying the discretizations
+to benchmark problems in two and three dimensions we have shown that the methods retain
+a high order of accuracy. Moreover, we have seen that by re-writing the equations in the
+vorticity-velocity-pressure form many convergence rates are improved compared to those
+obtained with a velocity-pressure scheme. However, useful properties of the corresponding
+divergence-conforming B-spline Galerkin method, such as pressure and Reynolds robustness,
+are not maintained in these collocation schemes. Finally, methods for problems posed in
+more complicated domains were created by mapping unknowns and equations between the
+physical and reference domains using structure-preserving transformations.
+There are many interesting directions for future work. Collocation schemes that do retain
+pressure and Reynolds robustness properties would be useful, as would developing a strategy
+for stabilization of these types of collocation schemes in advection-dominated flow regimes.
+The schemes proposed in this paper could also be extended to the multi-patch setting to
+allow for simulations posed on even more complicated domains. The use of locally adaptive
+splines would also aid in maximizing the ratio of accuracy to cost in which collocation already
+excels. Finally, while collocation improves upon the cost of numerical integration, unsteady,
+incompressible Navier-Stokes solution strategies will still likely involve the solution of linear
+systems during each time step, and thus reducing cost of linear system solution is also very
+important.
+Acknowledgements
+This material is based upon work supported by the National Science Foundation Graduate
+Research Fellowship Program under Grant No. DGE-1656518. Any opinions, findings, and
+conclusions or recommendations expressed in this material are those of the authors and do
+not necessarily reflect the views of the National Science Foundation.
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