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+SciPost Physics
+Submission
+Anomalies, Representations, and Self-Supervision
+Barry M. Dillon, Luigi Favaro, Friedrich Feiden, Tanmoy Modak, Tilman Plehn
+Institut für Theoretische Physik, Universität Heidelberg, Germany
+January 13, 2023
+Abstract
+We develop a self-supervised method for density-based anomaly detection using contrastive
+learning, and test it using event-level anomaly data from CMS ADC2021. The Anomaly-
+CLR technique is data-driven and uses augmentations of the background data to mimic
+non-Standard-Model events in a model-agnostic way. It uses a permutation-invariant Trans-
+former Encoder architecture to map the objects measured in a collider event to the represen-
+tation space, where the data augmentations define a representation space which is sensitive
+to potential anomalous features. An AutoEncoder trained on background representations
+then computes anomaly scores for a variety of signals in the representation space. With
+AnomalyCLR we find significant improvements on performance metrics for all signals when
+compared to the raw data baseline.
+Contents
+1
+Introduction
+2
+2
+Dataset
+4
+3
+AnomalyCLR
+5
+3.1
+Contrastive learning
+5
+3.2
+CLR for anomaly detection
+6
+4
+Application to event-level anomalies
+8
+5
+Anomaly scores
+10
+6
+Results
+11
+6.1
+Comparison of methods
+11
+6.2
+The effect of anomaly-augmentations
+12
+6.3
+The effect of representation dimension
+13
+7
+Summary & conclusions
+14
+References
+15
+1
+arXiv:2301.04660v1  [hep-ph]  11 Jan 2023
+
+SciPost Physics
+Submission
+1
+Introduction
+Model-agnostic new physics searches are one of the most interesting analysis prospects for the
+LHC and other colliders. Over the past decade the LHC has searched for new physics based on
+model-specific hypothesis testing. Despite these efforts there has been no strong evidence of new
+physics found. It is possible that new physics does exist at the scales probed by the LHC, and has
+not been uncovered due to the particular signal not being covered by previous analysis hypotheses.
+The ATLAS and CMS collaborations have both implemented model-agnostic new physics searches
+to deal with this [1, 2], however these methods suffer some drawbacks. For example scanning
+high-dimensional parameter spaces can lead to large look-elsewhere effects, or methods can lack
+the ability to make full use of the high-granularity low-level information collected in the experi-
+ments. Recent progress in machine learning based high-energy physics tools are making significant
+advances in solving many problems of such classical methods [3].
+The main machine learning tools to date for data-driven model-agnostic searches are based
+either on density-related scores, or on classification scores using a background-dominated control
+sample. The latter, typically known as CWoLa methods (Classification Without Labels) [4–6] have
+been shown to be very successful in applications such as bump hunting [7–14] and semi-visible jet
+searches [15], providing both anomaly scores and background estimates. However they run into
+difficulty when the dimension of the input space or number of observables becomes large, and so
+the question of whether or not they can be used on low level data is still uncertain. CWoLa tools
+have already been adopted by the ATLAS collaboration [16].
+Density-based methods use machine learning to estimate the density in the phase space, and
+then identify anomalies as those laying in the low density regions. These tools typically work
+on high-dimensional inputs and so can be used on low-level data. The first density-based meth-
+ods were the AutoEncoder studies [17, 18], where the network is optimised to compress and
+reconstruct the kinematics of a jet or event. While this is not strictly density estimation, the
+optimisation is highly aligned with learning a density, since regions of the phase space which
+are most populated are those which should be reconstructed the best and thus have the low-
+est anomaly score. There has been significant progress with the AutoEncoder tools and other
+density-based anomaly detection methods in recent years [19–33], with studies covering inter-
+pretability of AutoEncoders [34, 35], topic modelling [36, 37], null hypothesis tests for anomaly
+detection [38], ABCD methods [39], the Normalised AutoEncoder (NAE) [40], and normalising
+flow techniques [41–44]. For a comprehensive summary of many different anomaly detection
+methods we refer the reader to the community challenge papers in Refs [45,46].
+One issue with the density-based approaches [44,47] is that the score is not invariant under
+simple transformations in the phase space. This means that a simple re-mapping of the momenta
+or coordinates fundamentally changes what the anomaly score is. This poses the question of how
+to choose a representation of the data for use in density-based anomaly detection tasks. It is
+also worth noting that despite the great progress that more sophisticated neural network archi-
+tectures and the implementation of symmetries in networks has brought to supervised classifi-
+cation [48–51], they have not yet led to the same progress in anomaly detection. In this work
+we develop a new approach to density-based anomaly detection using self-supervision, which de-
+fines the representation of the data in a model-agnostic way using the power of highly expressive
+networks such as transformers or graph networks to boost anomaly detection performance.
+Supervised machine learning methods use the idea of a truth-label to optimise the neural net-
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+works, usually to classify between data with different truth labels. Unsupervised methods are
+those which do not require truth labels, instead optimising a network using a reconstruction loss
+or a negative log likelihood, for example. Self-supervision on the other hand uses ‘pseudo-labels’,
+labels generated from the data without knowledge of a truth label, to optimise the networks. In
+contrastive learning [52], these labels correspond to a link between an original event and an aug-
+mented event. We define the augmentation as some physical modification of the event kinematics.
+Contrastive learning uses the pseudo-labels to devise an auxiliary task for the network optimisation
+through the contrastive loss function. Now the network learns how to process high-dimensional
+correlations in the data, and thus the representations learned by these networks can be very useful
+for downstream tasks. We introduced the self-supervised JetCLR method in [53] and demonstrated
+its ability to construct highly expressive representations for classification tasks. In [54] this same
+technique was used to construct representations for CWoLa-based anomaly detection. In addition
+to these works, other self-supervised / representation learning techniques have been applied in
+particle physics [55,56] and in other scientific disciplines such as astrophysics [57–60]. In [53,54]
+the augmentations corresponded to transformations of the event to which the underlying physics
+should be invariant to rotations or translations, but also soft-collinear parton splittings.
+We introduce AnomalyCLR, a new method based on the idea of ‘anomaly-augmentations’.
+These anomaly-augmentations are modifications of the original event to which the underlying
+physics is not invariant. In fact these augmentations are chosen to mimic very general features
+that anomalous events might have, such as high multiplicity, large MET, or large pT. Despite choos-
+ing explicitly the augmentations, the approach does not target any specific new physics model, and
+we will see from the results that the approach is model agnostic. AnomalyCLR projects the kine-
+matics of each event to a representation vector, which we then use to train an AutoEncoder and
+define the anomaly scores. It enriches the representation space using known invariances in the
+data, such as invariance to azimuthal rotations, and known generic features of anomalies. Self-
+supervised anomaly detection methods have gained prominence in the machine learning literature
+recently [61–64], and while the approaches are necessarily domain specific, we have drawn on
+these methods. The anomaly score can be computed in different ways, and we opt for the Au-
+toEncoder approach. So the workflow is as follows; train AnomalyCLR to obtain a representation
+vector for each event in the dataset, then train an AutoEncoder on these representations to obtain
+the anomaly scores. This is in contrast to the typical approach of training the AutoEncoder directly
+on the raw kinematical data from the events. We test AnomalyCLR on the CMS Anomaly Detection
+Challenge dataset [65], and, compared to the raw data baseline, we find significant improvements
+on all signals.
+In Section 2 we will discuss the dataset and the different backgrounds and signals. In Section 3
+we will then introduce the AnomalyCLR idea, first discussing contrastive learning and then how
+this can be modified for use in anomaly detection. The specifics of the application to event-level
+collider data such as the CMS ADC dataset is given in Section 4. The discussion on how we estimate
+anomaly scores is given in Section 5, where the architecture and optimisation of the AutoEncoder
+we use is discussed. The results are presented in Section 6, along with an analysis of how different
+anomaly-augmentations and different representation dimensions affect the results. We conclude
+in Section 7 with a discussion of the results and future directions.
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+2
+Dataset
+To test the performance of the AnomalyCLR representations compared to raw data in an anomaly
+detection task we use the CMS anomaly detection challenge dataset [65], which contains simu-
+lated proton-proton collisions with a 13 TeV centre-of-mass energy. The events are selected to
+have at least one e or µ with transverse momenta pT >23. The pseudo-rapidity (|η|) is required
+to be <3 and <2.1 respectively for e and µ. Further, the events are allowed to have up to 10 jets
+with pT > 15 GeV and |η| < 4, up to 4 muons pT > 3 GeV and |η| < 2.1, up to 4 electrons pT > 3
+GeV and |η|<3 and missing transverse energy (MET). The dataset is generated with Pythia 8.240
+generator [66] with a fast detector simulation using Delphes 3.3.2 [67] with the Phase-II CMS
+detector card. The jets are reconstructed using anti-kt algorithm [68]. In the provided dataset
+each event is formatted such that the first entry is assigned for MET, next eight are assigned for
+electrons and muons respectively and, the final 10 entries are for jets. For each particle object
+the data set contains information of pT, η, φ and particle id such that the shape of an event in
+the data frame is [N,19,4] where N is the total number of events. Note that if an event has less
+than the maximum allowed of a type of object, the remaining entries in that case are zero padded.
+The background dataset consists of a number of Standard Model processes and to determine the
+performance of the anomaly detection algorithm four light BSM scenarios are considered.
+Backgrounds
+For the SM background a collection of events are generated from production channels with at
+least a single lepton in the final state. The fraction of events to be included in the SM for each
+process is fixed by its trigger efficiency and the LO cross section. Thus, four leading processes are
+considered: W and Z inclusive productions, QCD multijet contributions, and t¯t production. The
+proportions between the four processes are given in [69] as:
+pp → W ± + jets → ℓ±νℓ + jets
+(59.2%)
+pp → Z + jets → ℓ+ℓ− + jets
+(6.7%)
+pp → t¯t + jets
+(0.3%)
+pp → jets
+(33.8%) .
+(1)
+with ℓ = e,µ,τ. The QCD multijet production is by far the largest production process at the
+LHC. Although leptons in QCD multijet backgrounds are rarely present and mainly originate from
+decays of unstable hadrons, the sheer volume of QCD multijet production makes it one of the
+largest processes in the data stream for the challenge.
+New physics signals
+The signal datasets provided by the challenge consist of events simulated from the following signal
+models:
+• Leptoquark (LQ): A 80 GeV LQ decaying in to a b and τ.
+• Neutral scalar boson A: A 50 GeV neutral scalar boson A. The production mechanism
+pp → A+X → Z∗Z∗+X (with X is inclusive activity) followed by both Z∗ decaying into charged
+leptons.
+• Scalar boson h0: A scalar boson 60 GeV h0 with pp → h0 + X → τ+τ− + X production.
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+• A charged scalar h±: Charged scalar with 60 GeV mass and pp → h±+X → τν+X production.
+The most distinguishing high-level features of these signals when compared with the background
+processes are the electron, muon, and jet multiplicities and the pT and MET distributions † .
+3
+AnomalyCLR
+In this section we describe the AnomalyCLR method ‡. Contrastive learning of representations
+(CLR) [52] is a technique used to construct highly-expressive representations of data for use in
+downstream tasks, in our case this task is anomaly detection. It is self-supervised in that the
+technique does not require any ‘truth’ labels for the training data. The advantage of this from the
+collider physics perspective is that the technique could be run directly on experimental data rather
+than on simulation. Due to the ability of deep learning methods to learn non-trivial correlations
+in data that is not expected to be well-modelled by simulation, this is an important aspect of CLR
+for anomaly detection.
+3.1
+Contrastive learning
+The basic idea is that some function f (·) (typically a neural network) is used to map from the data
+space D to a representation space R, with the function being optimised to solve some auxiliary
+task which does not require truth labels. This auxiliary task is framed as an optimisation problem
+using ‘pseudo-labels’. In the anomaly detection scenario addressed in this work, the function that
+performs the mapping from D to R is optimised only on background data. Given that the collider
+events or objects such as jets typically consist of unordered sets of particles reconstructed by the
+experiment, we opt for a permutation-invariant function to perform the mapping from D to R.
+Specifically, we use a transformer encoder neural network, there are more details on this later in
+the section.
+The auxiliary task that our function is optimised to solve uses augmentations of the collider
+data. In the traditional contrastive learning approach these augmentations are used to define two
+types of pseudo-labels:
+1. Positive-pair labels
+These labels match each data point in the sample to an augmented version of itself.
+2. Negative-pair labels
+These labels match each data point in the sample to every other data point which is not itself
+or an augmented/transformed version of itself.
+The function f (·) is then trained to map from the raw data to the representation space such that
+positive-pairs are close together in R and negative-pairs are far apart in R. These two optimi-
+sation goals are referred to as alignment (of positive-pairs) and uniformity (of negative-pairs),
+respectively. The augmentations are chosen to be modifications of the data that should leave the
+underlying physics unchanged, for example a symmetry in the physical system or an augmentation
+that could mimic a detector resolution effect.
+†We note that since the publication of previous papers using this dataset, a bug fix in the simulation has resulted in
+a new dataset, and so it is difficult to make direct comparisons between new and old results.
+‡The code will be made available at https://github.com/bmdillon/AnomalyCLR.
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+Each data point is described by an array of data xi with the subscript labelling the specific data
+point. We denote an augmentation of a data point as x′
+i, with the positive-pairs and negative-pairs
+being defined as the sets {(xi, x′
+i)} and {(xi, x j)}∪{(xi, x′
+j)} for i ̸= j, respectively. The contrastive
+loss function that the network is trained to minimise then is
+LCLR = −log
+es(zi,z′
+i)/τ
+�
+j̸=i∈batch
+�
+es(zi,zj)/τ + es(zi,z′
+j)/τ� ,
+(2)
+where zi = f (xi) and z′
+i = f (x′
+i) are the outputs of the mapping function. The cosine similar-
+ity measure s(·,·) is used to compare events and measure distances between them in the new
+representation space,
+s(zi,zj) =
+zi · zj
+|zi||zj| = cosθi j .
+(3)
+In this way, s(·,·) projects each vector zi to the surface of a unit hypersphere and computes the
+cosine distance between each pair. As it stands, s(·,·) is not a proper distance metric, however we
+could form one by taking di j = θi j/π as the distance between each event in the representation
+space, although we do not explore this here. The numerator of the contrastive loss in Eq. (2)
+accounts for the positive-pair and alignment, where distances between events and their augmented
+counter-parts enter. While the denominator accounts for the negative-pairs and uniformity, where
+distances between completely different events are accounted for. The degree to which we trade
+off between the different tasks is determined by the temperature hyper-parameter τ in the loss
+function.
+3.2
+CLR for anomaly detection
+While contrastive learning has been shown to be very useful in generating representations for
+downstream classification tasks [53], there is a potential issue when using this approach for down-
+stream anomaly detection tasks. For the classification task, for example in [53], the function f (·)
+is optimised on data from both the background and signal classes, despite not using their truth-
+labels explicitly in the optimisation. Through the contrastive learning this allows the function to
+encode non-trivial features of both the background and signal data in the representations. When
+using contrastive learning for a downstream anomaly detection task however, the function f (·) is
+optimised on just the background data (or at least a significantly background-dominated dataset).
+This means that the representation learned by the function f (·) will focus solely on features rele-
+vant for the background data. This could mean that anomalous data is not out-of-distribution and
+so may not lead to competitive performance in downstream anomaly detection tasks. This will
+become evident when we look at the results in Section 6. To remedy this we introduce Anoma-
+lyCLR, a modified approach to contrastive learning for anomaly detection in particle physics. At
+the core of this approach is the introduction of ‘anomaly-augmentations’, such that we now have
+two categories for augmentations:
+1. Physical augmentations
+These are augmentations of the data that we would like the mapping to be invariant to.
+2. Anomaly-augmentations
+These are unphysical augmentations of the data that are supposed to mimic potential anoma-
+lies, we want the representations to be highly discriminative towards these augmentations.
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+We add a third pseudo-label:
+3. Anomaly-pair labels
+These labels match each data point in the sample to an anomaly-augmented version of itself.
+The advantage of anomaly-augmentations is that we can increase the sensitivity of the anomaly
+detection tools to anomalies using just the background data, potentially the data directly measured
+at colliders. This keeps the approach in line with the original data-driven CLR idea. We can then
+define the anomaly-augmented contrastive loss function as
+LAnomCLR = −log
+e[s(zi,z′
+i)−s(zi,z∗
+i )]/τ
+�
+j̸=i∈batch
+�
+es(zi,zj)/τ + es(zi,z′
+j)/τ� ,
+(4)
+where we denote the representations of the anomaly-augmented events by z∗, and so the anomaly-
+pair is defined as {(xi, x∗
+i )}. Note that the anomaly-augmentations only enter in the numerator
+of Eq. (4), and without these the loss function becomes the regular contrastive loss function.
+Introducing the anomaly-pairs we expose the network to data features that are outside of the
+background distribution. The CLR portion of the loss function still optimises for alignment and
+uniformity, however this uniformity is now disrupted by the anomaly-pair term. As a result the
+background data will not be uniformly distributed in the representation space, with some regions
+encoding features of the anomaly-augmented data. This means that anomalous data with features
+similar to those generated by the anomaly-augmentations should be out-of-distribution in this
+representation space.
+We did some minor testing on alternative forms of this loss function, for example including
+the anomaly-augmentations in the denominator of the loss function with the negative-pairs. How-
+ever since the anomaly-augmentations compute distances between a data point and its augmented
+counter-part, and not between other data points (i.e. i ̸= j), it is more intuitive to include this
+term in the numerator. The denominator in Eq. (4) is used to encode features in the represen-
+tation space that discriminate between the different data points used during training, which for
+anomaly detection is the background data. This is not necessary for anomaly detection, and the
+anomaly-pairs should provide the representations with all the discriminative power they need,
+so we experimented with removing the denominator in Eq. (4) altogether, and found that this is
+sufficient. In this case the loss function is written as
+L+
+AnomCLR = −log e[s(zi,z′
+i)−s(zi,z∗
+i )]/τ =
+s(zi,z∗
+i ) − s(zi,z′
+i)
+τ
+,
+(5)
+where the plus sign in L+
+AnomCLR indicates that only positive-pairs are used. This results in a much
+less computationally expensive loss function, since we no longer need to compute pair-wise cor-
+relations between each entry in a batch the complexity scales as Nbatch rather than N 2
+batch. We also
+remove the dependence on τ in L+
+AnomCLR, since there is no longer a trade-off between positive-
+and negative-pairs. We could of course introduce a term to control the trade-off between the
+physical and anomaly-augmentation terms, but we do not explore that here. In our results we will
+compare the performance of both loss functions.
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+Application to event-level anomalies
+The application of AnomalyCLR to different physical scenarios requires an understanding of the
+data and the physics in order to construct the physical and anomaly-augmentations. For the event-
+level dataset discussed in Section 2 we consider three physical augmentations the data:
+1. Azimuthal rotations
+The whole final state is rotated by an angle φ randomly sampled from [0,2π].
+2. η − φ smearing
+The (η,φ) coordinate of every object in the event is resampled according from a Normal
+distribution centred on the original coordinate and with a variance inversely proportional to
+the pT, i.e. η′ ∼ N (η,σ(pT)) and φ′ ∼ N (φ,σ(pT)).
+3. Energy smearing
+The pT of every object in the event is re-sampled according to p′
+T ∼ N (pT, f (pT)) with f (pT)
+determining the strength of the smearing.
+These augmentations reflect both the symmetries in the data and the experimental resolution of
+the detector. Detectors are imperfect, especially in measuring jet energies, and we encode this in
+the representations of the data through the energy-smearing augmentation. Here we re-sample
+the jet pT’s as p′
+T ∼ N (pT, f (pT)), where f (pT) =
+�
+0.052p2
+T + 1.502pT is the energy smearing
+applied by Delphes (the pT’s are normalised by 1GeV). If not explicitly mentioned, we always
+assume units of GeV for energy. For the anomaly-augmentation we consider some very simple
+scenarios:
+1. Multiplicity shift, x′
+i = m(xi)
+For each event m(·) adds a random number of electrons, muons, and jets to the event. The
+number is chosen randomly within the limits (ne,4− ne), (nµ,4− nµ), and (nj,10− nj) for
+electrons, muons, and jet, respectively. The azimuthal angle and pseudo-rapidities are also
+chosen randomly within the limits allowed, and the pT for each object is chosen as a random
+fraction of the maximum pT in the event. Once the objects have been added, the MET of the
+event is recalculated and updated.
+2. Multiplicity shift, keeping MET and the total pT constant, , x′
+i = m(xi)
+This is similar to the above augmentation, but now m(·) generates the extra objects by splitting
+the existing objects and smearing the η−φ coordinates using the function used in the physical
+augmentations above.
+3. pT and MET shifts, x′
+i = spT (xi)
+Here spT (·) shifts the pT’s in the event by the same random factor. We randomly choose whether
+we shift just the MET, just the reconstructed object pT’s, or both. And we ensure that the the
+trigger selection is not spoiled by these shifts.
+With the physical augmentations we apply all of them simultaneously, whereas for the anomaly-
+augmentations we apply just one augmentation to each event. The augmentation that is applied is
+selected randomly and uniformly. We do not apply both a physical augmentation and an anomaly-
+augmentation to the events in s(zi,z∗
+i ), since this would conflict with the optimisation goal of the
+s(zi,z′
+i) term. It would also be possible to have an anomaly-augmentation that removes objects
+from the event, however this effect is already captured by the augmentation that adds objects to
+the event. Many of the events in the background have the minimal multiplicity allowed by the
+applied cuts, so the effect of an anomaly-pair with a low multiplicity background event and the
+same event augmented to have more objects is the exact same as the effect of an anomaly-pair
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+with a high-multiplicity background event augmented to have less objects. This is because of the
+symmetry in the distance function s(zi,z∗
+i ). So the anomaly-augmentations here are as general
+as can be, and do not target any specific new physics scenario, therefore the technique should be
+model-agnostic.
+Architecture and training
+The collider event data being used has a well-defined structure:
+• MET: one entry with (pT,η,φ)
+• Electrons: four entries, each with (pT,η,φ)
+• Muons: four entries, each with (pT,η,φ)
+• Jets: ten entries, each with (pT,η,φ).
+This amounts to a 19 × 3 array, with the electrons, muons, and jets being ordered by pT and hav-
+ing zero-padded entries where there is less than the maximum allowed number of reconstructed
+objects. The multiplicity is typically much less than the maximum allowed, so the data for a
+single collider event can have many zeros. The transformer allows us to avoid this by having a
+permutation-invariant and variable length input format. Because the data is now processed in
+a permutation-invariant way, the information on which entry corresponds to which object (MET,
+electron, muon, or jet) is lost. We reinstate this information by adding a one-hot encoded ID vec-
+tor to (pT,η,φ), with a 1 indicating the correct ID. This means that each reconstructed object is
+now represented by a 7D vector. Before passing the kinematic data to the transformer we do some
+very minor preprocessing to make sure that the numbers the networks see are O(1). Specifically,
+we divide all MET and pT values by the average pT of all objects (electrons, muons, jets) in the
+background dataset, we do not shift the values to be centred on zero because the distribution is
+highly peaked at zero and we want the preprocessed data to have the same sparsity as the original
+data. We then divide all η and φ values by 4 and π, respectively. When training the AutoEncoder
+networks discussed in the next section we use the same preprocessing of the data, this ensures
+that any difference in the results can be attributed to AnomalyCLR.
+The transformer starts by projecting each object to a larger vector whose dimension is deter-
+mined by the embedding dimension. The embeddings for each object are then passed through the
+transformer, with a feed-forward network between each transformer layer. The output from the
+transformer has a dimension of (n× model dimension) with n being the number of objects in the
+event. The last steps are to sum over the n vectors in this output, which enforces the permutation-
+invariance, and to pass this vector through a fully-connected head network. The output of this
+head network is what is passed to the loss function. For more details on the architecture we re-
+fer the reader to [53], here we only list the hyper-parameters used in training the network in
+Table 1. The representation used in the anomaly detection task is taken from the output of the
+transformer network, before being passed through the head network. It is well documented in the
+machine learning literature that these intermediate representations from self-supervised networks
+generally contain more discriminating features, for example in [52].
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+hyper-parameter
+model (embedding) dimension 160
+feed-forward hidden dimension 160
+output dimension
+160
+# self-attention heads
+4
+# transformer layers (N)
+4
+# layers
+2
+dropout rate
+0.1
+hyper-parameter
+optimiser
+Adam(β1=0.9, β2=0.999)
+learning rate
+5 × 10−5
+batch size
+128
+# epochs
+500
+Table 1: Default setup of the transformer-encoder network and the AnomalyCLR train-
+ing, unless noted explicitly.
+5
+Anomaly scores
+The basic flow in an AutoEncoder involves two steps; (i) mapping high-dimensional input data
+to a compressed latent space using a neural network called an encoder, and (ii) mapping the
+compressed latent space representation to a reconstructed version of the input data using a neural
+network called a decoder. We refer to the encoder network as e(·) and the decoder network as d(·).
+With input data of dimension D, and a bottleneck of dimension B, the encoder maps e : �D → �B,
+while the decoder maps d : �B → �D, with the AutoEncoder defined as h = e ◦ d : �D → �D.
+Acting on a single input ⃗x, the AutoEncoder returns ⃗x′ = h(⃗x), and is optimised to minimise the
+mean-squared-error (MSE) loss function between the input and reconstructed input,
+L(⃗x,θ) =
+�
+⃗x − ⃗x′�2 ,
+(6)
+where θ represents the learnable parameters of the AutoEncoder. In the limit where the AutoEn-
+coder is able to reconstruct inputs perfectly, which is guaranteed to be possible when D = B, the
+function hθ(·) is simply the identity. But with B < D the AutoEncoder may not be able to perfectly
+reconstruct all features in the data, and therefore it should learn to reconstruct only the most
+common or prominent features in the data. This means that events containing rare or anomalous
+features should have a larger ‘reconstruction loss’, i.e. L(⃗x,θ), and this can then be used as the
+anomaly score.
+The encoder and decoder networks have 5 feed forward layers each with 256, 128, 64, 32, and
+16 neurons, connected by a 5-dimensional bottleneck. The activation function between layers is
+a LeakyReLU with default slope. The decoder is a mirrored version of the encoder. We don’t apply
+regularization techniques during training. The training is performed using Adam optimiser with
+learning rate 0.001 for 100 epochs, the batch size is 4096, and the number of SM events used is
+106. Note that we have not optimised the AutoEncoder architecture, simply choosing the same
+architecture used in [39]. Instead we have only ensured that they are trained to convergence and
+that the training is stable. The AutoEncoder is trained on both the representations obtained from
+contrastive learning and the raw data. In the case of the raw data we apply the same preprocessing
+to the data as is applied to the data in the contrastive learning network. In this way we ensure
+that any differences in the anomaly detection performance can be attributed to the contrastive
+learning methods.
+10
+
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+6
+Results
+In this section we present some results using the different techniques discussed in the preceding
+sections. The results here are three-fold; we first compare the different methods based on anomaly
+detection performance, we then study the effects of the different anomaly-augmentations on the
+AnomalyCLR performance, and lastly we look at the effect of the representation dimension on the
+performance.
+6.1
+Comparison of methods
+We compare the methods using the ROC (Receiver Operating Characteristic) curves, the SI (Signif-
+icance Improvement) curves, and the AUC (Area Under the ROC Curve). The baseline we compare
+to is the AutoEncoder trained on raw kinematic data. We present results using the CLR method
+without anomaly-augmentations (LCLR), and the CLR method with anomaly-augmentations (both
+LAnomCLR and L+
+AnomCLR). So we have 4 methods in total to compare. For all results on the raw
+data we have trained 5 AutoEncoder networks and taken the central value and the error estimation
+from the mean and standard deviation of the results. For the CLR methods we also aggregate over
+5 different runs, where in each run we train a different transformer network and a different Au-
+toEncoder. The CLR representations have a dimension of 160 and where anomaly-augmentations
+are used we have used them all as outlined in Section 4. In Fig. 1 we present AnomalyCLR results
+using L+
+AnomCLR and see that it leads to significant improvements over the raw data representations,
+not only in the AUC but also at all signal efficiencies. In the Significance Improvement (SI) curves
+we also see large improvements, with the SI being between ∼ 3.5−4 for A → 4l and h+. We can see
+from Table 2 that the L+
+AnomCLR loss function is clearly advantageous over LAnomCLR, beating it on
+all signals with the exception of A → 4l, where LAnomCLR achieves better performance at εs =0.3.
+A point of interest here is that the AutoEncoder on raw data outperforms the AutoEncoder on the
+CLR representations in most cases. This is likely due to the fact that traditional CLR optimises
+for uniformity, and since it is trained on background only, the mapping is not optimised to sepa-
+rate SM-like background events from any event which may look different to that. The benefit of
+Signal
+AE-Raw
+CLR
+AnomCLR
+AnomCLR+
+AUC
+A
+0.885(2)
+0.880(7)
+0.907(6)
+0.909(3)
+h0
+0.755(2)
+0.740(5)
+0.765(4)
+0.776(2)
+h+
+0.900(4)
+0.87(1)
+0.913(2)
+0.930(1)
+LQ
+0.856(2)
+0.841(9)
+0.854(3)
+0.880(1)
+ε−1
+b (εs =0.3)
+A
+47(2)
+80(22)
+156(34)
+139(20)
+h0
+14.9(7)
+11(1)
+18(1)
+23(1)
+h+
+60(10)
+28(6)
+98(9)
+171(7)
+LQ
+24.4(6)
+18(2)
+28(3)
+39(1)
+SI(εs =0.3)
+A
+2.05(5)
+2.7(4)
+3.7(4)
+3.5(2)
+h0
+1.16(3)
+0.99(4)
+1.26(4)
+1.44(3)
+h+
+2.3(2)
+1.6(2)
+3.0(1)
+3.9(1)
+LQ
+1.48(2)
+1.3(1)
+1.6(1)
+1.88(2)
+Table 2: Comparison of the different CLR loss functions, with and without anomaly-
+augmentations, and the AE trained on raw data.
+11
+
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+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ϵs
+100
+101
+102
+103
+ϵ−1
+b
+AE-Raw
+A, AUC=0.885(2)
+h0, AUC=0.755(2)
+h+, AUC=0.900(4)
+LQ, AUC=0.856(2)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ϵs
+100
+101
+102
+103
+ϵ−1
+b
+AnomCLR+
+A, AUC:0.909(3)
+h0, AUC:0.776(2)
+h+, AUC:0.930(1)
+LQ, AUC:0.880(1)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ϵs
+0.0
+1.0
+2.0
+3.0
+4.0
+5.0
+SI
+AE-Raw
+A
+h0
+h+
+LQ
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ϵs
+0.0
+1.0
+2.0
+3.0
+4.0
+5.0
+SI
+AnomCLR+
+A
+h0
+h+
+LQ
+Figure 1: Comparison between the AE on raw data and the AE on the CLR representa-
+tions trained with the L+
+AnomCLR loss function.
+anomaly-augmentations here is strikingly clear.
+6.2
+The effect of anomaly-augmentations
+We now want to study how the addition of the individual anomaly-augmentations affects the
+anomaly detection performance. For this we use just L+
+AnomCLR , however we expect the results
+with LAnomCLR to be similar. We use a representation dimension of 160 and obtain the error
+estimate from just 2 runs due to the computational cost of the scan.
+We can see from Fig. 2 that the affect of the augmentations together results in the best over-
+all performance. One thing we noticed is that it can be difficult to determine from the affect of
+individual augmentations, or subgroups of them, what the performance of all of them together
+will be. For example, in most cases if we take just the m(x) augmentation, i.e. the multiplicity
+augmentation that simply adds reconstructed objects, we see that it alone decreases performance
+below baseline for three out of four signals. However when used in combination with the others
+it increases the performance. This is most clear for the leptoquark signal, where all augmenta-
+12
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+0.800
+0.850
+0.900
+A
+AUC
+0.700
+0.750
+h0
+0.800
+0.850
+0.900
+0.950
+h+
+m(x)
+m(x)
+spT(x) m(x)/m(x)
+all
+anomaly augmentations
+0.750
+0.800
+0.850
+LQ
+Figure 2: Results of a scan on the anomaly-augmentations used with the L+
+AnomCLR loss
+function. The augmentations are defined in Section 4. The dashed lines here correspond
+to the AutoEncoder on raw data baseline performance.
+tions taken individually result in a performance which is at or below baseline, but when taken
+together we get a significant boost in the AUC. We also see the interplay between the m(x) and
+m(x) augmentations, since individually these augmentations do not seem to help much, but when
+they are both applied in the same optimisation we see a reduced error and in most cases better
+performance. When drawing conclusions here we should keep in mind that only two runs for each
+combination have been used to compute the mean and error estimation.
+6.3
+The effect of representation dimension
+With CLR we can project our raw data from D to a representation of any dimension we like.
+We would expect that the larger the representation dimension the more information that can be
+encoded in the space. However we also expect that this would plateau or even peak at some point,
+and this what we want to investigate here. For this we use just L+
+AnomCLR , however we expect the
+results with LAnomCLR to be similar. Here we also obtain the error estimate from just 2 runs due
+to the computational cost of the scan.
+In Fig. 3 we see that increasing the representation dimension certainly improves the perfor-
+mance of the anomaly detection, at least up until a certain point. The A, h0, and LQ signals
+all appear to achieve peak performance somewhere between dimensions 120 and 200, while the
+h+ signals performance increases right up until 400. There is no fundamental limitation related
+to the representation size which we would expect to cause a degradation at larger dimensions,
+however there are two points we should keep in mind here. The first is simple, these means and
+variances are calculated with only two runs, so more runs might present a clearer picture. The
+13
+
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+Submission
+0.850
+0.875
+0.900
+0.925
+A
+AUC
+0.740
+0.760
+0.780
+h0
+0.900
+0.920
+0.940
+h+
+4
+8
+12
+20
+40
+80
+120
+160
+200
+300
+400
+500
+representation dimension
+0.840
+0.860
+0.880
+LQ
+Figure 3: Results of a scan on the representation dimension used with the L+
+AnomCLR loss
+function. The dashed lines here correspond to the AutoEncoder on raw data baseline
+performance.
+second point is that we have not optimised the AutoEncoder architecture or hyper-parameters as
+the representation size increases. While it is beyond the scope of this paper, it is possible that
+an independent hyper-parameter optimisation for each representation dimension would improve
+these results, particularly at larger dimensions. What these results show is that there is a clear
+tendancy for the results to improve as we increase from dimensions of ∼ 4 to ∼ 100, as we would
+naturally expect.
+7
+Summary & conclusions
+In this paper we have introduced AnomalyCLR§, a new method for density-based anomaly detec-
+tion in high-energy physics. It makes use of anomalous augmentations of collider data to build a
+representation space from which to construct anomaly scores with a range of methods, for exam-
+ple using AutoEncoders. It is a self-supervised method, based on the contrastive learning idea. We
+tested this method on the CMS ADC dataset, and compared to the raw data baselines we find large
+improvements on all signals. At a fixed signal efficiency of 0.3 and a fixed representation dimen-
+sion of 160 we find significance improvements for the different signals in the range of 14−70%,
+and a decreased relative error on the significance improvement in each case. Allowing for varying
+signal efficiencies and representation dimensions would improve these performance markers even
+§The AnomalyCLR code, along with the event-level anomaly detection application, will be made available at
+https://github.com/bmdillon/AnomalyCLR.
+14
+
+SciPost Physics
+Submission
+further.
+Density-based anomaly detection, using AutoEncoders or normalising flows, suffer from the
+ambiguity that a change in the ‘coordinate system’ or representation of the data results in a fun-
+damental change in how the anomaly score is defined. This makes it difficult to choose a suitable
+representation by hand, for example a simple re-mapping of pT’s along with some re-scaling of
+numerical inputs. These simple choices are difficult to motivate from a physics perspective and
+can drastically change the results of the anomaly detection. This change can be for better or for
+worse, and typically depends on the signal models used to test the algorithm.
+AnomalyCLR addresses this by constructing a representation of the data using self-supervised
+contrastive learning with the addition of anomaly-augmented data. The anomaly-augmented data
+is constructed from the background data through feature augmentation, designed to emulate a
+generic anomaly. We have discussed in detail how we do this for the event-level anomalies in
+the CMS ADC dataset, however this would of course be different in different physics cases. We
+proposed a new loss function which we use to train a deep transformer-based neural network. This
+network projects the events to a new representation, in which the anomaly-augmented events are
+far from their original counterparts, while being close to events which are similar. The transformer
+network then learns a highly discriminative representation of the events which is sensitive to
+the presence of potential anomalies. We have seen that the choice of these augmentations is
+quite model agnostic. This model-agnostic nature of the approach can be seen in how the results
+improve across all four signals considered.
+We have shown the effectiveness of self-supervision and the idea of anomaly-augmentations
+in significantly enhancing anomaly detection performance in a model-agnostic way. This opens
+the door to further studies, such as improving the density-estimation portion of the method with a
+more sophisticated hyper-parameter optimisation of the AutoEncoders, using normalising flows,
+or even using the Normalised AutoEncoder. More generally, the use of anomaly-augmented data
+could be explored further in other anomaly detection approaches.
+Acknowledgements
+We would like to thank Jernej Kamenik and Ben Nachman for their helpful comments on the
+manuscript. BMD acknowledges funding from the Alexander von Humboldt Foundation. LF, TM,
+and TP are funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
+under grant 396021762 – TRR 257: Particle Physics Phenomenology after the Higgs Discovery
+and Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES Ex-
+cellence Cluster).
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+On the accuracy of one-way approximate models
+for nonlinear waves in soft solids
+Harold Berjamin a
+aSchool of Mathematical and Statistical Sciences, University of Galway, University Road, Galway, Republic of Ireland
+Abstract
+A simple strain-rate viscoelasticity model of isotropic soft solid is introduced. The constitutive equations account for
+finite strain, incompressibility, material frame-indifference, nonlinear elasticity, and viscous dissipation. A nonlinear
+viscous wave equation for the shear strain is obtained exactly, and a corresponding one-way Burgers-type equation
+is derived by making standard approximations. Analysis of the travelling wave solutions shows that the two partial
+differential equations produce distinct solutions, and that deviations are exacerbated when wave amplitudes are not
+arbitrarily small. In the elastic limit, the one-way approximate wave equation can be linked to simple wave theory,
+thus allowing direct error measurements.
+1
+Introduction
+In nonlinear acoustics, the Burgers equation is often viewed as the simplest model equation that includes nonlinear
+wave propagation and diffusion effects (Witham, 1999). This partial differential equation in space and time can be
+derived directly from the one-dimensional Navier–Stokes equation by dropping the pressure term, or as a special case
+of the Westerwelt equation. Besides Burgers’ equation, other one-way wave equations have been derived to describe
+wave propagation in fluids and solids at large amplitudes (Hamilton and Blackstock, 1998; Naugolnykh and Ostro-
+vsky, 1998). Based on an appropriate scaling of the wave amplitude, such approximate partial differential equations
+describe unidirectional wave motion for slowly-varying wave profiles of moderate amplitude.
+One-way approximate wave equations have found applications in various areas of nonlinear acoustics. For in-
+stance, works by Radostin et al. (2013) and Nazarov et al. (2017) describe compression wave propagation in solids with
+bimodular elastic behaviour. Another example is the Zabolotskaya equation that describes unidirectional plane shear
+wave propagation in soft solids such as gels and brain tissue (Zabolotskaya et al., 2004), see also Cormack and Hamil-
+ton (2018). In these latter cases, the underlying three-dimensional constitutive theories were revisited by Destrade
+et al. (2013) as well as Saccomandi and Vianello (2021) to enforce objectivity (i.e., invariance by change of observer),
+leading to slight modifications of the equations of motion.
+For these partial differential equations, not many analytical solutions are known. Nevertheless, it is sometimes
+possible to derive exact stationary wave solutions that keep an invariant wave profile throughout the motion, which
+occurs at a suitable constant speed. Those permanent waveforms result from the interaction between nonlinearity
+and dispersion (here of dissipative nature), a common feature that they share with solitary waves.
+One might wonder whether it is preferable to seek closed-form travelling wave solutions by using directly the
+full equations of motion, or by using their one-way approximation. As a matter of fact, both approaches have been
+considered separately in the above literature. The present study aims to provide evidence to advocate for a derivation
+of travelling waves based on the complete equations of motion, thus supporting a remark by Jordan and Puri (2005)
+in relation with the study by Catheline et al. (2003) — this remark led to the publication of an erratum that briefly
+discusses the validity of a particular one-way wave equation (Catheline et al., 2005).
+For this purpose, we consider the case of shear wave propagation in soft viscoelastic solids of strain rate type. We
+derive the simplest three-dimensional constitutive theory that accounts for finite strain, incompressibility, material
+frame-indifference, and viscous dissipation (Section 2). Then, this theory is applied to simple shear deformations, aka.
+transverse plane waves (Section 3), including the reduction to a one-way model described by a Burgers-type equation
+with cubic nonlinearity. Finally, we investigate the travelling wave solutions deduced from the full equations of motion
+as well as from the reduced wave equation (Section 4). Results show non-negligible discrepancies introduced by the
+reduction to unidirectional motion as soon as wave amplitudes are no longer infinitesimal. These comparisons are
+reconsidered in the lossless elastic limit where connections between the one-way model and simple wave theory are
+established (Section 5).
+1
+arXiv:2301.03284v1  [cond-mat.soft]  9 Jan 2023
+
+2
+Strain-rate model
+2.1
+Basic equations
+In what follows, we present the basic equations of Lagrangian dynamics for incompressible solids (Holzapfel, 2000).
+We consider a homogeneous and isotropic solid continuum on which no external body force is applied. Its motion
+in the Euclidean space is described by using an orthonormal Cartesian coordinate system (O,x, y,z). Thus, a particle
+initially located at some position X of the reference configuration moves to a position x of the current configuration.
+The deformation gradient is the second-order tensor defined as F = ∂x/∂X . Introducing the displacement field u =
+x − X and the identity tensor I = [δi j ] whose components are represented by Kronecker’s delta, we therefore have
+F = I +Gradu where Grad denotes the gradient operator with respect to the material coordinates X = (x, y,z).
+In incompressible solids, isochoricity
+J = detF ≡ 1
+(1)
+is prescribed. Thus, the mass density ρ is constant in time. It follows also that ˙J = JF −⊺ : ˙F ≡ 0, where the dot de-
+notes the material time derivative ∂/∂t and the colon indicates double contraction. Introducing the Eulerian velocity
+gradient L = ˙FF −1, this condition can be rewritten as trL = 0.
+Various strain tensors are defined as functions of F. Here, constitutive laws are expressed in terms of the Green–
+Lagrange strain tensor E = 1
+2(F ⊺F − I), which is often a preferred choice in physical acoustics. We introduce also its
+rate ˙E = F ⊺DF obtained by differentiation with respect to time, where D = 1
+2(L +L⊺) is the strain rate tensor. We note
+that D is trace-free due to incompressibility (1).
+The motion is governed by the conservation of linear momentum equation ρ ˙v = DivP, where v = ˙x is the velocity
+field and ρ is the mass density. The equation of motion involves the Lagrangian divergence of the first Piola–Kirchhoff
+stress tensor P = FS where S is the second Piola–Kirchhoff stress tensor. Those stress tensors are specified later on by
+the provision of a constitutive law.
+The present definitions are consistent with notations and conventions used in the monograph by Holzapfel (2000).
+In particular, the divergence of the tensor P reads [DivP]i = Pi j,j componentwise, where indices after the coma de-
+note spatial differentiation. In some other texts, a transposed definition of the divergence is used. Then, the equation
+of motion involves the material divergence of the nominal stress tensor P⊺ instead of P.
+2.2
+Generalities
+In the present study, we consider deformable solids whose constitutive behaviour is described by the state vari-
+ables S = {s,E}, where s is the specific entropy. The choice of variables S is coherent with the postulate of frame-
+indifference of the internal energy (Holzapfel, 2000). In fact, a change of observer specified by a superimposed rigid-
+body motion leaves S invariant, as well as the internal energy U. Note that the internal energy does not depend on
+rates of strain.
+The internal energy per unit volume U is a function of state to be specified. The thermodynamic temperature
+is defined as the conjugate variable of s in the partial Legendre transform of U/ρ with respect to s (Berjamin et al.,
+2021). However, the explicit dependence of U with respect to s is usually omitted in the definition of a strain energy
+density function W e such that U = W e(E). The strain energy W e is regarded as a scalar-valued isotropic function of
+its arguments. Thus, its dependence with respect to E can be reduced to a dependence with respect to three scalar
+invariants
+I1 = tr(E),
+I2 = tr(E2),
+I3 = tr(E3).
+(2)
+They can be used directly, or other physically meaningful scalar quantities might be defined from them.
+The first and second principles of thermodynamics yield the Clausius–Duhem inequality
+D = (S −Se) : ˙E = Sv : ˙E ≥ 0,
+(3)
+where D is the dissipation, S = Se +Sv is the total second Piola–Kirchhoff stress,
+Se = −pC −1 + ∂W e
+∂E
+(4)
+denotes the elastic part, and Sv is a viscous contribution to be specified subsequently. The scalar p is an arbitrary
+Lagrange multiplier for the incompressibility constraint (1), see Sec. 6.3 of Holzapfel (2000), and C = I +2E is the right
+Cauchy–Green strain tensor F ⊺F. Therefore, no dissipation occurs in the elastic case S = Se where the viscous stress
+tensor Sv is equal to zero.
+According to the dissipation inequality (3), the viscous stress Sv is a function of state and evolution variables, e.g.
+the set S ∪ { ˙E} which is a consistent choice to enforce frame-invariance (Antman, 1998; Ball, 2002). We introduce a
+dissipation potential W v(E, ˙E) such that
+Sv = ∂W v
+∂ ˙E
+(5)
+2
+
+defines the viscous stress (Maugin, 1999). In general, the dissipation potential is described by additional invariants
+(Pioletti and Rakotomanana, 2000)
+I4 = tr( ˙E),
+I5 = tr( ˙E2),
+I6 = tr( ˙E3),
+I7 = tr( ˙EE),
+I8 = tr( ˙EE2),
+I9 = tr( ˙E2E),
+I10 = tr( ˙E2E2).
+(6)
+In the present study, we consider Newtonian-type viscosity models whose dissipation potential is as simple as possi-
+ble.
+2.3
+Consequences of incompressibility
+First, let us investigate the consequences of the incompressibility constraint (1). As noted in Jacob et al. (2007), the
+invariants (2) of E are linked through
+I1 = I2 − 4
+3 I3 − I 2
+1 +2I1I2 − 2
+3 I 3
+1 ,
+(7)
+by virtue of incompressibility. This identity follows from the expression of the principal invariants of the unimodular
+tensor C = I +2E in terms of the invariants Ik, see the Appendix of Destrade et al. (2010). Using the differential version
+of the incompressibility constraint, the invariants (2)-(6) of E, ˙E satisfy the particular relationship
+1
+2 I4 = I7 −2I8 +2I1I7 −
+�
+I1 − I2 + I 2
+1
+�
+I4
+(8)
+deduced from the identity trD = 0, see Appendix.
+The relationship (7) means that the invariant I1 = tr(E) is no longer linear with respect to the components of the
+strain tensor E; instead, Eq. (7) shows that it has terms of polynomial order two and three with respect to the strain.
+Furthermore, due to the relationship (8), the invariant I4 = tr( ˙E) is still linear with respect to the components of the
+strain-rate tensor ˙E. However, Eq. (8) shows that I4 is no longer invariant on the strain tensor E; instead, it has terms
+of polynomial order one, two and three with respect to the Green–Lagrange strain.
+2.4
+Constitutive assumptions
+In weakly nonlinear elasticity, the strain energy density function is sought in the form of a polynomial of the invariants
+Ik with constant coefficients. Similarly to Zabolotskaya et al. (2004), we assume that the internal energyU has a fourth-
+order polynomial expression with respect to the components of the strain tensor E of the form
+W e = µI2 + 1
+3 AI3 +DI 2
+2,
+(9)
+where µ ≥ 0 is the shear modulus (in Pa), and the coefficients A, D are higher-order elastic constants.
+Now, let us propose an expression for the dissipation potential. To end up with a linear viscosity model similar to
+that by Destrade et al. (2013), we assume that the dissipation potential is a second-order polynomial expansion of the
+strain rate tensor ˙E, and a zeroth-order polynomial of E. This assumption amounts to selecting W v of second order
+in (E, ˙E), and to ignore the terms proportional to ˙E that produce elastic stresses. Due to the relationships (7)-(8), we
+therefore keep
+W v = ηI5,
+(10)
+where η ≥ 0 is the shear viscosity (in Pa.s). In the above expression, the absence of bulk viscosity “ζ” is due to the
+assumption on polynomial orders for the viscous part, and to the incompressibility property (8). Setting the bulk
+viscosity ζ = 2
+3η in Destrade et al. (2013) yields the same expressions as above.
+Computation of the tensor derivatives of the potentials (9)-(10) by means of the chain rule for W •(Ik,...) yields the
+following elastic (4) and viscous stress contributions (5)
+Se = −pC −1 +2(µ+2DI2)E + AE2,
+Sv = 2η ˙E.
+(11)
+Thermodynamic consistency (3) is ensured provided that the dissipation D = 2W v is non-negative. In fact, the present
+dissipation potential W v is a homogeneous function of degree two with respect to ˙E (Maugin, 1999). A sufficient
+condition for the restriction D ≥ 0 to be always satisfied is that the viscosity η is non-negative.
+3
+
+3
+Plane shear waves
+3.1
+Nonlinear viscous wave equation
+Similarly to Destrade et al. (2013), we consider simple shear deformations described by the displacement field u =
+[u,0,0]⊺ where u = u(z,t) denotes the particle displacement along the x-direction. Thus, the deformation gradient
+tensor reads
+F =
+�
+�
+1
+0
+γ
+0
+1
+0
+0
+0
+1
+�
+�,
+(12)
+where γ = ∂u/∂z is the shear strain. The velocity field takes the form v = [v,0,0]⊺ where v = ∂u/∂t is the shear velocity.
+In the equation of motion ρ ˙v = DivP, the relevant first Piola–Kirchhoff stress component P13 is deduced from
+the expression of the elastic and viscous parts Pe
+13 = µγ+Γγ3 and Pv
+13 = η(1+2γ2) ˙γ, where only terms up to order γ3
+have been kept. The non-negative constant Γ = µ+ A/2+D is a parameter of nonlinearity (Zabolotskaya et al., 2004).
+Hence, upon division by the shear modulus µ, the x-component of the equation of motion produces the nonlinear
+wave equation
+1
+c2
+∂2u
+∂t2 = ∂2u
+∂z2 + 2
+3β ∂
+∂z
+�∂u
+∂z
+�3
++τ ∂
+∂z
+��
+1+2
+�∂u
+∂z
+�2� ∂2u
+∂z∂t
+�
+,
+(13)
+describing transverse wave propagation along the z-direction, where we have introduced the notations
+c =
+�
+µ
+ρ ,
+β = 3
+2
+Γ
+µ,
+τ = η
+µ.
+(14)
+Spatial differentiation of Eq. (13) allows to write a similar wave equation for the strain
+1
+c2
+∂2γ
+∂t2 = ∂2γ
+∂z2 + 2
+3β ∂2
+∂z2 γ3 +τ ∂2
+∂z2
+��
+1+2γ2� ∂γ
+∂t
+�
+,
+(15)
+which will be used later on.
+According to the wave equations (13)-(15), shear waves of infinitesimal amplitude propagate at the shear wave
+speed c =
+�
+µ/ρ in the absence of nonlinearity and viscosity (β = 0, τ = 0). Typically, this sound velocity equals
+c ≈ 2 m/s in gels (Jacob et al., 2007), whereas β ≈ 10 and τ ≈ 0.12 ms at a loading frequency of 100 Hz. Here, we have
+obtained the same wave equations than those derived in Destrade et al. (2013) for the particular bulk viscosity ζ = 2
+3η.
+Note in passing the presence of a nonlinear viscous term which is absent in Zabolotskaya et al. (2004).1
+3.2
+Slow scale approximations
+Similarly to Zabolotskaya et al. (2004) and Pucci et al. (2019), we proceed now to a reduction of the above wave equa-
+tion (13) for one-way wave propagation with slowly varying profile. We present two approximations based either on a
+slow space variable or a slow time variable.
+Slow space
+Let us follow the scaling procedure in Zabolotskaya et al. (2004). For this purpose, we introduce the
+following scaling defined by the change of variables {˜z = ϵ2z, ˜t = t − z/c,u = ϵ ˜u}, where ϵ is a small parameter and
+˜u = ˜u(˜z, ˜t). Furthermore, we assume that τ is of order ϵ2. Note that this set of assumptions corresponds to a slowly-
+varying profile in space.
+This Ansatz is then substituted in the equation of motion (13). At leading (cubic) order in ϵ, the motion of soft
+viscous solids is governed by the scalar equation
+ϵ3c ∂2 ˜u
+∂˜z∂˜t = ϵ3 β
+c2
+�∂ ˜u
+∂˜t
+�2 ∂2 ˜u
+∂˜t2 +ϵτ
+2
+∂3 ˜u
+∂˜t3 .
+(16)
+Transforming back to the initial displacement u and physical coordinates (z,t) leads to a reduced wave equation
+c ∂v
+∂z +
+�
+1−βv2/c2� ∂v
+∂t = τ
+2
+∂2v
+∂t2 ,
+(17)
+for the velocity v = ∂u/∂t.
+1The wave equation proposed by Catheline et al. (2003) and analysed by Jordan and Puri (2005) cannot be obtained rigorously from the equations
+of motion unless time derivatives are (questionably) replaced by spatial derivatives.
+4
+
+Up to the choice of time variable used here (i.e., the physical time t instead of the retarded time ˜t), the partial
+differential equation (17) is identical to the cubic Burgers-type equation of Zabolotskaya et al. (2004). However, the
+underlying modelling assumptions are not equivalent, since the initial wave equation (13) includes the extra nonlinear
+viscosity term 2τ∂(γ2 ˙γ)/∂z. This additional term is lost in the rescaling procedure given that it is of higher order in ϵ
+than the leading-order viscous term τ∂ ˙γ/∂z. In the end, while the modelling efforts by Destrade et al. (2013) aimed
+at enforcing objectivity lead to a slight modification of the wave equation (more precisely, the addition of a nonlinear
+viscous term), they do not induce any modification of the transport equation (17).
+Slow time
+For later comparisons, let us derive a similar Burgers-type equation governing the evolution of the strain
+instead of the velocity by following Pucci et al. (2019). To do so, we introduce the slow-time scaling based on the
+change of variables {˜t = ϵ2t, ˜z = z −ct,u = ϵ ˜u} where ϵ is a small parameter. Proceeding in a similar fashion to above,
+we end up with the nonlinear transport equation
+∂γ
+∂t +c
+�
+1+βγ2� ∂γ
+∂z = τc2
+2
+∂2γ
+∂z2 ,
+(18)
+where γ = ∂u/∂z is the shear strain. Here too, after keeping leading order terms, we have transformed back to the
+initial physical coordinates (z,t). Therefore, the above partial differential equation may be viewed as a one-way ap-
+proximation of the wave equation (15). Their travelling wave solutions are compared in the next section.
+4
+Travelling wave solutions
+4.1
+Nonlinear viscous wave equation
+Let us seek travelling wave solutions to the wave equation (15), i.e. specific smooth waveforms that propagate at a
+constant velocity with a steady profile. In a similar fashion to Destrade et al. (2013), we first introduce the following
+rescaled dimensionless variables and coordinates
+g(¯z, ¯t) =
+�
+2
+3βγ(z,t),
+¯t = t/τ,
+¯z = z/(cτ),
+(19)
+in Eq. (15), such that
+∂2g
+∂¯t2 = ∂2g
+∂¯z2 + ∂2
+∂¯z2 g 3 + ∂2
+∂¯z2
+��
+1+ 3
+β g 2
+� ∂g
+∂¯t
+�
+.
+(20)
+Next, we seek travelling wave solutions of the form g =
+�
+ν2 −1G(ξ) where ξ = (ν2 − 1)(¯t − ¯z/ν) involves the dimen-
+sionless wave velocity ν ≥ 1. Injecting this Ansatz in the above partial differential equation and integrating twice with
+respect to ξ with vanishing integration constants yields a nonlinear differential equation for the strain:
+G = G3 +
+�
+1+αG2� d
+dξG,
+(21)
+where α = 3(ν2 −1)/β is a parameter.
+From the above differential equation, one observes that travelling wave solutions to the wave equation (15) should
+connect the equilibrium strains G = 0 and G = ±1 by following a smooth transition that depends on the parameter α.
+Solutions read (Destrade et al., 2013)
+ξ = −ln
+�
+1
+2G
+�4
+3(1−G2)
+� 1+α
+2
+�
+(22)
+in implicit form, where we have enforced G(0) = 1/2 without loss of generality. Illustrations are provided later on.
+4.2
+Slow time approximation
+In a similar fashion, let us now seek travelling wave solutions to the reduced wave equation (18). Thus, we first perform
+the substitutions (19) to get
+∂g
+∂¯t + ∂
+∂¯z
+�
+g + 1
+2 g 3
+�
+= 1
+2
+∂2g
+∂¯z2 .
+(23)
+In order to obtain wave solutions that correspond to the same strain values at infinity as in Sec. 4.1, we introduce a
+slightly different scaling. Indeed, let us inject the Ansatz g =
+�
+ν2 −1G(χ) with χ = (ν2 − 1)(ϑ¯t − ¯z) in Eq. (23), where
+ϑ = 1+ 1
+2(ν2 −1) is the new dimensionless velocity (Fig. 1). Thus, we arrive at the differential equation
+G = G3 + d
+dχG
+(24)
+5
+
+1
+1.2
+1.4
+1.6
+1.8
+1
+1.5
+2
+ν
+ϑ
+Figure 1: Scaled velocity ϑ = 1+ 1
+2(ν2 −1) for the ‘slow-time’ reduced model in terms of the scaled velocity ν for the full
+wave equation.
+of which the strain values 0 and 1 are steady states. Enforcing the initial value G = 1/2 at χ = 0 gives
+G =
+1
+�
+1+3e−2χ ,
+(25)
+which does not involve any extra parameter. One observes that this expression corresponds to the case α = 0 in
+Eqs. (21)-(22).
+Remark. One might proceed in a similar fashion with the Burgers-type equation (17) corresponding to the slow space
+approximation. Similarly to (19), we perform the substitutions r(¯z, ¯t) =
+�
+2β/3v(z,t)/c in Eq. (17) to get
+∂r
+∂¯z + ∂
+∂¯t
+�
+r − 1
+2r 3
+�
+= 1
+2
+∂2r
+∂¯t2 .
+(26)
+Next, we introduce r =
+�
+ν2 −1V (ψ) where ψ = (ν2 −1)(¯t − ¯z/κ) involves the dimensionless velocity κ defined by the
+relationship κ−1 = 1− 1
+2(ν2 −1). This way, we obtain the same differential equation V = V 3 + d
+dψV for the dimension-
+less velocity V as previously for the strain (24). Therefore, within the scope of the present study, the slow time and
+slow space approximations lead to related travelling wave solutions that describe the evolution of distinct kinematic
+variables (strain and velocity, respectively).
+4.3
+Comparison
+Let us compare the solutions (22)-(25) obtained for the full wave equation (15) and the one-way model (18). First, one
+observes that these travelling waves of same amplitude do not propagate at the same speed, as illustrated in Fig. 1.
+Indeed, given the expression of ϑ, we can express the relative error E = ϑ/ν−1 on the scaled velocity as a function of
+ν. To ensure that the latter remains less than 5% (respectively 1%), we obtain the requirement ν ≤ 1.3 (resp. ν ≤ 1.1)
+marked by dotted lines in the figure.
+Now, let us observe that for a unit kink covering the range 0 ≤ G ≤ 1, the corresponding shear strains satisfy
+0 ≤ γ ≤
+�
+α/2
+(27)
+where α = 3(ν2 − 1)/β was introduced earlier on, see Eq. (19). In other words, the coefficient α in the differential
+equation (21) is related to the maximum strain of travelling waves, and these bounds are valid for both models at hand
+due to application of the rescaling procedure (19). Thus, restrictions of the wave speed ν can be expressed in terms of
+the strain. To ensure that the velocity error E remains less than 5% (respectively 1%), we therefore require γ
+�
+β ≤ 1.0
+(resp. γ
+�
+β ≤ 0.56). Note that the parameter of nonlinearity can take such values as β ≈ 10 for gels (Jacob et al., 2007).
+Therefore, the slow scale approximation has a very restricted validity for such a soft viscoelastic material.
+This property is further illustrated in Fig. 2, where we have represented the evolution of the relative velocity ν−1
+(or ϑ−1) in terms of the maximum strain amplitude, both for the full wave equation and its one-way approximation.
+According to the expression of α above, we have the relationship ϑ−1 = 1
+3β(
+�
+α/2)2 in the case of the one-way approx-
+imate model, which produce lines of slope two in log-log coordinates (dashed lines in the figure). However, for the full
+wave equation, this relationship between the wave speed ν and the strain amplitude is not satisfied. For fixed values
+of the nonlinearity parameter β, differences between the one-way model and the full wave equation become visible
+at large strains.
+In Fig. 3, we display the evolution of the waveforms (22)-(25) in terms of the scaled coordinates ξ, χ. In the case
+of the full wave equation (21), the parameter α takes the values {0,1.2,3}. It appears that the waveforms so-obtained
+follow a drastically different evolution when parameters are modified. In particular, the wavefront deduced from
+the full wave equation (solid lines) does not exhibit the same invariance and symmetry properties as the wavefront
+deduced from the one-way model (dashed line).
+6
+
+10−1
+100
+10−3
+10−2
+10−1
+100
+β = 3
+β = 1
+Strain amplitude
+Relative velocity
+one-way
+wave eq.
+Figure 2: For the full wave equation (solid line) and the ‘slow-time’ reduced model (dashed line), we represent the
+evolution of the relative velocity ν − 1 (respectively, ϑ − 1) in terms of the strain amplitude
+�
+α/2. The axes have a
+logarithmic scale.
+−2
+0
+2
+0
+0.5
+1
+α
+ξ, χ
+G
+wave eq.
+one-way
+Figure 3: Steady waveforms deduced from Eqs. (22)-(25) for increasing values of the parameter 0 ≤ α ≤ 3 (arrow).
+Evolution of the scaled shear strain G in terms of the related dimensionless coordinate ξ or χ.
+7
+
+5
+Simple waves
+In the lossless case, exact one-way wave equations can be derived by using the method of Riemann invariants, see
+for instance the introductory example by John (1976). Such particular wave solutions called simple waves keep one
+Riemann invariant constant. In other words, the particle velocity v = R−−Q(γ) withQ(γ) = c
+�γ
+0
+�
+1+2βg 2 dg depends
+explicitly on the strain γ. The scalar R− is an arbitrary constant, for instance R− ≡ 0 in some specific boundary-value
+problems (Berjamin and Chockalingam, 2022), which will be assumed satisfied from now on. Spatial differentiation
+of the velocity then produces
+∂γ
+∂t +c
+�
+1+2βγ2 ∂γ
+∂z = 0,
+(28)
+where we have used the equality of mixed partials ∂v/∂z = ∂γ/∂t. Obviously, the lossless one-way wave equation (18)
+with τ = 0 is an approximation of (28) for 2βγ2 ≪ 1.
+Let us analyse this requirement in a more quantitative manner. To ensure that the relative error on the advection
+velocity E =
+1+a
+�
+1+2a −1 for a = βγ2 remains less than 5% (respectively 1%), we obtain the requirement a ≤ 0.44 (resp.
+a ≤ 0.16). Application of the square root leads to the restriction γ
+�
+β ≤ 0.66 (resp. γ
+�
+β ≤ 0.40) which is slightly more
+constraining than in the case of viscoelastic travelling waves (Sec. 4.3).
+Along a simple wave, computation of the partial derivative of the velocity v = R− −Q(γ) with respect to time pro-
+duces
+c ∂v
+∂z +
+�
+1+2βγ2�−1/2 ∂v
+∂t = 0,
+(29)
+where the strain γ = Q−1(−v) can be expressed formally as a function of the velocity, despite no analytical expression
+of the inverse function Q−1 of Q is known in the present case. If |v| is small, then we can use the approximation
+γ ≃ −v/c of the strain which follows from the asymptotic equivalence of Q ∼ cγ at small strains. Next, the (·)−1/2-
+factor in Eq. (29) can be approximated by the polynomial expression 1−βγ2 as long as 2βγ2 ≪ 1. This way, we have
+shown that the one-way wave equation (17) is an approximation of Eq. (29) obtained for R− = 0 and 2βv2/c2 ≪ 1 in
+the elastic limit τ = 0. This observation is consistent with the discussions in Catheline et al. (2005). In summary, the
+lossless ‘slow-space’ and ‘slow-time’ reductions (17)-(18) with τ = 0 are approximate governing equations for simple
+waves with small values of βv2/c2 and of βγ2, respectively.
+6
+Conclusion
+For a specific strain-rate viscoelasticity theory of soft solids, we have shown that one-way approximate wave propaga-
+tion models produce significantly different travelling wave solutions than the full equations of motion as soon as the
+wave amplitude is not infinitesimal. Similar observations are reported in the literature in relation with shear shock
+formation (Berjamin and Chockalingam, 2022). In the elastic limit, we have examined the validity of one-way approx-
+imations in relation with simple wave theory, thus leading to dedicated criteria of validity involving small velocity and
+strain amplitudes. We conclude that these approximations should be used with care given their limited accuracy, in
+general. Nevertheless, they might remain useful for the interpretation of experimental results where their validity is
+not always severely penalised (Catheline et al., 2003, 2005).
+Acknowledgments
+The author is grateful to Michel Destrade (Galway, Ireland) for support. This project has received funding from the
+European Union’s Horizon 2020 research and innovation programme under grant agreement TBI-WAVES — H2020-
+MSCA-IF-2020 project No. 101023950.
+A
+Consequence of incompressibility
+This Appendix is devoted to the derivation of Eq. (8). We start with the Cayley–Hamilton identity for the right Cauchy–
+Green tensor C = F ⊤F, which reads
+C 3 −I C 2 +II C −III I = 0,
+(30)
+where I, II, III are the principal invariants of C. In the case of volume-preserving motions (1), the tensor C is uni-
+modular, i.e. we have III = 1. Next, multiplication of (30) by C −1 ˙E on the right side, substitution of C = I + 2E and
+computation of the trace entails the relationship
+(I4 +4I7 +4I8)−(3+2I1)(I4 +2I7)
++(3+4I1 +2I 2
+1 −2I2)I4 = 0,
+(31)
+8
+
+where we have used the incompressibility property trD = tr(C −1 ˙E) = 0, the definition of the invariants (2)-(6), and the
+relationship between I, II and the invariants Ik used here (Destrade et al., 2010). Rearranging terms, we get the desired
+identity (8).
+References
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+9
+

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+1
+Can Continuous Aperture MIMO Achieve Much
+Better Performance than Discrete MIMO?
+Zhongzhichao Wan, Jieao Zhu, and Linglong Dai, Fellow, IEEE
+Abstract—The concept of continuous-aperture multiple-
+input multiple-output (CAP-MIMO) technology has been
+proposed recently, which aims at achieving high spectrum
+density by deploying extremely dense antennas or even
+continuous antennas in a given aperture. The fundamental
+question of CAP-MIMO is whether it can achieve much
+better performance than the traditional discrete MIMO
+system. In this paper, to model the CAP-MIMO, we use self-
+adjoint operators to depict the structural characteristics of
+the continuous random electromagnetic fields from physical
+laws. Then, we propose a non-asymptotic performance
+comparison scheme between continuous and discrete MIMO
+systems based on the analysis of mutual information. We
+show the consistency of the proposed scheme by proving
+that the mutual information between discretized transceivers
+converges to that between continuous transceivers. Numeri-
+cal analysis verifies the theoretical results, and suggests that
+the mutual information obtained from the discrete MIMO
+with widely adopted half-wavelength spaced antennas al-
+most achieves the mutual information obtained from CAP-
+MIMO.
+Index Terms—Multiple-input multiple-output (MIMO),
+Continuous-aperture MIMO (CAP-MIMO), mutual infor-
+mation, random fields, Fredholm determinant.
+I. INTRODUCTION
+The spectrum efficiency of wireless communication
+systems has been greatly improved from 3G to 5G
+because of the use of multiple-input multiple-output
+(MIMO) technology [1]–[3]. The MIMO systems utilize
+multiple antennas to exploit the spatial multiplexing gain
+[4], where the antennas are modeled as discrete points
+in the continuous space. Along with the tendency of
+increasing the number of antennas to achieve higher
+spectrum efficiency, people are considering deploying
+extremely dense antennas in a given aperture [5], [6].
+When the number of antennas in a given aperture tends to
+infinity, the traditional MIMO systems with transceivers
+composed of discrete point antennas are equivalent
+to the MIMO systems with continuously controllable
+transceivers. Therefore, the MIMO with extremely dense
+All authors are with the Department of Electronic Engineer-
+ing, Tsinghua University as well as Beijing National Research
+Center
+for
+Information Science
+and
+Technology
+(BNRist), Bei-
+jing 100084, China (E-mails: {wzzc20, zja21}@mails.tsinghua.edu.cn;
+[email protected]).
+This work was supported in part by the National Key Research
+and Development Program of China (Grant No. 2020YFB1807201), in
+part by the National Natural Science Foundation of China (Grant No.
+62031019).
+antennas is called continuous-aperture MIMO (CAP-
+MIMO), and is also called holographic MIMO [7]–[9]
+or large intelligent surface [5], [10] in the recent litera-
+ture1. It has attracted increasing interest in the research
+of MIMO technology. Recent works about CAP-MIMO
+include pattern optimization [6], antenna design [11],
+channel estimation [7], and so on. For CAP-MIMO, the
+fundamental question is whether the CAP-MIMO system
+can achieve much better performance than the traditional
+discrete MIMO system.
+A. Related works
+The structure of CAP-MIMO has been defined in the
+previous part but there are many structures for realizing
+the discrete MIMO. Therefore, we need to choose which
+structure of the discrete MIMO to compare with CAP-
+MIMO. A representative structure of discrete MIMO
+uses half-wavelength spaced antennas to compose the
+transceivers [12]–[14], because half-wavelength sampling
+of the electromagnetic field can reconstruct the original
+field according to the sampling theorem.
+There have been several works discussing the per-
+formance comparison between CAP-MIMO and discrete
+MIMO with half-wavelength spaced antennas. The perfor-
+mance comparison is from the degrees of freedom (DoF)
+perspective. Specifically, when discarding the evanescent
+wave components, the Fourier transform of the received
+field, which is in the wavenumber domain, is concentrated
+in a circle or a segment. This concentration phenomenon
+means that the field is bandlimited in the wavenumber
+domain, thus it can be perfectly recovered from the half-
+wavelength sampling points in the spatial domain [15]
+according to the Nyquist sampling theorem [16]. The
+above conclusion is based on the assumption that we can
+observe the received field in the infinitely large spatial
+domain. However, in practice, the destination where we
+can observe the field is in a finitely large aperture.
+For a rigorous analysis framework of the DoF in a
+finitely large aperture, the prolate spheroidal wave func-
+tion (PSWF) [17] is introduced to perform orthogonal
+expansion on the electromagnetic field. Specifically, to
+1The MIMO with extremely dense antennas can be accurately de-
+scribed by the name CAP-MIMO, while holographic MIMO and large
+intelligent surface do not focus on the continuity of the transceiver
+apertures. Therefore, in the rest part of the paper, we will prefer using
+the name CAP-MIMO rather than using other names like holographic
+MIMO.
+arXiv:2301.08411v1  [cs.IT]  20 Jan 2023
+
+2
+reconstruct the wavenumber-bandlimited electromagnetic
+field observed in a length-l spatial region, the PSWFs
+were used as the basis based on the Slepian’s concentra-
+tion problem [18]. Such an electromagnetic field can be
+perfectly reconstructed from infinite number of PSWFs,
+and approximately reconstructed from a finite number
+of PSWFs. If the reconstruction error can be controlled
+within a given threshold by using N0 PSWFs, the number
+of DoFs of the field can be approximated by N0 [19].
+This analyzing scheme is strict for arbitrary l, but can
+only provide the asymptotic result of the DoF, i.e., the
+quantitative result of N0 can be obtained only when
+the length l or the frequency tends to infinity. However,
+the practical systems are with finitely large aperture and
+finite frequency. The asymptotic result can not provide
+quantitative number of DoFs for practical systems. There-
+fore, a non-asymptotic performance comparison scheme
+between CAP-MIMO and discrete MIMO is required for
+the accurate performance comparison with finitely large
+apertures.
+B. Our contributions
+To solve this problem, in this paper, we provide a non-
+asymptotic performance comparison scheme between
+CAP-MIMO and discrete MIMO, and we further prove
+the rationality of the scheme2. Specifically, the contribu-
+tions of this paper can be summarized as follows:
+• We build models of CAP-MIMO and discrete MIMO
+based on electromagnetic theory. For CAP-MIMO
+with continuous transceivers, we model the structural
+characteristics of the continuous random electro-
+magnetic fields from physical laws by using self-
+adjoint operators. Based on this model, we can
+utilize the spectrum theory of operators to derive the
+information that can be obtained from the received
+field. The existing models of MIMO with discrete
+transceivers are spatially discretized from the contin-
+uous model. Moreover, signal-to-noise ratio (SNR)
+control schemes are introduced to ensure the fairness
+of the comparison between CAP-MIMO and discrete
+MIMO.
+• Then, before comparing the performance between
+CAP-MIMO with continuous transceivers and tra-
+ditional MIMO with discrete transceivers, we first
+utilize the simplified model with continuous trans-
+mitter and discrete receiver. Under this simplified
+model, the transmitter is continuous, which is the
+same as that in the CAP-MIMO system. By theo-
+retically analyzing the mutual information that can
+be obtained from the discrete receiver in this sim-
+plified model, we can obtain some insights about
+how the discretization of the receiver affects the
+mutual information. Moreover, the theoretical proof
+2Simulation
+codes
+will
+be
+provided
+to
+reproduce
+the
+re-
+sults in this paper: http://oa.ee.tsinghua.edu.cn/dailinglong/publications/
+publications.html.
+of the convergence of the mutual information in the
+simplified model can inspire the analysis of a more
+practical scenario, i.e., the discrete transceivers.
+• Finally, we extend the convergence proof from the
+model with discrete receiver to the model with
+discrete transceiver. We prove that the mutual in-
+formation between the discrete transceivers con-
+verges to the mutual information between continuous
+transceivers when the number of antennas of the
+discretized transceivers tends to infinity. Therefore,
+the fairness of the performance comparison is guar-
+anteed. Numerical results are provided to verify the
+theoretical analysis. Moreover, it shows the near-
+optimality of the half-wavelength sampling of the
+transceivers in traditional discrete MIMO.
+C. Organization and notation
+Organization: The rest of our paper is organized as
+follows. Section. II introduces the basic model of EIT and
+proposes models with continuous or discrete transceivers.
+The mutual information between the transceivers is also
+derived. Section. III proves the convergence of the mutual
+information between continuous transmitter and discrete
+receiver when the number of discrete antennas increases.
+Then, the convergence of the mutual information between
+discrete transceivers is illustrated in Section IV. Finally,
+we conclude the paper in Section V.
+Notation: bold characters denote matrices and vectors;
+j is the imaginary unit; E [x] denotes the mean of random
+variable x; x∗ denotes the conjugation of a number or
+a function x; XH denotes the conjugate transpose of
+a vector or a matrix X; µ0 is the permeability of a
+vacuum, Z0 is the free-space intrinsic impedance and
+c is the speed of light in a vacuum; ∇ is the nabla
+operator, and ∇× is the curl operator; |φ⟩ is the quantum
+mechanical notation of a function φ, where the inner
+product is denoted by ⟨ψ| φ⟩; det(·) denotes the matrix
+determinant or the Fredholm determinant; tr(·) denotes
+the trace of a matrix or an operator. Im denotes the m×m
+identity matrix, 1 denotes the indentity operator, δ(x)
+denotes the delta function, and 1i=j denotes the indicator
+function; |x| denotes the modulus of a complex variable,
+and ∥f(x)∥L∞(a,b) is the uniform norm of the function
+f(x) over the interval [a, b]. C∞(K) denotes the set of
+smooth functions supported on a compact set K.
+II. MODELS OF CONTINUOUS AND DISCRETE
+SYSTEMS
+In this section, we introduce the models of contin-
+uous and discrete systems for performance comparison
+between CAP-MIMO and discrete MIMO. We control
+the SNR at the receiver side to ensure the fairness of the
+comparison. The information obtained from these models
+is derived from operators and matrices.
+
+3
+A. Basic model of electromagnetic information theory
+To model the transceivers and the channel, we follow
+the approach of electromagnetic information theory (EIT).
+The EIT is an interdisciplinary subject that integrates the
+classical electromagnetic theory and information theory to
+build an analysis framework for the ultimate performance
+bound of wireless communication systems [20]. The anal-
+ysis framework of EIT is based on spatially continuous
+electromagnetic fields, which provides us the tool to
+model and analyze the continuous transceivers. Then, for
+the consistency, the model of discrete transceivers are
+viewed as the discretization of the continuous model from
+EIT.
+The model of EIT is built on the vector wave equa-
+tion [21] without boundary conditions, which is expressed
+by
+∇×∇×E (r)−κ2
+0E (r) = jωµ0J (r) = jκ0Z0J (r) , (1)
+where κ0 = ω√µ0ε0 is the wavenumber, and Z0 =
+µ0c = 120π [Ω] is the free-space intrinsic impedance.
+We assume that the transceivers are confined in two
+regions Vs and Vr, separately. The current density at the
+source is J(s), where s ∈ R3 is the coordinate of the
+source. The induced electric field at the destination is
+E(r), where r ∈ R3 is the coordinate of the field observer.
+To solve the linear partial differential equation (1), a
+general theoretical approach is to introduce the dyadic
+Green’s function G(r, s) ∈ C3×3. According to the
+linearity of (1), the electric field E(r) can be expressed
+by
+E(r) =
+�
+Vs
+G(r, s)J(s)ds,
+r ∈ Vr.
+(2)
+By exploiting the symmetric properties of the free space,
+the Green’s function in unbounded, homogeneous medi-
+ums at a fixed frequency point is [22]
+G(r, s) = jκ0Z0
+4π
+�
+I + ∇r∇H
+r
+κ2
+0
+� ejκ0∥r−s∥
+∥r − s∥
+= jκ0Z0
+4π
+ejκ0∥r−s∥
+∥r − s∥
+�
+�
+I − ˆpˆpH�
++
+j
+2π ∥r − s∥ /λ
+�
+I − 3ˆpˆpH�
+−
+1
+(2π ∥r − s∥ /λ)2
+�
+I − 3ˆpˆpH�
+�
+[Ω/m2],
+(3)
+where ˆp =
+p
+∥p∥ and p = r − s.
+Since there are some non-ideal factors at the receiver
+that corrupts the recieved field, we call them the noise
+field N(r). The received electric field can be expressed
+by Y(r) = E(r) + N(r). The above equations represent
+the deterministic model in the electromagnetic theory. To
+satisfy the demand of wireless communication, we need
+to convey information through the electromagnetic field.
+Specifically, the wireless communiation system encodes
+the information in the current J(s), and decodes the
+information from the noisy electric field Y(r). Due to
+the randomness of the transmitted bit source, the electro-
+magnetic fields are randomly excited by the transmitter
+equipments before being radiated into the propagation
+media. Therefore, the electromagnetic fields should be
+modeled as random fields [23], which are random func-
+tions with several arguments. We denote the autocorre-
+lation function of the current and the electric field as
+matrix-valued functions RJ(s, s
+′) = E[J(s)JH(s
+′)] and
+RE(r, r
+′) = E[E(r)EH(r
+′)]. The relationship between
+RJ and RE is determined by the Green’s function, which
+is
+RE(r, r′) =
+�
+Vs
+�
+Vs
+G(r, s)RJ(s, s′)GH(r, s)dsds′.
+(4)
+Similar definitions of the autocorrelation functions for
+the noise field and the noisy electric field are repre-
+sented as RN(r, r
+′) = E[N(r)NH(r
+′)] and RY(r, r
+′) =
+E[Y(r)YH(r
+′)].
+B. Continuous trasceivers
+In this part, we will build the model of CAP-MIMO
+with continuous transceivers based on the EIT model
+in the above subsection, and then derive the mutual
+information between the continuous transceivers. For sim-
+plicity, in the rest part of the paper, we assume that the
+transceivers are linear along the ˆz-direction. Moreover,
+since the current J can only exist on the linear source
+and we only observe the electric field on the linear
+receiver, we express all the physical quantities in a
+Cartesian coordinate system that satisfies s = (0, 0, s) and
+r = (d, 0, r), where d is the distance between the parallel
+source and destination line. This model corresponds to
+single-polarized linear antennas. Through this simplifica-
+tion scheme, we use J(s) and E(r) instead of J(s) and
+E(r). The relationship between them can be expressed by
+E(r) =
+� l
+0 G(r, s)J(s)ds, where G(r, s) is the upper left
+element of the matrix G(r, s). We can derive G(r, s) as
+G(r, s) =jZ0ej2π
+√
+x2+d2/λ
+2λ
+√
+x2 + d2
+�
+j
+2π
+√
+x2 + d2/λ
+d2 − 2x2
+x2 + d2
++
+d2
+x2 + d2 −
+1
+(2π/λ)2(x2 + d2)
+d2 − 2x2
+x2 + d2
+�
+,
+(5)
+where x = r − s and λ = 2π/κ0 is the wavelength.
+Here we consider the scenario with no channel state
+information, which means that the signals on the source
+are under equal power allocation. The second moments
+(autocorrelation) of J are denoted by RJ(s, s′) = Pδ(s−
+s′), s, s′ ∈ [0, l].
+Since the noiseless received field is uniquely deter-
+mined by the source and the deterministic channel, the
+autocorrelation function of the electric field is expressed
+
+4
+by the source autocorrelation RJ(s, s′) and the Green’s
+function G(r, s), written as
+RE(r, r′) =
+� l
+0
+� l
+0
+G(r, s)RJ(s, s′)G∗(r′, s′)dsds′
+= P
+� l
+0
+G(r, s)G∗(r′, s)ds.
+(6)
+The received field on the destination is Y (r)
+=
+E(r) + N(r), where N(r) is the noise field at the
+receiver. In this paper, we consider thermal noise model
+E [N(r)N ∗(r′)] =
+n0
+2 δ(r − r′). According to [24], we
+can perform Mercer expansion on the electric field E(r)
+to obtain a set of mutually independent random variables
+ξk. The expansion can be written as E(r) = �
+k ξkφk(r),
+where E[ξkiξ∗
+kj] = λki1i=j and ⟨φki(r), φkj(r)⟩ = δkikj.
+This expansion scheme has split the continuous field
+into independent components. Since the white noise field
+can be expanded under arbitrary orthogonal bases, the
+continuous channel is also decomposed into independent
+subchannels, which makes the mutual information of the
+subchannels summable.
+Next we will show that for the operator TE := φ(r) →
+� l
+0 KE(r, r′)φ(r′)dr′, where KE(r, r′) = RE(r, r′) =
+P
+� l
+0 G(r, s)G∗(r′, s)ds, all of its eigenvalues are real
+and nonnegative. Moreover, the sum of its eigenval-
+ues �∞
+i=1 λi equals P
+� l
+0
+� l
+0 G(r, s)G∗(r, s)drds. Notice
+that TE can be decomposed to T ∗T, where T
+:=
+φ(r) →
+√
+P
+� l
+0 G(r, s)φ(r)ds and T ∗
+:= φ(r) →
+√
+P
+� l
+0 G∗(r, s)φ(r)ds. This decomposition means that
+TE = T ∗
+E is a self-adjoint operator. We assume that λ
+is an eigenvalue of TE and φ(r) is the corresponding
+eigenfunction. Since
+λ = λ⟨φ(r), φ(r)⟩ = ⟨T ∗
+Eφ(r), φ(r)⟩
+= ⟨φ(r), TEφ(r)⟩ = λ∗,
+(7)
+we know that λ is real. From
+λ = λ⟨φ(r), φ(r)⟩ = ⟨T ∗Tφ(r), φ(r)⟩
+= ⟨Tφ(r), Tφ(r)⟩ ⩾ 0,
+(8)
+we know that λ is nonnegative.
+From [25] we know that an integral operator on [a, b]
+is a trace class operator if its kernel K(x, y) satisfies
+K(x, y) and ∂yK(x, y) are continuous on [a, b]2. There-
+fore TE is a trace class operator, which means that the
+sum of its eigenvalues is finite and can be expressed
+by [26]
+tr(TE) =
+� l
+0
+KE(r, r)dr = P
+� l
+0
+� l
+0
+G(r, s)G∗(r, s)drds.
+(9)
+Corollary 1. The non-negative values
+λk
+n0/2 represent
+the SNR of the independent subchannels. The mutual
+information between the noisy received field and the
+current on the source can be expressed by
+I0(J; Y ) =
++∞
+�
+k=1
+log
+�
+1 +
+λk
+n0/2
+�
+.
+(10)
+By introducing the Fredholm determinant which is the
+determinant of operators, we can express (10) by
+I0(J; Y ) = log det
+�
+1 + TE
+n0/2
+�
+,
+(11)
+where (TEφ)(r) :=
+� L
+0 RE(r, r′)φ(r′)dr′ and λk are the
+eigenvalues of TE.
+Remark 1. Our analysis here is based on the sim-
+plified model with uni-polarized linear antennas as the
+transceiver. This simplification reduces the dimension of
+the problem, where random fields degenerate to one-
+dimensional random processes. For the more general
+scenarios, such analyzing schemes are still effective. If
+the random field is defined in a region X, we can expand
+E(r) by E(r) = �
+k ξkΦk(r) and its autocorrelation
+function RE(r, r
+′) by RE(r, r
+′) = �
+k λkΦk(r)ΦH
+k (r
+′)
+[27]. The expansion satisfies that λk and Φk(r) are
+eigenvalues and eigenfunctions of the integral equation
+�
+X RE(r, r
+′)Φ(r
+′)dr
+′ = λkΦ(r). Similar expressions of
+the mutual information in (10) and (11) can be derived.
+C. Continuous transmitter and discrete receiver
+Before building the model with discrete transceivers,
+in this subsection, we will first build a simplfied model
+with continuous transmitter and discrete receiver. The
+simplfied model analyzed here can bring some insights
+about the discretization of both transceivers and the
+SNR control schemes. For the continuous transmitter,
+we still use the length-l linear transmitter along the ˆz-
+direction. For the discrete receiver, we build a model
+with m point antennas on a segment parallel to the
+linear transmitter in the destination region. The ith point
+antenna is placed on ri ∈ [0, l]. The correlation matrices
+of the received signals and received noise are denoted
+by K
+′
+E and K
+′
+N. For the received signals, we assume
+that it is the sampling of the continuous electric field
+on the point ri, which means that K
+′
+E = KE(ri, rj).
+However, for the received noise on the antenna, it can
+not directly be assumed as the point sampling of the
+noise field, because of the delta function. To solve this
+problem, we assume that K
+′
+N =
+n1
+2 Im is an identity
+matrix, and control the signal-to-noise ratio (SNR) of this
+model the same as that of the continuous model to ensure
+the fairness of the comparison. The SNR at the receiver
+of the continuous model is �∞
+i=1
+λi
+n0/2, where λi is the
+ith eigenvalue of the operator TE. From Lemma 1 we
+know that �∞
+i=1
+λi
+n0/2 =
+P
+n0/2
+� l
+0
+� l
+0 G(r, s)G∗(r, s)drds
+is finite. The SNR at the receiver of the discrete model is
+�m
+i=1
+λ
+′
+i
+n1/2, where λ
+′
+i is the ith eigenvalue of the matrix
+K
+′
+E.
+The SNR control scheme is necessary because if we
+do not control the SNR, the mutual information that can
+be obtained from the discrete antennas in the receiver
+may infinitely increase. Let us take a counter-example
+where the power of received signal and received noise on
+
+5
+each point antenna remain unchanged when the number
+of antennas in a given aperture increases. For dense
+antennas we can assume that N received signals of the
+antennas in a small aperture are nearly the same, while
+the corresponding noises are independent according to
+the model. Then, the SNR for the N antennas will
+keep near-linearity increasing with N, since when we
+perform combing of the N received signals we have
+SNR =
+E[(�N
+i=1 Ei)(�N
+i=1 E∗
+i )]
+E[(�N
+i=1 Ni)(�N
+i=1 N ∗
+i )] ≈ N E[E1E∗
+1 ]
+E[N1N ∗
+1 ]. Therefore
+the mutual information that can be obtained from the
+N antennas will keep near-logarithm increasing with N,
+which corresponds to the simulation in [28].
+According to (9), the noise power in the discrete
+receiver model can be controlled by
+n1 = n0
+�m
+i=1 KE(ri, ri)
+� l
+0 KE(r, r)dr
+.
+(12)
+We denote the determinant of matrix K ∈ Cm×m by
+det(Ki,j)m
+i,j=1. Then we can express the mutual informa-
+tion between the transceivers by
+I1 = log
+�
+det(K
+′
+N + K
+′
+E)
+det(K
+′
+N)
+�
+= logdet
+�
+1i=j + KE(ri, rj)
+n1/2
+�m
+i,j=1
+.
+(13)
+Remark 2. Here the SNR on each of the point antennas in
+the discrete model changes with the density of point anten-
+nas. Notice that L
+m
+�m
+i=1 KE(ri, ri) is the approximation
+of the integral
+� l
+0 KE(r, r)dr. When m approximates
+infinity, n1 will approximate mn0
+2l . This phenomenon has
+several annotations, including the increase of the noise
+power on each point antenna, the reduction of antenna
+efficiency, and the corollary of the discretization of EIT
+continuous models.
+From the perspective of noise power, we can explain it
+by spatial sampling. For the point antenna arrays, more
+antennas on a given aperture corresponds to a higher
+sampling rate in the spatial domain and a wider lowpass
+filter in the wavenumber domain. Since a wide lowpass
+filter can receive more noise power from the white noise
+field, the noise power should increase with the density of
+the antennas.
+From the perspective of antenna efficiency, the well-
+known Hannan’s efficiency shows that for both transmit-
+ting and receiving antennas, the antenna gain is propor-
+tional to lxly for two-dimensional surface antennas [29].
+Therefore, for the linear model we considered, the an-
+tenna gain will be inversely proportional to the sampling
+number when the antennas are dense enough.
+Besides these two annotations, another perspective is
+viewing the model of discrete point antennas as the
+discretization from the EIT continuous model. If we
+consider m linear continuous antennas instead of point
+antennas in the destination region. All the antennas are
+connected head to tail to occupy the [0, l] position in the
+space and detect the electric field by inner producting it
+with its eigenmode. This model fulfills the requirement of
+discretizing the continuous receiver to discrete receiving
+antennas. The signal received by the ith antenna is
+Yi =
+� ai+1
+ai
+Y (r)φ(r)dr, where [ai, ai+1] is the occupied
+region of the ith antenna, and φ(r) is the eigenmode of the
+antenna. If we assume φ(r) ≡ 1, the correlation matrix
+of the received electric field can be expressed by
+(KE)i,j = E
+�� ai+1
+ai
+� aj+1
+aj
+E(r)E∗(r′)drdr′
+�
+= (ai+1 − ai)(aj+1 − aj)KE(ri, rj),
+(14)
+where ri ∈ [ai, ai+1] and rj ∈ [aj, aj+1] according to
+the mean value theorem for integrals. For the noise field
+on the destination, we have
+(KN)i,j = E
+�� ai+1
+ai
+� aj+1
+aj
+N(r)N ∗(r′)drdr′
+�
+=
+�
+(ai+1 − ai) n0
+2
+i = j
+0
+i ̸= j .
+(15)
+Therefore, the SNR after the discretization will discreases
+by ai+1 − ai, which is the case when the antennas are
+dense enough.
+After explaining the rationality of the SNR control
+scheme, we will introduce the following lemma to show
+the convergence of the noise power on each discrete point
+antenna, which will be useful for the following proofs.
+Lemma 1. When the number of antennas m in a given
+aperture increases, the noise power on each antenna n1/2
+will approach
+mn0
+2l . The difference between them is at
+most inverse-proportional to m.
+Proof: From (12) and the middle point quadrature
+rule, we have
+����
+l
+mn1 − n0
+���� = n0
+���
+� l
+0 KE(r, r)dr − l/m �m
+i=1 KE(ri, ri)
+���
+���
+� l
+0 KE(r, r)dr
+���
+⩽
+n0l3 ���K
+′′
+E(r, r)
+���
+L∞(0,l)
+24m2
+���
+� l
+0 KE(r, r)dr
+���
+,
+(16)
+which completes the proof.
+D. Discrete transceivers
+The models discussed in the above subsections keep
+the transmitter continuous and only perform discretization
+on the receiver. However, the commonly used model
+to depict wireless communication is the discrete MIMO
+model, in which both the transceivers are modeled as
+discrete point antennas. Therefore, in this section, we
+will introduce a model which discretizes the transceivers
+simultaneously, which is the extension of the model with
+continuous transmitter and discrete receiver. Then, similar
+
+6
+d
+Continuous 
+Transmitter
+Continuous 
+Receiver
+d
+Discrete 
+Receiver
+l
+/ 2
+
+l
+d
+Discrete 
+Receiver
+l
+/ 2
+
+Discrete 
+Transmitter
+Continuous 
+Transmitter
+0I
+1I
+2I
+Fig. 1. Comparison between the three models in this section with continuous transceivers and the model with discrete transceivers.
+to the scheme in the above subsection, we will provide the
+corresponding SNR control scheme to ensure the fairness
+of the comparison.
+Specifically, we build a model with m point antennas
+on a length-l segment in the source region and m point
+antennas on a length-l segment in the destination region.
+Similar to the above subsection, we assume that the ith
+point antenna is placed at si in the source region and ri
+in the destination region. The correlation matrix of the
+signals in the source region is set to be an identity matrix
+K
+′′
+J = PIm, which corresponds to the power allocation
+scheme with no channel state information at the transmit-
+ter. The channel gain from the ith antenna in the source
+region and the jth antenna in the destination region can be
+expressed by Hi,j = G(si, rj). The correlation matrix of
+the received signal is denoted by K
+′′
+E = HK
+′′
+JHH. The
+noise matrix is denoted by K
+′′
+N = n2
+2 Im. Similar SNR
+control on the receiver side is used, which is expressed
+by
+n2 = n0
+�m
+i=1
+�m
+j=1 G(ri, sj)G∗(ri, sj)
+� l
+0
+� l
+0 G(r, s)G∗(r, s)drds
+.
+(17)
+The mutual information between the transceivers is ex-
+pressed as:
+I2 = log
+�
+det(K
+′′
+N + K
+′′
+E)
+det(K
+′′
+N)
+�
+= logdet
+�
+1i=j +
+�
+k G(ri, rk)G∗(rj, rk)
+n2/2
+�m
+i,j=1
+.
+(18)
+The comparison between the three models built in Sec-
+tion. II-B, Section. II-C and in this subsection is shown
+in Fig. 1. In the following two sections we will introduce
+the intermediate quantity I
+′
+0 and I
+′′
+0 to theoretically prove
+that I1 and I2 converge to I0. The flow chart of the proof
+is shown in Fig. 2
+III. PERFORMANCE COMPARISON BETWEEN DISCRETE
+AND CONTINUOUS RECEIVERS
+In the above section we have proposed the mod-
+els of continuous and discrete transceivers and derived
+the corresponding mutual information. Before compar-
+ing the performance between CAP-MIMO with contin-
+uous transceivers and traditional MIMO with discrete
+transceivers, we first utilize the simplified model with
+continuous transmitter and discrete receiver in this sec-
+tion. Under this simplified model, the transmitter is con-
+tinuous, which is the same as that in the CAP-MIMO
+system. The comparison is based on the convergence
+analysis of the mutual information when the number
+of antennas in the discrete receiver increases. Numer-
+ical analysis is provided to verify the correctness of
+the convergence analysis. The discussion about discrete
+transceivers inspired by the analysis in this part will be
+in the next section.
+A. Convergence analysis of the mutual information
+To compare the mutual information I0 and I1, we intro-
+duce an intermediate quantity I
+′
+0 = logdet
+�
+1 + mTE
+ln1/2
+�
+.
+We can bound |I0 −I1| by |I0 −I
+′
+0|+|I1 −I
+′
+0|. According
+to [30], I1 can be viewed as the approximation of I
+′
+0
+using a numerical integral scheme. In our discussion the
+point antennas in the destination region are evenly spaced,
+which means that ai = (i−1)l/m and ri = (i−0.5)l/m.
+To bound |I1 − I
+′
+0|, we introduce the following lemma
+from [30]:
+Lemma 2. We define d(z) := det(1+zT) and dQ(z) :=
+det (1i=j + wjzK(ri, rj))m
+i,j=1, where K is the kernel of
+the operator T. The difference between d(z) and dQ(z)
+
+7
+0I
+'
+0I
+1I
+2I
+''
+0I
+Discretize the receiver
+Discretize the transmitter
+Discretize the transceivers
+Lemma 1
+Lemma 2
+Lemma 3
+Lemma 4
+Lemma 5
+Theorem 1
+Theorem 2
+Fig. 2. Flow chart of the proof in this paper.
+is
+d(z) − dQ(z) =
+∞
+�
+n=1
+zn
+n!
+�
+Qn
+m(Kn)
+−
+�
+[a,b]n Kn(x1, · · · , xn)dx1 · · · dxn
+�
+,
+(19)
+where Kn(x1, · · · , xn)
+=
+det (K(xi, xj))n
+i,j=1, and
+Qn
+m(f) = �m
+j1=1,··· ,jn=1
+�n
+i=1 wjif(rj1, · · · , rjn).
+Lemma 2 provides a method to compare the difference
+between a Fredholm determinant of operator and a clas-
+sical determinant of matrix. In our model, the operator T
+corresponds to the integral operator TE, z equals 2m
+ln1 ,and
+wj = l/m according to the equally spaced antennas.
+Notice that Qn
+m(f) is the numerical approximation of
+the integral
+�
+[a,b]n Kn(x1, · · · , xn)dx1 · · · dxn, we need
+to use numerical integral theory to estiamte the approxi-
+mation error. For the model with equally spaced antennas,
+this expression corresponds to a multivariate m-point
+composite midpoint quadrature rule.
+For the error bound of a m-point composite midpoint
+quadrature [31], we have
+�����Qm(f) −
+� l
+0
+f(x)dx
+����� ⩽
+l3
+24m2 ∥f
+′′∥L∞(0,l)
+(20)
+According to [32], the numerical approximation error
+for multiple integrals in a n-dimensional unit cube can be
+bounded by
+�����
+�
+Gn
+f −
+� n
+�
+i=1
+�
+Qi(f)
+����� ⩽ E1 +
+n
+�
+i=2
+i−1
+�
+j=1
+WjEi,
+(21)
+where Qj(g) := �
+j wi,jg(xi,j), Wi = �
+j |wi,j| and
+Ei ⩾
+���Qi(f; xi) −
+� 1
+0 f(x1, · · · , xn)dxi
+���. According to
+the models in this paper, we have wi,j = l/m and Wi = l.
+By simple variation of the integral band, we can bound the
+approximation error of the multi-dimensional numerical
+integral quadrature rule by
+�����Qn
+m(Kn) −
+�
+[0,l]n Kn(x1, · · · , xn)dx1 · · · dxn
+�����
+⩽ ni−1
+n
+�
+i=1
+Ei,
+(22)
+where
+Ei =
+�����Qi(Kn; xi) −
+� l
+0
+Kn(x1, · · · , xn)dxi
+����� .
+(23)
+Therefore, we have
+�����Qn
+m(Kn) −
+�
+[0,l]n Kn(x1, · · · , xn)dx1 · · · dxn
+�����
+⩽ nln+2
+24m2 |Kn|2
+(24)
+where |Kn|2 = max
+i ∥ ∂2Kn
+∂x2
+i ∥L∞((0,l)n).
+Similar to [30, Lemma A.4], we can bound |Kn|k by
+using the Hadamard’s inequality, which leads to
+|Kn|k ⩽ 2knn/2
+�
+� max
+i+j⩽k
+�����
+∂i
+x∂j
+yK(x, y)
+∂xi∂yj
+�����
+L∞((0,l)2)
+�
+�
+n
+.
+(25)
+Next we will show that
+���
+∂i
+x∂j
+yK(x,y)
+∂xi∂yj
+��� is upper-bounded.
+Since we have K(x, y) =
+� l
+0 G(x, s)G∗(y, s)ds, we
+will first analyze the property of G(x, s). We decom-
+pose G(x, s) as G1(x, s) + jG2(x, s), where G1, G2 ∈
+C∞([0, l]2). The smoothness of G1, G2 in their domains
+
+8
+is trivial since they are compositions of polynomial func-
+tions, trigonometric functions and square root functions.
+Consider the integral kernel K(x, y) expressed in terms
+of G1, G2, i.e.,
+K(x, y) =
+� l
+0
+�
+G1(x, s)G1(y, s) + G2(x, s)G2(y, s)
+�
+ds
++ j
+� l
+0
+�
+G1(y, s)G2(x, s) − G1(x, s)G2(y, s)
+�
+ds.
+(26)
+Since G1(x, s) and G2(y, s) are smooth in [0, l]2,
+we can conclude that f1(x, y) = G1(x, s)G1(y, s) +
+G2(x, s)G2(y, s) and f2(x, y) = G1(y, s)G2(x, s) −
+G1(x, s)G2(y, s) are smooth in the same domain. Since
+compactly supported smooth functions attain their maxi-
+mum or minimum values, the partial derivatives of K(·, ·)
+are upper-bounded for any order i, j, i.e.,
+����
+∂i+jK(x, y)
+∂xi∂yj
+���� < ∞,
+∀i, j.
+(27)
+Therefore, by substituting (24) and (25) into Lemma
+2, we can bound the difference between the mutual
+information I
+′
+0 and I1 by the following lemma:
+Lemma 3. The mutual information I1 converges to the
+mutual information I
+′
+0. The difference
+���I1 − I
+′
+0
+��� is at most
+inverse-proportional to m2.
+Proof: From (6) we know that for the operator
+TE, the kernel function can be expressed by K(x, y) =
+� L
+0 g(x, s)g∗(y, s)ds. From (25) we have
+|Kn|2 ⩽ 4nn/2An,
+(28)
+where A = max
+��� ∂i+jK(x,y)
+∂xi∂yj
+���
+L∞((0,l)2) is a constant.
+Therefore we have
+|d(z) − dQ(z)| ⩽
+∞
+�
+n=1
+zn
+n!
+nln+2
+24m2 max
+i
+����
+∂2Kn
+∂x2
+����
+L∞((0,l)n)
+⩽
+∞
+�
+n=1
+zn
+n!
+nln+2
+6m2 nn/2An.
+(29)
+According to the Stirling’s approximation, we have n! ⩾
+nne−n√
+2πn, which leads to
+|d(z) − dQ(z)| ⩽
+l2
+6m2
+∞
+�
+n=1
+� n
+2π
+(Aezl)n
+nn/2
+.
+(30)
+Since it is obvious that �∞
+n=1
+� n
+2π
+(Aezl)n
+nn/2
+is convergent,
+the difference between d(z) and dQ(z) is proportional to
+m−2. For the difference between mutual information I1
+and I
+′
+0, we have
+|I1−I
+′
+0| ⩽
+|d(z) − dQ(z)|
+min(d(z), dQ(z)) <
+l2
+6m2
+∞
+�
+n=1
+� n
+2π
+(Aezl)n
+nn/2
+,
+(31)
+where z =
+m
+ln1/2. From Lemma 1 we know that
+l
+mn1 ⩾
+n0 −
+n0l3���K
+′′
+E(r,r)
+���
+L∞(0,l)
+24m2|
+� l
+0 KE(r,r)dr| . Therefore, z is upperbounded,
+which completes the proof of Lemma 3.
+According to Lemma 1 and Lemma 3, we have
+Theorem 1, which bounds the difference between I0 and
+I1.
+Theorem 1. The mutual information I1 that can be
+obtained from the discrete receiver converges to the
+mutual information I0 that can be obtained from the
+continuous receiver when the number of points increases.
+The convergence rate is at least inverse-proportional to
+the square of the sampling number m.
+Proof: Since the Fredholm determinant f(z) =
+det(1 + zTE) is an analytic function, we have
+|det(1 + zTE) − det(1 + z1TE)|
+= |z − z1|
+����
+∂det(1 + xTE)
+∂x
+����
+x∈[min(z,z1),max(z,z1)]
+.
+(32)
+In our assumption z
+=
+2
+n0
+and z1
+=
+2m
+ln1 . The
+analycity of det(1 + zTE) implies that
+∂det(1+xTE)
+∂x
+is also an anlytic function and is bounded on the
+interval
+[min(z, z1), max(z, z1)].
+We
+denote
+M
+=
+max
+x
+��� ∂det(1+xTE)
+∂x
+���, where x ∈ [min(z, z1), max(z, z1)].
+From Lemma 1 we have
+����
+l
+mn1 − n0
+���� ⩽
+n0l3 ���K
+′′
+E(r, r)
+���
+L∞(0,l)
+24m2 � l
+0 KE(r, r)dr
+.
+(33)
+Since n1/m → n0/l when m approximates infinity, we
+denote the minimum value of n1/m by c. Therefore,
+���det(1 +
+2
+n0 TE) − det(1 + 2m
+ln1 TE)
+��� can be bounded by
+����det(1 + 2
+n0
+TE) − det(1 + 2m
+ln1
+TE)
+����
+⩽
+Ml2 ���K
+′′
+E(r, r)
+���
+L∞(0,l)
+12m2c
+� l
+0 KE(r, r)dr
+.
+(34)
+Similar to the direvation of (31), we know that when m
+increases, I
+′
+0 will converge to I0. The convergence rate
+is at least inverse-proportional to m2. From Lemma 3
+we know that
+���I1 − I
+′
+0
+��� is at most inverse-proportional to
+m2. Since |I0 − I1| ⩽
+���I0 − I
+′
+0
+��� +
+���I1 − I
+′
+0
+���, Theorem 1
+is proved.
+Remark 3. Theorem 1 shows that the SNR control
+scheme between the discrete and continuous models is
+appropriate, since the limit of the mutual information of
+the discrete model is proved to be that of the continu-
+ous model. That is to say, our proposed model is self-
+consistent. Therefore, we can use the proposed model
+to compare the mutual information from the discrete
+and continuous receivers. Our analysis is based on
+
+9
+0
+50
+100
+150
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+continuous receiver
+discrete receiver
+5 10 15 20
+1.5
+2
+2.5
+3
+continuous receiver
+discrete receiver
+Fig. 3. The mutual information as a function of the sampling number.
+The transmitter is kept continuous and the receiver is discretized.
+RE(r, r′) = P
+� l
+0 G(r, s)G∗(r′, s)ds which corresponds
+to the scenario when no CSI can be obtained at the
+transmitter but not limited to this scenario. It can be eas-
+ily extended to other shapes of autocorrelation functions
+after power allocation at the transmitter, as long as the
+analyticity of RE(r, r′) is guaranteed.
+B. Numerical analysis about the mutual information
+As proven in the above subsection, the mutual infor-
+mation between the continuous transmitter and discrete
+receiver converges to the mutual information between
+continuous transceivers. Therefore, the model of the dis-
+crete receiver can be viewed as the discretization of the
+continuous receiver. In this subsection, we will use nu-
+merical analysis to show the correctness of the theoretical
+results. Moreover, we will show the near-optimality of the
+discrete receiver with half-wavelength sampling.
+We set the length l of the transceivers to 2 m. The trans-
+mitter is kept continuous, while the receiver is discretized
+to m point antennas. The wavelength of the electromag-
+netic field is fixed to 0.04 m, which correpsonds to the fre-
+quency of 7.5 GHz. The distance between the transceivers
+varies from 10 m to 0.1 m. The simulation results are
+shown in Fig. 3. From the simulation, we can observe
+the convergence of the mutual information between the
+continuous transmitter and the discrete receiver, which
+verifies the theoretical analysis. For the three distances
+between transceivers, the half-wavelength sampling al-
+most achieves the supremum mutual information between
+continuous transceivers. Therefore, half-wavelength sam-
+pling of the receiver is suboptimal. Moreover, when the
+distance between transceivers decreases, we can observe
+that the mutual information converges slower. When the
+distance equals 0.1 m, the half wavelength sampling is at
+the critical state of convergence. If the distance is less
+than 0.1 m, a performance gap between the model with
+the continuous receiver and that with the discrete receiver
+may be observed. This performance gap has theoretical
+meaning but may not be useful because the distance will
+be comparable to the wavelength in this scenario, where
+the evanescent wave components will hold a dominant
+position.
+IV. COMPARISON BETWEEN CONTINUOUS AND
+DISCRETE TRANSCEIVERS
+In the above section we have compared the mutual
+information between the models with continuous and
+discrete receivers. For both models the transmitter is kept
+continuous, which simplifies the analyzing procedure. In-
+spired by the analysis in the above section, in this section
+we will compare the mutual information between contin-
+uous transceivers and that between discrete transceivers.
+Numerical analysis is then provided to show the near-
+optimality of the half-wavelength sampling scheme.
+A. Convergence analysis of the mutual information
+The analysis in this section focuses on the difference
+between I0 and I2. It is an extension of the conver-
+gence analysis in the above section. We define I
+′′
+0
+=
+logdet
+�
+1 + m2TE
+l2n2/2
+�
+as an intermediate variable similar
+to I
+′
+0. First we will discuss the convergence of |I0 − I
+′′
+0 |
+in the following lemma:
+Lemma 4. The mutual information I
+′′
+0 converges to the
+mutual information I0. The difference
+���I0 − I
+′′
+0
+��� is at most
+inverse-proportional to m2.
+Proof: From the SNR control scheme of discrete
+transceivers (17) and the multivariate m-point composite
+midpoint quadrature rule, we have
+����n0 − l2
+m2 n2
+���� =
+n0
+� l
+0 K(r, r)dr
+�����
+� l
+0
+� l
+0
+g(r, r, z)dzdr
+− l2
+m2
+m
+�
+i=1,j=1
+g(ri, ri, rj)
+�����
+⩽
+n0l4
+24m2 � l
+0 K(r, r)dr
+� ����
+∂2g(r, r, z)
+∂r2
+����
+L∞((0,l)2)
++
+����
+∂2g(r, r, z)
+∂z2
+����
+L∞((0,l)2)
+�
+,
+(35)
+where g(x, y, z)
+:=
+G(x, z)G∗(y, z), ri
+=
+(i −
+0.5)l/m. It is obvious that n2/m2 converges to n0/l2
+when m
+→
+∞. We denote the minimum value
+of n2/m2 by c. Then, according to (32), we know
+that
+���det(1 +
+2
+n0 TE) − det(1 + 2m2
+l2n2 TE)
+��� converges to 0
+when m → ∞. Therefore, |I0 − I
+′′
+0 | converges to 0, and
+the convergence rate is at least inversely proportional to
+m2.
+
+10
+Then we will discuss the convergence of |I2 − I
+′′
+0 | in
+the following lemma:
+Lemma 5. The difference
+���I2 − I
+′′
+0
+��� approaches 0 when
+m approaches infinity. Moreover, it is at most inverse-
+proportional to m2.
+Proof: We denote the Fredholm determinant and
+its discretization by d(z) = det(1 + zT) and dV (z) =
+det (1i=j + wjz �m
+k=1 wkG(ri, rk)G∗(rj, rk))m
+i,j=1,
+where K is the kernel of the operator T. To bound
+the difference between d(z) and dV (z), we define
+gn(x1, · · · , xn, s1, · · · , sn) as
+gn(x1, · · · , xn, s1, · · · , sn)
+= det
+�
+�
+g(x1, x1, s1)
+· · ·
+g(x1, xn, s1)
+· · ·
+g(xi, xj, si)
+· · ·
+g(xn, x1, sn)
+· · ·
+g(xn, xn, sn)
+�
+� .
+(36)
+From the definition of g(x, y, z), we know that
+� l
+0 g(xi, xj, si)dsi
+=
+K(xi, xj). According
+to the
+property
+of
+determinants
+that
+det(ai,j)m
+i,j=1
+=
+�
+k1,··· ,km(−1)ka1,k1 · · · am,km,
+where
+k1 · · · km
+is
+the kth exchange of 1 · · · n, we can find that
+Kn(x1, · · · , xn) =
+�
+[0,l]n gn(x1, · · · , xn, s1, · · · , sn)
+ds1 · · · dsn.
+(37)
+If we define Cn
+m(gn) by (38) we have (39). Here
+wαi correspondes to the distance between antennas in
+the source region and s correpsonds to the location
+of the antennas in the source region. When further
+considering
+the
+discretization
+of
+the
+receiver
+as
+in (19), we should set xn
+1
+to the location of the
+antennas in the destination region, and add additional
+weights w which equals the distance between antennas
+in the destination region. Similar to the definition
+of Qn
+m in (19), we define V n
+m(gn) by V n
+m(gn)
+=
+�m
+j1,··· ,jn=1
+�
+i wjiCn
+m(gn(rj1, · · · , rjn, s1,α1, · · · , sn,αn)).
+When sj,αi = rαi, we have
+V n
+m(gn) =
+m
+�
+j1,··· ,jn=1
+m
+�
+α1,··· ,αn=1
+� n
+�
+i=1
+ji
+� � n
+�
+i=1
+αi
+�
+gn(rj1, · · · , rjn, rα1, · · · , rαn)
+=
+m
+�
+j1,··· ,j2n=1
+� 2n
+�
+i=1
+ji
+�
+gn(rj1, · · · , rj2n).
+(40)
+The difference between d(z) and dV (z) is
+d(z) − dV (z)
+=
+∞
+�
+n=1
+zn
+n!
+�
+V n
+m(gn) −
+�
+[0,l]n Kn(x1, · · · , xn)dx1 · · · dxn
+�
+=
+∞
+�
+n=1
+zn
+n!
+�
+V n
+m(gn) −
+�
+[0,l]2n gn(x1, · · · , x2n)dx1 · · · dx2n
+�
+(41)
+Note that V n
+m(gn) is the numerical discretization of the
+function gn with 2n variables, we can bound V n
+m(gn) −
+�
+[a,b]2n gn(x1, · · · , x2n)dx1 · · · dx2n by using the multi-
+variate numerical integration error bound:
+�����V n
+m(gn) −
+�
+[0,l]2n gn(x1, · · · , x2n)dx1 · · · dx2n
+�����
+⩽ l2n−1
+2n
+�
+i=1
+Ei,
+(42)
+where
+Ei =
+�����Qi(gn; xi) −
+� l
+0
+gn(x1, · · · , x2n)dxi
+����� .
+(43)
+According to the m-point composite midpoint quadrature
+rule, we have
+�����Qi(gn; xi) −
+� l
+0
+gn(x1, · · · , x2n)dxi
+�����
+⩽
+l3
+24m2
+����
+∂2gn
+∂x2
+i
+����
+L∞((0,l)2n)
+,
+(44)
+From the Hadamard’s inequality [33], we can further
+bound it by
+�����V n
+m(gn) −
+�
+[a,b]2n gn(x1, · · · , x2n)dx1 · · · dx2n
+�����
+⩽ 2nl2n+2
+24m2 max
+j
+�����
+∂2gn
+∂x2
+j
+�����
+L∞((0,l)2n)
+⩽ 2nl2n+2
+24m2 max
+�
+�nn/2
+�����
+∂2g(x, y, z)
+∂z2
+����
+L∞((0,l)3)
+�n
+,
+4nn/2
+�
+� max
+i+j⩽2
+�����
+∂i
+x∂j
+yg(x, y, z)
+∂xi∂yj
+�����
+L∞((0,l)3)
+�
+�
+n �
+�
+⩽ l2n+2
+3m2 n(n+2)/2
+�
+� max
+i+j+k⩽2
+�����
+∂i
+x∂j
+y∂k
+z g(x, y, z)
+∂xi∂yj∂zk
+�����
+L∞((0,l)3)
+�
+�
+n
+.
+(45)
+Similar to Lemma 3, we know that |I2 −I
+′
+0| converges
+to 0, and the error is at most inverse proportional to m2.
+Therefore, we have Theorem 2:
+Theorem 2. The mutual information I2 that can be
+obtained from the discrete transceivers converges to the
+mutual information I0 that can be obtained from the
+continuous transceivers when the number of antennas in
+the discrete transceivers increases. The difference |I0−I2|
+is at least inverse-proportional to the square of the
+sampling number m.
+Remark
+4. Similar to Remark
+3 in Section. III,
+the convergence analysis in this section is not lim-
+ited to the scenario with equal power allocation.
+
+11
+Cn
+m(gn) = det
+�
+�
+�
+α1 wα1g(x1, x1, s1,α1)
+· · ·
+�
+α1 wα1g(xn, xn, s1,α1)
+· · ·
+�
+αi wαig(xi, xj, si,αi)
+· · ·
+�
+αn wαng(xn, x1, sn,αn)
+· · ·
+�
+αn wαng(xn, xn, sn,αn),
+�
+� .
+(38)
+Cn
+m(gn) =
+�
+k1···kn
+(−1)k(
+m
+�
+α1=1
+wα1g(x1, xk1, s1,α1)) · · · (
+m
+�
+αn=1
+wαng(xn, xkn, sn,αn))
+=
+�
+k1···kn
+�
+(−1)k
+m
+�
+α1,··· ,αn=1
+n
+�
+i=1
+wαig(xi, xki, si,αi)
+�
+=
+m
+�
+α1,··· ,αn=1
+�� n
+�
+i=1
+wαi
+� �
+k1···kn
+(−1)k
+n
+�
+i=1
+g(xi, xki, si,αi)
+�
+=
+m
+�
+α1,··· ,αn=1
+�� n
+�
+i=1
+wαi
+�
+gn(x1, · · · , xn, s1,α1, · · · , sn,αn)
+�
+.
+(39)
+0
+50
+100
+150
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+continuous transceiver
+discrete receiver
+discrete transceiver
+10
+20
+0.1188
+0.119
+0.1192
+Fig. 4. The mutual information variation with different sampling num-
+bers. The mutual information that correpsonds to the three models with
+continuous and discrete transceivers is plotted. The distance between
+the transceivers is large.
+For arbitrary analytic function RJ(s, s′), the conver-
+gence of |I0 − I2| can be obtained. Instead of dis-
+cretizing
+�
+G(r, z)G∗(r′, z)dz to �
+i G(r, ri)G(r′, ri),
+we will discretize
+��
+G(r, z)RJ(z, z′)G∗(r′, z′)dzdz′ to
+�
+i,j G(r, ri)RJ(ri, rj)G(r′, rj) in the extended sce-
+narios
+with
+power
+allocation
+schemes.
+Then,
+in-
+stead of g(x, y, z) we need a four-variable function
+h(x, y, z, ω) := G(x, z)RJ(z, ω)G(y, ω) and the deriva-
+tion procedure of the convergence has no essential differ-
+ence with Theorem 2.
+B. Numerical analysis about the mutual information
+In this subsection, we will verify the correctness of
+the convergence analysis in the above subsection by
+simulations. The length l of the transceivers is fixed
+to 2 m. We have plotted the mutual information of the
+three models: continuous transceiver, continuous trans-
+mitter and discrete receiver, and discrete transceiver. The
+transceivers are both discretized to m point antennas.
+The wavelength of the electromagnetic field is fixed to
+0.04 m, which corresponds to the frequency of 7.5 GHz.
+First we will show the scenarios when the distance be-
+tween the transceivers is large. The distance between the
+transceivers varies from 50 m to 200 m. The simulation
+results are shown in Fig. 4.
+From the simulation results we find that the mutual
+information nearly keeps the same when the sampling
+number increases. The reason for this phenomenon is that
+the DoF of the channel is nearly inverse-proportional to
+the distance between transceivers. For example, the DoF
+when the distance equals 50 m can be approximated by
+l2/(dλ) = 2, which means that when the sampling num-
+ber is 5, the multiplexing gain is almost fully explored.
+Therefore, for large distances between transceivers, the
+dominant limitation is the channel DoF, which means that
+the suboptimal performance can be achieved by sampling
+sparser than half-wavelength.
+Moreover, we have shown the variation of the mutual
+information with the sampling number when the distance
+between transceivers is small. In Fig. 5 the distance
+between the transceiver is 0.1 m and 1 m. We can find that
+when the distance decreases, the dense sampling of the
+transceivers becomes important to fully explore the limit
+of the mutual information. However, the half-wavelength
+sampling of the transceivers still achieves suboptimal
+performance, which means that denser sampling schemes
+are not necessary.
+V. CONCLUSION
+In this paper, we proposed a comparison scheme be-
+tween continuous and discrete MIMO systems which is
+
+12
+0
+50
+100
+150
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+continuous transceiver
+discrete receiver
+discrete transceiver
+Fig. 5. The mutual information as a function of the sampling number.
+The mutual information values that correspond to the three models with
+continuous and discrete transceivers are plotted. The distance between
+the transceivers is small.
+based on a precise non-asymptotic analysis framework.
+Three information-theoretic models of the continuous
+and discrete transceivers were built, with the first model
+corresponds to the fully continuous electromagnetic infor-
+mation theory model, and the third model corresponds to
+the matrix-vector MIMO model. We proposed physically
+consistent SNR control schemes to ensure the fairness of
+the comparison, and proved that the mutual information
+between discrete MIMO transceivers converges to that
+of continuous electromagnetic transceivers. Numerical
+results verified the theoretical analysis and showed the
+near-optimality of the half-wavelength sampling scheme.
+Further works can be done by extending the lin-
+ear transceivers to rectangular or other two-dimensional
+transceivers for generality. The analysis based on the
+capacity after water-filling of the mutual information also
+remains to be explored.
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+

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+Quantum interference in the resonance fluorescence of a J = 1/2 − J′ = 1/2
+atomic system: Quantum beats, nonclassicality, and non-Gaussianity
+H. M. Castro-Beltr´an,1, ∗ O. de los Santos-S´anchez,2 L. Guti´errez,3 and A. D. Alcantar-Vidal1
+1Centro de Investigaci´on en Ingenier´ıa y Ciencias Aplicadas and Instituto de Investigaci´on en Ciencias B´asicas y Aplicadas,
+Universidad Aut´onoma del Estado de Morelos, Avenida Universidad 1001, 62209 Cuernavaca, Morelos, M´exico
+2Tecnologico de Monterrey, Escuela de Ingenier´ıa y Ciencias,
+Ave.
+Carlos Lazo 100, Santa Fe, Mexico City, Mexico, 01389
+3Instituto de Ciencias F´ısicas, Universidad Nacional Aut´onoma de M´exico,
+62210 Cuernavaca, Morelos, M´exico
+(Dated: January 10, 2023)
+We study the resonance fluorescence of a system with angular momentum J = 1/2−J′ = 1/2 level
+structure driven by a single, linearly polarized, monochromatic laser field. Quantum interference
+among the two, antiparallel, π transitions leads to rich results. We develop the article around two
+broad overlapping themes: (i) the observation of quantum beats in the intensity and the dipole-
+dipole, intensity-intensity, and quadrature-intensity correlations, when the atom is subject to a
+strong laser and large Zeeman splittings. The mean and modulation frequencies of the beats are
+given by the average and difference, respectively, among two close generalized Rabi frequencies
+related to a Mollow-like spectrum with two pairs of sidebands.
+(ii) The nonclassical and non-
+Gaussian properties of phase-dependent fluorescence for the cases of weak to moderate excitation
+and in the regime of beats. The fluorescence in the beats regime is nonclassical, mainly from the
+third-order dipole fluctuations, which reveal them to be also strongly non-Gaussian. For weak to
+moderate driving laser and small detunings and Zeeman splittings the nonclassicality is an interplay
+of second- (squeezing) and third-order dipole noise.
+I.
+INTRODUCTION
+Recently, the properties of the resonance fluorescence
+of a single atomic system with angular momentum transi-
+tion J = 1/2−J′ = 1/2 driven by a monochromatic laser
+have been the subject of great interest due to the possi-
+bility of observing vacuum-induced coherence effects due
+to interference among the two antiparallel π transitions,
+emitting into the same frequency range of the electro-
+magnetic vacuum. Here, the π transitions are incoher-
+ently coupled, mediated by spontaneous emision in the σ
+transitions and then excited by the laser. The antiparal-
+lel dipoles of the transitions makes it realistic to observe
+interference effects, while V and Λ three-level systems re-
+quire additional preparation because the transitions are
+perpendicular [1, 2]. Particular attention has been de-
+voted to the spectrum [3–6], time-energy complementar-
+ity [4, 5], Young’s interference [7], photon correlations
+[8], frequency-resolved photon correlations [9], squeezing
+[10], phase shifts [11], and cooperative effects in photon
+correlations [12]. The case of additional laser excitation
+of one of the σ transitions on the spectrum and squeezing
+has been studied in [13–15].
+Quantum beats are among the more familiar mani-
+festations of quantum interference. They appear in the
+modulation of the decay by spontaneous emission of mul-
+tilevel systems due to the energy difference among transi-
+tions [2]. So far, few experiments of quantum interference
+experiments have been performed on the J = 1/2 − J′ =
+∗ [email protected]
+1/2, in this case observing Young-type fringes [7]. Hence,
+further experiments are desirable. Quantum beats in the
+intensity are the result of the inability to tell the path
+of a particular photon when observed by a broadband
+detector. The beats can also occur in two-time correla-
+tions. As a general rule, initial conditions should be a
+superposition state.
+In this paper we investigate theoretically effects of
+quantum interference on the total intensity and two-time
+correlations such as dipole-dipole (to calculate spectra),
+intensity-intensity, intensity-amplitude correlations, and
+variance of the light emitted into the π transitions of the
+J = 1/2−J′ = 1/2 atomic system driven by a linearly po-
+larized laser and a magnetic field to break the degeneracy.
+While we put emphasis on the regime of observation of
+quantum beats, the nonclassical and non-Gaussian prop-
+erties of the fluorescence are also investigated.
+After describing the main features of the model in
+Section II, we discuss the basic dynamic and stationary
+properties of the atomic expectation values in Section
+III. Here, we analyze the previously overlooked time-
+dependent behavior of the atomic populations.
+Those
+of the excited states, for instance, although equal in the
+steady state, evolve with different Rabi frequencies and
+amplitudes. This is at the root of the formation of beats
+in the intensity and the correlations. In the regime of
+strong laser and magnetic fields these beats are character-
+ized by well-defined oscillations at the average frequency
+among two generalized Rabi frequencies, modulated at
+the difference of those frequencies.
+To observe beats
+in the intensity both ground state populations must be
+nonzero initially, ideally equal [1]. Similarly, for the two-
+time correlations, the vector of initial conditions must
+arXiv:2301.03061v1  [quant-ph]  8 Jan 2023
+
+2
+have at least two nonzero terms.
+In Section IV we describe the scattered field intensity
+and quadratures. Here, beats depend only on the inter-
+ference of the two upper populations in the nondegener-
+ate case, with both lower populations initially nonzero.
+Cross terms of the oppposite π transitions represent in-
+terference in the steady state intensity. Then, In Section
+V, using the dressed states approach, we show that the
+double sideband spectrum [5] stems from a dipole-dipole
+correlation with beats, where the terms of addition of sin-
+gle π transitions dominate over those of the cross terms.
+In Section VI we study Brown-Twiss photon-photon
+correlations [16, 17], extending the work of Ref.[8] to the
+nondegenerate case. Besides the ubiquitous antibunching
+effect, for weak to moderate laser drivings the interplay
+of parameters, together with detuning and Zeeman split-
+tings, can make for somewhat involved evolutions, e.g.,
+long decays due to optical pumping in the non-degenerate
+case. Again, cross terms are minor contributors to the
+full correlation in the beats regime.
+Section VII is devoted to a study of phase-dependent
+fluctuations by conditional homodyne detection (CHD)
+[18, 19] in both the temporal and spectral domains. The
+CHD method is characterized by amplitude-intensity cor-
+relations (AIC), which are of third order in the field am-
+plitude. When the atomic operators are decomposed into
+a mean plus a noise operator the AIC is split into a
+second-order term which would be a measure of squeezing
+if the third-order one were negligible. But the latter is
+not negligible outside the weak field regime of resonance
+fluorescence, which make the fluctuations non-Gaussian
+and also nonclassical by the violation of classical inequal-
+ities [20]. We obtain the spectra of the total, second- and
+third-order terms of the AIC. Narrow peaks in the spec-
+tra reveal population trapping when detunings favour the
+long term population or optical pumping of the ground
+state of the more detuned transition, which in the time
+domain show the above mentioned long decays.
+The
+third-order terms make up most of the beats and thus
+they are non-Gaussian and nonclassical but not squeezed.
+In Section VIII we consider squeezing by means of the
+variance of fluctuations. As usual, squeezing in resonance
+fluorescence is small and restricted to weak or moderate
+Rabi frequencies.
+Finally, in Section IX we provide a
+discussion and conclusions, and two Appendices give de-
+tails on solution methods, initial conditions, and optimal
+appearance of beats.
+II.
+MODEL
+The system, illustrated in Fig. 1, consists of a two-level
+atom with transition J = 1/2 – J = 1/2 and states with
+magnetic quantum number m = ±J,
+|1⟩ = |J, −1/2⟩,
+|2⟩ = |J, 1/2⟩,
+|3⟩ = |J, −1/2⟩,
+|4⟩ = |J, 1/2⟩.
+(1)
+The matrix elements are
+FIG. 1.
+Scheme of the J = 1/2 – J = 1/2 atomic system
+interacting with a laser driving the |1⟩ − |3⟩ and |2⟩ − |4⟩
+transitions with Rabi frequency Ω and detuning ∆.
+There
+are spontaneous decay rates γ1, γ2 and γσ, vacuum-induced
+coherence γ12, and Zeeman frequency splittings Bℓ and Bu.
+d1 = ⟨1|ˆd|3⟩ = − 1
+√
+3Dez,
+d2 = ⟨2|ˆd|4⟩ = −d1,
+d3 = ⟨2|ˆd|3⟩ =
+�
+2
+3De−,
+d4 = ⟨1|ˆd|4⟩ = d∗
+3,
+(2)
+where D is the reduced dipole matrix element. We choose
+the field polarization basis {ez, e−, e+} (linear, left cir-
+cular, right circular), where e± = ∓(ex ± iey)/2.
+The π transitions, |1⟩ − |3⟩ and |2⟩ − |4⟩ (m = m′), are
+coupled to linearly polarized light and have their dipole
+moments antiparallel. On the other hand, the σ tran-
+sitions, |1⟩ − |4⟩ and |2⟩ − |3⟩ (m ̸= m′), are coupled
+to circularly polarized light. This configuration can be
+found, for example, in 198Hg+ [3], and 40Ca+ [12].
+The level degeneracy is removed by the application of
+a static magnetic field Bz along the z direction, the Zee-
+man effect.
+Note that the energy splittings gµBBz of
+the upper (u) and lower (ℓ) levels are different due to
+unequal Land´e g factors, gu and gℓ, respectively; µB is
+Bohr’s magneton. The difference Zeeman splitting is
+δ = (gu − gℓ)µBBz
+¯h
+= gu − gℓ
+gℓ
+Bℓ,
+(3)
+where Bℓ = glµBBz/¯h. For 198Hg+ gu = 2/3 and gℓ = 2,
+so ¯hδ = −(4/3)µBBz = −(2/3)¯hBℓ.
+The atom is driven by a monochromatic laser of fre-
+quency ωL, linearly polarized in the z direction, propa-
+gating in the x direction,
+EL(x, t) = E0ei(ωLt−kLx)ez + c.c.,
+(4)
+thus driving only the π transitions.
+The free atomic, H0, and interaction, V , parts of the
+Hamiltonian are, respectively:
+H0 = ¯hω13A11 + ¯h(ω24 + Bℓ)A22 + ¯hBℓA44,
+(5)
+V = ¯hΩ(A13 − A24)eiωLt + h.c.
+(6)
+
+3
+where Ajk = |j⟩⟨k| are atomic operators, ω13 and ω24 =
+ω13 + δ are the frequencies of the |1⟩ − |3⟩ and |2⟩ − |4⟩
+transitions, respectively, and Ω = E0D/
+√
+3 ¯h is the Rabi
+frequency. The frequencies of the other transitions are
+ω23 = ω13 − δ and ω14 = ω13 − Bℓ. Using the unitary
+transformation
+U = exp [(A11 + A22)iωLt],
+(7)
+the Hamiltonian in the frame rotating at the laser fre-
+quency is
+H = U †(H0 + V )U,
+= −¯h∆A11 − ¯h(∆ − δ)A22 + ¯hBℓ(A22 + A44)
++¯hΩ [(A13 − A24) + h.c.] ,
+(8)
+where ∆ = ωL −ω13 is the detuning of the laser from the
+|1⟩ − |3⟩ resonance transition, and ∆ − δ is the detuning
+on the |2⟩ − |4⟩ transition.
+The excited states decay either in the π transitions
+emitting photons with linear polarization at rates γ1 =
+γ2, or in the σ transitions emitting photons of circular
+polarization at rate γσ. There is also a cross-coupling
+of the excited states by the reservoir, responsible for the
+quantum interference we wish to study. In general, the
+decay rates are written as
+γij = di · d∗
+j
+|di||dj|
+√γiγj,
+i, j = 1, 2.
+(9)
+In particular, we have γii = γ1 = γ2 and γ13 = γ24 = γσ.
+Also, given that d1 and d2 are antiparallel, γ12 = γ21 =
+−√γ1γ2 = −γ1.
+The total decay rate is
+γ = γ1 + γσ = γ2 + γσ.
+(10)
+The decays for the π and σ transitions occur with the
+branching fractions bπ and bσ [5], respectively,
+γ1 = γ2 = bπγ,
+bπ = 1/3,
+(11a)
+γσ = bσγ,
+bσ = 2/3.
+(11b)
+III.
+MASTER EQUATION
+The dynamics of the atom-laser-reservoir system is de-
+scribed by the master equation for the reduced atomic
+density operator, ρ. In a frame rotating at the laser fre-
+quency (˜ρ = UρU †) it is given by
+˙˜ρ = − i
+¯h[H, ˜ρ] + Lγ ˜ρ,
+(12)
+where −(i/¯h)[H, ˜ρ] describes the coherent atom-laser in-
+teraction and Lγ ˜ρ describes the damping due to sponta-
+neous emission [5, 21]. Defining
+S−
+1 = A31,
+S−
+2 = A42,
+S−
+3 = A32,
+S−
+4 = A41,
+S+
+i = (S−
+i )†,
+(13)
+the dissipative part is written as
+Lγ ˜ρ = 1
+2
+2
+�
+i,j=1
+γij
+�
+2S−
+i ˜ρS+
+j − S+
+i S−
+j ˜ρ − ˜ρS+
+i S−
+j
+�
++γσ
+2
+4
+�
+i=3
+�
+2S−
+i ˜ρS+
+i − S+
+i S−
+i ˜ρ − ˜ρS+
+i S−
+i
+�
+. (14)
+We now define the Bloch vector of the system as
+Q ≡ (A11, A12, A13, A14, A21, A22, A23, A24,
+A31, A32, A33, A34, A41, A42, A43, A44)T . (15)
+The equations for the expectation values of the atomic
+operators, ⟨Ajk⟩ = ˜ρkj, are the so-called Bloch equations,
+which we write as
+d
+dt⟨Q(t)⟩ = MB⟨Q(t)⟩,
+(16)
+where MB is a matrix of coeficients of the full master
+equation, and the formal solution is
+⟨Q(t)⟩ = eMBt⟨Q(0)⟩.
+(17)
+Since we are interested only in properties of the fluores-
+cence emitted in the π transitions we use the simplifying
+fact, already noticed in [8], that these Bloch equations
+can be split into two decoupled homogeneous sets. Set 1
+contains the equations for the populations and the coher-
+ences of the coherently driven π transitions; these are
+⟨ ˙A11⟩ = −γ⟨A11⟩ + iΩ(⟨A31⟩ − ⟨A13⟩),
+⟨ ˙A13⟩ = −
+�γ
+2 + i∆
+�
+⟨A13⟩ − iΩ(⟨A11⟩ − ⟨A33⟩),
+⟨ ˙A22⟩ = −γ⟨A22⟩ − iΩ(⟨A42⟩ − ⟨A24⟩),
+⟨ ˙A24⟩ = −
+�γ
+2 + i(∆ − δ)
+�
+⟨A24⟩ + iΩ(⟨A22⟩ − ⟨A44⟩),
+⟨ ˙A31⟩ = −
+�γ
+2 − i∆
+�
+⟨A31⟩ + iΩ(⟨A11⟩ − ⟨A33⟩),
+⟨ ˙A33⟩ = γ1⟨A11⟩ + γσ⟨A22⟩ − iΩ(⟨A31⟩ − ⟨A13⟩),
+⟨ ˙A42⟩ = −
+�γ
+2 − i(∆ − δ)
+�
+⟨A42⟩ − iΩ(⟨A22⟩ − ⟨A44⟩),
+⟨ ˙A44⟩ = γσ⟨A11⟩ + γ2⟨A22⟩ + iΩ(⟨A42⟩ − ⟨A24⟩).
+(18)
+with Bloch vector
+R ≡ (A11, A13, A22, A24, A31, A33, A42, A44)T
+(19)
+and a corresponding matrix M, Eq. (A3). Equations (18)
+do not depend on γ12, the vacuum-induced coupling of
+the upper levels, but on the applied magnetic field only
+through the difference of Zeeman splittings, δ.
+The steady state solutions, for which we introduce the
+
+4
+0
+2
+4
+6
+8
+10
+12
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+2
+4
+6
+8
+10
+12
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+2
+4
+6
+8
+10
+12
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+2
+4
+6
+8
+10
+12
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+γt
+γt
+(d) ∆ = −2γ, δ = −4γ
+(b) ∆ = 2γ, δ = −2γ
+(
+⟨A22 (t)⟩
+⟨A44 (t)⟩
+⟨A11 (t)⟩
+⟨A33 (t)⟩
+(a) ∆ = 0, δ = 0
+FIG. 2.
+Time-dependent populations ⟨A11(t)⟩ (solid-black),
+⟨A22(t)⟩ (dashed-red), ⟨A33(t)⟩ (dots-green), and ⟨A44(t)⟩
+(dashed-dots-blue), with the atom initially in state |3⟩. The
+parameters are: Ω = γ and (a) ∆ = δ = 0; (b) ∆ = 2γ,
+δ = −2γ; (c) ∆ = δ = −2γ; (d) ∆ = −2γ, δ = −4γ.
+short notation αjk = ⟨Ajk⟩st, are
+α11 = α22 = Ω2
+2D,
+(20a)
+α33 = Ω2 + γ2/4 + ∆2
+2D
+,
+(20b)
+α44 = Ω2 + γ2/4 + (∆ − δ)2
+2D
+,
+(20c)
+α13 = Ω(∆ + iγ/2)
+2D
+,
+(20d)
+α24 = Ω(δ − ∆ − iγ/2)
+2D
+,
+(20e)
+αkj = α∗
+jk.
+where
+D = 2Ω2 + γ2 + δ2
+4
++
+�
+∆ − δ
+2
+�2
+.
+(21)
+Note also that in the degenerate system (δ = 0) α33 =
+α44 and that α31 = −α42, where the minus sign arises
+from the fact that the dipole moments d1 and d2 are
+antiparallel.
+Set 2 contains the equations for the coherences of the σ
+transitions and those among both upper and both lower
+levels,
+R2 ≡ (A12, A14, A21, A23, A32, A34, A41, A43)T . (22)
+The equations for their expected values do depend on Bℓ
+and γ12. These coherences vanish because the σ tran-
+sitions are driven incoherently (⟨{A14, A23, A32, A41}⟩),
+i.e., by spontaneous emission, or because they are medi-
+ated by those σ transitions (⟨{A12, A21, A34, A43}⟩). For
+completeness, we write the steady state results:
+α12 = α34 = α14 = α23 = 0,
+αkj = α∗
+jk.
+(23)
+0
+1
+2
+3
+4
+5
+6
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+1
+2
+3
+4
+5
+6
+0
+0.2
+0.4
+0.6
+0.8
+0
+1
+2
+3
+4
+5
+6
+0
+0.2
+0.4
+0.6
+0.8
+0
+1
+2
+3
+4
+5
+6
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+α11 = α22
+α33
+α44
+(d) ∆ = −2γ, δ = −4γ
+(b) ∆ = 2γ, δ = −2γ
+(a) ∆ = 0, δ = 0
+(
+Ω/γ
+Ω/γ
+FIG. 3.
+Steady-state populations as a function of Rabi
+frequency: α11 = α22 (dashed-red), α33 (solid-black), and
+α44 (dots-blue). All other parameters as in Fig. 2.
+The properties of the fluorescence of the π transitions,
+the subject matter of this article, do not depend on the
+equations for Set 2. Only the second- and third-order
+amplitude-intensity correlations and the dipole correla-
+tion for the spectrum of the σ transitions would require
+the full set of Bloch equations.
+We gain valuable information on the nontrivial dynam-
+ics of the atomic system from single-time expectation val-
+ues, apparently ignored in the previous literature on the
+system. In Fig. 2 we show the populations for several
+particular cases, all with the atom initially in state |3⟩.
+In the degenerate case, δ = 0, the upper populations
+reach opposite phases by the end of the first Rabi cycle,
+Fig. 2(a). This is understandable since the electron oc-
+cupation of, say, state |1⟩ implies not to be in state |2⟩,
+and viceversa. The same occurs for the lower popula-
+tions. Next, we show three situations for the nondegen-
+erate case with δ < 0 (as it is for 198Hg+). In Fig. 2(b)
+the laser is slightly detuned above the |1⟩−|3⟩ transition,
+but highly detuned from the |2⟩−|4⟩ transition; the oscil-
+lations get out of phase and most of the population ends
+up in state |4⟩ by optical pumping. In Fig. 2(c) the laser
+is detuned below the |1⟩−|3⟩ transition, and the |2⟩−|4⟩
+transition is now on resonance with the laser; again, the
+oscillations are out of phase but most of the population
+ends up now in state |3⟩. In Fig. 2(d) we extend the pre-
+vious case but with stronger applied magnetic field, thus
+the non-degeneracy is more evident; the large detuning
+on both transitions makes it recover the opposite phases
+of the degenerate case.
+In Fig. 3 we show the steady state populations as a
+function of the Rabi frequency; the other parameters are
+the same as in Fig. 2. For strong fields the populations
+tend to be equal (1/4), but arrive at that limit at dif-
+ferent rates; for instance, for large detunings on both
+transitions, Fig. 3(d), it takes larger fields, as compared
+to the degenerate case, Fig. 3(a). On the other hand,
+for small detunings and weak-moderate fields, when one
+
+5
+transition is closer to resonance than the other, the lower
+state of the more detuned transition is more populated,
+as seen in Figs. 3 (b) and (c).
+IV.
+THE SCATTERED FIELD
+In this Section we present the main dynamical and
+stationary properties of the field scattered by the atom,
+with emphasis on the π transitions.
+A.
+Single-Time and Stationary Properties
+The positive-frequency part of the emitted field oper-
+ator is [5, 21]
+ˆE+(r, t) = ˆE+
+free(r, t) + ˆE+
+S (r, ˆt),
+(24)
+where ˆE+
+free(r, t) is the free-field part, which does not con-
+tribute to normally ordered correlations, hence we omit
+it in further calculations, and
+ˆE+
+S (r, t) = −η
+r
+4
+�
+i=1
+ω2
+i ˆr × (ˆr × di)S−
+i (ˆt)
+(25)
+is the dipole source field operator in the far-field zone,
+where ˆt = t−r/c is the retarded time and η = (4πϵ0c2)−1.
+Since ωi ≫ δ, we may approximate the four transition as
+a single one ω0 in Eq. (25, but cannot do so at the level
+of decay rates, Rabi frequencies, and splittings.
+Making ˆr = ey the direction of observation and using
+Eq. (2) we have
+ˆE+
+S (r, ˆt) = ˆE+
+π (r, ˆt) ez + ˆE+
+σ (r, ˆt) ex,
+(26)
+i.e., the fields scattered from the π and σ transitions are
+polarized in the ez and ex directions, respectively, where
+ˆE+
+π (r, ˆt) = fπ(r)
+�
+A31(ˆt) − A42(ˆt)
+�
+,
+(27a)
+ˆE+
+σ (r, ˆt) = fσ(r)
+�
+A32(ˆt) − A41(ˆt)
+�
+,
+(27b)
+are the positive-frequency source field operators of the π
+and σ transitions, and
+fπ(r) = −ηω2
+1D/
+√
+3r,
+fσ(r) =
+√
+2fπ(r),
+(28)
+are their geometric factors.
+The intensity in the π transitions is given by
+Iπ(r, ˆt) = ⟨ ˆE−
+π (r, ˆt) · ˆE+
+π (r, ˆt)⟩
+= f 2
+π(r)⟨A13(ˆt)A31(ˆt) + A24(ˆt)A42(ˆt)⟩
+= f 2
+π(r)⟨A11(ˆt) + A22(ˆt)⟩,
+(29a)
+while in the steady state is
+Ist
+π = f 2
+π(r) [α11 + α22] = Ω2
+D .
+(29b)
+0
+2
+4
+6
+8
+10
+0
+0.05
+0.10
+0.15
+0.20
+0
+2
+4
+6
+8
+10
+0
+0.1
+0.2
+0.3
+0
+2
+4
+6
+8
+10
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+2
+4
+6
+8
+10
+0
+0.1
+0.2
+0.3
+0.4
+0
+2
+4
+6
+8
+10
+0
+0.05
+0.10
+0.15
+0.20
+0.25
+0
+2
+4
+6
+8
+10
+0
+0.05
+0.10
+0.15
+0
+2
+4
+6
+8
+10
+0
+0.05
+0.10
+0.15
+0
+2
+4
+6
+8
+10
+0
+0.05
+0.10
+0.15
+0.20
+0.25
+Iπ (r, t) �f 2
+π (r)
+Iπ (r, t) �f 2
+π (r)
+⟨A11 (t)⟩
+⟨A22 (t)⟩
+⟨A11 (t)⟩
+⟨A22 (t)⟩
+⟨A11 (t)⟩
+⟨A22 (t)⟩
+⟨A11 (t)⟩
+⟨A22 (t)⟩
+γt
+γt
+γt
+γt
+γt
+γt
+(d)
+(c)
+(a)
+(b)
+FIG. 4.
+Fluorescence intensity of the π transitions with equal
+initial ground state populations, ⟨A33(0)⟩ = ⟨A44(0)⟩ = 1/2.
+The other parameters are as in Fig. 2: Ω = γ and (a) ∆ =
+δ = 0; (b) ∆ = 2γ, δ = −2γ; (c) ∆ = δ = −2γ; (d) ∆ = −2γ,
+δ = −4γ.
+The insets show the excited states populations:
+⟨A11(t)⟩ (solid), ⟨A22(t)⟩ (dashed).
+Adding the excited state populations with the atom
+initially in the single state |3⟩ in Eq. 29a gives simply
+Iπ(r, ˆt) = f 2
+π(r)⟨A11(ˆt)⟩, i.e., without the contribution
+of ⟨A22(ˆt)⟩. More interesting is the case where the ini-
+tial condition is ⟨A33(0)⟩ = ⟨A44(0)⟩ = 1/2, shown in
+Fig. 4 (see the populations ⟨A11(t)⟩ and ⟨A22(t)⟩ in the
+insets). The modulation in the intensity is reminiscent
+of the quantum beats in the spontaneous decay in the V
+three-level system [1, 2]. These beats are basically due
+to the inability to tell from which of the π transitions
+a photon comes from. This is the standard Young-type
+interference [4, 5, 7]. The main requirement is that the
+initial condition for both ground states are nonzero (see
+Appendix B.
+More interesting, though, is the case of strong resonant
+laser and magnetic fields and the laser is detuned far from
+the |2⟩ − |4⟩ resonance frequency, shown in Fig. 5. Due
+to the laser detuning, the population ⟨A22(t)⟩ has larger
+frequency and smaller amplitude than that of ⟨A11(t)⟩, as
+seen in the insets. Remarkably well-defined wave-packets
+or beats are observed due to the interference of the flu-
+orescence of both π transitions with close Rabi frequen-
+cies, with clear average and modulation frequencies (see
+Fig. 5a). The beats get scrambled with larger frequency
+and amplitude differences, Fig. 5b.
+Save for the decay, these beats are more like the classic
+textbook ones, described by a modulation and an average
+frequency, unlike the beats from spontaneous emission or
+weak resonance fluorescence from two or more closely sep-
+arated levels. Henceforth, we reserve the moniker beats
+to those due to strong applied fields. Further analyses of
+the beats are given in the next Sections, as they show up
+
+6
+0
+2
+4
+6
+8
+10
+0
+0.2
+0.4
+0.6
+0
+2
+4
+6
+8
+10
+0
+0.2
+0.4
+0.6
+0.8
+0
+1
+2
+3
+4
+5
+0
+0.1
+0.2
+0.3
+0.4
+0
+1
+2
+3
+4
+5
+0
+0.1
+0.2
+0.3
+0.4
+Iπ (r, t)
+�
+f2
+π (r)
+Iπ (r, t)
+�
+f2
+π (r)
+⟨A22 (t)⟩
+⟨A11 (t)⟩
+⟨A22 (t)⟩
+⟨A11 (t)⟩
+γt
+γt
+γt
+γt
+(b)
+(a)
+FIG. 5.
+Fluorescence intensity for Ω = 9γ, ∆ = 0, and (a)
+δ = −8γ and (b) δ = −15γ. The insets show the excited state
+populations ⟨A11⟩ (solid line) and ⟨A22⟩ (dotted line). The
+initial conditions are ⟨A33(0)⟩ = ⟨A44(0)⟩ = 1/2, ⟨A11(0)⟩ =
+⟨A22(0)⟩ = 0.
+also in two-time correlations with particular features.
+Similarly, for the σ transitions we have
+Iσ(r, ˆt) = ⟨ ˆE−
+σ (r, ˆt) · ˆE+
+σ (r, ˆt)⟩
+= f 2
+σ(r)[⟨A23(ˆt)A32(ˆt) + A14(ˆt)A41(ˆt)⟩]
+= f 2
+σ(r)[⟨A11(ˆt) + A22(ˆt)⟩],
+(30a)
+Ist
+σ = f 2
+σ(r) [α11 + α22] ,
+(30b)
+also showing beats with intensity twice that of the π tran-
+sitions given that f 2
+σ(r) = 2f 2
+π(r).
+The field quadrature operator at any time is
+ˆEπ,φ(r, ˆt) = 1
+2
+�
+E−
+π (r, ˆt)e−iφ + E+
+π (r, ˆt)eiφ�
+= fπ(r)(S1,φ − S2,φ),
+(31)
+where φ = 0, π/2 are the quadrature phases we consider,
+and
+S1,φ = 1
+2
+�
+A13e−iφ + A31eiφ�
+,
+(32a)
+S2,φ = 1
+2
+�
+A24e−iφ + A42eiφ�
+.
+(32b)
+The mean quadrature field is given by
+⟨ ˆEπ,φ⟩st = fπ(r)
+2
+�
+(α13 − α24) e−iφ + (α31 − α42) eiφ�
+= fπ(r)Re
+�
+(α13 − α24) e−iφ�
+(33)
+= fπ(r)Re
+�Ω (∆ + (iγ − δ)/2)
+D
+e−iφ
+�
+,
+B.
+Intensity and Quadrature Fluctuations
+Here we introduce the intensity and quadratures of the
+field in terms of atomic fluctuation operators ∆Ajk =
+Ajk − ⟨Ajk⟩st, such that
+⟨AklAmn⟩ = αklαmn + ⟨∆Akl∆Amn⟩.
+(34)
+Only the π transitions have nonvanishing coherence
+terms (α13, α24 ̸= 0). The fluorescence in the σ transi-
+tions is fully incoherent (α14 = α23 = 0), so its intensity
+is given by Eq. (30b). In the remainder of this section
+we deal only with the π transition. The quadrature op-
+erators are then written as
+ˆEπ,φ(r, ˆt) = fπ(r)[απ,φ + ∆Sπ,φ(ˆt)],
+(35a)
+where
+απ,φ = 1
+2(α31 − α42)eiφ + 1
+2(α13 − α24)e−iφ,
+(35b)
+= Re
+�Ω (∆ + (iγ − δ)/2)
+D
+e−iφ
+�
+,
+∆Sπ,φ = 1
+2(∆A31 − ∆A42)eiφ + 1
+2(∆A13 − ∆24)e−iφ.
+(35c)
+From Eqs. (29b) and (34) we write the steady state
+intensity in terms of products of dipole and dipole fluc-
+tuation operator expectation values,
+Ist
+π (r) = f 2
+π(r)
+�
+Icoh
+π,0 + Iinc
+π,0 + Icoh
+π,cross + Iinc
+π,cross
+�
+,(36)
+where
+Icoh
+π,0 = |⟨A13⟩st|2 + |⟨A24⟩st|2,
+(37a)
+Iinc
+π,0 = ⟨∆A13∆A31⟩ + ⟨∆A24∆A42⟩,
+(37b)
+Icoh
+π,cross = −⟨A13⟩st⟨A42⟩st − ⟨A24⟩st⟨A31⟩st
+= −2Re (⟨A13⟩st⟨A42⟩st) ,
+(37c)
+Iinc
+π,cross = −⟨∆A13∆A42⟩ − ⟨∆A24∆A31⟩
+= −2Re (⟨∆A13∆A42⟩) .
+(37d)
+Superindices coh and inc stand, respectively, for the co-
+herent (depending on mean dipoles) and incoherent (de-
+pending on noise terms) parts of the emission. Subindex
+0 stands for terms with the addition of single transition
+products, giving the total intensity, while subindex cross
+stands for terms with products of the two π transitions,
+the steady state interference part of the intensity.
+In
+
+7
+terms of atomic expectation values these intensities are:
+Icoh
+π,0 = |α13|2 + |α24|2
+(38a)
+= Ω2
+4D2
+�γ2
+2 + ∆2 + (δ − ∆)2
+�
+,
+Iinc
+π,0 = α11 + α22 − |α13|2 − |α24|2
+(38b)
+= Ω2
+D2
+�
+2Ω2 − γ2
+4 − ∆2 − δ2
+�
+,
+Icoh
+π,cross = −2Re (α13α42)
+(38c)
+= Ω2
+2D2
+�γ2
+4 + ∆(∆ − δ)
+�
+,
+Iinc
+π,cross = 2Re (α13α42) = −Icoh
+π,cross,
+(38d)
+The sum of these terms is, of course, the total intensity,
+Eq. (29a). As usual in resonance fluorescence, the coher-
+ent and incoherent intensities are similar only in the weak
+field regime, Ω ≤ γ. Here, in particular, the term Iinc
+π,0
+(no interference) becomes much larger than the others
+for strong driving.
+C.
+Degree of Interference - Coherent Part
+In Ref. [5], a measure of the effect of interference in
+the coherent part of the intensity was as
+Icoh
+π,0 + Icoh
+π,cross = Icoh
+π,0 (1 + C(δ)),
+C(δ) = Icoh
+π,cross
+Icoh
+π,0
+=
+γ2/4 + ∆(∆ − δ)
+γ2/4 + δ2/4 + (∆ − δ/2)2 , (39)
+independent
+of
+the
+Rabi
+frequency
+and
+shown
+in
+Fig. 6(a).
+Some special cases are found analytically:
+C(0) = 1,
+δ = 0,
+(40a)
+C(δ0) = 0,
+δ0 = ∆[1 + (γ/2∆)2],
+(40b)
+C(δmin) =
+−1
+1 + γ2/2∆2 ,
+δmin = 2∆[1 + (γ/2∆)2],
+(40c)
+C(δ±
+1/2) = 1/2,
+δ±
+1/2 = −∆ ±
+�
+3∆2 + (γ2/2).
+(40d)
+In the degenerate case, C(δ = 0) = 1 means perfect
+constructive interference. That is because at δ = 0 both
+π transitions (and both σ transitions) share the same
+reservoir environment. Increasing δ the reservoir overlap
+decreases, so is the interference. Negative values of C
+indicate destructive interference; its minimum is given
+by δmin. For large detunings, ∆2 ≫ γ2 we have
+δ0 = ∆,
+δmin = 2∆,
+δ±
+1/2 = −∆ ±
+√
+3 |∆|.
+(40e)
+We have used the special cases δ = {0, δ0, δmin} as a
+guide to obtain many of the figures in this paper.
+-1.0
+-0.5
+0
+0.5
+1.0
+-20
+-10
+0
+10
+20
+0
+0.5
+1.0
+δ/γ
+∆ = −5γ
+K(δ)
+C(δ)
+∆ = −2γ
+∆ = 0
+(a)
+(b)
+FIG. 6.
+Relative weight of the interference terms C(δ) (a)
+and K(δ) (b). In (b) Ω = γ/4. For 198Hg+, δ ≤ 0.
+D.
+Degree of Interference - Incoherent Part
+Likewise, we define a measure, K(δ), of the effect of
+interference in the intensity’s incoherent part,
+Iinc
+π,0 + Iinc
+π,cross = Iinc
+π,0(1 + K(δ)),
+K(δ) = Iinc
+π,cross
+Iinc
+π,0
+=
+γ2/4 + ∆(∆ − δ)
+2 [γ2/4 + δ2 + ∆2 − 2Ω2]. (41)
+Unlike C(δ), K(δ) also depends on the Rabi frequency
+as Ω−2, since fluctuations increase with laser intensity.
+Special cases are:
+K(0) =
+γ2/4 + ∆2
+2 [γ2/4 + ∆2 − 2Ω2],
+δ = 0,
+(42a)
+K(δ) = 0,
+δ = ∆ + γ2
+4∆
+or
+Ω ≫ γ, ∆, δ.
+(42b)
+The behavior of K(δ) with ∆ is more subtle. It is ba-
+sically required that ∆ ∼ Ω in order to preserve the
+shape seen in Fig. 6(b), in which case the minima for
+C(δ) and K(δ) are very similar. On-resonance, for ex-
+ample, Ω should be no larger than 0.35γ. Also, we can
+infer that the beats are little affected by the interference
+term unless ∆ >∼ Ω ≫ γ.
+V.
+TWO-TIME DIPOLE CORRELATIONS AND
+POWER SPECTRUM
+The resonance fluorescence spectrum of the J = 1/2 →
+J = 1/2 atomic system was first considered in [3] and
+then very thoroughly in [4, 5]. Thus, here we only con-
+sider basic definitions and issues related to the observa-
+tion of beats.
+The stationary Wiener-Khintchine power spectrum is
+given by the Fourier transform of the field autocorrelation
+function
+Sπ(ω) = Re
+� ∞
+0
+dτe−iωτ⟨ ˆE−
+π (0) ˆE+
+π (τ)⟩,
+(43)
+
+8
+0
+2
+4
+6
+8
+10
+0
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+-20
+-10
+0
+10
+20
+ω/γ
+Sinc
+π (ω) (arb. units)
+γτ
+�
+∆E−
+π (0) ∆E+
+π (τ)
+� �
+f2
+π (r)
+FIG. 7.
+Dipole correlation function ⟨∆ ˆE−
+π (0)∆ ˆE+
+π (τ)⟩ for
+Ω = 9γ, δ = −8γ, and ∆ = 0. The inset shows the corre-
+sponding incoherent spectrum Sinc
+π
+(ω).
+such that
+� ∞
+−∞ Sπ(ω)dω = Ist
+π . By writing the atomic
+operators in Eq. (27a) as Ajk(t) = αjk + ∆Ajk(t), we
+separate the spectrum in two parts: a coherent one,
+Scoh
+π
+(ω) = Re
+� ∞
+0
+e−iωτdτ
+�
+Icoh
+π,0 + Icoh
+π,cross
+�
+= π
+�
+Icoh
+π,0 + Icoh
+π,cross
+�
+δ(ω)
+= πΩ2
+D2
+�
+γ2
+4 +
+�
+∆ − δ
+2
+�2�
+δ(ω),
+(44)
+due to elastic scattering, where Icoh
+π,0 and Icoh
+π,cross are given
+by Eqs. (38) (a) and (c), respectively; and an incoherent
+part,
+Sinc
+π (ω) = Re
+� ∞
+0
+dτe−iωτ⟨∆ ˆE−
+π (0)∆ ˆE+
+π (τ)⟩,
+specifically,
+Sinc
+π (ω) = Re
+� ∞
+0
+dτe−iωτ [⟨∆A13(0)∆A31(τ)⟩
++⟨∆A24(0)∆A42(τ)⟩ − ⟨∆A13(0)∆A42(τ)⟩
+−⟨∆A24(0)∆A31(τ)⟩] ,
+(45)
+due to atomic fluctuations. An outline of the numerical
+calculation is given in Appendix A.
+The dipole correlation ⟨ ˆE−
+π (0) ˆE+
+π (τ)⟩ and the incoher-
+ent spectrum in the strong driving regime and strong
+nondegeneracy (large δ) are shown in Fig. 7. The spec-
+trum (inset) displays a central peak and two pairs of
+Mollow-like-sidebands [22] with peaks at the Rabi side-
+bands ±Ω1 and ±Ω2, while the correlation features de-
+caying quantum beats due to the closeness of the Rabi
+peaks.
+As usual in the strong-field regime, the dressed system
+approach allows to discern the origin of the peaks from
+the transitions among the dressed states, to find their
+positions [5], and thus find the frequencies of the beats.
+The generalized Rabi frequencies are
+Ω1 = E+
+1 − E−
+1 =
+�
+4Ω2 + ∆2,
+(46a)
+Ω2 = E+
+2 − E−
+2 =
+�
+4Ω2 + (δ − ∆)2,
+(46b)
+TABLE I. Eigenvalues of matrix M/γ and initial conditions
+of the correlations in Eq. (45) for Ω = 9γ and ∆ = 0.
+Eigenvalues
+δ = −8γ
+δ = −15γ
+λ1
+−0.749386 + 0i
+−0.836531 + 0i
+λ2
+−0.583099 − 18.0094i
+−0.583308 − 17.9981i
+λ3
+−0.583099 + 18.0094i
+−0.583308 + 17.9981i
+λ4
+−0.569785 − 19.6808i
+−0.5492 − 23.4257i
+λ5
+−0.569785 + 19.6808i
+−0.5492 + 23.4257i
+λ6
+−0.5 + 0i
+−0.5 + 0i
+λ7
+−0.444846 + 0i
+−0.398452 + 0i
+λ8
+0 + 0i
+0 + 0i
+Init. cond.
+⟨∆A13∆A31⟩
+0.20836 + 0i
+0.14734 + 0i
+⟨∆A24∆A42⟩
+0.174014 + 0i
+0.086982 + 0i
+⟨∆A13∆A42⟩
+0.000134 + 0.002146i
+0.000067 + 0.002011i
+⟨∆A24∆A31⟩
+0.000134 − 0.002146i
+0.000067 − 0.002011i
+where
+E±
+1 = −∆
+2 ± 1
+2
+�
+4Ω2 + ∆2,
+(47a)
+E±
+2 = Bℓ + δ − ∆
+2
+± 1
+2
+�
+4Ω2 + (δ − ∆)2,
+(47b)
+are the eigenvalues of the Hamiltonian (8). Due to the
+spontaneous decays these frequencies would have to be
+corrected, but they are very good in the relevant strong
+field limit. Indeed, we notice that Ω1 and Ω2 are very
+close to the imaginary parts of the eigenvalues λ2,3 and
+λ4,5, respectively, of matrix M, shown in Table I.
+The beats are the result of the superposition of waves
+at the frequencies Ω1 and Ω2 of the spectral sidebands,
+with average frequency
+Ωav = Ω2 + Ω1
+2
+=
+�
+4Ω2 + (δ − ∆)2 +
+√
+4Ω2 + ∆2
+2
+,
+(48)
+and beat or modulation frequency
+Ωbeat = Ω2 − Ω1
+2
+=
+�
+4Ω2 + (δ − ∆)2 −
+√
+4Ω2 + ∆2
+2
+.
+(49)
+Now, we can identify the origin and modulation fre-
+quency of the beats in the time-dependent intensity,
+Eq. (29a), since the excited state populations ⟨A11(t)⟩
+and ⟨A22(t)⟩ oscillate at the generalized Rabi frequen-
+cies Ω1 and Ω2, respectively, with initial conditions
+given by a nonzero superposition of ground state pop-
+ulations at t = 0. In the case of the dipole correlation
+⟨ ˆE−
+π (0) ˆE+
+π (τ)⟩, however, the initial conditions are given
+by products of stationary atomic expectation values,
+most of them the coherences α13, α24, which become very
+small in the regime of beats. Thus, as seen in Table I,
+the terms ⟨∆A13(0)∆A31(τ)⟩ and ⟨∆A24(0)∆A42(τ)⟩ are
+
+9
+0
+3
+6
+9
+12
+15
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0
+3
+6
+9
+12
+15
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+0
+3
+6
+9
+12
+15
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+γτ
+∆ = −2γ, δ = −4γ
+∆ = −2γ, δ = −2γ
+∆ = 2γ,
+δ = −2γ
+∆ = 0,
+δ = 0
+(
+(a) Ω = γ/4
+(b) Ω = γ/2
+g(2)
+π (τ)
+g(2)
+π (τ)
+g(2)
+π (τ)
+FIG. 8.
+Photon correlations for (a) Ω = 0.25γ, (b) Ω = 0.5γ
+and (c) Ω = γ. The pairs of values (∆, δ) are the same as
+those in Fig. 2.
+much larger than the cross terms ⟨∆A13(0)∆A42(τ)⟩ and
+⟨∆A24(0)∆A31(τ)⟩, so the beats are basically due to the
+interference of those dominant terms.
+VI.
+PHOTON-PHOTON CORRELATIONS
+The standard method to investigate intensity fluctua-
+tions of a light source uses Brown-Twiss photon-photon
+correlations [16, 17]. The conditional character of this
+type of measurement makes it nearly free of detector in-
+efficiencies, unlike a single-detector measurement of the
+photoelectron probability distribution.
+In Ref. [8] the
+correlations of two photons from the π transitions were
+studied, albeit only for the degenerate case. In this paper
+we extend it to the case of nondegenerate states. These
+correlations are defined as
+g(2)
+π (τ) =
+G(2)
+π (τ)
+G(2)
+π (τ → ∞)
+(50)
+where, using Eq. (27a) for the field operators,
+G(2)
+π (τ) = ⟨ ˆE−
+π (0) ˆE−
+π (τ) ˆE+
+π (τ) ˆE+
+π (0)⟩
+= f 4
+π(r)⟨[A13(0) − A24(0)][A11(τ) + A22(τ)]
+×[A31(0) − A42(0)]⟩,
+(51a)
+and
+G(2)
+π (τ → ∞) =
+�
+Ist
+π
+�2 = f 4
+π(r) (α11 + α22)2 (51b)
+is the normalization factor. G(2)
+π (τ) can be further re-
+duced, since ⟨A13Ajk(τ)A42(0)⟩ = ⟨A24Ajk(τ)A31(0)⟩ =
+0, due to having vanishing initial conditions.
+Figure 8 shows g(2)
+π (τ) for moderate values of the Rabi
+frequency (near saturation) and the same sets of detun-
+ings ∆ and δ of Fig. 2. As usual in resonance fluores-
+cence, the correlation shows antibunching, g(2)
+π (0) = 0,
+0
+2
+4
+6
+8
+10
+0
+0.5
+1.0
+1.5
+2.0
+0
+2
+4
+6
+8
+10
+0
+0.5
+1.0
+1.5
+2.0
+0
+2
+4
+6
+8
+10
+0
+0.5
+1.0
+1.5
+2.0
+0
+2
+4
+6
+8
+10
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+g(2)
+π (τ)
+γτ
+γτ
+g(2)
+π (τ)
+(b) δ = −10γ
+(d) δ = −15γ
+(
+(a) δ = −8γ
+FIG. 9.
+Photon-photon correlations showing beats in the
+strong field limit, Ω = 9γ, ∆ = 0, and large Zeeman splittings.
+The horizontal line helps to see that the wave packet is slightly
+rised.
+that is, a single atom cannot emit two photons simultane-
+ously. Unlike the two-level atom resonance fluorescence,
+the correlation is not simply the normalized population
+of the excited state, nor it is only the sum of the cor-
+relations of each single π transition. Besides the terms
+⟨A13(0)A11(τ)A31(0)⟩ and ⟨A24(0)A22(τ)A42(0)⟩, which
+are also out of phase, as seen from the time-dependent
+populations of their excited states (Fig. 2), there are six
+cross terms in the full correlation. In the nondegenerate
+case the multiple contributions cause in some cases quite
+irregular evolution. For instance, as we will see in the
+next Section, the slow decay of the correlation when the
+laser drives the atom near saturation, but below the ω13
+resonance transition, is related to a very narrow peak in
+the spectrum.
+The case of strong driving and large nondegeneracy
+is shown in Fig. 9, featuring quantum beats. There are
+several effects resulting from the increase of the nonde-
+generacy factor δ: (i) the larger number of visible wave
+packets; (ii) both average and beat frequencies approach
+one another, so the wave packets get shorter for larger
+photon-pair intervals τ, containing very few of the fast
+oscillations, as seen in Fig. 9(d); (iii) the wavepackets are
+initially slightly lifted above the g(2)(τ) = 1 value.
+VII.
+QUADRATURE FLUCTUATIONS
+Squeezing, the reduction of noise in one quadrature
+below that of a coherent state at the expense of the
+other, is the hallmark of phase-dependent fluctuations
+of the electromagnetic field [cite]. It is usually measured
+by balanced homodyne detection (BHD), but low quan-
+tum detector efficiency degrade the weak squeezing pro-
+duced in resonance fluorescence and cavity QED systems.
+One alternative our group has used is conditional homo-
+dyne detection (CHD) [18, 19], which correlates a quadra-
+ture amplitude on the cue of an intensity measurement.
+CHD measures a third-order amplitude-intensity corre-
+
+10
+lation (AIC) which, in the weak driving limit is reduced
+to the second-order one and that allows for measuring
+squeezing. Being a conditional measurement it is nearly
+free of detector inefficiencies.
+While the original goal of CHD was to measure the
+weak squeezing in cavity QED [18, 19], it was soon re-
+alized that nonzero third-order fluctuations of the am-
+plitude provide clear evidence of non-Gaussian fluctua-
+tions and higher-order field nonclassicality. In the present
+work the fluctuations are mainly third-order ones, due to
+near and above saturation excitation, and violate classi-
+cal bounds. We thus explore the phase-dependent fluctu-
+ations under conditions of quantum interference following
+our recent work [20, 23, 24].
+A.
+Amplitude-Intensity Correlations
+In CHD a quadrature’s field Eφ is measured by BHD
+on the cue of photon countings in a separate detector,
+where φ = 0, π/2 is the phase of the local oscillator. This
+is characterized by a correlation among the amplitude
+and the intensity of the field,
+hπ,φ(τ) =
+Hπ,φ(τ)
+Hπ,φ(τ → ∞),
+(52)
+where
+Hπ,φ(τ) = ⟨: ˆE−
+π (0) ˆE+
+π (0) ˆEπ,φ(τ) :⟩,
+(53a)
+the dots :: indicating time and normal operator orderings,
+and
+Hπ,φ(τ → ∞) = Ist
+π ⟨Eπ,φ⟩st
+(53b)
+= f 3
+π(r) [α11 + α22] Re
+�
+(α13 − α24) e−iφ�
+= f 3
+π(r) Ω3
+D2 Re
+�
+(∆ + (iγ − δ)/2) e−iφ�
+is the normalization factor.
+For the sake of concreteness, in this Section we limit
+our discussion to the out-of-phase quadrature, φ = π/2,
+which is the one that features squeezing when ωL = ω13,
+that is ∆ = 0. We do consider, however, squeezing in the
+in-phase quadrature φ = 0 in Sect. VIII on the variance.
+In several atom-laser systems hπ,φ(τ) has been proven
+to be time-asymmetric [20, 24]. This is not the case with
+the J = 1/2 → J = 1/2 system so we limit the analysis
+to positive intervals τ ≥ 0.
+Omitting the geometrical
+factor f 3
+π(r), which is later cancelled by normalization,
+we have
+Hπ,φ(τ) = ⟨ ˆE−
+π (0) ˆEπ,φ(τ) ˆE+
+π (0)⟩
+= Re
+�
+e−iφ⟨A13(0)[A13(τ) − A24(τ)]A31(0)
++A24(0)[A13(τ) − A24(τ)]A42(0)⟩} .
+(54)
+Note that Hπ,φ(0) = 0 meaning that, like antibunching
+in g(2), the atom has to build a new photon wavepacket
+after one has been emitted.
+The AIC suggests nontrivial behavior when we take
+dipole fluctuations into account, that is, when the atomic
+operators are split into their mean plus noise, Ajk =
+αjk + ∆Ajk; upon substitution in Eq. (54) we get
+Hπ,φ(τ) = Ist
+π ⟨Eπ,φ⟩st + H(2)
+π,φ(τ) + H(3)
+π,φ(τ),
+(55)
+or in normalized form as
+hπ,φ(τ) = 1 +
+H(2)
+π,φ(τ)
+Ist
+π ⟨Eπ,φ⟩st
++
+H(3)
+π,φ(τ)
+Ist
+π ⟨Eπ,φ⟩st
+,
+(56)
+where
+H(2)
+π,φ(τ) = 2Re
+�
+⟨ ˆE+
+π ⟩st⟨∆ ˆE−
+π (0)∆ ˆEπ,φ(τ)⟩
+�
+= Re
+�
+(α31 − α42) [⟨(∆A13(0) − ∆A24(0))
+�
+∆A13(τ) − ∆A24(τ))⟩e−iφ
++⟨(∆A13(0) − ∆A24(0))
+�
+∆A31(τ)⟩ − ∆A42(τ))⟩eiφ��
+,
+(57)
+H(3)
+π,φ(τ) = ⟨∆ ˆE−
+π (0)∆ ˆEπ,φ(τ)∆ ˆE+
+π (0)⟩
+= Re
+�
+eiφ⟨[∆A13(0) − ∆A24(0)] [∆A31(τ) − ∆A42(τ)] [∆A31(0) − ∆A42(0)]⟩
+�
+.
+(58)
+The initial conditions of the correlations are given in Ap-
+pendix A.
+From hπ,π/2(0) = 0 we can obtain analytically the ini-
+tial values of the second- and third-order terms,
+h(2)
+π,π/2(0) = 1 − (2∆ − δ)2 + γ2
+2D
+,
+(59)
+h(3)
+π,π/2(0) = (2∆ − δ)2 + γ2
+2D
+− 2,
+(60)
+where D is given by Eq. (21).
+
+11
+Being the AIC a function of odd-order in the field am-
+plitude we rightly expect a richer landscape than that
+of the intensity correlations, more so when one considers
+quantum interference and the complex parameter space.
+For instance, the correlation can take on not only nega-
+tive values but break classical bounds [18, 19]:
+0 ≤ hφ(τ) − 1 ≤ 1 ,
+(61a)
+|h(2)
+φ (τ) − 1| ≤ |h(2)
+φ (0) − 1| ≤ 1 ,
+(61b)
+where the second line is valid only for weak fields such
+that h(3)
+φ (τ) ∼ 0. These classical bounds are stronger cri-
+teria for nonclassicality of the emitted field than squeezed
+light measurements, the more familiar probing of phase-
+dependent fluctuations. A detailed hierarchy of nonclas-
+sicality measures for higher-order correlation functions is
+presented in Refs. [25, 26]. In Ref. [20] an inequality was
+obtained that considers the full hφ(τ) by calculating the
+AIC for a field in a coherent state,
+−1 ≤ hφ(τ) ≤ 1 .
+(62)
+For a meaningful violation of Poisson statistics, hφ(τ)
+must be outside these bounds.
+Also, hφ(τ) is a measure of non-Gaussian fluctuations,
+here of third-order in the field fluctuations. Resonance
+fluorescence is a particularly strong case of non-Gaussian
+noise by being a highly nonlinear stationary nonequilib-
+rium process [20, 23, 24, 27, 28], thanks also to its small
+Hilbert space. This makes resonance fluorescence unsuit-
+able to a quasiprobability distribution approach.
+B.
+Fluctuations Spectra
+Since quadrature fluctuations, such as squeezing, are
+often studied in the frequency domain we now define the
+spectrum of the amplitude-intensity correlations:
+Sπ,φ(ω) = 8γ1
+� ∞
+0
+dτ cos (ωτ) [hπ,φ(τ) − 1]
+(63)
+which, following Eqs. (52) and (55), can be decomposed
+into terms of second- and third-order in the dipole fluc-
+tuations
+S(q)
+π,φ(ω) = 8γ1
+� ∞
+0
+dτ cos (ωτ)h(q)
+π,φ(τ),
+(64)
+where q = 2, 3, so that Sπ,φ(ω) = S(2)
+π,φ(ω) + S(3)
+π,φ(ω).
+As mentioned above, the AIC was devised initially to
+measure squeezing without the issue of imperfect detec-
+tion efficiencies. Obviously, hπ,φ(τ) and Sπ,φ(ω) are not
+measures of squeezing. They measure a third-order mo-
+ment in the field’s amplitude, while squeezing is a second-
+order one in its fluctuations.
+The so-called spectrum
+of squeezing is the one for q = 2, with the advantage
+of the AIC of not depending on the efficiency of detec-
+tion. Squeezing is signaled by frequency intervals where
+0
+2
+4
+6
+8
+10
+12
+0
+2
+4
+6
+-10
+-5
+0
+5
+10
+0
+1
+2
+0
+2
+4
+6
+8
+10
+12
+0
+2
+4
+6
+-10
+-5
+0
+5
+10
+-1
+0
+1
+2
+0
+2
+4
+6
+8
+10
+12
+0
+2
+4
+6
+-10
+-5
+0
+5
+10
+-1
+0
+1
+2
+Sπ,π/2 (ω)
+hπ,π/2 (τ)
+(b) Ω = γ/2
+γτ
+ω/γ
+(a) Ω = γ/4
+(d) Ω = γ/4
+(e) Ω = γ/2
+(f
+) Ω = γ
+(
+FIG. 10.
+Amplitude-intensity correlations (left panel) and
+spectra (right panel) for the φ = π/2 quadrature in the weak-
+moderate field limit. Parameters and line styles are the same
+as in Fig. 8: ∆ = δ = 0 (solid-black); ∆ = 2γ and δ = −2γ
+(dots-red); ∆ = −2γ and δ = −2γ (dashed-green); ∆ = −2γ
+and δ = −4γ (dot-dashed-blue).
+S(2)
+π,φ(ω) < 0. As a further note, the full incoherent spec-
+trum, Eq. (45), can be obtained by adding the squeezing
+spectra of both quadratures [29],
+Sinc
+π (ω) =
+1
+8γ1
+�
+S(2)
+π,0(ω) + S(2)
+π,π/2(ω)
+�
+.
+(65)
+C.
+Results
+We now show plots of the AICs and their spectra in
+Figs. 10-12 for the φ = π/2 quadrature and the same sets
+of detunings ∆, δ of Fig. 2, and weak to moderate Rabi
+frequencies, γ/4 < Ω < γ. With the three parameters Ω,
+∆, and δ, the landscape of effects is vast.
+We first notice a few general features seen in hπ,π/2(τ),
+Fig. 10.
+With increasing Rabi frequencies, detunings,
+and Zeeman splittings we observe the clear breakdown of
+the classical inequalities besides the one at τ = 0. Cor-
+respondingly, in the spectra, the extrema get displaced
+and broadened.
+Now, we want to single out the case
+of nondegeneracy with small detuning on the |1⟩ − |3⟩
+transition but large on the |2⟩ − |4⟩ one, ∆ = −δ = 2γ
+(green-dashed line). For weak field, Ω = γ/4, the AIC
+does not have a regular evolution for short times but it
+does decay very slowly, with a correponding very narrow
+spectral peak. The slow decay is also clearly visible in the
+photon correlation, Fig. 8a. As we mentioned in Sect. III
+regarding Fig. 2b, state |4⟩ ends up with a large portion
+of the steady state population due to optical pumping;
+not quite a trapping state, so there is no electron shelv-
+ing per se, as argued in [5]. This effect is washed out for
+larger Rabi frequencies, which allow for faster recycling
+of the populations. To a lesser degree, slow decay and
+sharp peak occur for opposite signs of ∆ and δ.
+
+12
+0
+2
+4
+6
+8
+10
+-2
+-1
+0
+1
+2
+3
+4
+-10 -8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+0
+2
+4
+6
+8
+10
+-2
+-1
+0
+1
+2
+3
+4
+-10 -8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+0
+2
+4
+6
+8
+10
+-1
+0
+1
+2
+3
+-10 -8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+-0.5
+0
+0.5
+1.0
+(f
+) Ω = γ
+(
+(b) Ω = γ/2
+(e) Ω = γ/2
+(a) Ω = γ/4
+(d) Ω = γ/4
+h(2)
+π,π/2 (τ)
+ω/γ
+γτ
+S(2)
+π,π/2 (ω)
+FIG. 11.
+Second-order component of the AIC and spectra of
+Fig. 10.
+0
+2
+4
+6
+8
+10
+-1
+0
+1
+2
+-10 -8 -6 -4 -2
+0
+2
+4
+6
+8
+10
+0
+1
+2
+0
+2
+4
+6
+8
+10
+-1
+0
+1
+2
+-10 -8 -6 -4 -2
+0
+2
+4
+6
+8
+10
+-1
+0
+1
+2
+0
+2
+4
+6
+8
+10
+-1
+0
+1
+2
+3
+-10 -8 -6 -4 -2
+0
+2
+4
+6
+8
+10
+-2
+-1
+0
+1
+(f
+) Ω = γ
+(
+(b) Ω = γ/2
+(e) Ω = γ/2
+(a) Ω = γ/4
+(d) Ω = γ/4
+h(3)
+π,π/2 (τ)
+ω/γ
+γτ
+S(3)
+π,π/2 (ω)
+FIG. 12.
+Third-order component of the AIC and spectra of
+Fig. 10.
+The splitting of the AIC and spectra into components
+of second and third order in the fluctuations, Figs. 11, 12,
+helps to understand better the quadrature fluctuations.
+For the second-order ones we have the squeezing spectra:
+around ω = 0 for ∆ = 0 and small Rabi frequencies,
+Ω < γ/4; and in sidebands for larger detunings, Rabi
+frequencies and Zeeman splittings. In h(2)
+π,π/2(τ) there is a
+reduction in amplitudes and nonclassicality for increasing
+Rabi frequencies except for the case of oppposite signs of
+detuning and difference Zeeman splitting. Note that the
+sharp spectral peak in the latter case takes up most of
+the corresponding peak in Fig. 10. This is because both
+π transitions are largely detuned from the laser, keeping
+Ω small.
+Increasing the laser strength the third-order effects
+overcome the second-order ones. For instance, regarding
+the size of the features. Also, a comparison of Figs. 11
+0
+2
+4
+6
+8
+10
+-10
+0
+10
+20
+-40
+-20
+0
+20
+40
+-10
+-5
+0
+5
+10
+0
+2
+4
+6
+8
+10
+-10
+0
+10
+20
+-40
+-20
+0
+20
+40
+-10
+-5
+0
+5
+10
+0
+2
+4
+6
+8
+10
+-10
+0
+10
+20
+-40
+-20
+0
+20
+40
+-10
+-5
+0
+5
+10
+0
+2
+4
+6
+8
+10
+-10
+0
+10
+20
+-40
+-20
+0
+20
+40
+-5
+0
+5
+hπ,π/2 (τ)
+h
+(2)
+π,π/2 (τ)
+h
+(3)
+π,π/2 (τ)
+S
+(3)
+π,π/2 (τ)
+S
+(2)
+π,π/2 (τ)
+Sπ,π/2 (τ)
+γτ
+ω/γ
+(b)
+(f
+)
+(
+(g)
+(d)
+(h)
+(a)
+(e)
+FIG. 13.
+AIC and spectra for Ω = 9γ, ∆ = 0, (a,e) δ = −8γ,
+(b,f) δ = −10γ, (c,g) δ = −12γ, (d,h) δ = −15γ. Lines are:
+full AIC and spectra (solid-black), second-order (dots-red),
+and third-order (dashed-blue).
+and 12 shows that h(3)
+φ (τ) is mainly responsible for the
+breakdown of the classical bounds when the driving field
+is on or above saturation.
+Moreover, we see that the
+slow-decay–sharp-peak is mainly a third-order effect.
+To close this Section, the AIC and spectra for very
+strong fields and large Zeeman splittings, Ω, |δ| ≫ γ are
+shown in Fig. 13. The AIC shows beats as in the photon
+correlations.
+Unlike those in g(2)(τ), these wavepack-
+ets oscillate around h(τ) = 1.
+Because the regime
+is that of strong excitation the third-order component
+clearly dominates, making the fluorescence notably non-
+Gaussian, and clearly violates the classical inequalities.
+The spectral peaks are localized around the Rabi frequen-
+cies ±Ω1, ±Ω2. Studies of the spectrum of squeezing for
+the J = 1/2 − J = 1/2 system were reported in [10].
+Those authors choose ±Ω1, ±Ω2 with a less strong laser
+but large detuning and large Zeeman splittings, observ-
+ing the double sidebands, but no mention or hint of beats
+was made.
+VIII.
+VARIANCE
+The variance is a measure of the total noise in a
+quadrature; it is defined as
+Vφ = ⟨: (∆Eφ)2 :⟩ = Re
+�
+e−iφ⟨∆ ˆE−∆ ˆEφ⟩st
+�
+, (66)
+
+13
+and is related to the spectrum of squeezing as
+Vφ =
+1
+4πγη
+� ∞
+−∞
+dωS(2)
+φ (ω).
+(67)
+where η is the detector efficiency. The maximum value
+of Vφ is 1/4, obtained when there is very strong driving,
+when almost all the emitted light is incoherent. Negative
+values of the variance are a signature of squeezing but,
+unlike the quadrature spectra, the squeezing is the total
+one in the field, independent of frequency.
+For the π transitions we have
+Vπ,φ = f 2
+π(r)
+2
+Re
+�
+−(α13 − α24)2e−2iφ
++(α11 + α22 − |α13 − α24|2)
+�
+,
+(68)
+= f 2
+π(r)
+2
+Ω2
+D
+�
+1 − [(2∆ − δ) cos φ + γ sin φ]2
+2D
+�
+.
+(69)
+For φ = π/2 and φ = 0 we have, respectively,
+Vπ,π/2 = f 2
+π(r)
+2
+Ω2
+D
+�
+1 − γ2
+2D
+�
+,
+(70a)
+Vπ,0 = f 2
+π(r)
+2
+Ω2
+D
+�
+1 − (2∆ − δ)2
+2D
+�
+,
+(70b)
+where D is given by Eq. (21).
+In Fig. 14 we plot the variances of the out-of-phase
+φ = π/2 (left panel) and in-phase φ = 0 (right panel)
+quadratures. The interplay of parameters is a complex
+one, but we mostly use the ones of previous figures. For
+φ = π/2 and ∆ = 0, as usual in resonance fluorescence
+systems, squeezing is restricted to a small range of Rabi
+frequencies, detunings, and Zeeman splittings. For φ = 0
+nonzero laser or Zeeman detunings are necessary to pro-
+duce squeezing, with a strong dependence on their sign:
+on-resonance (not shown) there is no squeezing, as for
+a two-level atom; in Fig. 14(d) the laser is tuned be-
+low that transition, ∆ = −2γ, and there is no squeezing
+(positive variance) but the variance is reduced for large
+δ; in Fig. 14(e) the laser is tuned above the transition,
+∆ = −2γ, and there is squeezing for larger Rabi frequen-
+cies. Large values of δ tend to reduce the variance, be it
+positive or negative.
+A.
+Out-of-phase quadrature
+We now discuss a complementary view of the variance.
+For φ = π/2 we can identify the Rabi frequency interval
+within which squeezing takes place,
+0 < Ω < 1
+2
+�
+γ2/2 − δ2/2 − 2(∆ − δ/2)2,
+(71)
+and the Rabi frequency for maximum squeezing is
+˜Ωπ/2 = 1
+2
+�
+γ4/2 − 2[(δ − ∆)2 + ∆2]2
+3γ2 + 2[(δ − ∆)2 + ∆2]2 .
+(72)
+0
+0.5
+1.0
+1.5
+2.0
+-0.05
+0
+0.05
+0.10
+0.15
+0.20
+0
+0.5
+1.0
+1.5
+2.0
+0
+0.04
+0.08
+0.12
+0.16
+0.20
+0.24
+0
+0.5
+1.0
+1.5
+2.0
+0
+0.04
+0.08
+0.12
+0.16
+0
+0.5
+1.0
+1.5
+2.0
+-0.02
+0
+0.02
+0.04
+-3
+-2
+-1
+0
+1
+2
+3
+-0.03
+-0.02
+-0.01
+0
+0.01
+-3
+-2
+-1
+0
+1
+2
+3
+-0.04
+0
+0.04
+0.08
+0.12
+0.16
+0.2
+Vπ,π/2/f2
+π(r)
+Vπ,0/f2
+π(r)
+Ω/γ
+Ω/γ
+Ω/γ
+∆/γ
+∆/γ
+Ω/γ
+(
+(a)
+(b)
+(d)
+(e)
+(f
+)
+FIG. 14.
+Variance of the quadratures of the fluorescence of
+the π transitions: left panel for φ = π/2 and right panel for
+φ = 0. (a,b,d,e) as a function of Rabi frequency and (c,f)
+as a function of detuning.
+In all cases δ = 0 is given by
+a solid-black line, and δ = −0.5γ by a dashed-red line; the
+dotted-blue line is δ = −2γ in (a,b,d,e) and δ = −γ in (c,f).
+Additionally, (a) ∆ = 0, (b) ∆ = −2γ, (c) Ω = 0.2γ, (d)
+∆ = 0, (e) ∆ = 2γ, (f) Ω = 0.8γ.
+Thus, the variance at ˜Ωπ/2 is
+V
+(˜Ωπ/2)
+π,π/2 (∆ = 0, δ) = f 2
+π(r)
+16
+(γ4/2 − 2δ4)(δ2 − γ2)
+γ2(γ2 + 2δ2)(δ2 + γ2),
+(73a)
+for ∆ = 0 and |δ/γ| < 1/
+√
+2;
+V
+(˜Ωπ/2)
+π,π/2 (∆, δ = 0) = f 2
+π(r)
+16
+(γ4/2 − 8∆4)(4∆2 − γ2)
+γ2(γ2 + 4∆2)2
+,
+(73b)
+for δ = 0 and |∆/γ| < 1/
+√
+2; and the maximum total
+squeezing is obtained at ∆ = δ = 0,
+V
+(˜Ωπ/2)
+π,π/2 (0, 0) = −f 2
+π(r)
+32 ,
+˜Ωπ/2 =
+γ
+2
+√
+6.
+(73c)
+For φ = π/2 squeezing is limited to elliptical regions
+of weak driving and small detunings ∆ and δ:
+2δ2 + 8Ω2 < γ2,
+∆ = 0,
+(74a)
+4∆2 + 8Ω2 < γ2,
+δ = 0.
+(74b)
+B.
+In-phase quadrature
+For φ = 0, squeezing is obtained in the Rabi frequency
+interval, for δ = 0,
+0 < Ω <
+1
+√
+2
+�
+∆2 − γ2/4,
+|∆| > γ/2,
+(75)
+
+14
+with maximum squeezing at the Rabi frequency
+˜Ω0 =
+1
+2
+√
+2
+�
+16∆2 − γ2
+12∆2 + γ2 ,
+(76)
+requiring finite detuning from both π transitions (∆ ̸= 0)
+and stronger driving, Ω ∼ γ [see Fig. 14(d)-(f)].
+Thus, the variance at ˜Ω0 is
+V (˜Ω0)
+π,0 (δ) = −f 2
+π(r)
+128
+4∆2 − γ2
+∆2(4∆2 + γ2),
+|∆| ≥ γ/2. (77)
+This expression gets the asymptotic value
+lim
+∆→∞ V (˜Ω0)
+π,0
+= −f 2
+π(r)
+32 ,
+(78)
+which is the same as that for the π/2 quadrature. The
+region for squeezing obeys the relation
+4∆2 − 8Ω2 < γ2.
+(79)
+So, to obtain squeezing in this quadrature it is necessary
+to have detunings ∆ > γ/4 for any Rabi frequency.
+IX.
+DISCUSSION AND CONCLUSIONS
+We have studied several properties of the resonance
+fluorescence of the π transitions in a J = 1/2 − J = 1/2
+angular momentum atomic system driven by a linearly
+polarized laser field and a magnetic field along the π tran-
+sition to lift the level degeneracies. Interference among
+the various transition amplitudes create a rich landscape
+of effects. Most notable among our results is the observa-
+tion of quantum beats when the atom is subject to large
+laser and magnetic fields. In this regime, two close Rabi
+frequencies interfere, giving rise to a well-defined modu-
+lation of the fast oscillations. These Rabi frequencies are
+the source of the two pairs of sidebands in the incoherent
+part of the power spectrum [5] and in the squeezing spec-
+trum [10]. We studied beats in the total intensity and
+two-time functions such as the dipole-dipole, intensity-
+intensity and intensity-amplitude correlations.
+In the
+beats’ regime the role of vacuum-induced coherence is
+small because the upper levels are very separated due to
+very large difference Zeeman splitting.
+Before the beats we considered the previously over-
+looked time-dependent populations and reviewed aspects
+of the known stationary ones. The fact that the upper
+state populations evolve out of phase should not be a
+surprise.
+This, and nonzero initial population of both
+ground states (in contrast to nonzero populations of ex-
+cited states for spontaneous emission), are major factors
+in the interference among the terms in the intensity. Ex-
+cept for very strong laser fields, the steady state popula-
+tions depend strongly on the difference Zeeman splitting.
+The AIC also permits to quantify the degree of non-
+Gaussianity; the fluctuations of third-order in the field
+quadrature amplitude due to strong atom-laser nonlin-
+earity dominate over the second-order ones with strong
+driving.
+The beats are in the strongly non-Gaussian
+regime.
+The correlations show nonclassical features of the fluo-
+rescence light such as antibunching, g(2)(0) = 0, and vio-
+lation of classical inequalities in the amplitude-intensity
+correlations, Eqs. (61 -62). We studied squeezing using
+the variance, i. e., the total noise in a quadrature, as
+well as using the second-order part of the spectrum. In
+the regime of beats there is squeezing, near the effective
+Rabi frequencies, but none in the total noise.
+For a system with many parameters the interplay
+among them is a complex one, making the interpretation
+of results nontrivial. Thus, for most of our plots we chose
+parameters in two groups: i) where they are relatively
+small, Ω, ∆, δ ∼ γ, chosen to illustrate several degrees of
+vacuum-induced coherence; and ii) where they are large,
+Ω, ∆, δ ≫ γ, and quantum beats are revealed. Overall,
+particular care must be taken regarding detunings. On
+the one hand, large difference Zeeman splitting means
+that the excited levels would be very separated and in-
+teract with different frequency portions of the reservoir,
+hence diminishing the vacuum-induced coherence.
+On
+the other, large laser-atom detunings, which might in-
+crease the VIC, mean reduced fluorescence rates, which
+may also be detrimental in measurements. The beats,
+then, would be better observed if ∆ ≤ γ and δ of just
+several γ in the strong field regime.
+X.
+ACKNOWLEDGMENTS.
+The authors thank Dr. Ricardo Rom´an-Ancheyta and
+Dr. Ir´an Ramos-Prieto for useful comments at an early
+stage of the project. ADAV thanks CONACYT, Mexico,
+for scholarship No. 804318.
+ORCID
+numbers:
+H´ector
+M.
+Castro-Beltr´an
+https://orcid.org/0000-0002-3400-7652, Octavio de los
+Santos-S´anchez https://orcid.org/0000-0002-4316-0114,
+Luis Guti´errez https://orcid.org/0000-0002-5144-4782,
+Appendix A: Time-Dependent Matrix Solutions and
+Spectra
+The two-time photon correlations under study have
+the general form ⟨W(τ)⟩ = ⟨O1(0)R(τ)O2(0)⟩, where
+R is the Bloch vector and O1,2 are system operators.
+The same applies to correlations of fluctuation operators
+∆R, ∆O1,2. Using the quantum regression formula [30],
+the correlations obey the equation
+⟨ ˙W(τ)⟩ = M⟨W(τ)⟩,
+(A1)
+which has the formal solution
+⟨W(τ)⟩ = eMτ⟨W(0)⟩,
+(A2)
+where M is given by
+
+15
+M =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−γ
+−iΩ
+0
+0
+iΩ
+0
+0
+0
+−iΩ −
+� γ
+2 + i∆
+�
+0
+0
+0
+iΩ
+0
+0
+0
+0
+−γ
+iΩ
+0
+0
+−iΩ
+0
+0
+0
+iΩ
+−
+� γ
+2 + i(∆ − δ)
+�
+0
+0
+0
+−iΩ
+iΩ
+0
+0
+0
+−
+� γ
+2 − i∆
+�
+−iΩ
+0
+0
+γ1
+iΩ
+γσ
+0
+−iΩ
+0
+0
+0
+0
+0
+−iΩ
+0
+0
+0
+−
+� γ
+2 − i(∆ − δ)
+�
+iΩ
+γσ
+0
+γ2
+−iΩ
+0
+0
+iΩ
+0
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(A3)
+Also, spectra of stationary systems can be evaluated
+more effectively using the above formal approach.
+Be
+g(τ) = ⟨W(τ)⟩. Then, a spectrum is calculated as
+S(ω) ∝
+� ∞
+0
+cos ωτ g(τ) dτ =
+� ∞
+0
+cos ωτ eMτg(0) dτ
+= Re
+� ∞
+0
+e−(iω1−M)τg(0) dτ
+= Re
+�
+(iω1 − M)−1g(0)
+�
+,
+(A4)
+where 1 is the identity matrix. For example, the inco-
+herent spectrum requires calculations of the type
+Sinc(ω) = Re
+� ∞
+0
+dτe−iωτeMτ⟨∆Aij(0)∆Akl(0)⟩st
+= Re
+�
+(M − iω1)−1⟨∆Aij(0)∆Akl(0)⟩st
+�
+. (A5)
+For the initial conditions of the correlations we use the
+following operator products and correlations in compact
+form:
+AklAmn = Aknδlm ,
+(A6a)
+⟨AklAmn⟩ = αknδlm,
+(A6b)
+AijAklAmn = Ainδjkδlm,
+(A6c)
+⟨AijAklAmn⟩ = αinδjkδlm.
+(A6d)
+Hence, the relevant initial conditions are:
+⟨A13R⟩ = (0, 0, 0, 0, α11, α13, 0, 0)T ,
+(A7a)
+⟨A24R⟩ = (0, 0, 0, 0, 0, 0, α22, α24)T ,
+(A7b)
+⟨A13RA31⟩ = (0, 0, 0, 0, 0, α11, 0, 0)T ,
+(A7c)
+⟨A24RA42⟩ = (0, 0, 0, 0, 0, 0, 0, α22)T ,
+(A7d)
+⟨A13RA42⟩ = ⟨A24RA31⟩ = 0,
+(A7e)
+where R = (A11, A13, A22, A24, A31, A33, A42, A44)T is
+the Bloch vector. For correlations with fluctuation oper-
+ator products, ∆Aij = Aij − αij, we have
+⟨∆Akl∆Amn⟩ = αknδlm − αklαmn,
+(A8)
+⟨∆Aij∆Akl∆Amn⟩ = αinδlmδjk − αilαmnδjk
+−αinαklδjm − αijαknδlm
++2αijαklαmn.
+(A9)
+Now, recalling that α12 = α14 = α23 = α34 = 0, we
+write the detailed initial conditions of the correlations
+(Set 1 of Bloch equations and quantum regression for-
+mula):
+⟨∆A13∆R⟩ =
+�
+−α13α11, −α2
+13, −α13α22, −α13α24, α11 − |α13|2, α13 − α13α33, −α13α42, −α13α44
+�T ,
+(A10a)
+⟨∆A24∆R⟩ =
+�
+−α24α11, −α24α13, −α24α22, −α2
+24, −α24α31, −α24α33, α22 − |α24|2, α24 − α24α44
+�T ,
+(A10b)
+⟨∆A13∆R∆A31⟩ =
+�
+2|α13|2α11 − α2
+11, 2|α13|2α13 − 2α11α13,
+2|α13|2α22 − α11α22, 2|α13|2α24 − α11α24,
+2|α13|2α31 − 2α11α31, 2|α13|2α33 + α11 − 2|α13|2 − α11α33,
+2|α13|2α42 − 2α11α42, 2|α13|2α44 − α11α44
+�T .
+(A10c)
+⟨∆A24∆R∆A42⟩ =
+�
+2|α24|2α11 − α11α22, 2|α24|2α13 − α22α13,
+2|α24|2α22 − α2
+22, 2|α24|2α24 − 2α22α24,
+2|α24|2α31 − α22α31, 2|α24|2α33 − α22α33,
+2|α24|2α42 − 2α22α42, 2|α24|2α44 + α22 − 2|α24|2 − α22α44
+�T .
+(A10d)
+
+16
+⟨∆A13∆R∆A42⟩ =
+�
+2α13α11α42, 2α2
+13α42, 2α13α22α42, (2|α24|2 − α22)α13,
+(2|α13|2 − α11)α42, (2α13α33 − α13)α42, 2α13α2
+42, (2α13α44 − α13)α42
+�T ,
+(A10e)
+⟨∆A24∆R∆A31⟩ =
+�
+2α24α11α31, (2|α13|2 − α11)α24, 2α24α22α31, 2α2
+24α31,
+2α24α2
+31, (2α24α33 − α24)α31, (2|α24|2 − α22)α31, (2α24α44 − α24)α31
+�T .
+(A10f)
+Appendix B: Condition for Optimal Appearance of
+Beats in the Intensity
+We consider a simplified, unitary, model to estimate
+the optimal initial population of the ground states to
+make well-formed beats. First, we diagonalize the Hamil-
+tonian Eq. (8). The eigenvalues and eigenstates are
+E±
+1 = −∆
+2 ± 1
+2
+�
+4Ω2 + ∆2,
+(B1a)
+E±
+2 = Bℓ + δ − ∆
+2
+± 1
+2
+�
+4Ω2 + (δ − ∆)2,
+(B1b)
+and
+|u1⟩ = sin Θ1|1⟩ + cos Θ1|3⟩,
+|u2⟩ = − cos Θ1|1⟩ + sin Θ1|3⟩,
+|u3⟩ = sin Θ2|2⟩ + cos Θ2|4⟩,
+|u4⟩ = − cos Θ2|2⟩ + sin Θ2|4⟩,
+(B2)
+respectively, where
+sin Θ1 =
+2Ω
+��
+∆ +
+√
+∆2 + 4Ω2�2 + 4Ω2
+,
+cos Θ1 =
+∆ +
+√
+∆2 + 4Ω2
+��
+∆ +
+√
+∆2 + 4Ω2�2 + 4Ω2
+,
+sin Θ2 =
+2Ω
+��
+(δ − ∆) +
+�
+(δ − ∆)2 + 4Ω2
+�2
++ 4Ω2
+,
+cos Θ2 =
+(δ − ∆) +
+�
+(δ − ∆)2 + 4Ω2
+��
+(δ − ∆) +
+�
+(δ − ∆)2 + 4Ω2
+�2
++ 4Ω2
+.
+(B3)
+It is now straightforward to obtain the excited-state
+populations. If the initial state of the system is ρ(0) =
+⟨A33(0)⟩|3⟩⟨3| + ⟨A44(0)⟩|4⟩⟨4| we get
+⟨A33(t)⟩ = 1
+2⟨A33(0)⟩ sin2 (2Θ1)(1 − cos (Ω1t)), (B4a)
+⟨A44(t)⟩ = 1
+2⟨A44(0)⟩ sin2 (2Θ2)(1 − cos (Ω2t)), (B4b)
+and the intensity of the field is
+Iπ(r, t)
+f 2π(r) = ⟨A33(0)⟩ sin2 (2Θ1) + A44(0)⟩ sin2 (2Θ2)
+−⟨A33(0)⟩ sin2 (2Θ1) cos (Ω1t)
+−⟨A44(0)⟩ sin2 (2Θ2) cos (Ω2t).
+(B5)
+A necessary condition for the beating behavior to oc-
+cur is that the initial ground-state populations are both
+nonvanishing in the nondegenerate case. Now, assuming
+the relation
+⟨A33(0)⟩
+⟨A44(0)⟩ = sin2 (2Θ2)
+sin2 (2Θ1)
+(B6)
+is satisfied by chossing appropriate parameter values
+(Ω, δ, ∆) for given values of initial ground state popu-
+lations we would get
+Iπ(r, t) = f 2
+π(r)⟨A33(0)⟩ sin2 (2Θ1)
+× [1 − cos (Ωbeatt) cos (Ωavt)] ,
+(B7)
+where Ωbeat = (Ω2 − Ω1)/2 and Ωav = (Ω2 + Ω1)/2.
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+

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+arXiv:2301.05304v1  [math.RT]  12 Jan 2023
+A characterization of the L2-range of the Poisson transforms on a class of
+vector bundles over the quaternionic hyperbolic spaces
+Abdelhamid Boussejra ∗Achraf Ouald Chaib†
+Department of Mathematics, Faculty of Sciences
+University Ibn Tofail, Kénitra, Morocco
+Abstract
+We study the L2-boundedness of the Poisson transforms associated to the homogeneous vector bundles
+Sp(n, 1)×Sp(n)×Sp(1) Vτ over the quaternionic hyperbolic spaces Sp(n, 1)/Sp(n)× Sp(1) associated with irreducible
+representations τ of Sp(n)×Sp(1) which are trivial on Sp(n). As a consequence, we describe the image of the section
+space L2(Sp(n, 1)×Sp(n)×Sp(1) Vτ) under the generalized spectral projections associated to a family of eigensections
+of the Casimir operator.
+Keywords: Vector Poisson transform, Fourier restriction estimate, Strichartz conjecture.
+1
+Introduction
+Let G be a connected real semisimple noncompact Lie group with finite center, and K a maximal compact subgroup.
+Then X = G/K is a Riemannian symmetric space of noncompact type. Let G = KAN be an Iwasawa decomposition
+of G, and let M be the centralizer of A in K. We write g = κ(g)eH(g)n(g), for each g ∈ G according to G = KAN.
+A central result in harmonic analysis (see [17]) asserts that all joint eigenfunctions F of the algebra D(X) of invariant
+differential operators, are Poisson integrals
+F(g) = Pλf(g) :=
+�
+K
+e(iλ+ρ)H(g−1k)f(k) dk,
+of a hyperfunction f on K/M, for a generic λ ∈ a∗
+c (the complexification of a∗ the real dual of a).
+Since then a characterization of the Lp-range of the Poisson transform was developed in several articles such as [3],
+[5], [6], [7], [15], [20], [21], [22], [24], [25].
+The problem of characterizing the image of the Poisson transform Pλ of L2(K/M) with real and regular spectral
+parameter λ is intimately related to Strichartz conjecture [[25], Conjecture 4.5] on the uniform L2-boundedness of
+the generalized spectral projections associated with D(X).
+To be more specific, consider the generalized spectral
+projections Qλ defined initially for F ∈ C∞
+c (X) by
+QλF(x) =| c(λ) |−2 Pλ(FF(λ, .)(x),
+λ ∈ a∗,
+(1.1)
+where FF is the Helgason Fourier transform of F and c(λ) is the Harish-Chandra c-function.
+Conjecture (Strichartz [[25], Conjecture 4.5]). There exists a positive constant C such that for any Fλ = QλF with
+∗e-mail: [email protected]
+†e-mail:[email protected]
+1
+
+F ∈ L2(X) we have
+C−1 ∥ F ∥2
+L2(X)≤
+sup
+R>0,y∈X
+�
+a∗
++
+1
+Rr
+�
+B(y,R)
+| Fλ(x) |2 dx dλ ≤ C ∥ F ∥2
+L2(X),
+(1.2)
+and
+∥ F ∥2
+L2(X)= γr lim
+R→∞
+�
+a∗
++
+1
+Rr
+�
+B(y,R)
+| Fλ(x) |2 dx dλ.
+(1.3)
+Conversely, if Fλ is any family of joint eigenfunctions for which the right hand side of (1.2) or (1.3) is finite, then there
+exists F ∈ L2(X) such that Fλ = QλF for a.e. λ ∈ a∗
++.
+Here r = rank X, and B(y, R) denotes the open ball in X of radius R about y. The constant γr depends on the
+normalizations of the measures dx and dλ.
+The strichartz conjecture has been recently settled by Kaizuka, see [16]. Most of the proof consists in proving a
+uniform estimate for the Poisson transform. More precisely, the following was proved by Kaizuka [[16], Theorem 3.3]:
+Let F be a joint eigenfunction with eigenvalue corresponding to a real and regular spectral parameter λ . Then F is
+the Poisson transform by Pλ of some f ∈ L2(K/M) if and only if
+sup
+R>1
+1
+Rr
+�
+B(0,R)
+| F(x) |2 dx < ∞.
+Moreover there exists a positive constant C independent of such λ,
+C−1 | c(λ) |2∥ f ∥2
+L2(K/M)≤ sup
+R>1
+1
+Rr
+�
+B(0,R)
+| Pλf(x) |2 dx ≤ C | c(λ) |2∥ f ∥2
+L2(K/M) .
+The generalization of these results to vector bundles setting has only just begin. In [8] we extend Kaizuka result to
+homogeneous line bundles over non-compact complex Grassmann manifolds (See also [4]).
+Our aim in this paper is to generalize theses results to a class of homogeneous vector bundles over the quaternionic
+hyperbolic space G/K, where G is the symplectic group Sp(n, 1) with maximal compact subgroup K = Sp(n)×Sp(1).
+To state our results in rough form, let us first introduce the class of the homogenous vector bundles that we consider
+in this paper. Let τν be a unitary irreducible representation of Sp(1) realized on a (ν + 1)-dimensional Hilbert space
+(V, (., .)ν). We extend τν to a representation of K by setting τν ≡ 1 on Sp(n). As usual the space of sections of the
+homogeneous vector bundle G ×K V associated with τν will be identified with the space Γ(G, τν) of vector valued
+functions F : G → Vν which are right K-covariant of type τν, i.e.,
+F(gk) = τν(k)−1F(g),
+∀g ∈ G,
+∀k ∈ K.
+(1.4)
+We denote by C∞(G, τν) and C∞
+c (G, τν) the elements of Γ(G, τν) that are respectively smooth, smooth with compact
+support in G, and by L2(G, τν) the elements of Γ(G, τν) such that
+∥ F ∥L2(G,τν)=
+��
+G/K
+∥ F(g) ∥2
+ν dgK
+� 1
+2
+< ∞.
+In above ∥ . ∥ν is the norm in Vν and ∥ F(gK) ∥ν=∥ F(g) ∥ν is well defined for F satisfying (1.4).
+Let σν denote the restriction of τν to the group M ≃ Sp(n−1)×Sp(1). Over K/M we have the associated homogeneous
+vector bundle K ×M Vν with L2-sections identified with L2(K, σν) the space of all functions f : K → Vν which are
+M-covariant of type σν and satisfy
+∥ f ∥2
+L2(K,σν)=
+�
+K
+∥ f(k) ∥2
+ν dk < ∞,
+2
+
+where dk is the normalized Haar measure of K.
+For λ ∈ C and f ∈ L2(K, σν), the Poisson transform Pν
+λf is defined by
+Pν
+λf(g) =
+�
+K
+e−(iλ+ρ)H(g−1k)τν(κ(g−1k))f(k) dk
+Let Ω denote the Casimir element of the Lie algebra g of G, viewed as a differential operator acting on C∞(G, τ).
+Then the image Pν
+λ(L2(K, σν)) is a proper closed subspace of Eλ(G, τν) the space of all F ∈ C∞(G, τν) satisfying
+Ω F = −(λ2 + ρ2 − ν(ν + 2))F.
+For more details see section 2.
+For λ ∈ R \ {0}, we define a weighted L2-space E2
+λ(G, τν) consisting of all F in Eλ(G, τν) that satisfy
+∥ F ∥∗= sup
+R>1
+�
+1
+R
+�
+B(R)
+∥F(g)∥2
+ν dgK
+� 1
+2
+< ∞.
+Our first main result is an image characterization of the Poisson transform Pν
+λ of L2(K, σν) for λ ∈ R \ {0}.
+Theorem 1.1. Let λ ∈ R\{0} and ν a nonnegative integer.
+(i) There exists a positive constant Cν independent of λ such that for f ∈ L2(K, σν) we have
+C−1
+ν |cν(λ)| ∥f∥L2(K,σν) ≤ ∥Pν
+λf∥∗ ≤ Cν| | cν(λ) | ∥f∥L2(K,σν),
+(1.5)
+with
+cν(λ) = 2ρ−iλ
+Γ(ρ − 1)Γ(iλ)
+Γ( iλ+ν+ρ
+2
+)Γ( iλ+ρ−ν−2
+2
+)
+.
+Furthermore we have the following Plancherel type formula for the Poisson transform
+lim
+R→+∞
+1
+R
+�
+B(R)
+∥Pν
+λf(g)∥2
+ν dgK = 2 | cν(λ) |2 ∥f∥2
+L2(K,σν) .
+(1.6)
+ii) Pν
+λ is a topological isomorphism from L2(K, σν) onto E2
+λ(G, τν).
+This generalizes the result of Kaizuka [[16], (i) and (ii) in Theorem 3.3] which corresponds to τν trivial.
+Consequence
+For λ ∈ R we define the space
+E∗
+λ(G, τν) = {F ∈ Eλ(G, τν) : M(F) < ∞},
+where
+M(F) = lim sup
+R→∞
+�
+1
+R
+�
+B(R)
+| F(g) |2 dgK
+� 1
+2
+.
+Then as an immediate consequence of Theorem 1.1 we obtain the following result which generalizes a conjecture of
+W. Bray [10] which corresponds to τν trivial.
+Corollary 1.1. If λ ∈ R \ {0} then E∗
+λ(G, τν), M) is a Banach space.
+Remark 1.1. In the case of the trivial bundle (the scalar case) the conjecture of Bray was proved by Ionescu [15] for
+all rank one symmetric spaces . It was generalized to Riemannian symmetric spaces of higher rank by Kaizuka, see
+[16].
+3
+
+Next, let us introduce our second main result on the L2-range of the generalized spectral projections.
+For F ∈ C∞
+c (G, τν) the vector valued Helgason-Fourier transform FνF is given by (see [11])
+Fν F(λ, k) =
+�
+G
+e(iλ−ρ)H(g−1k)τν(κ(g−1k)−1)F(g) dg
+λ ∈ C,
+Then the following inversion formula holds (see section 4)
+F(g) = 1
+2π
+� ∞
+0
+�
+K
+e−(iλ+ρ)H(g−1k)τν(κ(g−1k))FνF(λ, k) | cν(λ) |−2 dλ dk
++
+�
+λj∈Dν
+dν(λj)
+�
+K
+e−(iλj+ρ)H(g−1k)τν(κ(g−1k))FνF(λj, k) dk.
+(1.7)
+In above dν(λ) = −iResµ=λ(cν(µ)cν(−µ))−1, λ ∈ Dν and Dν is a finite set in {λ ∈ C; ℑ(λ) > 0} which parametrizes
+the τν-spherical functions arising from the discrete series of G. It is empty if ν ≤ ρ − 2.
+The formula (1.7) gives rise to the decomposition of L2(G, τν) into a continuous part and a discrete part:
+L2(G, τν) = L2
+cont(G, τν) ⊕ L2
+disc(G, τν)
+Our aim here is to study the operator Qν
+λ, λ ∈ R, defined for F ∈ L2
+cont(G, τν) ∩ C∞
+c (C, τν) by
+Qν
+λF(g) =| cν(λ) |−2 Pν
+λ[Fν F(λ, .)](g),
+(1.8)
+More precisely, following Strichartz idea, we are interested in the following question:
+Characterize those Fλ ∈ Eλ(G, τν) (λ ∈ (0, ∞)) for which there exists F ∈ L2
+cont(G, τν) such that Fλ = Qν
+λF.
+To do so, we introduce the space E2
++(G, τν) consisting of all Vτν-valued measurable functions ψ on (0, ∞) × G such
+that
+(i) Ω ψ(λ, .) = −(λ2 + ρ2 − ν(ν + 2)) ψ(λ, .) a.e. λ ∈ (0, ∞)
+(ii) ∥ ψ ∥+< ∞.
+where
+∥ ψ ∥2
++= sup
+R>1
+� ∞
+0
+1
+R
+�
+B(R)
+∥ ψ(λ, g) ∥2
+ν dgK dλ.
+The second main result we prove in this paper can be stated as follows
+Theorem 1.2.
+(i) There exists a positive constant C such that for F ∈ L2(G, τν) we have
+C−1 ∥ F ∥L2(G,τν)≤∥ Qν
+λF ∥+≤ C ∥ F ∥L2(G,τν)
+(1.9)
+Furthermore we have
+lim
+R→∞
+� ∞
+0
+1
+R
+�
+B(R)
+∥ Qν
+λF ∥2
+ν dgK dλ = 2 ∥ F ∥2
+L2(G,τν)
+(1.10)
+(ii) The linear map Qν
+λ is a topological isomorphism from L2
+cont(G, τν) onto E2
++(G, τν).
+This extends Kaizuka result [ [16], (i) and (ii) in Theorem 3.6] on the Strichartz conjecture (see [25] Conjecture
+4.5] to the class of vector bundles considered here.
+Before giving the outline of the paper, let us mention that a number of authors have obtained an image characterization
+for the Poisson transform Pλ (λ ∈ a∗ \ {0}) of L2-functions on K/M in the rank one case, see [[3], [5], [7], [15]].
+Nevertheless, the obtained characterization is weaker than the one conjectured by Strichartz. The approach taken in
+4
+
+the quoted papers is based on the theory of Calderon-Zygmund singular integrals (see also [21]). Using a different
+approach based on the techniques used in the scattering theory, Kaizuka [16] settled the Strichartz conjecture on
+Riemannian symmetric spaces of noncompact type, of arbitrary rank.
+We now describe the contents of this paper. The proofs of our results are a generalisation of Kaizuka’s method [16]. In
+section 2 we recall some basic facts on the quaternionc hyperbolic spaces and introduce the vector Poisson transforms.
+In section 3, we define the Helgason-Fourier transform on the vector bundles G ×K Vν and give the inversion and
+Plancherel Theorem. The proof of Theorem 1.2 follows from the Plancherel formula and Theorem 1.1. The main
+ingredients in proving Theorem 1.1 are a Fourier restriction estimate for the vector valued Helgason-Fourier transform
+(Proposition 4.1 in section 4) and an asymptotic formula for the vector Poisson transform in the framework of Agmon-
+Hörmander spaces [2] (Theorem 5.1). The proof of Theorem 5.1 will be derived from the Key lemma of this paper
+giving the asymptotic behaviour of the translate of the τν-spherical functions. Section 6 is devoted to the proof of our
+main results. In section 7 we prove the Key Lemma.
+2
+Preliminaries
+2.1
+The quaternionic hyperbolic space
+Let G = Sp(n, 1) be the group of all linear transformations of the right H-vector space Hn+1 which preserve the
+quadratic form
+n
+�
+j=1
+| uj |2 − | un+1 |2. Let K = Sp(n) × Sp(1) be the subgroup of G consisting of pairs (a, d)
+of unitaries.
+Then K is a maximal compact subgroup of G.
+The quaternionic hyperbolic space is the rank one
+symmetric space G/K of the noncompact type. It can be realized as the unit ball B(Hn) = {x ∈ Hn; | x |< 1}.
+The group G acts on B(Hn) by the fractional linear mappings x �→ g.x = (ax + b)(cx + d)−1, if g =
+�
+a
+b
+c
+d
+�
+, with
+a ∈ Hn×n, b ∈ Hn×1, c ∈ H1×n and d ∈ H.
+Denote by g the Lie algebra of G; g = k ⊕ p the Cartan decomposition of g, where p is a vector space of matrices of
+the form
+��
+0
+x
+x∗
+0
+�
+, x ∈ Hn
+�
+, and k =
+��
+X
+0
+0
+q
+�
+, X∗ + X = 0, q + q = 0
+�
+, where X∗ is the conjugate transpose
+of the matrix X and q ∈ H.
+Let H =
+�
+0n
+e1
+te1
+0
+�
+∈ p with te1 = (1, 0, · · · , 0). Then a = R H is a Cartan subspace in p, and the corresponding
+analytic subgroup A = {at = exp t H; t ∈ R}, where at =
+
+
+
+cht
+0
+sht
+0
+0n−1
+0
+sht
+0
+cht
+
+
+ . With A determined we then have that
+M =
+
+
+
+
+
+g =
+
+
+
+q
+0
+0
+0
+m
+0
+0
+0
+q
+
+
+ , m ∈ Sp(n − 1), | q |= 1
+
+
+
+
+
+≃ Sp(n − 1) × Sp(1).
+Let α ∈ a∗ be defined by α(H) = 1. Then a system Σ of restricted roots of the pair (g, a) is Σ = {±α, ±2α} if n ≥ 2
+and Σ = {±2α} if n = 1, with Weyl group W ≃ {±Id}. A positive subsystem of roots corresponding to the positive
+Weyl chamber a+ ≃ (0, ∞) in a is Σ+ = {α, 2α} if n ≥ 2 and Σ+ = {2α} if n = 1.
+Let n = gα + g2α be the direct sum of the positive root subspaces, with dim gα = 4(n − 1) and dim g2α = 3 and N the
+corresponding analytic subgroup of G. Then the half sum of the positive restricted roots with multiplicities counted
+ρ equals to (2n + 1)α, and shall be viewed as a real number ρ = 2n + 1 by the identification a∗
+c ≃ C via λα ↔ λ.
+Let A+ = {at ∈ A;
+t ≥ 0}. Then we have the Cartan decomposition G = KA+K, that is any g ∈ G can be written
+g = k1(g) eA+(g) k2(g),
+k1(g), k2(g) ∈ K and A+(g) ∈ a+.
+5
+
+If we write g ∈ G in (n + 1) × (n + 1) block notation as g =
+�
+a
+b
+c
+d
+�
+. Then a straightforward computation gives
+cosh A+(g) =| d |
+and
+H(g) = log | ce1 + d | .
+(2.1)
+We normalize the invariant measure dgK on G/K so that the following integral formula holds: for all h ∈ L1(G/K),
+�
+G/K
+h(gK)dgK =
+�
+G
+h(g.0)dg =
+�
+K
+� ∞
+0
+h(k at)∆(t) dk dt,
+(2.2)
+where dt is the Lebesgue measure, ∆(t) = (2 sinh t)4n−1(2 cosh t)3, and dk is the Haar measure of K with
+�
+K
+dk = 1.
+2.2
+The vector Poisson transform
+In this subsection we define the Poisson transform associated to the vector bundles G×KVν over Sp(n, 1)/Sp(n)×Sp(1)
+and derive some results referring to [23], [27], and [28] for more informations on the subject.
+Let σν denote the restriction of τν to M. For λ ∈ C we consider the representation σν,λ of P = MAN on Vν defined
+by σν,λ(man) = aρ−iλσν(m). Then σν,λ defines a principal series representations of G on the Hilbert space
+Hν,λ := {f : G → Vν | f(gman) = σ−1
+ν,λ(man)f(g) ∀man ∈ MAN, f|K ∈ L2},
+where G acts by the left regular representation. We shall denote by C−ω(G, σν,λ) the space of its hyperfunctions
+vectors. By the Iwasawa decomposition, the restriction map from G to K gives an isomorphism from Hν,λ onto the
+space L2(K, σν). This yields, the so-called compact picture of Hν,λ, with the group action given by
+πσν,λ(g)f(k) = e(iλ−ρ)H(g−1k)f(κ(g−1k)).
+By C−ω(K, σν) we denote the space of its hyperfunctions vectors.
+A Poisson transform is the continuous, linear, G-equivariant map Pν
+λ from C−ω(G, σν,λ) to C∞(G, τν) defined by
+Pν
+λ f(g) =
+�
+K
+τν(k)f(gk) dk.
+In the compact picture the Poisson transform is given by
+Pν
+λ f(g) =
+�
+K
+e−(iλ+ρ)H(g−1k)τν(κ(g−1k)) f(k) dk.
+Let D(G, τν) denote the algebra of left invariant differential operators on C∞(G, τν). Let Eν,λ(G) be the space of all
+F ∈ C∞(G, τν) such that Ω F = −(λ2 + ρ2 − ν(ν + 2)) F.
+Proposition 2.1. (i) D(G, τν) is the algebra generated by the Casimir operator Ω of g.
+(ii) For λ ∈ C, ν ∈ N, the Poisson transform Pν
+λ maps C−ω(G, σν,λ) to Eν,λ(G).
+Proof. (i) Let U(a) be the universal enveloping algebra of the complexification of a. Since the restriction of τν to M
+is irreducible, then D(G, τν) ≃ U(a)W . As a is one dimensional, then D(G, τν) ≃ C[s2], symmetric functions of one
+variable . Thus D(G, τν) is generated by the Casimir element Ω of the Lie algebra g of G, viewed as a differential
+operator acting on C∞(G, τν).
+(ii) Since σν is irreducible, the image of Pν
+λ consists of joint eigenfunctions with respect to the action of Ω. Moreover
+Ω acts by the infinitesimal character of the the principal series representations πσν,λ. It follows from Proposition 8.22
+and Lemma 12.28 in [18], that
+πσν,λ(Ω) = −(λ2 + ρ2 − c(σν))Id
+on
+C−ω(G, σν,λ),
+(2.3)
+where c(σν) is the Casimir value of σν given by c(σν) = ν(ν + 2).
+6
+
+Let Φν,λ be the τν-spherical function associated to σν. Then Φν,λ admits the following Eisenstein integral repre-
+sentation (see [[11], Lemma 3.2]):
+Φν,λ(g) =
+�
+K
+e−(iλ+ρ)H(g−1k)τν(κ(g−1k)k−1) dk.
+Note that Φν,λ lies in C∞(G, τν, τν) the space of smooth functions F : G → End(Vτν) satisfying
+F(k1gk2) = τν(k−1
+2 )F(g)τν(k−1
+1 ),
+the so called τν-radial functions. Being τν-radial, Φν,λ is completely determined by its restriction to A, by the Cartan
+decomposition G = KAK. Moreover, since σν is irreducible, it follows that Φν,λ(at) ∈ EndM(Vν) ≃ CIdVν, ∀at ∈ A.
+Therefore there exists ϕν : R → C such that Φν,λ(at) = ϕν(t).IdVν. We have
+ϕν,λ(t) =
+1
+ν + 1
+�
+K
+e−(iλ+ρ)H(g−1k)χν(κ(g−1k)k−1) dk,
+(2.4)
+where χν is the character of τν.
+This so-called trace τν-spherical function has been computed explicitly in [12] using the radial part of the Casimir
+operator Ω (see also [26] ). We have ϕν,λ(t) = (cosh t)νφ(ρ−2,ν+1)
+λ
+(t), where φ(ρ−2,ν+1)
+λ
+(t) is the Jacobi function (cf.
+[19])
+φ(ρ−2,ν+1)
+λ
+(t) = 2F1(iλ + ρ + ν
+2
+, −iλ + ρ + ν
+2
+; ρ − 1; − sinh2 t).
+We deduce from (A4) the asymptotic behaviour of ϕν,λ
+ϕλ,ν(at) = e(iλ−ρ)t[cν(λ) + ◦(1)], as t → ∞
+if
+ℑ(λ) < 0.
+(2.5)
+where
+cν(λ) =
+2ρ−iλΓ(ρ − 1)Γ(iλ)
+Γ( iλ+ρ+ν
+2
+)Γ( iλ+ρ−ν−2
+2
+)
+.
+(2.6)
+For λ ∈ C the c-function of Harish-Chandra associated to τν is defined by
+c(τν, λ) =
+�
+N
+e−(iλ+ρ)H(n)τν(κ(n)) dn.
+The integral converges for λ such that ℜ(iλ) > 0 and it has a meromorphic continuation to C.
+In above dn is the Haar measure of N = θ(N), θ being the Cartan involution.
+We may use formula (2.6) to give explicitly c(τν, λ). Indeed, one easily check that c(τν, λ) ∈ EndM(Vν) = CIdVν.
+Then using the following result on the behaviour of Φν,λ(at) ([28], Proposition 2.4)
+Φν,λ(at) = e(iλ−ρ)t(c(τν, λ) + ◦(1))as
+t → ∞,
+together with Φν,λ(at) = ϕν,λ(t).Id, we find then from (2.5) that c(τν, λ) = cν(λ)IdVν.
+We end this section by recalling a result of Olbrich [23] on the range of the Poisson transform on vector bundles which
+reads in our case as follows
+Theorem 2.1. [23] Let ν ∈ N and λ ∈ C such that
+(i) −2iλ /∈ N
+(ii) iλ + ρ /∈ −2N − ν ∪ −2N + ν + 2.
+Then the Poisson transform Pν
+λ is a K-isomorphism from C−ω(K, σν) onto Eν,λ(G).
+7
+
+3
+The vector-valued Helgason-Fourier transfrorm
+In this section we give the inversion and the Plancherel formulas for the Helgason-Fourier transform on the vector
+bundle G ×K Vν.
+According to [11] the vector-valued Helgason-Fourier transform of f ∈ C∞
+c (G, τν) is the Vν-valued function on C × K
+defined by:
+Fνf(λ, k) =
+�
+G
+eλ,ν(k−1g) f(g)dg,
+where eλ,ν is the vector valued function eλ,ν : G → End(Vν) given by
+eλ,ν(g) = e(iλ−ρ)H(g−1)τ −1
+ν (κ(g−1)).
+Notice that our sign on "λ" is the opposite of the one in [11].
+In order to state the next theorem, we introduce the finite set in {λ, ℑ(λ) ≥ 0}
+Dν = {λj = i(ν − ρ + 2 − 2j), j = 0, 1, · · · , ν − ρ + 2 − 2j > 0}.
+Note that Dν is empty if ν ≤ ρ − 2. It parametrizes the discrete series representation of G containing τν, see [12].
+Let
+dν(λj) = 2−2(ρ−ν−1)(ν − ρ − 2j + 2)(ρ − 2 + j)!(ν − j)!
+Γ2(ρ − 1)j!(ν − ρ − j + 2)!
+,
+λj ∈ Dν
+For λj ∈ Dν, we define the operators Qν
+j
+L2(G, τν) → Eν,λj(G, τν)
+F �→ dν(λj) Φν,λj ∗ F
+We denote the image by A2
+j. We set
+L2
+disc(G, τν) =
+�
+j; ν−ρ+2−2j>0
+A2
+j,
+and denote by L2
+cont(G, τν) its orthocomplement. Let L2
+σν(R+ × K, | cν(λ) |−2 dλ dk) be the space of vector functions
+φ : R+ × K → Vν satisfying
+(i) For each fixed λ, φ(λ, km) = σν(m)−1φ(λ, k), ∀m ∈ M
+(ii)
+�
+R+×K ∥ Fνφ(λ, k) ∥2 | cν(λ) |−2 dλ dk < ∞.
+Theorem 3.1. (i) For F ∈ C∞
+c (G, τν) we have the following inversion and Plancherel formulas
+F(g) = 1
+2π
+� ∞
+0
+�
+K
+e∗
+λ,ν(k−1g)FνF(λ, k) | cν(λ) |−2 dλ dk +
+�
+λj∈Dν
+dν(λj)
+�
+K
+e∗
+λj,ν(k−1g)FνF(λj, k) dk,
+(3.1)
+�
+G
+∥ F(g) ∥2
+ν dgK = 1
+2π
+� ∞
+0
+�
+K
+∥ FνF((λ, k) ∥2
+ν| cν(λ) |−2 dλ dk+
+�
+λj∈Dν
+dν(λj)
+�
+K
+< FνF(λj, k), FνF(−λj, k) >ν dk
+(3.2)
+(ii) The Fourier transform Fν extends to an isometry from L2
+cont(G, τν) onto the space L2
+σν(R+ ×K, | cν(λ) |−2 dλ dk).
+The first part of Theorem 3.1 can be easily deduced from the inversion and Plancherel formulas for the spherical
+transform.
+8
+
+Let C∞
+c (G, τν, τν) denote the space of smooth compactly supported τν-radial functions. The spherical transform of
+F ∈ C∞
+c (G, τν, τν) is the C-valued function HνF defined by:
+HνF(λ) =
+1
+ν + 1
+�
+G
+T r[Φν,λ(g−1)F(g))]dg,
+λ ∈ C.
+The inversion and the Plancherel formulas for the τ-spherical transform have been given explicitly in [12]. For the
+convenience of the reader we give an elementary proof by using the Jacobi transform.
+Theorem 3.2. For F ∈ C∞
+c (G, τν, τν) we have the following inversion and Plancherel formulas
+F(g) = 1
+2π
+� +∞
+0
+Φν,λ(g)HνF(λ) | cν(λ) |−2 dλ +
+�
+λj∈Dν
+Φν,λj(g)Hνf(λj) dν(λj),
+(3.3)
+�
+G
+∥ F(g) ∥2
+HS dg = ν + 1
+2π
+� +∞
+0
+| HνF((λ) |2| cν(λ) |−2 dλ + (ν + 1)
+�
+λj∈Dν
+dν(λj) | HνF((λj) |2,
+(3.4)
+In above ∥ ∥HS stands for the Hilbert-Schmidt norm.
+Proof. Let F ∈ C∞
+c (G, τν, τν) and let fν be its scalar component.
+Using the integral formula (2.2), the identity
+Φν,λ(at) = Φν,λ(a−t) = (cosh t)νφ(ρ−2,ν+1)
+λ
+(t) and the fact that ∆(t) = (2 cosh t)−2ν∆ρ−2,ν+1, we have
+HνF(λ) =
+� ∞
+0
+fν(t)(cosh t)νφ(ρ−2,ν+1)
+λ
+(t) ∆(t) dt
+=
+� ∞
+0
+fν(t)(22 cosh t)−νφ(ρ−2,ν+1)
+λ
+(t) ∆ρ−2,ν+1(t) dt.
+(3.5)
+Thus the τν-spherical transform HνF may be written in terms of the Jacobi transform J α,β, with α = ρ − 2 and
+β = ν + 1. Namely, we have
+HνF(λ) = J ρ−2,ν+1[(22 cosh t)−νfν](λ).
+We refer to (A5) in the Appendix for the definition of the Jacobi transform.
+Now the theorem follows from the inversion and the Plancherel formulas for the Jacobi transform (A6), (A6’) and
+(A7) in the Appendix.
+For the proof of the surjectivity statement in Theorem 3.1 we shall need the following result
+Proposition 3.1. Let F ∈ C∞
+c (G, τν) and Φ ∈ C∞(G, τν, τν). Then we have
+Fν(F ∗ Φ)(λ, k) = HνΦ(λ)FνF(λ, k),
+λ ∈ C, k ∈ K,
+where the convolution is defined by
+(Φ ∗ F)(g) =
+�
+G
+Φν,λ(x−1g)F(x) dx.
+Proof. Let Φ ∈ C∞(G, τν, τν), v ∈ Vν, and set Fv = Φ(. )v. Then we have the following relation between the Fourier
+transform and the spherical transform
+FνFv(λ, k) = HνΦ(λ)τ(k−1)v.
+(3.6)
+By definition
+Fν(F ∗ Φ)(λ, k) =
+�
+G
+�
+G
+eν
+λ(k−1g)Φ(x−1g)F(x)dxdg
+=
+�
+G
+dx
+�
+G
+eν
+λ(k−1xy)Φ(y)F(x)dy
+9
+
+Using the following cocycle relations for the Iwasawa function H(x)
+H(xy) = H(xκ(y)) + H(y),
+and
+κ(xy) = κ(xκ(y)),
+for all x, y ∈ G, we get the following identity
+eν
+λ(k−1xy) = e(iλ−ρ)H(x−1k)eν
+λ(κ−1(x−1k)y),
+from which we obtain
+Fν(Φ ∗ F)(λ, k) =
+�
+G
+e(iλ−ρ)H(x−1k)
+��
+G
+eλ,ν(κ−1(x−1k)y)Φ(y)F(x) dy
+�
+dx.
+Next, put hv(y) = Φ(y)v, v ∈ Vτν. Then (3.6) implies
+�
+G
+eλ,ν(κ−1(x−1k)y)Φ(y)F(x) dy = Fν(hF (x))(λ, κ−1(x−1k))
+= H(Φ)(λ)τν(κ−1(x−1k))F(x),
+from which we deduce
+Fν(Φ ∗ F)(λ, k) = H(Φ)(λ)
+�
+G
+e(iλ−ρ)H(x−1k)τν(κ−1(x−1k))F(x)dx,
+and the proposition follows.
+We now come to the proof of Theorem 3.1.
+Proof. (i) We may follow the same method as in [11] to prove the inversion formula (3.1) and the Plancherel formula
+(3.2) from Theorem 3.2. We give an outline of the proof.
+Let F ∈ C∞
+c (G, τν) and consider the τν-radial function defined for any g ∈ G by
+Fg,v(x).w =
+�
+K
+< τν(k)w, v >ν F(gkx) dk,
+v being a fixed vector in Vν. Then a straightforward calculation shows that
+HνFg,v(λ) =
+1
+ν + 1 < (Φν,λ ∗ F)(g), v >ν .
+The inversion formula for the spherical transform together with T rFg,v(e) =< F(g), v >ν imply
+F(g) = 1
+2π
+� ∞
+0
+(Φν,λ ∗ F)(g) | cν(λ) |−2 dλ +
+�
+λj∈Dν
+(Φν,λj ∗ F)(g)dν(λj).
+To conclude use the following result for the translated spherical function ( see [11] Proposition 3.3)
+Φν,λ(x−1y) =
+�
+K
+e−(iλ+ρ)H(y−1k)e(iλ−rho)H(x−1k)τν(κ(y−1k))τν(κ−1(x−1k)) dk,
+(3.7)
+to get
+(Φν,λ ∗ F)(g) =
+�
+K
+e−(iλ+ρ)H(g−1k)τν(κ(g−1k))FνF(λ, k) dk,
+10
+
+and the inversion formula (3.1) follows.
+The proof of the Plancherel formula (3.2) is essentially the same as in the scalar case, so we omit it.
+Note that as a consequence of the Plancherel formula not involving the discrete series, we have
+�
+G
+∥ F(g) ∥2 dgK = 1
+π
+� ∞
+0
+�
+K
+∥ FνF(λ, k) ∥2 | cν(λ) |−2 dλ dk,
+for every F ∈ L2
+cont(G, τν).
+(ii) We prove the surjectivity statement. Suppose that there exists a function f in L2
+σν(R+ × K, | cν(λ) |−2 dλ dk)
+such that
+� ∞
+0
+�
+K
+< f(λ, k), FνF(λ, k) >| cν(λ) |−2 dλ dk = 0
+for all F ∈ C∞
+c (G, τν). Changing F into F ∗ Φ where Φ ∈ C∞(G, τν, τν) and using Proposition 3.1, we have
+� ∞
+0
+�
+K
+< f(λ, k), FνF(λ, k) > Hνφ(λ) | cν(λ) |−2 dλ dk = 0
+By the Stone-Weierstrass theorem, the algebra {HνΦ, Φ ∈ C∞(G, τν, τν)} is dense in C∞
+e (R) the space of even
+continuous functions on R vanishing at infinity. Therefore for every F ∈ C∞
+c (G, τν) there is a set EF of measure zero
+in R such that
+�
+K
+< f(λ, k), FνF(λ, k) > dk = 0
+for all λ not in EF . The rest of the proof is based on an adaptation of the arguments given in [14] Theorem 1.5, for
+the scalar case, and the proof of Theorem 3.1 is completed.
+4
+Fourier restriction estimate
+The main result of this section is the following uniform continuity estimate for the Fourier-Helgason restriction operator.
+Proposition 4.1. Let ν ∈ N. There exists a positive constant Cν such that for λ ∈ R\{0} and R > 1, we have
+� �
+K
+∥FνF(λ, k)∥2
+νdk
+�1/2
+≤ Cν|cν(λ)|R1/2
+� �
+G/K
+∥F(g)∥2
+ν dgK
+�1/2
+,
+(4.1)
+for every F ∈ L2(G, τν) with suppF ⊂ B(R).
+To prove this result we shall need estimates of the Harish-Chandra c-function.
+To this end we introduce the
+function bν(λ) defined on R by
+bν(λ) =
+
+
+
+cν(λ)
+if
+ν−ρ+2
+2
+∈ Z+
+λ cν(λ)
+if
+ν−ρ+2
+2
+/∈ Z+
+Lemma 4.1. Assume ν > ρ − 2.
+(i) The function bν(λ) has no zero in R.
+(ii) There exists a positive constant C such that for λ ∈ R, we have
+C−1(1 + λ2)
+2ρ−4−ε(ν)
+4
+≤| bν(λ) |−1≤ C(1 + λ2)
+2ρ−4−ε(ν)
+4
+,
+(4.2)
+11
+
+with ε(ν) = ±1 according to ν−ρ+2
+2
+/∈ Z+ or ν−ρ+2
+2
+∈ Z+
+Proof.
+(i) If ν−ρ+2
+2
+/∈ Z+, then bν(λ) = 2ρ+ν−iλΓ(ρ−1)Γ(iλ+1)
+Γ( iλ+ρ+ν
+2
+)Γ( iλ+ρ−ν−2
+2
+), and clearly bν(λ) has no zero on R.
+If ν−ρ+2
+2
+∈ Z+ then bν(λ) a priori can have zero and pole at λ = 0. This is not the case, since
+lim
+λ→0 bν(λ) = (−1)
+ν−ρ+2
+2
+2ρ+νΓ(ρ − 1)( ν−ρ+2
+2
+)!
+Γ( ρ+ν
+2 )
+.
+(ii) To prove the estimate (4.2) we shall use the following property of the Γ-function
+lim
+|z|→∞
+Γ(z + a)
+Γ(z)
+z−a = 1, | arg(z) |< π − δ,
+(4.3)
+where a is any complex number, and log is the principal value of the logarithm and δ > 0.
+Assume first that ν−ρ+2
+2
+/∈ Z+. Using the duplicata formula for the function gamma
+Γ(2z) = 22z−2
+√π Γ(z)Γ(z + 1
+2),
+we rewrite bν(λ) as
+bν(λ) = 2ρ+ν−1
+√π
+Γ( iλ+1
+2
+)Γ( iλ+2
+2
+)
+Γ( iλ+ρ+ν
+2
+)Γ( iλ+ρ−ν−2
+2
+)
+.
+It follows from (4.3) that for every λ ∈ R, we have
+| bν(λ) |≤ C(1 + λ2)− 2ρ−5
+4
+and
+| bν(λ) |−1≤ C(1 + λ2)
+2ρ−5
+4 .
+The proof for the case ν−ρ+2
+2
+∈ Z+ follows the same line as in the case ν−ρ+2
+2
+/∈ Z+, so we omit it.
+This finishes the proof of the Lemma.
+Let us recall from [1] an auxiliary lemma which will be useful for the proof of Proposition 4.1.
+Let η be a positive Schwartz function on R whose Fourier transform has a compact support. For m ∈ R, set
+ηm(x) =
+�
+R
+η(t)(1 + |t − x|)m/2 dt.
+Lemma 4.2.
+i) ηm is a positive C∞-function with
+C−1(1 + t2)
+m
+2 ≤ ηm(t) ≤ C(1 + t2)
+m
+2 ,
+(4.4)
+for some positive constant C.
+ii) The Fourier transform of ηm has a compact support.
+In order to prove the Fourier restriction Theorem, we need to introduce the bundle valued Radon transform, see
+[9] for more informations.
+The Radon transform for F ∈ C∞
+c (G, τν) is defined by
+RF(g) = eρH(g)
+�
+N
+F(gn)dn.
+12
+
+We set RF(t, k) = RF(kat). Then, using the Iwaswa decomposition G = NAK, we may rewrite the Helgason-Fourier
+transform as
+FνF(λ, k) = FR(RF(·, k))(λ),
+where
+FRφ(λ) =
+�
+R
+e−iλtφ(t) dt,
+is the Euclidean Fourier transform of φ a Vν-valued smooth function with compact support in R.
+We define on p the scalar product < X, Y >= 1
+2T r(XY ) and denote by | | the corresponding norm. It induces a distance
+function d on G/K. By the Cartan decomposition G = K exp p, any g ∈ G may be written uniquely as g = k exp X,
+so that d(0, gK) =| X |. Define the open ball centred at 0 and of radius R by B(R) = {gK ∈ G/K;
+d(0, gK) < R}.
+Lemma 4.3. Let F ∈ C∞
+0 (G, τν). If supp F ⊂ B(R), then supp RF ⊂ [−R, R] × K.
+Proof. As (see [[13], page 476]
+d(0, ketHnK) ≥| t |,
+k ∈ K, n ∈ N, t ∈ R
+it follows that supp RF ⊂ [−R, R] × K if supp F ⊂ B(R)
+Proof of Proposition 4.1. It suffices to prove the estimate (4.1) for functions F ∈ C∞
+c (G, τν) supported in B(R).
+It follows from the Plancherel formula (3.2) that
+�
+B(R)
+∥ F(g) ∥2
+ν dgK ≥
+�
+K
+�
+R
+∥ FνF(λ, k) ∥2
+ν | cν(λ) |−2 dλ dk
+Therefore it is sufficient to show
+�
+K
+�
+R
+∥ FνF(λ, k) ∥2
+ν | cν(λ) |−2 dλ dk ≥ C | cν(λ) |−2
+R
+�
+R
+∥ FνF(λ, k) ∥2
+ν dk,
+(4.5)
+fir some positive constant C.
+By (4.2) we have | cν(λ) |−1≍ η 2ρ−3
+2 (λ). Therefore (4.5) is equivalent to
+η 2ρ−3
+2 (λ)
+R
+�
+K
+∥ FνF(λ, k) ∥2
+ν dk ≤
+�
+K
+�
+R
+∥ FνF(λ, k) ∥2
+ν η 2ρ−3
+2 (λ)dλ dk
+(4.6)
+Let T be the tempered distribution on R defined by T := F−1
+R η 2ρ−3
+2 . By Lemma 4.2, T is compactly supported . Let
+R0 > 1 such that supp T ⊂ [−R0, R0]. Then (4.6) is equivalent to
+�
+K
+∥ FR(T ∗ RF(. , k))(λ) ∥2
+ν dk ≤ CR
+�
+K
+�
+R
+FR(T ∗ RF(. , k))(λ) ∥2
+ν dλ dk,
+(4.7)
+where ∗ denotes the convolution on R.
+From suppT ⊂ [−R0, R0] and Lemma 4.3, it follows that for any k ∈ K, supp (T ∗ RF(. , k)) ⊂ [−(R + R0), R + R0].
+Thus
+�
+K
+∥ FR(T ∗ RF(. , k)(λ) ∥2
+ν dk ≤ 2(R + R0)
+�
+K
+�
+R
+∥ (T ∗ RF(. k))(t) ∥2
+ν dt dk
+Next use the Euclidean Plancherel formula to get (4.7), and the proof is finished.
+As a consequence of Proposition 4.1, we obtain the uniform continuity estimate for the Poisson transform Pν
+λ.
+Corollary 4.1. Let ν ∈ N. There exists a positive constant Cν such that for λ ∈ R\{0}, we have
+sup
+R>1
+�
+1
+R
+�
+B(R)
+∥ Pν
+λf(g) ∥2
+ν dgK
+�1/2
+≤ Cν |cν(λ)| ∥ f ∥L2(K,σν)
+(4.8)
+for every f ∈ L2(K, σν).
+13
+
+Proof. Let F ∈ L2(G, τν) with supp F ⊂ B(R), and let f ∈ L2(K, σν). Since λ is real and τν is unitary, the Poisson
+transform and the restriction Fourier transform are related by the following formula
+�
+B(R)
+< Pν
+λf(g), F(g) >ν dg =
+�
+K
+< f(k), FνF(λ, k) >ν dk.
+Thus
+|
+�
+B(R)
+< Pν
+λf(g), F(g) >ν dg | ≤ ∥f∥L2(K,σν)(
+�
+K
+∥ FνF(λ, k) ∥2
+ν dk)
+1
+2
+≤ Cν|cν(λ)|R1/2 ∥ f ∥L2(K,τν)∥ F ∥L2(G,τν),
+by the restriction Fourier theorem. Taking the supermum over all F with ∥ F ∥L2(G,τν)= 1, the corollary follows.
+5
+Asymptotic expansion for the Poisson transform
+In this section we give an asymptotic expansion for the Poisson transform.
+We first start by establishing some
+intermediate results.
+Let L2
+λ(K, σν) denote the finite linear span of the functions
+f g
+λ,v : k �−→ f g
+λ,v(k) = e(iλ−ρ)H(g−1k)τ −1
+ν (κ(g−1k))v,
+g ∈ G, v ∈ Vν.
+Lemma 5.1. For λ ∈ R \ {0}, ν ∈ N the space L2
+λ(K, σν) is a dense subspace of L2(K, σν).
+Proof. As λ ∈ R \ {0}, the density is just a reformulation of the injectivity of the Poisson transform Pν,λ.
+Lemma 5.2. Let λ ∈ R \ {0}, ν ∈ N. Then there exists a unique unitary isomorphism U ν
+λ on L2(K, σν) such that :
+U ν
+λ f g
+λ,v = f g
+−λ,v,
+g ∈ G.
+Moreover, for f1, f2 ∈ L2(K, σν), we have Pν
+λF1 = Pν
+−λF2 if and only if U ν
+λF1 = F2 ( i.e. U ν
+λ = (Pν
+−λ)−1 ◦ Pν
+λ).
+Proof. The proof is the same as in the scalar case so we omit it.
+We now introduce the function space B∗(G, τν) on G, consisting of functions F in L2
+loc(G, τν) satisfying
+∥ F ∥B∗(G,τν)= sup
+j∈N
+[2− j
+2
+�
+Aj
+∥ F(g) ∥2
+ν dgK] < ∞,
+where A0 = {g ∈ G; d(0, g.0) < 1} and Aj = {g ∈ G; 2j−1 ≤ d(0, g.0) < 2j}, for j ≥ 1.
+One could easily show that ∥ F ∥B∗(G,τν)≤∥ F ∥∗≤ 2 ∥ F ∥B∗(G,τν).
+We define an equivalent relation on B∗(G, τν). For F1, F2 ∈ B∗(G, τν) we write F1 ≃ F2 if
+lim
+R→+∞
+1
+R
+�
+B(R)
+∥ F1(g) − F2(g) ∥2
+ν dg = 0.
+Note that by using the polar decomposition we see that F1 ≃ F2 if
+lim
+R→+∞
+1
+R
+�
+K×[0,R]
+∥ F1(ketH)) − F2(ketH)) ∥2
+ν ∆(t) dt dk = 0.
+We now state the main result of this section
+14
+
+Theorem 5.1. Let ν ∈ N, λ ∈ R\{0}. For f ∈ L2(K, σν) we have the following asymptotic expansions for the Poisson
+transform in B∗(G, τν)
+Pλ,νf(x) ≃ τ −1
+ν (k2(x))[cν(λ)e(i��−ρ)(A+(x)f(k1(x)) + cν(−λ)e(−iλ−ρ)(A+(x))U ν
+λf(k1(x))],
+(5.1)
+where x = k1(x)eA+(x)k2(x).
+Most of the proof of the above theorem consists in proving the following Key Lemma, giving the asymptotic ex-
+pansion for the translates of the τν-spherical function.
+KEY LEMMA. For λ ∈ R \ {0}, g ∈ G and v ∈ Vν, we have the following asymptotic expansion in B∗(G, τν)
+Φν,λ(g−1x). v ≃ τ −1
+ν (k2(x))
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)A+(x)f g
+sλ,v(k1(x)),
+x = k1(x)eA+(x)k2(x).
+Proof of Theorem 5.1. We first note that both side of (5.1) depend continuously on f ∈ L2(K, σν). This can
+be proved in the same manner as in [8]. Therefore we only have to prove that the asymptotic expansion (5.1) holds
+for f ∈ L2
+λ(K, σν). Let f = f g
+λ,v. Then according to [[11], Proposition 3.3], we have
+Pν
+λf(x) = Φν,λ(g−1x)v.
+The theorem follows from the Key lemma.
+As a consequence of Theorem 5.1 we obtain the following result giving the behaviour of the Poisson integrals.
+Proposition 5.1.
+1. For any f ∈ L2(K, σν) we have the Plancherel-Poisson formula
+lim
+R→+∞
+1
+R
+�
+B(R)
+∥ Pν
+λf(g) ∥2
+ν dgK = 2 | cν(λ) |2 ∥ f ∥2
+L2(K,σν)
+(5.2)
+2. Let ν ∈ N. There exists a positive constant Cν such that for any λ ∈ R \ {0}, we have
+C−1
+ν
+| cν(λ) | ∥ f ∥L2(K,σν)≤∥ Pλ
+ν f ∥∗≤ Cν | cν(λ) | ∥ f ∥L2(K,σν),
+(5.3)
+for every f ∈ L2(K, σν).
+Proof.
+1. We define for f ∈ L2(K, σν)
+Sν
+λf(x) := τ −1
+ν (k2(x))[cν(λ)e(iλ−ρ)(A+(x)f(k1(x)) + cν(−λ)e(−iλ−ρ)(A+(x))U ν
+λf(k1(x))],
+x = k1(x)eA+(x)k2(x).
+By the unitarity of Uλ, we have
+1
+R
+�
+B(R)
+∥Sν
+λf(g)∥2dgK = 2|cν(λ)|2∥f∥2
+L2(K,τν)
+�
+1
+R
+� R
+0
+e−2ρt∆(t)dt
+�
++ 2|cν(λ)|2ℜ
+�
+< f, Uλf >L2(K,σν)
+1
+R
+� R
+0
+e2(iλ−ρ)t∆(t)dt
+�
+.
+From
+lim
+R→+∞
+1
+R
+� R
+0
+e−2ρt∆(t)dt = 1, and
+lim
+R→+∞
+1
+R
+� R
+0
+e2(iλ−ρ)t∆(t)dt = 0, we deduce that
+lim
+R→+∞
+1
+R
+�
+B(R)
+∥ Sν
+λf(g) ∥2
+ν dgK = 2 | cν(λ) |2∥ f ∥2
+L2(K,σν) .
+(5.4)
+15
+
+Next write
+1
+R
+�
+B(R)
+∥ Pν
+λf(g) ∥2
+ν dgK = 1
+R
+�
+B(R)
+(∥ Sν
+λf(g) ∥2
+ν + ∥ Pν
+λf(g) − Sν
+λf(g) ∥2
+ν
++ 2Re[< Pν
+λf(g) − Sν
+λf(g), Sν
+λf(g) >])dgK.
+The estimate (5.2) then follows from (5.4), Theorem 5.1 and the Schwarz inequality.
+2. The right hand side of the estimate (5.3) has already been proved, see corollary 4.1.
+The left hand side of the estimate (5.3) obviously follows from the estimate (5.2). This finishes the proof of the
+proposition.
+Remark 5.1. Let f1, f2 ∈ L2(K, σν). Then using the polarization identity as well as the estimate (5.2), we get
+lim
+R→+∞
+1
+R
+�
+B(R)
+< Pν
+λf1(g), Pν
+λf2(g) >ν dgK = 2 | cν(λ) |2< f1, f2 >L2(K,σν)
+(5.5)
+6
+Proof of the main results
+In this section we shall prove Theorem 1.1 on the L2-range of the vector Poisson transform and Theorem 1.2 charac-
+terizing the image Qν
+λ(L2(G, τν).
+6.1
+The L2-range of the Poisson transform
+We first recall some results of harmonic analysis on the homogeneous vector bundle K ×M Vν associated to the
+representation σν of M.
+Let �K be the unitary dual of K. For δ ∈ �K let Vδ denote a representation space of δ with dδ = dim Vδ. We denote by
+�K(σν) the set of δ ∈ �K such that σν occurs in δ |M with multiplicity mδ > 0.
+The decomposition of L2(K, σν) under K (the group K acts by left translations on this space) is given by the Frobenius
+reciprocity law
+L2(K, σν) =
+�
+δ∈�
+K(σν)
+Vδ ⊗ HomM(Vν, Vδ),
+where v ⊗L, for v ∈ Vδ, L ∈ HomM(Vν, Vδ) is identified with the function (v ⊗L)(k) = L∗(δ(k−1)v), where L∗ denotes
+the adjoint of L.
+For each δ ∈ �K(σν) let (Lj)mδ
+j=1 be an orthonormal basis of HomM(Vν, Vδ) with respect to the inner product
+< L1, L2 >=
+1
+ν + 1T r(L1L∗
+2).
+Let {v1, · · · , vdδ} be an orhonormal basis of Vδ. Then
+f δ
+ij : k →
+�
+dδ
+ν + 1L∗
+i δ(k−1)vj,
+1 ≤ i ≤ mδ,
+1 ≤ j ≤ dδ,
+δ ∈ �K(σ)
+form an orthonormal basis of L2(K, σν).
+For f ∈ L2(K, σν) we have the Fourier series expansion f(k) =
+�
+δ∈�
+K(σ)
+mδ
+�
+i=1
+dδ
+�
+j=1
+aδ
+ijf δ
+ij(k) with
+∥ f ∥2
+L2(K,σ)=
+�
+δ∈�
+K(σ)
+mδ
+�
+i=1
+dδ
+�
+j=1
+| aδ
+ij |2 .
+16
+
+We define for δ ∈ �K(σ) and λ ∈ C, the generalized Eisenstein integral
+ΦL
+λ,δ(g) =
+�
+K
+e−(iλ+ρ)H(g−1k)τν(κ(g−1k))L∗δ(k−1)dk,
+L ∈ HomM(Vν, Vδ).
+It is easy to see that ΦL
+λ,δ satisfies the following identity
+ΦL
+λ,δ(k1gk2) = τν(k−1
+2 )ΦL
+λ,δ(g)δ(k−1
+1 ),
+k1, k2 ∈ K, g ∈ G.
+We now prove an asymptotic estimate for the generalized Eisenstein integrals.
+Proposition 6.1. Let ν ∈ N, λ ∈ R \ {0}. Then for δ ∈ �K(σν), T, S ∈ HomM(Vν, Vδ) we have
+lim
+R→+∞
+1
+R
+�
+B(R)
+Tr
+�
+ΦT
+λ,δ(g)∗ΦS
+λ,δ(g)
+�
+dgK = 2 | cν(λ) |2 Tr(T S∗).
+(6.1)
+Proof. By definition we have
+lim
+R→+∞
+1
+R
+�
+B(R)
+Tr
+�
+ΦT
+λ,δ(g)∗ΦS
+λ,δ(g)
+�
+dgK =
+dδ
+�
+j=1
+lim
+R→+∞
+1
+R
+�
+B(R)
+< ΦS
+λ,δ(g)vj, ΦT
+λ,δ(g)vj >ν dgK
+Noting that ΦT
+λ,δ(g)vj is the Poisson transform of the function k �→ L∗δ(k−1)vj and using (5.5), we get
+lim
+R→+∞
+1
+R
+�
+B(R)
+Tr
+�
+ΦT
+λ,δ(g)∗ΦS
+λ,δ(g)
+�
+dgK = 2 | cν(λ) |2
+dδ
+�
+j=1
+�
+K
+< S∗δ(k−1)vj, T ∗δ(k−1)vj >ν dk.
+Hence Schur Lemma lead us to conclude that
+lim
+R→+∞
+1
+R
+�
+B(R)
+Tr
+�
+ΦT
+λ,δ(g)∗ΦS
+λ,δ(g)
+�
+dgK = 2 | cν(λ) |2 Tr(T S∗), and
+the proof is finished.
+Remark 6.1. Noting that
+T r(
+�
+ΦT
+λ,δ(g)∗ΦS
+λ,δ(g)
+�
+= T r(
+�
+ΦT
+λ,δ(a)∗ΦS
+λ,δ(a)
+�
+,
+g = k1 a k2,
+it follows from (6.1) that
+lim
+R→+∞
+1
+R
+� R
+0
+T r
+�
+ΦT
+λ,δ(at)∗ΦS
+λ,δ(at)
+�
+∆(t)dt =| cν(λ) |2 Tr(T S∗).
+(6.2)
+Proof of Theorem 1.1.
+(i) The estimate (5.3) implies that the Poisson transform Pλ,ν maps L2(K, σν) into Eλ(G, τν) and that the estimate
+(1.5) holds.
+(ii) We now prove that the Poisson transform maps L2(K, σν) onto E2
+λ(G, τν). Let F ∈ E2
+λ(G, τν). Since λ ∈ R \ {0},
+we know by Theorem 2.1 that there exists a hyperfunction f ∈ C−ω(K, σν) such that F = Pλ,νf.
+Let f =
+�
+δ∈�
+K(σ)
+dδ
+�
+j=1
+mδ
+�
+i=1
+aδ
+ijf δ
+ij, be the Fourier series expansion of f. Then we have
+F(g) =
+�
+δ∈�
+K(σ)
+�
+dδ
+ν + 1
+dδ
+�
+j=1
+mδ
+�
+i=1
+aδ
+ijΦLi
+λ,δ(g)vj
+in
+C∞(G, V ).
+By the Schur relations, we have
+�
+K
+< ΦLi
+λ,δ(kat)vj, ΦLm
+λ,δ′(kat)vn >ν dk =
+�
+0
+if δ ≁ δ′
+1
+dδ T r(ΦLm
+λ,δ′ (at))∗ΦLi
+λ,δ(at) < vj, vn > if
+δ′ = δ
+17
+
+Therefore
+�
+K
+∥ F(kat) ∥2 dk =
+1
+ν + 1
+�
+δ∈�
+K(σ)
+dδ
+�
+j=1
+�
+1≤i,j≤mδ
+aδ
+ijaδ
+mjT r[(ΦLm
+λ,δ (at))∗ΦLi
+λ,δ(at)]
+=
+1
+ν + 1
+�
+δ∈�
+K(σ)
+dδ
+�
+j=1
+T r
+
+
+�
+1≤i,m≤mδ
+(aδ
+mjΦLm
+λ,δ (at))∗(aδ
+ijΦLi
+λ,δ(at)
+
+
+=
+1
+ν + 1
+�
+δ∈�
+K(σ)
+dδ
+�
+j=1
+∥
+mδ
+�
+i=1
+aδ
+ijΦLi
+λ,δ(at) ∥2
+HS,
+Let Λ be a finite subset in �K(σ). Since ∥ F ∥∗< ∞, it follows that, for any R > 1 we have
+∞ >∥ F ∥2
+∗≥
+1
+ν + 1
+�
+δ∈Λ
+dδ
+�
+j=1
+1
+R
+� R
+0
+∥
+mδ
+�
+i=1
+aδ
+ijΦLi
+λ,δ(at) ∥2
+HS ∆(t) dt
+By (6.2) we have
+lim
+R→∞
+1
+R
+� R
+0
+∥
+mδ
+�
+i=1
+aδ
+ijΦLi
+λ,δ(at) ∥2
+HS ∆(t) dt = lim
+R→∞
+�
+1≤i,m≤mδ
+aδ
+ijaδ
+mj
+1
+R
+� R
+0
+T r[(ΦLm
+λ,δ (at))∗ΦLi
+λ,δ(at)] ∆(t)dt
+= 2 | cν(λ) |2
+�
+1≤i,m≤mδ
+aδ
+ijaδ
+mjT r(LiL∗
+m)
+= 2(ν + 1) | cν(λ) |2
+mδ
+�
+i=1
+| aδ
+ij |2 .
+Thus ∞ >∥ F ∥2
+∗≥| cν(λ) |2 �
+δ∈Λ
+dδ
+�
+j=1
+mδ
+�
+i=1
+| aδ
+ij |2. Since Λ is arbitrary, it follows that
+| cν(λ) |2
+�
+δ∈�
+K(σ)
+dδ
+�
+j=1
+mδ
+�
+i=1
+| aδ
+ij |2≤∥ F ∥2
+∗ .
+This shows that f ∈ L2(K, σν) with | cν(λ) |∥ f ∥L2(K,σν)≤∥ Pν
+λf ∥∗ and the proof of the theorem is completed.
+6.2
+The L2-range of the generalized spectral projections
+We now proceed to the poof of the second main result of this paper.
+Proof of Theorem 1.2.
+Let F ∈ L2
+c(G, τν) ∩ C∞(G, τν). It follows from the definition ( see (1.8)) that the operator Qν
+λ may be written as
+Qν
+λF(g) =| cν(λ) |−2 Pν
+λ(FνF(λ, .))(g).
+(6.3)
+Using Theorem 1.1 we deduce that
+sup
+R>1
+1
+R
+�
+B(R)
+∥ Qν
+λF(g) ∥2
+ν dgK ≤ Cν | cν(λ) |−2
+�
+K
+∥ FνF(λ, k) ∥2
+ν dk.
+The above inequality and the Plancherel formula (3.4) imply
+� ∞
+0
+(sup
+R>1
+1
+R
+�
+B(R)
+∥ Qν
+λF(g) ∥2
+ν dgK) dλ ≤ Cν
+� ∞
+0
+�
+K
+∥ FνF(λ, k) ∥2
+ν| cν(λ) |−2 dk dλ
+≤ Cν ∥ F ∥2
+L2(G,τ) .
+18
+
+This prove the right hand side of the inequality (1.9).
+From (6.3) and (1.6) we have
+lim
+R→∞
+1
+R
+�
+B(R)
+∥ Qν
+λF(g) ∥2
+ν dgK = 2 | cν(λ) |−2
+�
+K
+∥ FνF(λ, k) ∥2
+ν dk,
+and since for all R > 1
+1
+R
+�
+B(R)
+∥ Qν
+λF(g) ∥2 dgK ≤ Cν | cν(λ) |−2
+�
+K
+∥ FνF(λ, k) ∥2 dk,
+a.e. λ ∈ (0, ∞),
+we may apply the Lebesgue’s dominated convergence theorem to get
+lim
+R→∞
+� ∞
+0
+�
+1
+R
+�
+B(R)
+∥ Qν
+λF(g) ∥2
+ν dgK
+�
+dλ = 2 ∥ F ∥2
+L2(G,τν) .
+It follows from the above equality that
+C ∥ F ∥2
+L2(G,τν)≤
+� ∞
+0
+(sup
+R>1
+�
+B(R)
+∥ Qν
+λF(x) ∥2 dx) dλ.
+This complete the proof of the inequality (1.9).
+We now prove that Qν
+λ maps L2
+c(G, τν) onto E2
+λ(G, τν). Let Fλ ∈ E2
+λ(G, τν). Then we have
+sup
+R>1
+1
+R
+�
+B(R)
+∥ Fλ(g) ∥2
+ν dgK < ∞,
+for a.e.
+λ ∈ (0, ∞).
+By Theorem 1.1, there exists fλ ∈ L2(K, σν) such that Fλ(g) =| cν(λ) |−2 Pν
+λfλ(g) with
+sup
+R>1
+1
+R
+�
+B(R)
+∥ Fλ(g) ∥2
+ν dgK ≥ C−1
+ν
+| cν(λ) |−2
+�
+K
+∥ fλ(k) ∥2 dk
+Integrating the both side of the above inequality over (0, ∞), we get
+∞ >∥ Fλ ∥2
+∗≥ C−1
+ν
+� ∞
+O
+�
+K
+∥ fλ(k) ∥2
+ν | cν(λ) |−2 dk dλ.
+It now follows from Theorem 3.1, that there exists F ∈ L2
+c(G, τν) such that FνF(λ, k) = fλ(k).
+Henceforth Fλ(g) =| cν(λ) |−2 Pλ,ν(FνF(λ, .)(g). This finishes the proof of Theorem 1.2.
+7
+Proof of the Key Lemma
+In this section we prove the Key Lemma of this paper. To this end we need to establish some auxiliary results. We
+first prove an asymptotic formula for the τν-spherical function.
+Proposition 7.1. Let λ ∈ R \ {0}. For any v ∈ Vν we have
+Φν,λ(g). v ≃
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)A+(g)τ −1
+ν (κ1(g)κ2(g)). v,
+(7.1)
+g = κ1(g)eA+(g)κ2(g)
+Proof. Since ∆(t) ≤ e2ρ t, we get
+1
+R
+�
+B(R)
+∥ e(iλ−ρ)A+(g)τ −1
+ν (κ1(g)κ2(g)). v ∥2 dg = 1
+R ∥ v ∥2
+� R
+0
+e−2ρ t∆(t)dt
+≤∥ v ∥2 .
+19
+
+This shows that the right hand side of (7.1) belongs to B∗(G, τν).
+Since λ ∈ R \ {0}, we may use the identity (A3) to write
+ϕν,λ(t) −
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)t =
+�
+s∈{±1}
+cν(sλ)
+�
+(2 cosh t)νΨρ−2,ν+1
+sλ
+(t) − e(isλ−ρ)t�
+=
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)t �
+(1 + e−2t)νe(ρ+ν−isλ)tΨρ−2,ν+1
+sλ
+(t) − 1
+�
+.
+It follows from (A2’) that
+ϕν,λ(t) −
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)t =
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)t �
+(1 + e−2t)ν − 1) + e−2tEsλ(t)
+�
+,
+where | Esλ(t) |≤ 2νC if t ≥ 1. Therefore
+| ϕν,λ(t) −
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)t |≤ Cν,λe−ρe−2t,
+if t ≥ 1. This together with
+| ϕν,λ(t) −
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)t |≤ Cν,λe−ρt,
+for t ∈ [0, 1], imply that
+lim
+R→∞
+1
+R
+�
+B(R)
+∥ Φν,λ(g). v −
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)A+(g)τ −1(κ1(g)κ2(g)). v ∥2
+ν dgK =
+=∥ v ∥2 lim
+R→∞
+1
+R
+� R
+0
+| ϕν,λ(t) −
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)t |2 ∆(t) dt = 0,
+and the proof is finished.
+Lemma 7.1. Let g ∈ G, k ∈ K and t a non negative real number . Then we have
+0 ≤ A+(g−1k exp(tH)) − H(g−1k exp(tH)) ≤ 1+ | g.0 |
+1− | g.0 |e−2t,
+(7.2)
+Proof. Let g−1 =
+�
+a
+b
+c
+d,
+�
+and k ==
+�
+u
+0
+O
+v,
+�
+, where a, b, c and d are n×n, n×1, 1×n and 1×1 matrices respectively.
+A direct computation yields
+g−1k exp(tH) =
+�
+∗
+∗ ∗
+c1
+d1
+�
+,
+where c1 = c u
+�
+cosh t
+0
+0
+In−1
+�
+and d1 = sinh t cue1 + cosh t dv.
+By (2.1) we have
+eH(g−1k exp(tH)) = et | cue1 + dv |,
+and
+eA+(g−1k exp(tH)) =| sinh t cue1 + cosh t dv | +(| sinh t cue1 + cosh t dv |2 −1)
+1
+2 .
+20
+
+From
+eA+(g−1k exp(tH))−H(g−1k exp(tH)) =
+e−t
+| cue1 + dv |[| sinh t cue1 + cosh t dv | +(| sinh t cue1 + cosh t dv |2 −1)
+1
+2 ],
+together with
+| sinh t cue1 + cosh t dv | +(| sinh t cue1 + cosh t dv |2 −1)
+1
+2 ≤ 2 | sinh t cue1v−1 + cosh t d |
+≤| cue1v−1 + d | et+ | d − cue1v−1 | e−t
+we deduce that
+e(A+(g−1k exp(tH))−H(g−1k exp(tH)) ≤ 1 + | d − cue1v−1 |
+| cue1v−1 + d |e−2t.
+Noting that (g.0)∗ = −(d−1c), and k.e1 = ue1v−1, we get
+e(A+(g−1k exp(tH))−H(g−1k exp(tH)) ≤ 1 + | 1+ < g.0, k.e1 >|
+| 1− < g.0, k.e1 >|e−2t
+≤ 1 + 1+ | g.0 |
+1− | g.0 |e−2t,
+from which we deduce (7.2), and the proof of the lemma is finished.
+Proof of the Key Lemma. Since B∗(G, τν) is G-invariant, we may apply Proposition 7.1 to get
+Φν,λ(g−1x)v ≃ τ −1
+ν (κ1(g−1x)κ2(g−1x)
+�
+s∈{±}
+cν(sλ)e(isλ−ρ)A+(g−1x)v.
+Thus it suffices to show that
+τ −1
+ν (κ1(g−1x)κ2(g−1x)
+�
+s∈{±}
+cν(sλ)e(isλ−ρ)A+(g−1x)v ≃ τ −1
+ν (k2(x))
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)A+(x)f g
+sλ,v(k1(x)),
+(7.3)
+Note that
+τ −1
+ν [k1(g−1k1(x)eA+(x)k2(x))k2(g−1k1(x)eA+(x)k2(x))] = τ −1
+ν [k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))k2(x))],
+x = k1(x)eA+(x)k2(x).
+Henceforth (7.3) is equivalent to
+τ −1
+ν [k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))]
+�
+s∈{±1}
+cν(λ)e(isλ−ρ)A+(g−1k1(x)eA+(x)) v
+≃
+�
+s∈{±1}
+cν(λ)e(isλ−ρ)A+(x)f g
+sλ,v(k1(x))
+(7.4)
+We write the left hand side of (7.4) as
+�
+s∈{±1}
+cν(λ)e(isλ−ρ)A+(x)f g
+sλ,v(k1(x)) + rg(x)v,
+where
+rg(x) =τ −1
+ν [k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))]
+�
+s∈{±1}
+cν(λ)e(isλ−ρ)A+(g−1k1(x)eA+(x))
+−
+�
+s∈{±1}
+cν(λ)e(isλ−ρ)[A+(x)+H(g−1k1(x))]τ −1
+ν (κ(g−1k1(x)),
+x ∈ G
+(7.5)
+21
+
+To finish the proof we show that for each g ∈ G, rg ≃ 0.
+Noting that
+H(g−1k1(x)eA+(x)) = H(g−1k1(x)) + A+(x),
+we rewrite rg as
+rg(x) = [τ −1
+ν (k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))) − τ−1
+ν (κ(g−1k1(x))]
+�
+s∈{±1}
+cν(λ)e(isλ−ρ)H(g−1k1(x)eA+(x))
++ τ −1
+ν (k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x)))
+
+ �
+s∈{±1}
+cν(λ)[e(isλ−ρ)A+(g−1k1(x)eA+(x))) − e(isλ−ρ)H((g−1k1(x)eA+(x))]
+
+
+=: Ig(x) + Jg(x).
+Using the following
+Lemma 7.2. Let g =
+�
+a
+b
+c
+d
+�
+∈ Sp(n, 1). Then we have
+τν(κ1(g)κ2(g)) = τν( d
+| d |)
+(7.6)
+τν(κ(g)) = τν( ce1 + d
+| ce1 + d |)
+(7.7)
+lim
+R→∞ τν(κ1(g exp(RH))κ2(g exp(RH))) = τν(κ(g)).
+(7.8)
+we easily see that Igv ≃ 0.
+We have
+Jg(x) =τ −1
+ν (k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x)))e(isλ−ρ)H(g−1k1(x)eA+(x))
+�
+s∈{±1}
+cν(λ)
+�
+e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1
+�
+As τν is unitary we have
+1
+R
+�
+K×[0,R]
+∥ Jg(ketH)v ∥2
+ν ∆(t)dt dk
+≤∥ v ∥2 2 | cν(λ) |2
+R
+�
+K×[0,R]
+e−2ρH(g−1ketH) | e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |2
+From
+| e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |≤ C(| λ | +ρ) | A+(g−1k1(x)eA+(x)) − H(g−1k1(x)eA+(x) |
+together with Lemma 7.2 we get
+�
+K×[0,R]
+e−2ρH(g−1ketH) | e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |2
+≤
+�
+C(| λ | +ρ)1+ | g.0 |
+1− | g.0 |
+�2 1
+R
+�
+K×[0,R]
+e−2ρH(g−1k)e−2(ρ+2t)∆(t) dk dt.
+22
+
+As
+�
+K
+e−2ρH(g−1k) dk = 1 and ∆(t) ≤ 2ρe2ρt we obtain
+lim
+R→∞
+1
+R
+�
+K×[0,R]
+e−2ρH(g−1ketH) | e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |2= 0.
+This shows that Jg ≃ 0. Therefore we have proved that for each g ∈ G, rg ≃ 0 as to be shown.
+It remain to prove Lemma 7.2.
+Proof of Lemma 7.2. If g =
+�
+a
+b
+c
+d
+�
+=
+�
+u1
+0
+0
+v1
+�
+at
+�
+u2
+0
+0
+v2
+�
+with respect to the Cartan decomposition G =
+KAK. Then we easily see that d = cosh t v1v2 and (7.6) follows. Analogously if g =
+�
+a
+b
+c
+d
+�
+=
+�
+u
+0
+0
+v
+�
+at n with
+respect to the Iwasawa decomposition. Then from g.e1 =
+�
+ae1 + b
+ce1 + d
+�
+= et
+�
+u
+v
+�
+we get et v = ce1 + d and (7.7) follows.
+We have
+g exp(RH) =
+�
+∗
+∗∗
+∗ ∗ ∗
+sinh Re1 + cosh Rd
+�
+Then (7.6) imply that τν(κ1(g)κ2(g)) = τν( tanh Rce1+d
+|tanh Rce1+d|). Thus limR→∞ τν(κ1(g)κ2(g)) = τν( ce1+d
+|ce1+d|). This finishes
+the proof of Lemma 7.2, and the proof of the Key Lemma is completed.
+8
+Appendix
+In this section we collect some results on the Jacobi functions, referring to [19] for more details.
+For α, β, λ ∈ C; α ̸= −1, −2, · · · and t ∈ R, the Jacobi function is defined by
+φ(α,β)
+λ
+(t) = 2F1(iλ + ρα,β
+2
+, −iλ + ρα,β
+2
+; α + 1; − sinh2 t),
+where 2F1 is the Gauss hypergeometric function and ρα,β = α + β + 1.
+The Jacobi function φ(α,β)
+λ
+is the unique even smooth function on R which satisfy φ(α,β)
+λ
+(0) = 1 and the differential
+equation
+{ d2
+dt2 + [(2α + 1) coth t + (2β + 1) tanh t] d
+dt + λ2 + ρ2
+α,β}φ(α,β)
+λ
+(t) = 0.
+(A1)
+For λ /∈ −iN another solution Ψα,β
+λ
+of (A1) such that
+Ψα,β
+λ
+(t) = e(iλ−ρα,β)t(1 + ◦(1)),
+as
+t → ∞
+(A2)
+is given by
+Ψα,β
+λ
+(t) = (2 sinh t)iλ−ρα,β 2F1(ρα,β − iλ
+2
+, β − α + 1 − iλ
+2
+; 1 − iλ; −
+1
+sinh2 t).
+Moreover there exists a constant C > 0 such that for all λ ∈ R and all t ≥ 1 we have
+Ψα,β
+λ
+(t) = e(iλ−ρα,β)t(1 + e−2tΘλ(t)),
+with
+| Θλ(t) |≤ C.
+(A2’)
+For λ /∈ iZ, we have
+φ(α,β)
+λ
+(t) =
+�
+s=±1
+cα,β(sλ)Ψα,β
+sλ (t)
+(A3)
+23
+
+where
+cα,β(λ) = 2ρα,β−iλ Γ(α + 1)Γ(iλ)
+Γ( iλ+ρα,β
+2
+)Γ( iλ+α−β+1
+2
+)
+.
+For ℜ(iλ) > 0, the asymptotic behaviour of φ(α,β)
+λ
+as t → ∞ is then given by
+lim
+t→∞ e(ρα,β−iλ)tφ(α,β)
+λ
+(t) = cα,β(λ).
+(A4)
+Let De(R) denote the space of even smooth function with compact support on R. For f ∈ De(R), the Fourier-Jacobi
+transform J α,βf (λ ∈ C) is defined by
+J α,βf(λ) =
+� ∞
+0
+f(t)φ(α,β)
+λ
+(t)∆α,β(t) dt,
+(A5)
+where ∆α,β(t) = (2 sinh t)2α+1(2 cosh t)2β+1.
+In the sequel, we assume that α > −1, β ∈ R. Then the meromorphic function cα,β(−λ)−1 has only simple poles for
+ℑλ ≥ 0 which occur in the set
+Dα,β = {λk = i(| β | −α − 1 − 2k); k = 0, 1, · · · , | β | −α − 1 − 2k > 0}.
+(If | β |≤ α + 1, then Dα,β is empty).
+The following inversion and Plancherel formulas for the Jacobi transform hold for every f ∈ De(R):
+f(t) = 1
+2π
+� ∞
+0
+(J α,βf)(λ) φ(α,β)
+λ
+(t) | cα,β(λ) |−2 dλ +
+�
+λk∈Dα,β
+dk(J α,βf)(λk) φ(α,β)
+λk
+(t),
+(A6)
+� ∞
+0
+| f(t) |2 ∆(t) dt = 1
+2π
+� ∞
+0
+| (J α,βf)(λ) |2 | cα,β(λ) |−2 dλ +
+�
+λk∈Dα,β
+dk | (J α,βf)(λk) |2
+(A6’)
+where dk = −i Resλ=λk(cα,β(λ)cα,β(−λ))−1, is given explicitly by
+dk = (β − α − 2k − 1)2−2(α+β)Γ(α + k + 1)Γ(β − k)
+Γ2(α + 1)Γ(β − α − k)k!
+.
+(A7)
+References
+[1] Anker,J. P.: A basis inequality for Scattering Theory for Riemannian Symmetric Spaces of the Noncompact Type.
+Amer. J. Math. 113 (3), 391-398 (1991)
+[2] A. Agmon, L. Hormander, Asymptotic properties of solutions of differential equations with simple characteristics,
+J. Analyse Math, 30 (1976), 1-38.
+[3] A. Boussejra, A. Intissar, Caractérisation des integrales de Poisson-Szego de L2(∂Bn) dans la boule de Bergman
+Bn, n ≥ 2. C. R. Acad. Sci. 318 (1994).
+[4] A. Boussejra, A. Intissar, L2-concrete Spectral Analysis of the Invariant Laplacian in the Unit Complex Ball. J.
+Funct. Anal. 160 (1998), 115-140.
+[5] A. Boussejra, H. Sami Characterization of the Lp-range of the Poisson transform in Hyperbolic spaces. J. Lie
+Theory. 12 (2002), 1-14.
+24
+
+[6] A. Boussejra, Boundary behavior of Poisson integrals on Boundaries of Symmetric Spaces, J. Lie Theory. 21
+(2011), 243-261.
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+Climate change heterogeneity:
+A new quantitative approach
+∗
+Mar´ıa Dolores Gadea Rivas †
+University of Zaragoza
+Jes´us Gonzalo ‡
+U. Carlos III de Madrid
+July 10, 2022
+Abstract
+Climate change is a non-uniform phenomenon.
+This paper proposes a new
+quantitative methodology to characterize, measure and test the existence of
+climate change heterogeneity. It consists of three steps. First, we introduce a
+new testable warming typology based on the evolution of the trend of the whole
+temperature distribution and not only on the average. Second, we define the
+concepts of warming acceleration and warming amplification in a testable for-
+mat. And third, we introduce the new testable concept of warming dominance
+to determine whether region A is suffering a worse warming process than region
+B. Applying this three-step methodology, we find that Spain and the Globe ex-
+perience a clear distributional warming process (beyond the standard average)
+but of different types. In both cases, this process is accelerating over time and
+asymmetrically amplified. Overall, warming in Spain dominates the Globe in
+all the quantiles except the lower tail of the global temperature distribution
+that corresponds to the Artic region. Our climate change heterogeneity results
+open the door to the need for a non-uniform causal-effect climate analysis that
+goes beyond the standard causality in mean as well as for a more efficient design
+of the mitigation-adaptation policies. In particular, the heterogeneity we find
+suggests that these policies should contain a common global component and a
+clear local-regional element. Future climate agreements should take the whole
+temperature distribution into account.
+JEL classification: C31, C32, Q54
+Keywords:
+Climate change; Climate heterogeneity; Global-Local Warming;
+Functional stochastic processes; Distributional characteristics; Trends; Quan-
+tiles; Temperature distributions.
+∗The authors gratefully acknowledge the financial support from the Gobierno de Aragon and FEDER
+funds (grant, LMP71-18), the Spanish Ministerio de Ciencia y Tecnolog´ıa, Agencia Espa˜nola de Investi-
+gaci´on (AEI) and European Regional Development Fund (ERDF, EU) under grants PID2019-104960GB-
+IOO, ECO2017-83255-C3-1-P (AEI/ERDF, EU) and ECO2016-81901-REDT, and Bank of Spain (ER grant
+program). We thank Rodrigo Gonzalez Laiz for excellent research assistance.
+† Department of Applied Economics, University of Zaragoza. Gran V´ıa, 4, 50005 Zaragoza (Spain). Tel:
++34 9767 61842, fax: +34 976 761840 and e-mail: [email protected]
+‡ Department of Economics, University Carlos III, Madrid 126 28903 Getafe (Spain).
+Tel: +34 91
+6249853, fax: +34 91 6249329 and e-mail: [email protected] (corresponding author)
+1
+arXiv:2301.02648v1  [econ.EM]  6 Jan 2023
+
+Climate change heterogeneity
+2
+1
+Introduction
+All the assessment reports (AR) published by the Intergovernmental Panel of Cli-
+mate Change (IPCC) show that there is overwhelming scientific evidence of the
+existence of global warming (GW). It is also well known that climate change (CC)
+is a non-uniform phenomenon. What is not so clear is the degree of heterogeneity
+across all the regions in our planet. In fact, an important part of the Sixth Assess-
+ment Report (AR6) published by the IPCC in 2021-2022 is dedicated to this issue:
+climate (warming) heterogeneity. This is reflected in the chapters studying regional
+climate change. Our paper introduces a new quantitative methodology that builds
+on that described in Gadea and Gonzalo 2020 (GG2020) to characterize, measure
+and test the existence of such climate change heterogeneity (CCH). This is done in
+three steps. First, we introduce a warming typology (W1, W2 and W3) based on
+the trending behavior of the quantiles of the temperature distribution of a given ge-
+ographical location. Second, we define in a testable format the concepts of warming
+acceleration and warming amplification. These concepts help to characterize (more
+ordinally than cardinally) the warming process of different regions. And third, we
+propose the new concept of warming dominance (WD) to establish when region A
+suffers a worse warming process than region B.
+We have chosen Spain as a benchmark geographical location because, as the AR6
+report states “. . . Spain is fully included in the Mediterranean (MED) Reference
+Region, but is one of the most climatically diverse countries in the world. . . ”.
+This fact opens up the possibility of studying warming heterogeneity (WH) from
+Spain to the Globe (outer heterogeneity, OWH) and also from Spain to some of its
+regions represented by Madrid and Barcelona (inner heterogeneity, IWH).
+The three steps rely on the results reported in GG2020, where the different
+distributional characteristics (moments, quantiles, inter quantile range, etc.) of the
+temperature distribution of a given geographical location are converted into time
+series objects. By doing this, we can easily implement and test all the concepts
+involved in the three steps.
+A summary of the results is as follows. Spain and the Globe present a clear
+warming process; but it evolves differently. Spain goes from a warming process where
+lower and upper temperatures share the same trend behavior (IQR is maintained
+constant over time, warming type W1) to one characterized by a larger increase in
+the upper temperatures (IQR increases over time, warming type W3). In contrast,
+
+Climate change heterogeneity
+3
+the Globe as a whole maintains a stable warming type process characterized by lower
+temperatures that increase more than the upper ones (IQR decreases in time).1 In
+our typology, this constitutes a case of warming type W2. Climate heterogeneity
+can go further.
+For instance, within Spain we find that Madrid is of type W3
+while the warming process of Barcelona is of type W1. This is in concordance with
+the Madrid climate being considered a Continental Mediterranean climate while
+Barcelona is more a pure Mediterranean one.
+The proposed warming typology (W1, W2 and W3), although dynamic, is more
+ordinal than cardinal. In this paper, the strength of a warming process is captured
+in the second step by analyzing its acceleration and its amplification with respect
+to a central tendency measure of the temperature distribution. Acceleration and
+amplification contribute to the analysis of warming heterogeneity. The acceleration
+in the Globe is present in all the quantiles above q30 while in Spain it already
+becomes significant above the 10th quantile. We find an asymmetric behavior of
+warming amplification; in Spain (in comparison with the Globe mean temperature)
+this is present in the upper temperatures (above the 80th and 90th quantiles) while
+in the Globe the opposite occurs (below the 20th and 30th quantiles). Within Spain,
+Madrid and Barcelona also behave differently in terms of acceleration and amplifi-
+cation. Overall, warming in Spain dominates that of the Globe in all the quantiles
+except for the lower quantile q05, and between Madrid and Barcelona there is a par-
+tial WD. Madrid WD Barcelona in the upper part of the distribution and Barcelona
+WD Madrid in the lower one.
+The existence of a clear heterogeneous warming process opens the door to the
+need of a new non-uniform causal (effect) research. One that goes beyond the stan-
+dard causality in mean analysis (see Tol, 2021). CCH also suggests that in order
+for the mitigation-adaptation policies to be as efficient as possible they should be
+designed following a type of common factor structure: a common global compo-
+nent plus an idiosyncratic local element.
+This goes in the line with the results
+found in Brock and Xepapadeas (2017), D’Autume et al. (2016) and Peng et al.
+(2021). Future climate agreements should clearly have this CCH into account. An
+important by-product of our warming heterogeneity results is the increase that this
+heterogeneity can generate in the public awareness of the GW process. A possible
+explanation for that can be found in the behavioral economics work by Malmendier
+1Similar results for Central England are found in GG2020 and for the US in Diebold and Rude-
+bush, 2022.
+
+Climate change heterogeneity
+4
+(2021), in the results of the European Social Survey analyzed in Nowakowski and
+Oswald (2020) or in the psychology survey by Maiella et al. (2020).
+The rest of the paper is organized as follows. Section 2 describes our basic climate
+econometrics methodology. Section 3 presents a brief description of the temperature
+data from Spain and the Globe. Section 4 addresses the application of our quantita-
+tive methodology in the cross-sectional version (temperatures measured monthly by
+stations in an annual interval) to Spain and (versus) the Globe. It also reports the
+results of applying the methodology using a purely temporal dimension (local daily
+temperature on an annual basis) for two representative stations in Spain (Madrid
+and Barcelona, empirical details in the Appendix). Section 5 offers a comparison
+and interpretation of the results. Finally, Section 6 concludes the paper.
+2
+Climate Econometrics Methodology
+In this section, we briefly summarize the novel econometric methodology introduced
+in GG2020 to analyze Global and Local Warming processes. Following GG2020,
+Warming is defined as an increasing trend in certain characteristics of the temper-
+ature distribution. More precisely:
+Definition 1. (Warming):
+Warming is defined as the existence of an increas-
+ing trend in some of the characteristics measuring the central tendency or position
+(quantiles) of the temperature distribution.
+An example is a deterministic trend with a polynomial function for certain val-
+ues of the β parameters Ct = β0 + β1t + β2t2 + ... + βktk.
+In GG2020 temperature is viewed as a functional stochastic process X = (Xt(ω), t ∈
+T), where T is an interval in R, defined in a probability space (Ω, ℑ, P). A conve-
+nient example of an infinite-dimensional discrete-time process consists of associating
+ξ = (ξn, n ∈ R+) with a sequence of random variables whose values are in an appro-
+priate function space. This may be obtained by setting
+Xt(n) = ξtN+n, 0 ≤ n ≤ N, t = 0, 1, 2, ..., T
+(1)
+so X = (Xt, t = 0, 1, 2, ..., T). If the sample paths of ξ are continuous, then we have
+a sequence X0, X1, .... of random variables in the space C[0, N]. The choice of the
+period or segment t will depend on the situation in hand. In our case, t will be the
+
+Climate change heterogeneity
+5
+period of a year, and N represents cross-sectional units or higher-frequency time
+series.
+We may be interested in modeling the whole sequence of G functions, for instance
+the sequence of state densities (f1(ω), f2(ω), ..., fT (ω) ) as in Chang et al. (2015,
+2016) or only certain characteristics (Ct(w)) of these G functions, for instance, the
+state mean, the state variance, the state quantile, etc. These characteristics can
+be considered time series objects and, therefore, all the econometric tools already
+developed in the time series literature can be applied to Ct(w). With this charac-
+teristic approach we go from Ω to RT , as in a standard stochastic process, passing
+through a G functional space:
+Ω
+(w)
+X
+−→
+G
+Xt(w)
+C−→
+R
+Ct(w)
+Going back to the convenient example and abusing notation, the stochastic struc-
+ture can be summarized in the following array:
+X10(w) = ξ0(w)
+X11(w) = ξ1(w)
+. . .
+X1N(w) = ξN(w)
+C1(w)
+X20(w) = ξN+1(w)
+X21(w) = ξN+2(w)
+. . .
+X2N(w) = ξ2N(w)
+C2(w)
+.
+.
+.
+.
+.
+.
+. . .
+. . .
+. . .
+.
+.
+.
+.
+.
+.
+XT0(w) = ξ(T−1)N+1(w)
+XT1(w) = ξ(T−1)N+2(w)
+. . .
+XTN(w) = ξTN(w)
+CT (w)
+(2)
+The objective of this section is to provide a simple test to detect the existence of
+a general unknown trend component in a given characteristic Ct of the temperature
+process Xt.
+To do this, we need to convert Definition 1 into a more practical
+definition.
+Definition 2. (Trend test): Let h(t) be an increasing function of t. A characteristic
+Ct of a functional stochastic process Xt contains a trend if β ̸= 0 in the regression
+Ct = α + βh(t) + ut, t = 1, ..., T.
+(3)
+The main problem of this definition is that the trend component in Ct as well
+as the function h(t) are unknown. Therefore this definition can not be easily imple-
+mented. If we assume that Ct does not have a trend component (it is I(0))2 and
+2Our definition of an I(0) process follows Johansen (1995). A stochastic process Yt that satisfies
+Yt − E(Yt) =
+∞
+�
+i=1
+Ψiεt−i is called I(0) if
+∞
+�
+i=1
+Ψ izi converges for |z| < 1 + δ, for some δ > 0 and
+∞
+�
+i=1
+Ψ
+i ̸= 0, where the condition εt ∼ iid(0,σ2) with σ2 > 0 is understood.
+
+Climate change heterogeneity
+6
+h(t) is linear, then we have the following well known result.
+Proposition 1. Let Ct = I(0). In the regression
+Ct = α + βt + ut
+(4)
+the OLS estimator
+�β =
+T�
+t=1
+(Ct − C)(t − t)
+T�
+t=1
+(t − t)2
+(5)
+satisfies
+T 3/2 �β = Op(1)
+(6)
+and asymptotically (T → ∞)
+tβ=0 is N(0, 1).
+In order to analyze the behavior of the t-statistic tβ = 0, for a general trend
+component in Ct, it is very convenient to use the concept of Summability (Berenguer-
+Rico and Gonzalo, 2014)
+Definition 3. (Order of Summability):
+A trend h(t) is said to be summable of
+order “δ” (S(δ)) if there exists a slowly varying function L(T),3 such that
+ST =
+1
+T 1+δ L(T)
+T
+�
+t=1
+h(t)
+(8)
+is O(1), but not o(1).
+Proposition 2. Let Ct = h(t) + I(0) such that h(t) is S(δ) with δ ≥ 0, and such
+that the function g(t) = h(t)t is S(δ + 1). In the regression
+Ct = α + βt + ut
+(9)
+the OLS �β estimator satisfies
+T (1−δ) �β = Op(1).
+(10)
+Assuming that the function h(t)2 is S(1 + 2δ − γ) with 0 ≤ γ ≤ 1 + δ, then
+3A positive Lebesgue measurable function, L, on (0, ∞) is slowly varying (in Karamata’s sense)
+at ∞ if
+L(λn)
+L(n) → 1 (n → ∞) ∀λ > 0.
+(7)
+(See Embrechts et al., 1999, p. 564).
+
+Climate change heterogeneity
+7
+tβ=0 =
+� Op(T γ/2) for 0 ≤ γ ≤ 1
+Op(T 1/2) for 1 ≤ γ ≤ 1 + δ
+(11)
+Examples of how this proposition applies for different particular Data Generat-
+ing Processes (DGP) can be found in GG.
+A question of great empirical importance is how our trend test (TT) of Proposi-
+tion 2 behaves when Ct = I(1) (accumulation of an I(0) process). Following Durlauf
+and Phillips (1988), T 1/2 �β = Op(1); however, tβ=0 diverges as T→∞. Therefore,
+our TT can detect the stochastic trend generated by an I(1) process. In fact, our
+test will detect trends generated by any of the three standard persistent processes
+considered in the literature (see Muller and Watson, 2008): (i) fractional or long-
+memory models; (ii) near-unit-root AR models; and (iii) local-level models. Let
+Ct = µ + zt, t = 1, ..., T.
+(12)
+In the first model, zt is a fractional process with 1/2 < d < 3/2. In the second
+model, zt follows an AR, with its largest root close to unity, ρT = 1 − c/T. In the
+third model, zt is decomposed into an I(1) and an I(0) component. Its simplest
+format is zt = υt + ϵt with υt = υt−1 +ηt, where ϵt is ID(0, q ∗ σ2), ηt is ID(0, σ2),
+σ2 > 0 and both disturbances are serially and mutually independent. Note that the
+pure unit-root process is nested in all three models: d = 1, c = 0, and q = 0.
+The long-run properties implied by each of these models can be characterized
+using the stochastic properties of the partial sum process for zt.
+The standard
+assumptions considered in the macroeconomics or finance literature assume the ex-
+istence of a “δ,” such that T −1/2+δ �T
+t=1 zt −→ σ H(.), where “δ” is a model-specific
+constant and H is a model-specific zero-mean Gaussian process with a given covari-
+ance kernel k(r, s). Then, it is clear that the process Ct = µ + zt is summable (see
+Berenguer-Rico and Gonzalo, 2014). This is the main reason why Proposition 3
+holds for these three persistent processes.
+Proposition 3. Let Ct = µ + zt, t = 1, ..., T, with zt any of the following three
+processes: (i) a fractional or long-memory model, with 1/2 < d < 3/2; (ii) a near-
+unit-root AR model; or (iii) a local-level model. Furthermore, T −1/2+δ �T
+t=1 zt −→ σ
+H(.), where “δ” is a model-specific constant and H is a model-specific zero-mean
+Gaussian process with a given covariance kernel k(r, s). Then, in the LS regression
+Ct = α + βt + ut,
+
+Climate change heterogeneity
+8
+the t-statistic diverges,
+tβ=0 = Op(T 1/2).
+After the development of the theoretical core, we are in a position to design
+tools to approach the empirical strategy. The following subsection describes each of
+them.
+2.1
+Empirical tools: definitions and tests
+From Propositions 2 and 3, Definition 2 can be simplified into the following testable
+and practical definition.
+Definition 4. (Practical definition 2):
+A characteristic Ct of a functional stochas-
+tic process Xt contains a trend if in the LS regression,
+Ct = α + βt + ut, t = 1, ..., T,
+(13)
+β = 0 is rejected.
+Several remarks are relevant with respect to this definition: (i) regression (13)
+has to be understood as the linear LS approximation of an unknown trend function
+h(t) (see White, 1980); (ii) the parameter β is the plim of �βols; (iii) if the regression
+(13) is the true data-generating process, with ut ∼ I(0), then the OLS �β estimator
+is asymptotically equivalent to the GLS estimator (see Grenander and Rosenblatt,
+1957); (iv) in practice, in order to test β = 0, it is recommended to use a robust
+HAC version of tβ=0 (see Busetti and Harvey, 2008); and (v) this test only detects
+the existence of a trend but not the type of trend.
+For all these reasons, in the empirical applications we implement Definition 4
+by estimating regression (13) using OLS and constructing a HAC version of tβ=0
+(Newey and West, 1987).
+These linear trends can be common across characteristics indicating similar pat-
+ters in the time evolution of these characteristics.
+Definition 5. (Co-trending): A set of m distributional characteristics (C1t,C2t,...,Cmt)
+do linearly co-trend if in the multivariate regression
+�
+�
+C1t
+...
+Cmt
+�
+� =
+�
+�
+α1
+...
+αm
+�
+� +
+�
+�
+β1
+...
+βm
+�
+� t +
+�
+�
+u1t
+...
+umt
+�
+�
+(14)
+
+Climate change heterogeneity
+9
+all the slopes are equal, β1 = β2 = ... = βm. 4
+This co-trending hypothesis can be tested by a standard Wald test.
+When m = 2 an alternative linear co-trending test can be obtained from the
+regression
+Cit − Cjt = α + βt + ut
+i ̸= j i, j = 1, ..., m by testing the null hypothesis of β = 0 vs β ̸= 0 using a simple
+tβ=0 test.
+Climate classification is a tool used to recognize, clarify and simplify the existent
+climate heterogeneity in the Globe. It also helps us to better understand the Globe’s
+climate and therefore to design more efficient global warming mitigation policies.
+The prevalent climate typology is that proposed by K¨oppen (1900) and later on
+modified in K¨oppen and Geiger (1930). It is an empirical classification that divides
+the climate into five major types, which are represented by the capital letters A
+(tropical zone), B (dry zone), C (temperate zone), D (continental zone), and E
+(polar zone). Each of these climate types except for B is defined by temperature
+criteria. More recent classifications can been found in the AR6 of the IPCC (2021,
+2022) but all of them share the spirit of the original one of K¨oppen (1900).
+The climate classification we propose in this section is also based on temperature
+data and it has three simple distinctive characteristics:
+• It considers the whole temperature distribution and not only the average
+• It has a dynamic nature: it is based on the evolution of the trend of the
+temperature quantiles (lower and upper).
+• It can be easily tested
+Definition 6. (Warming Typology): We define four types of warming processes:
+• W0: There is no trend in any of the quantiles (No warming).
+• W1: All the location distributional characteristics have the same positive trend
+(dispersion does not contain a trend)
+• W2: The Lower quantiles have a larger positive trend than the Upper quantiles
+(dispersion has a negative trend)
+• W3: The Upper quantiles have a larger positive trend than the Lower quantiles
+(dispersion has a positive trend).
+4This definition is slightly different from the one in Carrion-i-Silvestre and Kim (2019).
+
+Climate change heterogeneity
+10
+Climate is understood, unlike weather, as a medium and long-term phenomenon
+and, therefore, it is crucial to take trends into account. Notice that this typology
+can be used to describe macroclimate as well as microclimate locations.
+Most of the literature on Global or Local warming only considers the trend
+behavior of the central part of the distribution (mean or median). By doing this, we
+are losing very useful information that can be used to describe the whole warming
+process. This information is considered in the other elements of the typology W1,
+W2 and W3. This typology does not say anything about the intensity of the warming
+process and its dynamic. Part of this intensity is captured in the following definitions
+of warming acceleration and warming amplification.
+Definition 7. (Warming Acceleration):
+We say that there is warming acceler-
+ation in a distributional temperature characteristic Ct between the time periods
+t1 = (1, ..., s) and t2 = (s + 1, ..., T) if in the following two regressions:
+Ct = α1 + β1t + ut, t = 1, ..., s, ..., T,
+(15)
+Ct = α2 + β2t + ut, t = s + 1, ..., T,
+(16)
+the second trend slope is larger than the first one: β2 > β1.
+In practice, we implement this definition by testing in the previous system the
+null hypothesis β2 = β1 against the alternative β2 > β1 An alternative warming
+acceleration test can be formed by testing for a structural break at t = s. Neverthe-
+less, we prefer the approach of Definition 7 because it matches closely the existent
+narrative on warming acceleration in the climate literature.
+Definition 8. (Warming Amplification with respect to the mean):
+We say that
+there is a warming amplification in distributional characteristic Ct with respect the
+mean if in the following regression:
+Ct = β0 + β1meant + ϵt
+(17)
+the mean slope is greater than one: β1 > 1.
+When the mean, meant, and Ct come from the same distribution, we name this
+“inner” warming amplification. Otherwise, the mean may come from an external
+environment and, in that case, we call it “outer” warming amplification.
+Both concepts, acceleration and amplification, introduce a quantitative dimen-
+sion to the ordinarily defined classification. For example, the acceleration, which
+
+Climate change heterogeneity
+11
+has a dynamic character, allows us to observe the transition from one type of cli-
+mate to another. Amplification, on the other hand, makes it possible to compare
+the magnitude of the trends that define each type of climate. It should be noted
+that, although static in nature, it can be computed recursively at different points in
+time.
+In the previous definitions, we classify the warming process of different regions
+which is crucial in the design of local mitigation and adaptation policies. But we,
+also, need to compare the different climate change processes of two regions in order
+to characterize climate heterogeneity independently of the type of warming they are
+experimenting. For this purpose, we propose the following definition that shares the
+spirit of the stochastic dominance concept used in the economic-finance literature.
+Definition 9. (Warming Dominance (WD): We say that the temperature distri-
+butions of Region A warming dominates (WD) the temperature distributions of
+Region B if in the following regression
+qτt(A) − qτt(B) = ατ + βτt + uτt,
+(18)
+βτ ≥ 0 for all 0 < τ < 1 and there is at least one value τ ∗ for which a strict
+inequality holds.
+It is also possible to have only partial (WD). For instance, in the lower or upper
+quantiles.
+3
+The data
+3.1
+Spain
+The measurement of meteorological information in Spain started in the eighteenth
+century. However, it was not until the mid-nineteenth century that reliable and reg-
+ular data became available. In Spain, there are four main sources of meteorological
+information: the Resumen Anual, Bolet´ın Diario, Bolet´ın Mensual de Climatolog´ıa
+and Calendario Meteorol´ogico. These were first published in 1866, 1893, 1940 and
+1943, respectively. A detailed explanation of the different sources can be found in
+Carreras and Tafunell (2006).
+Currently, AEMET (Agencia Estatal de Meterolog´ıa) is the agency responsible
+for storing, managing and providing meteorological data to the public. Some of the
+historical publications, such as the Bolet´ın Diario and Calendario Meteorol´ogico can
+
+Climate change heterogeneity
+12
+be found in digital format in their respective archives for whose use it is necessary
+to use some kind of Optical Character Recognition (OCR) software.5
+In 2015, AEMET developed AEMET OpenData, an Application Programming
+Interface (API REST) that allows the dissemination and reuse of Spanish meteoro-
+logical and climatological information. To use it, the user needs to obtain an API
+key to allow access to the application. Then, either through the GUI or through
+a programming language such as Java or Python, the user can request data. More
+information about the use of the API can be found on their webpage.6
+In this paper, we are concerned with Spanish daily station data, specifically
+temperature data. Each station records the minimum, maximum and average tem-
+perature as well as the amount of precipitation, measured as liters per square meter.
+The data period ranges from 1920 to 2019. However, in 1920 there were only 13
+provinces (out of 52) who had stations available. It was not until 1965 that all the
+52 provinces had at least one working station. Moreover, it is important to keep in
+mind that the number of stations has increased substantially from only 14 stations in
+1920 to more than 250 in 2019. With this information in mind, we select the longest
+span of time that guarantees a wide sample of stations so that all the geographical
+areas of peninsular Spain are represented. For this reason, we decided to work with
+station data from 1950 to 2019. There are 30 stations whose geographical distri-
+bution is displayed in the map in Figure 1. The original daily data are converted
+into monthly data, so that we finally work with a total of 30x12 station-month units
+corresponding to peninsular Spain and, consequently, we have 360 observations each
+year with which to construct the annual distributional characteristics.
+3.2
+The Globe
+In the case of the Globe, we use the database of the Climate Research Unit (CRU)
+that offers monthly and yearly data of land and sea temperatures in both hemi-
+spheres from 1850 to the present, collected from different stations around the world.7
+Each station temperature is converted to an anomaly, taking 1961-1990 as the base
+5http : //www.aemet.es/es/conocermas/recursosenlinea/calendarios?n = todos and https :
+//repositorio.aemet.es/handle/20.500.11765/6290.
+6https : //opendata.aemet.es/centrodedescargas/inicio. The use of AEMET data is regulated
+in the following resolution https : //www.boe.es/boe/dias/2016/01/05/pdfs/BOE − A − 2016 −
+111.pdf.
+7We
+use
+CRUTEM
+version
+5.0.1.0,
+which
+can
+be
+downloaded
+from
+(https://crudata.uea.ac.uk/cru/data/temperature/).
+A recent revision of the methodology
+can be found in Jones et al. (2012).
+
+Climate change heterogeneity
+13
+period, and each grid-box value, on a five-degree grid, is the mean of all the station
+anomalies within that grid box. This database (in particular, the annual temper-
+ature of the Northern Hemisphere) has become one of the most widely used to
+illustrate GW from records of thermometer readings. These records form the blade
+of the well-known “hockey stick” graph, frequently used by academics and other
+institutions, such as, the IPCC. In this paper, we prefer to base our analysis on raw
+station data, as in GG2020.
+The database provides data from 1850 to nowadays, although due to the high
+variability at the beginning of the period it is customary in the literature to begin
+in 1880. In this work, we have selected the stations that are permanently present
+in the period 1950-2019 according to the concept of the station-month unit. In this
+way, the results are comparable with those obtained for Spain. Although there are
+10,633 stations on record, the effective number fluctuates each year and there are
+only 2,192 stations with data for all the years in the sample period, which yields
+19,284 station-month units each year (see this geographical distribution in the map
+in Figure 1).8 In summary, we analyze raw global data (stations instead of grids)
+for the period 1950 to 2019, compute station-month units that remain all the time
+and with these build the annual distributional characteristics.
+4
+Empirical strategy
+In this section we apply our three-step quantitative methodology to show the ex-
+istent climate heterogeneity between Spain and the Globe as well as within Spain,
+between Madrid and Barcelona. Because all our definitions are written in a test-
+ing format, it is straightforward to empirically apply them. First, we test for the
+existence of warming by testing the existence of a trend in a given distributional
+characteristic. How common are the trends of the different characteristics (revealed
+by a co-trending test) determine the warming typology. Second, the strength of
+the warming process is tested by testing the hypothesis of warming acceleration
+and warming amplification. And third, independently of the warming typology, we
+determine how the warming process of Spain compares with that of the Globe as
+a whole (we do the same for Madrid and Barcelona). This is done by testing for
+warming dominance.
+8In the CRU data there are 115 Spanish stations. However, after removing stations not present
+for the whole 1880 to 2019 period, only Madrid-Retiro, Valladolid and Soria remain. Since 1950,
+applying the same criteria, only 30 remain.
+
+Climate change heterogeneity
+14
+(a) Spain. Selected stations, AEMET data 1950-2019
+(b) The Globe. Selected stations, CRU data 1950-2019
+Figure 1
+Geographical distribution of stations
+The results are presented according to the following steps: first, we apply our
+trend test (see Definition 4) to determine the existence of local or global warming
+and test for any possible warming acceleration; second, we test different co-trending
+
+45
+18d:w 135°W 90
+45°
+S
+90Climate change heterogeneity
+15
+hypotheses to determine the type of warming of each area; thirdly, we test the
+warming amplification hypothesis for different quantiles with respect to the mean
+(of Spain as well as of the Globe): H0 : β1 = 1 versus Ha : β1 > 1 in (17); and
+finally, we compare the CC of different regions, for Spain and the Globe, and within
+Spain, between Madrid and Barcelona, with our warming dominance test (see 18).9
+4.1
+Local warming: Spain
+The cross-sectional analysis is approached under two assumptions. First, choosing a
+sufficiently long and representative period of the geographical diversity of the Span-
+ish Iberian Peninsula, 1950-2019. Second, we work with month-station units from
+daily observations to construct the annual observations of the time series object from
+the data supplied by the stations, following a methodology similar to that carried
+out for the whole planet in GG2020.10 The study comprises the steps described in
+the previous section. The density of the data and the evolution of characteristics
+are displayed, respectively in Figures 2 and 3.
+We find positive and significant trends in the mean, max, min and all the quan-
+tiles. Therefore from definition 1, we conclude there exists a clear local warming
+(see Table 1).
+The recursive evolution for the periods 1950-2019 and 1970-2019 shows a clear
+increase in the trends of the mean, some dispersion measures and higher quantiles
+(see the last column of Table 1). More precisely, there is a significant trend acceler-
+ation in most of the distributional characteristics except the lower quantiles (below
+q20). These quantiles, q05 and q10, remain stable.
+The co-trending tests for the full sample 1950-2019 show a similar evolution of
+the trend for all the quantiles with a constant iqr (see Table 2). This indicates
+that in this period the warming process of Spain can be considered a W1 type.
+More recently, 1970-2019, the co-trending tests (see Table 3) indicate the upper
+quantiles grow faster than the lower ones. This, together with a positive trend in
+the dispersion measured by the iqr shows that Spain has evolved from a W1 to a
+9Before testing for the presence of trends in the distributional characteristics of the data, we
+test for the existence of unit roots. To do so, we use the well-known Augmented Dickey-Fuller test
+(ADF; Dickey and Fuller, 1979), where the number of lags is selected in accordance with the SBIC
+criterion. The results, available from the authors on request, show that the null hypothesis of a
+unit root is rejected for all the characteristics considered.
+10The results with daily averages are very similar.
+The decision to work with monthly data
+instead of daily in the cross-sectional approach has been based on its compatibility with the data
+available for the Globe.
+
+Climate change heterogeneity
+16
+W3 warming type process
+Finally, no evidence of “inner” amplification during the period 1950-2019 is found
+in the lower quantiles. Regarding the upper quantiles, we found both “inner” and
+“outer” amplification in the second period, which supports the previous finding of
+a transition from type W1 to type W3 (see Table 4).
+Summing up, with our proposed tests for the evolution of the trend of the whole
+temperature distribution, we conclude that Spain has evolved from a W1 type to a
+much more dangerous W3 type. The results of acceleration and dynamic amplifi-
+cation reinforce the finding of this transition to type W3.
+Figure 2
+Spain annual temperature density calculated with monthly data across stations
+
+0.06
+0.05
+0.04
+0.03 -
+0.02 -
+0.01 ~
+0
+30.6096
+26.691
+22.7724
+18.8538
+14.9352
+11.0166
+7.09796
+3.17936
+1970
+-0.739249
+1960
+-4.65786
+1950
+temperature in degrees Celsius (month-station units)2010
+2000
+1990
+1980
+vears0.07Climate change heterogeneity
+17
+Figure 3
+Characteristics of temperature data in Spain with stations selected since 1950
+(monthly data across stations, AEMET, 1950-2019)
+
+30
+15
+mean
+max
+13
+25
+1950
+1970
+1990
+2010
+1950
+1970
+1990
+2010
+0
+-6
+min
+std
+1950
+1970
+1990
+2010
+1950
+1970
+1990
+2010
+35
+range
+30
+iqr
+8 
+25
+1950
+1970
+1990
+2010
+1950
+1970
+1990
+2010
+0.2
+2.5
+kurtosis
+skewness
+2
+0.2
+1950
+1970
+1990
+2010
+1950
+1970
+1990
+2010
+20 
+10
+0
+1950
+1970
+1990
+2010
+q5
+q10
+q20
+q30
+q40
+q50
+q60
+q70
+q80
+q90
+q95Climate change heterogeneity
+18
+Table 1
+Trend acceleration hypothesis (Spain monthly data across stations, AEMET,
+1950-2019)
+Trend test by periods
+Acceleration test
+names/periods
+1950-2019
+1970-2019
+1950-2019, 1970-2019
+mean
+0.0242
+0.0389
+3.0294
+(0.0000)
+(0.0000)
+(0.0015)
+max
+0.0312
+0.0526
+2.7871
+(0.0000)
+(0.0000)
+(0.0030)
+min
+0.0289
+0.0251
+-0.2557
+(0.0000)
+(0.0654)
+(0.6007)
+std
+0.0036
+0.0098
+1.7952
+(0.0518)
+(0.0021)
+(0.0374)
+iqr
+0.0051
+0.0158
+1.8197
+(0.1793)
+(0.0028)
+(0.0355)
+rank
+0.0023
+0.0276
+1.2705
+(0.8249)
+(0.1127)
+(0.1030)
+kur
+-0.0010
+-0.0018
+-0.9191
+(0.0203)
+(0.0198)
+(0.8202)
+skw
+0.0011
+-0.0002
+-1.5989
+(0.0271)
+(0.7423)
+(0.9439)
+q5
+0.0227
+0.0206
+-0.2559
+(0.0000)
+(0.0059)
+(0.6008)
+q10
+0.0200
+0.0203
+0.0406
+(0.0000)
+(0.0077)
+(0.4838)
+q20
+0.0209
+0.0300
+1.4158
+(0.0000)
+(0.0000)
+(0.0796)
+q30
+0.0221
+0.0333
+2.0100
+(0.0000)
+(0.0000)
+(0.0232)
+q40
+0.0213
+0.0366
+2.4867
+(0.0000)
+(0.0000)
+(0.0071)
+q50
+0.0211
+0.0404
+3.2496
+(0.0000)
+(0.0000)
+(0.0007)
+q60
+0.0246
+0.0446
+3.1147
+(0.0000)
+(0.0000)
+(0.0011)
+q70
+0.0273
+0.0478
+3.3143
+(0.0000)
+(0.0000)
+(0.0006)
+q80
+0.0275
+0.0471
+2.6949
+(0.0000)
+(0.0000)
+(0.0040)
+q90
+0.0321
+0.0548
+3.2441
+(0.0000)
+(0.0000)
+(0.0007)
+q95
+0.0335
+0.0526
+3.3568
+(0.0000)
+(0.0000)
+(0.0005)
+Note:
+OLS estimates and HAC p-values in parenthesis of the tβ=0 test from regression:
+Ct = α + βt + ut, for two
+different time periods. For the acceleration hypothesis we run the system: Ct = α1 + β1t + ut, t = 1, ..., s, ..., T, Ct =
+α2 + β2t + ut, t = s + 1, ..., T, and test the null hypothesis β2 = β1 against the alternativeβ2 > β1. We show the value of
+the t-statistic and its HAC p-value.
+Table 2
+Co-trending analysis (Spain monthly data across stations, AEMET, 1950-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+13.235
+0.211
+Lower quantiles (q05, q10, q20, q30)
+0.310
+0.958
+Medium quantiles (q40, q50, q60)
+0.438
+0.803
+Upper quantiles (q70, q80, q90, q95)
+1.515
+0.679
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+0.771
+0.993
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+8.331
+0.215
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+11.705
+0.111
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+-0.002
+0.786
+q95-q50
+0.012
+0.000
+q95-q05
+0.011
+0.096
+q75-q25 (iqr)
+0.005
+0.179
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+
+Climate change heterogeneity
+19
+Table 3
+Co-trending analysis (Spain monthly data across stations, AEMET, 1970-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+38.879
+0.000
+Lower quantiles (q05, q10, q20, q30)
+3.121
+0.373
+Medium quantiles (q40, q50, q60)
+1.314
+0.518
+Upper quantiles (q70, q80, q90, q95)
+1.719
+0.633
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+12.771
+0.047
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+10.675
+0.099
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+37.892
+0.000
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+0.020
+0.029
+q95-q50
+0.012
+0.050
+q55-q05
+0.032
+0.002
+q75-q25 (iqr)
+0.016
+0.003
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+Table 4
+Amplification hypothesis (Spain monthly data, AEMET 1950-2019
+periods/variables
+1950-2019
+1970-2019
+1950-2019
+1970-2019
+Inner
+Outer
+q05
+0.80
+0.56
+0.55
+0.39
+(0.866)
+(0.998)
+(0.990)
+(0.996)
+q10
+0.83
+0.65
+0.62
+0.52
+(0.899)
+(0.994)
+(0.992)
+(0.986)
+q20
+0.94
+0.90
+0.76
+0.81
+(0.816)
+(0.890)
+(0.993)
+(0.899)
+q30
+0.93
+0.91
+0.77
+0.87
+(0.935)
+(0.929)
+(0.997)
+(0.834)
+q40
+0.97
+1.03
+0.80
+0.97
+(0.744)
+(0.318)
+(0.978)
+(0.566)
+q50
+0.98
+1.10
+0.83
+1.12
+(0.612)
+(0.067)
+(0.944)
+(0.212)
+q60
+1.09
+1.15
+0.96
+1.23
+(0.103)
+(0.051)
+(0.619)
+(0.056)
+q70
+1.11
+1.16
+1.05
+1.30
+(0.040)
+(0.006)
+(0.350)
+(0.028)
+q80
+1.11
+1.14
+1.06
+1.29
+(0.083)
+(0.071)
+(0.325)
+(0.060)
+q90
+1.14
+1.16
+1.19
+1.45
+(0.101)
+(0.118)
+(0.078)
+(0.007)
+q95
+1.10
+1.09
+1.18
+1.36
+(0.089)
+(0.191)
+(0.051)
+(0.008)
+Note: OLS estimates and HAC p-values of the t-statistic of testing H0 : βi = 1 versus Ha : βi > 1
+in the regression: Cit = βi0 + βi1meant + ϵit. mean refers to the average of the Spanish Global
+temperature distribution for the “inner” and “outer”cases, respectively.
+
+Climate change heterogeneity
+20
+4.2
+Global warming: the Globe
+In this section, we carry out a similar analysis to that described in the previous
+subsection for Spain. Figures 4 and 5 show the time evolution of the Global temper-
+ature densities and their different distributional characteristics from 1950 to 2019.
+The data in both figures are obtained from stations that report data throughout the
+sample period.
+Table 5 shows a positive trend in the mean as well as in all the quantiles. This
+indicates the clear existence of Global warming, more pronounced (larger trend) in
+the lower part of the distribution (a negative trend in the dispersion measures). The
+warming process suffers an acceleration in all the quantiles above q30.
+From the co-trending analysis (see Tables 6 and 7) we can determine the type
+of warming process characterizing the whole Globe. Table 6 indicates that in the
+period 1950-2019 the Globe experimented a W2 warming type (the lower part of
+the temperature distribution grows faster than the middle and upper part, implying
+iqr and std have a negative trend). Similar results are maintained for the period
+1970-2019 (in this case only the dispersion measure std has a negative trend).
+The asymmetric amplification results shown in Table 8 reinforce the W2 typology
+for the whole Globe: an increase of one degree in the global mean temperature
+increases the lower quantiles by more than one degree. This does not occur with
+the upper part of the distribution. Notice that this amplification goes beyond the
+standard Artic amplification (q05) affecting also q10, q20 and q30.
+Summing up, the results from our different proposed tests for the evolution of
+the trend of the whole temperature distribution indicate that the Globe can be
+cataloged as a undergoing type W2 warming process.
+This warming type may
+have more serious consequences for ice melting, sea level increases, permafrost, CO2
+migration, etc. than the other types.
+
+Climate change heterogeneity
+21
+Figure 4
+Global annual temperature density calculated with monthly data across stations
+
+0.03
+0.025
+density
+0.02
+0.015
+0.01 -
+0.005 ~
+0
+33.369
+21.5248
+9.68063
+-2.16358
+-14.0078
+-25.852
+-37.6962
+-49.5404
+1970
+-61.3846
+1960
+-73.2288
+1950
+temperature in degrees Celsius (month-station units)2010
+2000
+1990
+1980
+vears0.04
+0.035Climate change heterogeneity
+22
+Figure 5
+Characteristics of temperature data in the Globe (monthly data across stations, CRU,
+1950-2019)
+
+12
+40
+11
+38
+10
+mean
+max
+1950196019701980199020002010
+1950196019701980
+199020002010
+13
+-44
+-46
+-48
+50
+-52
+std
+11
+1950196019701980199020002010
+1950196019701980199020002010
+8
+90
+85
+16
+iqr
+range
+80
+19501960 1970198019902000 2010
+1950196019701980199020002010
+G
+kur
+-0.8
+W
+-0.9
+skw
+195019601970:1980199020002010
+195019601970198019902000:2010
+20 
+10
+0
+10
+1950196019701980199020002010
+q5
+q10
+q20
+q30
+q40
+q50
+q60
+q70
+q80
+q90
+q95Climate change heterogeneity
+23
+Table 5
+Trend acceleration hypothesis (CRU monthly data across stations, 1950-2019)
+Trend test by periods
+Acceleration test
+names/periods
+1950-2019
+1970-2019
+1950-2019, 1970-2019
+mean
+0.0213
+0.0300
+2.2023
+(0.0000)
+(0.0000)
+(0.0147)
+max
+0.0361
+0.0523
+1.1217
+(0.0000)
+(0.0001)
+(0.1320)
+min
+0.0423
+-0.0109
+0.5016
+(0.0000)
+(0.5867)
+(0.3084)
+std
+-0.0070
+-0.0057
+0.1776
+(0.0000)
+(0.0570)
+(0.4296)
+iqr
+-0.0067
+-0.0043
+0.2454
+(0.0435)
+(0.4183)
+(0.4033)
+rank
+-0.0062
+0.0632
+0.2181
+(0.5876)
+(0.0005)
+(0.4138)
+kur
+-0.0010
+0.0001
+0.0445
+(0.5205)
+(0.9566)
+(0.4823)
+skw
+0.0006
+0.0003
+0.0301
+(0.0577)
+(0.5726)
+(0.4880)
+q5
+0.0404
+0.0468
+0.7035
+(0.0000)
+(0.0000)
+(0.2415)
+q10
+0.0305
+0.0406
+0.9273
+(0.0000)
+(0.0001)
+(0.1777)
+q20
+0.0253
+0.0342
+1.0156
+(0.0000)
+(0.0000)
+(0.1558)
+q30
+0.0215
+0.0280
+1.2056
+(0.0000)
+(0.0000)
+(0.1150)
+q40
+0.0192
+0.0293
+1.9873
+(0.0000)
+(0.0000)
+(0.0245)
+q50
+0.0179
+0.0268
+1.8614
+(0.0000)
+(0.0000)
+(0.0324)
+q60
+0.0185
+0.0291
+2.1971
+(0.0000)
+(0.0000)
+(0.0149)
+q70
+0.0185
+0.0288
+2.5770
+(0.0000)
+(0.0000)
+(0.0055)
+q80
+0.0160
+0.0257
+2.2460
+(0.0000)
+(0.0000)
+(0.0132)
+q90
+0.0146
+0.0243
+2.0848
+(0.0005)
+(0.0000)
+(0.0195)
+q95
+0.0143
+0.0239
+1.7520
+(0.0001)
+(0.0000)
+(0.0410)
+Note:
+OLS estimates and HAC p-values in parenthesis of the tβ=0 test from regression:
+Ct = α + βt + ut, for two
+different time periods. For the acceleration hypothesis we run the system: Ct = α1 + β1t + ut, t = 1, ..., s, ..., T, Ct =
+α2 + β2t + ut, t = s + 1, ..., T, and test the null hypothesis β2 = β1 against the alternativeβ2 > β1. We show the value of
+the t-statistic and its HAC p-value.
+Table 6
+Co-trending analysis (CRU montly data, 1950-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+25.143
+0.005
+Lower quantiles (q05, q10, q20, q30)
+9.545
+0.023
+Medium quantiles (q40, q50, q60)
+0.078
+0.962
+Upper quantiles (q70, q80, q90, q95)
+1.099
+0.777
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+17.691
+0.007
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+2.041
+0.916
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+24.683
+0.001
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+-0.022
+0.000
+q95-q50
+-0.004
+0.193
+q95-q05
+-0.026
+0.000
+q75-q25 (iqr)
+-0.007
+0.043
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+
+Climate change heterogeneity
+24
+Table 7
+Co-trending analysis (CRU montly data, 1970-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+18.478
+0.047
+Lower quantiles (q05, q10, q20, q30)
+5.523
+0.137
+Medium quantiles (q40, q50, q60)
+0.569
+0.752
+Upper quantiles (q70, q80, q90, q95)
+2.667
+0.446
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+7.606
+0.268
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+6.714
+0.348
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+14.520
+0.043
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+-0.020
+0.047
+q95-q50
+-0.003
+0.462
+q95-q05
+-0.023
+0.048
+q75-q25 (iqr)
+-0.004
+0.418
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+Table 8
+Amplification hypotheses (CRU monthly data across stations, 1950-2019)
+periods/variables
+1950-2019
+1970-2019
+q05
+2.00
+1.83
+(0.000)
+(0.000)
+q10
+1.79
+1.73
+(0.000)
+(0.001)
+q20
+1.41
+1.37
+(0.000)
+(0.000)
+q30
+1.07
+1.00
+(0.089)
+(0.502)
+q40
+0.88
+0.91
+(0.999)
+(0.973)
+q50
+0.74
+0.81
+(1.000)
+(0.997)
+q60
+0.74
+0.85
+(0.999)
+(0.973)
+q70
+0.77
+0.85
+(1.000)
+(0.988)
+q80
+0.72
+0.78
+(1.000)
+(1.000)
+q90
+0.69
+0.70
+(1.000)
+(1.000)
+q95
+0.60
+0.64
+(1.000)
+(1.000)
+Note: OLS estimates and HAC p-values of the t-statistic of testing H0 : βi = 1 versus Ha : βi > 1
+in the regression: Cit = βi0 +βi1meant +ϵit. mean refers to the average of the Global temperature
+distribution.
+
+Climate change heterogeneity
+25
+4.3
+Micro-local warming: Madrid and Barcelona
+The existence of warming heterogeneity implies that in order to design more ef-
+ficient mitigation policies, they have to be developed at different levels: global,
+country, region etc. How local we need to go will depend on the existing degree of
+micro-warming heterogeneity. In this subsection, we go to the smallest level, cli-
+mate station level . We analyze, within Spain, the warming process in two weather
+stations corresponding to two cities: Madrid (Retiro station) and Barcelona (Fabra
+station).
+11 Obviously, the data provided by these stations is not cross-sectional
+data but directly pure time series data. Our methodology can be easily applied to
+higher frequency time series, in this case daily data, to compute the distributional
+characteristics (see Figures A1 and A2)12.
+The results are shown in the Appendix. These two stations, Madrid-Retiro and
+Barcelona-Fabra clearly experience two different types of warming.
+First, there
+is evidence of micro-local warming, understood as the presence of significant and
+positive trends, in all the important temperature distributional characteristics of
+both stations. The acceleration phenomenon is also clearly detected, in other words,
+the warming increases as time passes (see Tables A1 and A5). Secondly, from the
+co-trending tests (Tables A2-A3 and A6-A7), it can be concluded that the warming
+process of Madrid-Retiro is type W3 while for Barcelona-Fabra it is type W1. In
+both cases the warming typology is stable through both sample periods (1950-2019
+and 1970-2019). Thirdly, as expected, Madrid-Retiro presents “inner” and “outer”
+amplification for the upper quantiles, while Barcelona-Fabra does so only for the
+center part of its temperature distribution (see Tables A4 and A8).
+Summing up, even within Spain we find evidence of warming heterogeneity.
+While Madrid (Continental Mediterranean climate) has a similar pattern as that
+of peninsular Spain (1970-2019) W3, Barcelona (Mediterranean coastline climate)
+maintains a W1 typology. Thus there are two different warming processes which
+require mitigation policies at the country as well as the very local level.
+11From Madrid and Barcelona there is data since 1920’s, nevertheless we began the study in 1950
+for consistency with the previous analysis of Spain and the Globe.
+12See the application to Central England in GG2020 and in Gadea and Gonzalo (2022) to Madrid,
+Zaragoza and Oxford.
+
+Climate change heterogeneity
+26
+5
+Comparing results
+The goal of this section is to show the existence of climate heterogeneity by com-
+paring the results obtained from applying our three-step methodology to different
+regions. These results are summarized in Table 10. It is clear that there is distribu-
+tional warming in all the analyzed areas; but this warming follows different patterns
+and sometimes the warming type is not even stable. In the case of Spain, it depends
+on the period under consideration. Figure 6 captures graphically the different trend
+behavior and intensity of the distributional characteristics by regions (Spain and the
+Globe and Madrid and Barcelona).13 The graphical results in this figure coincide
+with the results of the warming typology tests shown in Table 10.
+The middle of Table 10 shows that warming acceleration is detected in all the
+locations. This acceleration is more general in Spain than in the Globe (see also the
+heatmap in Figure 7) and in Barcelona than in Madrid. Apart from these differences,
+the acceleration shares certain similarities across regions. This is not the case for
+the warming amplification that is clearly asymmetric. Spain suffers an amplification
+in the upper quantiles while the Globe does so in the lower ones. Notice that the
+latter amplification goes beyond the standard results found in the literature for the
+Arctic region (q05). We detect amplification also for the regions corresponding to
+the quantiles q10-q30. In the case of Madrid and Barcelona, Madrid suffers a wider
+warming amplification than Barcelona.
+The results of the first two steps of our methodology are obtained region by region
+(Spain, the Globe, Madrid and Barcelona). It is the last step, via the warming
+dominance test (see the numerical results in Table 9) where we compare directly
+one region with another. Warming in Spain dominates that of the Globe in all the
+quantiles except the lower q05.14 This would support the idea held in European
+institutions and gathered in international reports on the greater intensity of climate
+13The analysis of other characteristics such as the third and fourth order moments can contribute
+to the temperature distributions. In the case of Spain, the kurtosis is always negative with a mean
+value of -0.8 and a significant negative trend, which means that we are dealing with a platykurtic
+distribution with tails less thick than Normal, a shape that is accelerating over time. However, it
+is ot possible to draw conclusions about symmetry given its high variability over time. Conversely,
+the temperature distribution in the Globe is clearly leptokurtic with an average kurtosis of 0.9
+and a negative but not significant trend. The global temperature observations are therefore more
+concentrated around the mean and their tails are thicker than in a Normal distribution.
+The
+skewness is clearly negative although a decreasing and significant trend points to a reduction of the
+negative skewness.
+14A more detailed analysis of the warming process suffered in the Artic region can be found in
+Gadea and Gonzalo (2021).
+
+Climate change heterogeneity
+27
+change in the Iberian Peninsula. Warming in Madrid dominates that of Barcelona
+in the upper quantiles, while the reverse is the case in the lower quantiles. This
+latter result coincides with the idea that regions close to the sea have milder upper
+temperatures.
+Further research (beyond the scope of this paper) will go in the direction of
+finding the possible causes behind the warming types W1, W2, and W3. Following
+the literature, on diurnal temperature asymmetry (Diurnal Temperature Range =
+DTR = Tmax − Tmin) we can suggest as possible causes for W2 the cloud coverage
+(Karl et al. 1993) and the planetary boundary layer (see Davy et al. 2017). For
+W3, the process of desertification (see Karl et al. 1993).
+Summarizing, in this section we describe, measure and test the existence of
+warming heterogeneity in different regions of the planet. It is important to note
+that these extensive results can not be obtained by the standard analysis of the
+average temperature.
+Table 9
+Warming dominance
+Spain-Globe
+Madrid-Barcelona
+Quantile
+β
+t-ratio
+β
+t-ratio
+q05
+-0.018
+(-2.770)
+-0.013
+(-3.730)
+q10
+-0.010
+(-1.504)
+-0.013
+(-4.215)
+q20
+-0.004
+(-0.950)
+-0.012
+(-2.988)
+q30
+0.001
+(0.180)
+-0.013
+(-4.164)
+q40
+0.002
+(0.788)
+-0.009
+(-2.909)
+q50
+0.003
+(1.025)
+-0.003
+(-0.701)
+q60
+0.006
+(1.933)
+-0.001
+(-0.219)
+q70
+0.009
+(3.266)
+0.006
+(1.252)
+q80
+0.012
+(3.203)
+0.016
+(3.331)
+q90
+0.017
+(3.862)
+0.010
+(1.869)
+q95
+0.019
+(4.930)
+0.014
+(1.993)
+Note: The slopes (t-statistic) of the following regression
+qτt(A) − qτt(B) = ατ + βτt + uτt
+In the first column A=Spain, B=Globe and in the second A=Madrid, B=Barcelona.
+
+Climate change heterogeneity
+28
+Table 10
+Summary of results
+Cross analysis
+Sample
+Period
+Type
+Acceleration
+Amplification
+Dominance
+Inner
+Outer
+Spain
+1950-2019
+W1
+[mean, std, iqr, rank,
+[q70, q80, q95]
+[q90, q95]
+[q60,..., q95]
+q20,..., q95]
+1970-2019
+W3
+[q50,..., q80]
+[q60,..., q95]
+The Globe
+1950-2019
+W2
+[mean
+[q05,..., q30]
+[q05]
+q40,..., q95]
+1970-2019
+W2
+[q05,..., q20]
+Time analysis
+Sample
+Period
+Type
+Acceleration
+Amplification
+Dominance
+Madrid, Retiro Station
+1950-2019
+W3
+[mean, std, rank,
+[q50,..., q95]
+[ q40,..., q95]
+[q80,..., q95]
+q40, ..., q95]
+1970-2019
+W3
+[q50,..., q95]
+[q40,..., q95]
+Barcelona, Fabra Station
+1950-2019
+W1
+[mean,
+-
+[q30,..., q90]
+[q05,..., q40]
+q20,..., q95]
+1970-2019
+W1
+[q60, q70]
+[q30,..., q70]
+Note: For Spain and the Globe we build characteristics from station-months units. For Madrid and Barcelona we use daily
+frequency time series. A significance level of 10% is considered for all tests and characteristics.
+
+Climate change heterogeneity
+29
+-0.01
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+mean  
+max   
+min   
+std   
+iqr   
+rank  
+kur   
+skw   
+q5    
+q10   
+q20   
+q30   
+q40   
+q50   
+q60   
+q70   
+q80   
+q90   
+q95   
+Globe-montly-1950
+Spain-montly-1950
+-0.01
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+mean  
+max   
+min   
+std   
+iqr   
+rank  
+kur   
+skw   
+q5    
+q10   
+q20   
+q30   
+q40   
+q50   
+q60   
+q70   
+q80   
+q90   
+q95   
+Spain-montly-1950
+Madrid-daily-1950
+Barcelona-daily-1950
+Note: The bars represent the intensity of the trends found in each characteristic measured through
+the value of the β-coefficient estimated in the regression Ct = α + βt + ut.
+Figure 6
+Trend evolution of different temperature distributional characteristics
+
+Climate change heterogeneity
+30
+1950-2019
+1955-2019
+1960-2019
+1965-2019
+1970-2019
+1975-2019
+1980-2019
+1985-2019
+1990-2019
+1995-2019
+2000-2019
+mean  
+max   
+min   
+std   
+iqr   
+rank  
+kur   
+skw   
+q5    
+q10   
+q20   
+q30   
+q40   
+q50   
+q60   
+q70   
+q80   
+q90   
+q95   
+-0.08
+-0.06
+-0.04
+-0.02
+    0
+ 0.02
+ 0.04
+ 0.06
+ 0.08
+  0.1
+(a) Globe
+Spain
+1950-2019
+1955-2019
+1960-2019
+1965-2019
+1970-2019
+1975-2019
+1980-2019
+1985-2019
+1990-2019
+1995-2019
+2000-2019
+mean  
+max   
+min   
+std   
+iqr   
+rank  
+kur   
+skw   
+q5    
+q10   
+q20   
+q30   
+q40   
+q50   
+q60   
+q70   
+q80   
+q90   
+q95   
+-0.08
+-0.06
+-0.04
+-0.02
+    0
+ 0.02
+ 0.04
+ 0.06
+ 0.08
+  0.1
+(b) Spain
+Note: The color scale on the right side of the figure shows the intensity of the trend, based on the
+value of the β-coefficient estimated in the regression Ct = α + βt + ut.
+Figure 7
+Comparing heatmaps
+
+Climate change heterogeneity
+31
+6
+Conclusions
+The existence of Global Warming is very well documented in all the scientific reports
+published by the IPCC. In the last one, the AR6 report (2022), special attention is
+dedicated to climate change heterogeneity (regional climate). Our paper presents a
+new quantitative methodology, based on the evolution of the trend of the whole tem-
+perature distribution and not only on the average, to characterize, to measure and
+to test the existence of such warming heterogeneity. It is found that the local warm-
+ing experienced by Spain (one of most climatically diverse areas) is very different
+from that of the Globe as a whole. In Spain, the upper-temperature quantiles tend
+to increase more than the lower ones, while in the Globe just the opposite occurs.
+In both cases the warming process is accelerating over time. Both regions suffer an
+amplification effect of an asymmetric nature: there is warming amplification in the
+lower quantiles of the Globe temperature (beyond the standard well-known results
+of the Arctic zone) and in the upper ones of Spain. Overall, warming in Spain domi-
+nates that of the Globe in all the quantiles except the lower q05. This places Spain in
+a very difficult warming situation compared to the Globe. Such a situation requires
+stronger mitigation-adaptation policies. For this reason, future climate agreements
+should take into consideration the whole temperature distribution and not only the
+average.
+Any time a novel methodology is proposed, new research issues emerge for future
+investigation. Among those which have been left out of this paper (some are part
+of our current research agenda), three points stand out as important:
+• There is a clear need for a new non-uniform causal-effect climate change anal-
+ysis beyond the standard causality in mean.
+• In order to improve efficiency, mitigation-adaptation policies should be de-
+signed containing a common global component and an idiosyncratic regional
+element.
+• The relation between warming heterogeneity and public awareness of climate
+change deserves to be analyzed.
+
+Climate change heterogeneity
+32
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+an extensive revision and an update to 2010. Journal of Geophysical Research
+117, 1-29.
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+La Malva,
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+The Psychological Distance and Climate Change:
+A Systematic Review
+on the Mitigation and Adaptation Behaviors. Frontiers in Psychology 11,
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+dasticity and autocorrelation consistent covariance matrix. Econometrica 55,
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+Discussion Papers 13660, Institute of Labor Economics (IZA).
+[30] Peng, W., Iyer, G., Binsted, M. et al.,2021. The surprisingly inexpensive cost
+of state-driven emission control strategies. Nat. Clim. Chang. 738–745.
+[31] Tol, R. S. J., 2021. The distributional impact of climate change. Annals of the
+New York. Academy of Sciences 1504 (1), 63-75.
+
+Climate change heterogeneity
+35
+[32] White, H., 1980. Using least square to approximate unknown regression func-
+tion. International Economic Review 21, 149–170.
+
+Climate change heterogeneity
+36
+7
+Appendix: Climate change of Madrid and Barcelona
+7.1
+Madrid-Retiro
+Figure A1
+Characteristics of temperature data in Madrid-Retiro (AEMET daily data, 1950-2019)
+
+16
+32
+30
+mean
+28
+max
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+5
+8
+wmw
+min
+std
+5
+6
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+15
+35
+rank
+w
+30
+10
+igr
+25
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+kur
+0.4
+0.2
+1.5
+0
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+30
+20
+10
+0
+1950
+1970
+1990
+2010 2019
+q5
+q10
+q20
+q30
+q40
+q50
+q60
+q70
+q80
+q90
+q95Climate change heterogeneity
+37
+Table A1
+Trend acceleration hypothesis (Madrid, daily data, AEMET, 1950-2019)
+Trend test by periods
+Acceleration test
+names/periods
+1950-2019
+1970-2019
+1950-2019, 1970-2019
+mean
+0.0326
+0.0447
+2.0972
+(0.0000)
+(0.0000)
+(0.0189)
+max
+0.0477
+0.0636
+1.2043
+(0.0000)
+(0.0000)
+(0.1153)
+min
+0.0362
+0.0087
+-1.5077
+(0.0011)
+(0.5859)
+(0.9330)
+std
+0.0112
+0.0197
+2.1160
+(0.0000)
+(0.0000)
+(0.0181)
+iqr
+0.0270
+0.0399
+1.1110
+(0.0000)
+(0.0004)
+(0.1343)
+rank
+0.0115
+0.0549
+2.0160
+(0.3666)
+(0.0045)
+(0.0229)
+kur
+-0.0016
+-0.0022
+-0.4449
+(0.0278)
+(0.0660)
+(0.6714)
+skw
+0.0012
+-0.0013
+-1.7769
+(0.1538)
+(0.2695)
+(0.9611)
+q5
+0.0248
+0.0183
+-0.5712
+(0.0000)
+(0.0774)
+(0.7156)
+q10
+0.0220
+0.0174
+-0.5815
+(0.0000)
+(0.0162)
+(0.7191)
+q20
+0.0200
+0.0187
+-0.1777
+(0.0000)
+(0.0099)
+(0.5704)
+q30
+0.0181
+0.0235
+0.6959
+(0.0000)
+(0.0019)
+(0.2438)
+q40
+0.0236
+0.0362
+1.6625
+(0.0000)
+(0.0000)
+(0.0494)
+q50
+0.0299
+0.0545
+2.8801
+(0.0000)
+(0.0000)
+(0.0023)
+q60
+0.0334
+0.0604
+3.1655
+(0.0000)
+(0.0000)
+(0.0010)
+q70
+0.0388
+0.0550
+1.7385
+(0.0000)
+(0.0000)
+(0.0422)
+q80
+0.0519
+0.0712
+1.9750
+(0.0000)
+(0.0000)
+(0.0251)
+q90
+0.0494
+0.0687
+1.7956
+(0.0000)
+(0.0000)
+(0.0374)
+q95
+0.0527
+0.0710
+1.7839
+(0.0000)
+(0.0000)
+(0.0383)
+Note:
+OLS estimates and HAC p-values in parenthesis of the tβ=0 test from regression:
+Ct = α + βt + ut, for two
+different time periods. For the acceleration hypothesis we run the system: Ct = α1 + β1t + ut, t = 1, ..., s, ..., T, Ct =
+α2 + β2t + ut, t = s + 1, ..., T, and test the null hypothesis β2 = β1 against the alternativeβ2 > β1. We show the value of
+the t-statistic and its HAC p-value.
+Table A2
+Co-trending analysis (Madrid-Retiro daily data, AEMET 1950-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+77.046
+0.000
+Lower quantiles (q05, q10, q20, q30)
+1.360
+0.715
+Medium quantiles (q40, q50, q60)
+2.036
+0.361
+Upper quantiles (q70, q80, q90, q95)
+3.944
+0.268
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+6.707
+0.349
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+31.822
+0.000
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+74.967
+0.000
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+0.005
+0.505
+q95-q50
+0.023
+0.000
+q05-q95
+-0.028
+0.000
+q75-q25 (iqr)
+0.027
+0.000
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+
+Climate change heterogeneity
+38
+Table A3
+Co-trending analysis (Madrid-Retiro daily data, AEMET, 1970-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+81.371
+0.000
+Lower quantiles (q05, q10, q20, q30)
+0.424
+0.935
+Medium quantiles (q40, q50, q60)
+8.111
+0.017
+Upper quantiles (q70, q80, q90, q95)
+3.214
+0.360
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+45.687
+0.000
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+18.851
+0.004
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+71.094
+0.000
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+0.036
+0.004
+q95-q50
+0.017
+0.051
+q05-q95
+-0.053
+0.000
+q75-q25 (iqr)
+0.040
+0.000
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+Table A4
+Amplification hypothesis (Madrid daily data, AEMET 1950-2019)
+periods/variables
+1950-2019
+1970-2019
+1950-2019
+1970-2019
+Inner
+Outer
+q05
+0.66
+0.43
+0.83
+0.56
+(0.993)
+(1.000)
+(0.802)
+(0.990)
+q10
+0.58
+0.42
+0.73
+0.54
+(1.000)
+(1.000)
+(0.974)
+(1.000)
+q20
+0.66
+0.53
+0.81
+0.65
+(1.000)
+(1.000)
+(0.961)
+(0.999)
+q30
+0.72
+0.74
+0.94
+0.90
+(1.000)
+(0.996)
+(0.758)
+(0.836)
+q40
+0.90
+1.02
+1.15
+1.21
+(0.887)
+(0.436)
+(0.072)
+(0.041)
+q50
+1.08
+1.29
+1.38
+1.53
+(0.188)
+(0.001)
+(0.001)
+(0.000)
+q60
+1.14
+1.31
+1.44
+1.54
+(0.040)
+(0.000)
+(0.000)
+(0.000)
+q70
+1.22
+1.23
+1.46
+1.38
+(0.012)
+(0.019)
+(0.000)
+(0.002)
+q80
+1.45
+1.36
+1.70
+1.52
+(0.000)
+(0.003)
+(0.000)
+(0.002)
+q90
+1.31
+1.29
+1.48
+1.38
+(0.004)
+(0.041)
+(0.005)
+(0.064)
+q95
+1.31
+1.33
+1.46
+1.39
+(0.001)
+(0.021)
+(0.007)
+(0.073)
+Note: OLS estimates and HAC p-values of the t-statistic of testing H0 : βi = 1 versus Ha : βi > 1
+in the regression: Cit = βi0 + βi1meant + ϵit. mean refers to the average of the Madrid or Spanish
+temperature distribution for the “inner” and “outer”cases, respectively.
+
+Climate change heterogeneity
+39
+7.2
+Barcelona-Fabra
+Figure A2
+Characteristics of temperature data in Barcelona-Fabra (AEMET daily data,
+1950-2019)
+
+16
+30
+15
+14
+mean
+25
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+8
+5
+std
+www
+min
+-5
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+30
+25
+igr
+20
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+2.5
+kur
+0.4
+skw
+0.2
+M
+0
+-0.2
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+20
+10
+0
+1950
+1970
+1990
+2010 2019
+q5
+q10
+q20
+q30
+q40
+q50
+q60
+q70
+q80
+q90
+q95Climate change heterogeneity
+40
+Table A5
+Trend acceleration hypothesis (Barcelona, daily data, AEMET, 1950-2019)
+Trend test by periods
+Acceleration test
+names/periods
+1950-2019
+1970-2019
+1950-2019, 1970-2019
+mean
+0.0340
+0.0512
+3.2979
+(0.0000)
+(0.0000)
+(0.0006)
+max
+0.0394
+0.0531
+0.7280
+(0.0000)
+(0.0038)
+(0.2339)
+min
+0.0397
+0.0231
+-0.7411
+(0.0011)
+(0.2654)
+(0.7700)
+std
+0.0013
+0.0057
+0.9146
+(0.6185)
+(0.1787)
+(0.1810)
+iqr
+0.0042
+0.0113
+0.7351
+(0.4418)
+(0.1892)
+(0.2318)
+rank
+-0.0004
+0.0300
+0.9299
+(0.9806)
+(0.3322)
+(0.1770)
+kur
+-0.0013
+-0.0018
+-0.2693
+(0.1555)
+(0.2075)
+(0.6060)
+skw
+0.0011
+-0.0022
+-1.7869
+(0.2678)
+(0.1942)
+(0.9619)
+q5
+0.0374
+0.0358
+-0.1381
+(0.0000)
+(0.0015)
+(0.5548)
+q10
+0.0350
+0.0385
+0.4361
+(0.0000)
+(0.0000)
+(0.3317)
+q20
+0.0317
+0.0439
+1.7009
+(0.0000)
+(0.0000)
+(0.0456)
+q30
+0.0308
+0.0488
+2.4813
+(0.0000)
+(0.0000)
+(0.0072)
+q40
+0.0324
+0.0537
+2.9244
+(0.0000)
+(0.0000)
+(0.0020)
+q50
+0.0325
+0.0548
+2.7535
+(0.0000)
+(0.0000)
+(0.0034)
+q60
+0.0344
+0.0636
+3.0915
+(0.0000)
+(0.0000)
+(0.0012)
+q70
+0.0330
+0.0583
+2.9241
+(0.0000)
+(0.0000)
+(0.0020)
+q80
+0.0357
+0.0551
+2.4081
+(0.0000)
+(0.0000)
+(0.0087)
+q90
+0.0394
+0.0567
+2.0957
+(0.0000)
+(0.0000)
+(0.0190)
+q95
+0.0390
+0.0525
+1.3435
+(0.0000)
+(0.0000)
+(0.0907)
+Note:
+OLS estimates and HAC p-values in parenthesis of the tβ=0 test from regression:
+Ct = α + βt + ut, for two
+different time periods. For the acceleration hypothesis we run the system: Ct = α1 + β1t + ut, t = 1, ..., s, ..., T, Ct =
+α2 + β2t + ut, t = s + 1, ..., T, and test the null hypothesis β2 = β1 against the alternativeβ2 > β1. We show the value of
+the t-statistic and its HAC p-value.
+Table A6
+Co-trending analysis (Barcelona-Fabra daily data, AEMET, 1950-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+3.368
+0.971
+Lower quantiles (q05, q10, q20, q30)
+1.036
+0.792
+Medium quantiles (q40, q50, q60)
+0.073
+0.964
+Upper quantiles (q70, q80, q90, q95)
+0.784
+0.853
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+1.171
+0.978
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+1.901
+0.929
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+2.969
+0.888
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+-0.005
+0.528
+q95-q50
+0.006
+0.233
+q05-q95
+-0.002
+0.856
+q75-q25 (iqr)
+0.004
+0.442
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+
+Climate change heterogeneity
+41
+Table A7
+Co-trending analysis (Barcelona-Fabra daily data, AEMET, 1970-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+13.165
+0.215
+Lower quantiles (q05, q10, q20, q30)
+1.904
+0.593
+Medium quantiles (q40, q50, q60)
+1.267
+0.531
+Upper quantiles (q70, q80, q90, q95)
+0.384
+0.943
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+10.103
+0.120
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+1.642
+0.949
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+9.693
+0.207
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+0.019
+0.192
+q95-q50
+-0.002
+0.821
+q05-q95
+-0.017
+0.241
+q75-q25 (iqr)
+0.011
+0.189
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+Table A8
+Amplification hypothesis (Barcelona daily data, AEMET 1950-2019)
+periods/variables
+1950-2019
+1970-2019
+1950-2019
+1970-2019
+Inner
+Outer
+q05
+0.99
+0.76
+1.19
+0.87
+(0.523)
+(0.918)
+(0.225)
+(0.720)
+q10
+0.90
+0.79
+1.10
+0.94
+(0.824)
+(0.980)
+(0.263)
+(0.668)
+q20
+0.89
+0.85
+1.09
+1.04
+(0.931)
+(0.964)
+(0.192)
+(0.318)
+q30
+0.96
+0.98
+1.22
+1.25
+(0.813)
+(0.585)
+(0.000)
+(0.000)
+q40
+0.99
+1.04
+1.27
+1.33
+(0.570)
+(0.300)
+(0.000)
+(0.000)
+q50
+1.01
+1.07
+1.27
+1.32
+(0.466)
+(0.224)
+(0.002)
+(0.003)
+q60
+1.09
+1.23
+1.29
+1.42
+(0.175)
+(0.005)
+(0.014)
+(0.001)
+q70
+1.09
+1.17
+1.26
+1.31
+(0.128)
+(0.012)
+(0.022)
+(0.008)
+q80
+1.06
+1.04
+1.22
+1.17
+(0.191)
+(0.338)
+(0.052)
+(0.117)
+q90
+1.09
+1.08
+1.22
+1.20
+(0.125)
+(0.241)
+(0.047)
+(0.121)
+q95
+1.06
+1.03
+1.16
+1.12
+(0.304)
+(0.432)
+(0.192)
+(0.298)
+Note: OLS estimates and HAC p-values of the t-statistic of testing H0 : βi = 1 versus Ha : βi > 1 in
+the regression: Cit = βi0 + βi1meant + ϵit. mean refers to the average of the Barcelona or Spanish
+temperature distribution for the “inner” and “outer”cases, respectively.
+

CdFRT4oBgHgl3EQfwDjC/content/tmp_files/2301.13637v1.pdf.txt

@@ -0,0 +1,1335 @@
+Tricking AI chips into Simulating the Human Brain:
+A Detailed Performance Analysis
+Lennart P. L. Landsmeer∗
+Quantum & Computer
+Engineering Department
+Delft University of Technology
+Delft, The Netherlands
+Dept. of Neuroscience
+Erasmus Medical Center
+Rotterdam, The Netherlands
+ORCID 0000-0003-0010-7249
+Max C.W. Engelen∗
+Dept. of Neuroscience
+Erasmus Medical Center
+Rotterdam, The Netherlands
+&
+Maxeler IoT Labs
+Delft, Netherlands
+ORCID 0000-0002-5762-1276
+Rene Miedema
+Quantum & Computer
+Engineering Department
+Delft University of Technology
+Delft, The Netherlands
+&
+Dept. of Neuroscience
+Erasmus Medical Center
+Rotterdam, The Netherlands
+ORCID 0000-0002-0447-1083
+Christos Strydis
+Quantum & Computer
+Engineering Department
+Delft University of Technology
+Delft, The Netherlands
+&
+Dept. of Neuroscience
+Erasmus Medical Center
+Rotterdam, The Netherlands
+ORCID 0000-0002-0935-9322
+Abstract—Challenging the Nvidia monopoly, dedicated AI-
+accelerator chips have begun emerging for tackling the compu-
+tational challenge that the inference and, especially, the training
+of modern deep neural networks (DNNs) poses to modern
+computers. The field has been ridden with studies assessing
+the performance of these contestants across various DNN model
+types. However, AI-experts are aware of the limitations of current
+DNNs and have been working towards the fourth AI wave
+which will, arguably, rely on more biologically inspired models,
+predominantly on spiking neural networks (SNNs). At the same
+time, GPUs have been heavily used for simulating such models
+in the field of computational neuroscience, yet AI-chips have not
+been tested on such workloads. The current paper aims at filling
+this important gap by evaluating multiple, cutting-edge AI-chips
+(Graphcore IPU, GroqChip, Nvidia GPU with Tensor Cores and
+Google TPU) on simulating a highly biologically detailed model of
+a brain region, the inferior olive (IO). This IO application stress-
+tests the different AI-platforms for highlighting architectural
+tradeoffs by varying its compute density, memory requirements
+and floating-point numerical accuracy. Our performance analysis
+reveals that the simulation problem maps extremely well onto the
+GPU and TPU architectures, which for networks of 125,000 cells
+leads to a 28x respectively 1,208x speedup over CPU runtimes.
+At this speed, the TPU sets a new record for largest real-time IO
+simulation. The GroqChip outperforms both platforms for small
+networks but, due to implementing some floating-point operations
+at reduced accuracy, is found not yet usable for brain simulation.
+Index Terms—AI accelerator, GPU, Brain simulation, com-
+puter architecture
+I. INTRODUCTION
+To date, GPUs have achieved spectacularly better perfor-
+mance in deep learning (DL) than CPUs [1]. Recently, novel,
+specialized AI hardware platforms have begun to emerge,
+holding the promise of accelerating training and inference
+even further. The workloads targeted mainly are artificial, and
+specifically, deep neural networks (DNNs), which have shown
+great potential in recent years. On the other hand, highly
+biologically plausible models such as conductance-based (e.g.,
+Hodgkin-Huxley) neurons have not attracted similar atten-
+tion from AI-chip manufacturers and analysts alike. This is
+strange, given that biological brains – the inspiration behind
+these DNNs – are modeled using equations built on similar
+elementary functions. What is more, high-detail models are
+touted as the next AI wave, which is intended to be more
+biologically inspired than its predecessors [2], [3]. Therefore,
+it makes sense both for neuroscientists and for AI researchers
+to reach for these AI accelerators and deploy them for brain
+simulations; yet no performance studies exist.
+In this work, we evaluate multiple, cutting-edge AI chips
+(Graphcore IPU [4], GroqChip [5], TensorRT-capable GPU [6]
+and Google TPU v3 [7]) on simulating a highly biologically
+detailed model of a brain region, the Inferior-Olivary nucleus
+(IO). Biologically detailed brain models, such as the IO,
+chiefly involve addition, multiplication, division and expo-
+nential operations, arranged as sparse computations. There
+is, thus, a large operation overlap with artificial networks.
+Therefore, new AI chips seem a good fit for these types
+of models. This IO application represents timely, relevant
+research and is constructed as an extended-Hodgkin-Huxley
+model. It is very suitable for stress-testing the different AI
+platforms and highlighting architectural tradeoffs by adjusting
+the compute density, memory requirements and numerical
+accuracy of the IO model. Evaluation is performed using the
+application encoded as a TensorFlow 2 [8] kernel, which in the
+case of the GroqChip, is necessarily compiled to its ONNX [9]
+equivalent. ONNX is an intermediary tool used to convert
+models between different machine-learning (ML) frameworks.
+In the analysis of the different accelerators, the exact same
+TensorFlow model is used, ensuring a fair comparison across
+the board. This is either used directly or ported to ONNX
+via the Python package tf2onnx. TensorFlow is a high-level
+API, requiring little to moderate intervention from the user
+and is therefore suitable for a wide user base. A schematic
+overview of this setup is presented in Fig. 1.
+While all accelerators in this study support chip-to-chip
+communication, this work constrains the application to single-
+chip performance comparisons; multi-chip is left as future
+1
+arXiv:2301.13637v1  [cs.LG]  31 Jan 2023
+
+IO-model 
+TensorFlow
+CPU
+GPU
+IPU
+ONNX 
+tf2onnx
+CPU
+GPU
+Groq
+TPU
+Fig. 1: Overview of the performance-analysis strategy followed in this work
+work. The contributions of this work are:
+• We take a deep dive into four cutting-edge AI architec-
+tures with a focus on biologically plausible spiking neural
+networks (SNNs).
+• We build the first ML-library-based, efficient implemen-
+tation of a detailed brain model, the Inferior Olive (IO).
+• We deploy the IO model onto the four AI platforms and
+benchmark their performance and numerical accuracy.
+• We demonstrate that modern ML libraries are seman-
+tically able to model classical problems in scientific
+computing, offering large performance gains and reduced
+development times while remaining hardware-agnostic.
+• Lastly, this work is the first to ever simulate a realistic
+mouse-sized IO model with real-time performance.
+The paper is organized as follows: Section II presents
+related works in the field. Section III introduces the IO
+model used as our benchmarking application, while Section IV
+briefly presents the four AI architectures under evaluation and
+attempts some performance predictions. Section V ensures
+experiment reproducibility by detailing the experiment param-
+eters and platform configurations used to acquire our results
+presented in Section VI. A general discussion of our findings
+is included in Section VII and conclusions in Section VIII.
+II. RELATED WORKS
+Models of biological neurons come in various levels of
+detail, ranging from population-level dynamics, from simpli-
+fied models of single neurons to highly detailed biophysically
+realistic neurons [10]. Coarse models of single neurons, no-
+tably leaky-integrate-and-fire (LIF) type models have seen a
+renewed interest in the DL-community (often referred here
+to as SNNs) as an alternative to artificial neural networks
+(ANNs) [11]. In contrast, computational neuroscience is usu-
+ally interested in biophysically accurate models that model the
+underlying biological processes in a way that makes it possible
+to gain insights about these processes. These conductance
+based models can be made more realistic by modeling of
+their 3-dimensional structure (the morphology) using multiple
+discretized compartments. Multi-compartmental conductance-
+based neurons are then simulated by explicit calculation of
+electrical currents flowing within, between and into discretized
+compartments [12].
+Due to the computional resources needed for large-scale
+conductance level brain simulations, computational neuro-
+science was an early adopter of general-purpose GPU (GP-
+GPU) in the HPC environment. Notable GPU-based ex-
+amples of large-scale, biologically detailed brain simulators
+include CoreNeuron [13], which enabled porting of exist-
+ing conductance-level NEURON [14] models to the GPU,
+and more recently, Arbor [15], a library-based approach to
+performance-portable, large-scale brain simulation. Their suc-
+cess shows that the computational problems of neuroscience
+map well to GP-GPU platforms and result in significant
+speedups for large-scale brain models. Still, even with hand-
+optimized CUDA code [16], the IO application (to be detailed
+in the next section) at biological sizes runs order-of-magnitude
+slower than the biological brain, hampering research.
+With respect to TensorFlow-based implementations of
+conductance-level models, there is PymoNNto [17], an attempt
+to bring the Brian [18] API of neural models to TensorFlow.
+While faster than the Brian simulator on a GTX1080 GPU,
+performance was not a primary goal and the architecture pro-
+hibits optimizations using TensorFlow’s JIT compiler backend,
+by scattering the computational definitions across the code-
+base. Although this shows that TensorFlow does express the
+right API surface for neural models, no efficient ML-library
+based conductance-level GP-GPU brain simulators exist.
+Simplified SNN models have readily available GP-GPU
+implementations of LIF and similar models as well. High-
+level ML-libraries like TensorFlow and PyTorch allowed for
+the hardware-agnostic implementation of their neural dy-
+namics, considerably lowering development efforts to build
+SNN simulators for GP-GPU simulation. For example, Nengo
+DL [19] allows for the GPU-based simulation of existing SNN
+models defined in the Nengo framework using TensorFlow.
+Beyond just simulating neural networks on the GPU, novel
+developments in surrogate gradients for event-based SNNs
+and automatic gradient calculation provided by ML-libraries
+allowed for the nearly simultaneous appearance of similar
+SNN deep-learning libraries Norse [20], snnTorch [21] and
+SpikingJelly [22]. BindsNET [23] is another, efficient SNN
+implementation in PyTorch with a focus on reinforcement
+learning. Again, these project show that not only ML-libraries
+have the expressive power and performance needed to run
+large-scale SNN models, also that this arguably can be devel-
+oped faster than hardware-specific low-level code. As these
+libraries had DL-oriented goals in mind, none of these imple-
+ments multi-compartmental, conductance-level neural models.
+Simplified SNN models also led to the development of
+specialized neuromorphic hardware to simulate them. Numer-
+ous publications show the benefits of using these chips for
+simplified-SNN simulation. For a short review of the various
+chips, we point the reader to [24]. While exciting with respect
+to low-power inference of SNN-based deep-learning models,
+these chips, due to their hardwired dynamics, lack the ability
+to simulate conductance level neural models.
+On AI chips that have the semantic power to capture
+more general HPC workloads, little has been published about
+both simplified and conductance-level SNNs. With respect
+to simplified SNN simulation, we find just one preprint tar-
+geting an AI chip, introducing an IPU-optimized version of
+2
+
+snnTorch [25]. Training throughput of a dense 3-layer LIF
+network on an image classification task is 3.4x higher on the
+IPU than on the A100. The reported performance benefits
+decrease if the network size is increased, with the A100
+apparently underutilized throughout the entire application.
+This shows the potential of using the IPU for simple SNN
+workloads, but the performance characteristics of other AI
+chips or more complex SNNs are not yet obvious.
+No works have been published targeting AI chips with
+conductance-level models or other biologically realistic brain
+simulation scenarios, neither using high-level ML libraries or
+hardware-specific SDKs. To the authors’ knowledge, this is the
+first work to implement an efficient, conductance-level, multi-
+compartmental neuron in an ML library and also the first to
+benchmark multiple AI chips on this workload class.
+III. THE INFERIOR-OLIVE APPLICATION
+The IO is a intrinsically oscillating brain region located
+in the brainstem, and is key to motor control and learn-
+ing [26]. The estimated neuron population for the mouse
+brain is approx. 104 neurons [27] and for humans between
+106 − 107 neurons [28]. These numbers will be referred to
+during hardware-performance evaluation (Section VI). In this
+work, we will capture in TensorFlow 2 the IO nucleus as
+an extended Hodgkin-Huxley (eHH) model, conductance-level
+brain model, first published in [29]. The model is a good
+example of the computational load of realistic brain models
+and, also, a good fit for our benchmarking purposes, since
+it captures complex neuron dynamics and fast interneural
+communication (in the form of gap junctions), as will be
+shown next.
+We restate the IO-neuron main equations in this section, but
+refer the reader to [29] for more details. In addition, we model
+connectivity based on the network described in [30].
+1) The cable model:
+Cm
+dV (i)
+dt
+= −
+�
+k∈Channels
+I(i)
+k
+−
+�
+i∈Compartments
+Ik,j
+−
+�
+i∈Gap junctions
+Igj,k,j − I(i)
+app
+(1)
+The eHH model describes the membrane that envelops the
+neurons as a capacitor. The cell internal voltage can thus be
+calculated by integrating currents flowing into and out of the
+cell (eq. 1). Here, Iapp is an optional term describing externally
+applied currents by the experimenter.
+2) Channel currents: Channels (CaL, h, KCa, Na, Kdr, K,
+CaH, Na, K) allow currents to flow through the cell membrane.
+They produce this current as function of internal state variables
+changing over time. In general, this current (eq. 2) results from
+the potential difference to an channel specific reversal potential
+E multiplied by the product of one or more internal gating
+variables, each optionally raised to an integer power (eq. 2).
+The gating variables follow an Ordinary Differential Equation
+(ODE), that brings them to a certain cell-voltage dependent
+steady state at a given speed (eq. 3). These latter equations
+Listing 1 Axonal sodium-channel current
+m inf = 1/(1+ t f . exp ( −(V axon+30) / 5 . 5 ) )
+h
+inf = 1/(1+ t f . exp ( ( V axon+60) / 5 . 8 ) )
+tau h = 1.5* t f . exp ( −(V axon+40) /33)
+dh dt = ( h inf −h ) / tau h
+I na
+= g Na * ( V axon−V Na) * m inf **3* h
+Listing 2 Sparse gap-junction current
+V d i f f
+=
+t f . gather ( V dend ,
+gj
+src ) \
+−
+t f . gather ( V dend ,
+g j
+t g t )
+I
+per
+gj = V d i f f
+*
+g gj
+*
+(0.2 + \
+0.8
+*
+t f . exp ( −0.01* V d i f f * V d i f f ) )
+I gapp
+=
+t f . tensor scatter nd add (
+t f . zeros
+like (V) ,
+t f . reshape ( gj
+tgt ,
+( −1 ,1) ) ,
+I
+per
+gj )
+are usually gaussian or sigmoidal functions of the voltage.
+For certain fast operating channels we set n(t) = n∞(V ) as
+a numerical stability optimization.
+Ii = ¯gi
+��
+k
+ni,k(t)mk
+�
+(V − Ei)
+(2)
+τn (V ) dn
+dt = n∞ (V ) − n (t)
+(3)
+3) Compartmental currents: A single IO cell consist of
+three separate compartments, the axon, soma and dendrite.
+Currents flowing between different compartments are modeled
+resistively as: Ii,j = gi,j (Vj − Vi)
+4) Gap-junction currents: Gap junctions are direct electri-
+cal connections between different IO cells and allow current
+to flow between them. They follow experimentally determined
+Connexin-36 protein dynamics:
+Igj = ggj∆V
+�
+0.2 + 0.8 exp
+�
+−∆V 2/100
+��
+(4)
+with ∆V the potential-difference between two connected cells.
+5) Topology: The real IO looks like a large, folded sheet
+with mostly local connectivity. As approximating this structure
+is not a focus of this paper, our model neurons are assumed
+to exist on a discrete 3-D grid with wrap-around connectivity
+(i.e., a hypertorus). This should exhibit the same non-local
+memory-access patterns as a more realistic model. Connec-
+tions are sampled as a function of inter-neuron distance r on
+a radially symmetric distribution: p(r) ∝ u(rmax − r)(e−r2 −
+e−r2
+max)n(r), where n(r) is the density of neurons in the
+volume shell around r. This distribution is sampled until we
+have 10 connections per neuron on average.
+6) TensorFlow Translation: The previous equations sum up
+to a total of 14 ODEs per neuron. This system of ODEs is
+translated to a series of TensorFlow operators in Python. By
+defining the model in TensorFlow instead of using platform-
+specific APIs, we make sure all platforms have equal op-
+timization opportunities. Furthermore, TensorFlow naturally
+translates to ONNX models, which is the only high-level
+API available for GroqChip. Straightforward translation to
+TensorFlow is achieved by storing all state in a large 2d-array
+and direct substitution of mathematical expressions by their
+3
+
+TensorFlow counterparts (see Listing 1). When certain model
+parameters need to be user-specified (e.g., gi or Iapp), these
+are passed to the TensorFlow kernel, which then needs to be
+recompiled before running again.
+Translating gap junctions to both TensorFlow and ONNX
+in a performant way requires expressing them as vector
+operations, as opposed to more traditional for-loop-based
+approaches [16]. With just 10 connections per IO neuron on
+average, cell-to-cell communication is sparse. The effective
+operation from a TensorFlow perspective is two sparse-matrix
+(SM) multiplications. As a novel contribution in computa-
+tional neuroscience, we model those as tf.gather and
+tf.tensor_scatter_nd_add operations (see Listing 2).
+Apart from being more specific and memory-efficient in
+describing SM multiplications, these functions have a direct
+mapping to ONNX operators as Gather and ScatterND since
+ONNX specification opset 11, contrary to SM multiplica-
+tions which currently are not possible in ONNX.
+At each timestep, ODEs are integrated using Forward-Euler
+to produce the next state array, resulting in a hardware-
+agnostic timestepping function. For TensorFlow backends, a
+JIT-compilable TensorFlow function is constructed that exe-
+cutes 40 timesteps at a ∆t of 0.025ms, resulting in a 1ms
+sampling accuracy. For ONNX backends, the timestep function
+is converted to an ONNX model and either the public onnx-
+runtime library or Groq compiler is used to compile this
+into executable code. This does not lead to the best possible
+performance by default, thus hardware-specific optimizations
+are discussed in Section V-B.
+IV. TARGET PLATFORMS
+Hardware platforms were selected from the top-performing
+AI accelerators in the MLCommons MLPerf training bench-
+mark v2.0 [31]. From this, the Intel Habana Gaudi was not
+available to us. The GroqChip was included as it was already
+available through academic channels. An overview of all AI
+chips is given in Tab. I and will be presented next. A modern,
+server-grade CPU is also included as a baseline for our
+subsequent performance and numerical-accuracy comparisons.
+1) Nvidia GPU [6]: These are well-established accelerators
+in the HPC world. With the introduction of the Tensor Cores
+in Nvidia GPUs, they also became well-known for their AI
+capabilities. Tensor Cores are capable of matrix multiplications
+in a very efficient manner. The current generation of tensor
+cores can support up to TensorFloat-32 (TF32) precision TF32
+is a floating point with float 32 dynamic range but float 16
+precision. There are multiple ways of interacting with them;
+e.g., via cuBLAS and TensorRT.
+2) GroqChip [5]: This is a deterministic Tensor Streaming
+Processor (TSP), resembling a modified systolic array archi-
+tecture. The chip layout is a conventional 2D mesh of cores,
+each with its own dedicated functionality. A column of these
+cores – all of the same type – is called a functional slice.
+Data travels horizontally, executing 320 SIMD-style lanes. A
+single instruction can control 16 lanes, effectively creating 20
+superlanes that can all be operated independently from each
+other. The functional slices consist of one vector processor
+(VXM), two matrix execution modules (MXM), switch ex-
+ecution modules (SXM) and memory modules (MEM). Each
+functional unit (core) accepts a set of instructions; for example,
+the MEM unit could receive the instruction to put a vector onto
+one of the data streams or store the results from the data stream
+in its available SRAM. As soon as data is loaded onto a data
+stream, it automatically ‘flows’ in the direction of the stream,
+which can be either EAST-bound or WEST-bound. When an
+addition needs to be performed, both inputs need to arrive
+at the same time as the add instruction at the corresponding
+VXM core. This design choice puts the burden of optimization
+on the software generating the instructions. This is either done
+by the Groq compiler automatically from an ONNX-graph
+input or manually controlled by a user through the exposed
+Groq-API, which has different levels of abstraction on top of
+the Groq-ISA. To support the creation of large-scale systems,
+the GroqChip has dedicated Chip-to-Chip modules that are
+capable of performing off-chip communication without losing
+their determinism [32]. For this work, we will mainly utilize
+the VXM and MEM units, The memory modules add up
+to a total of 220 MiB of on-chip SRAM. Each superlane
+implements a 4x4 mesh of vector ALUs capable of doing
+x16-SIMD. Each ALU has a 32-bit input operand but with
+the exception of additions and multiplications, instructions are
+done in a reduced-precision FP32 format.
+3) Graphcore IPU [4]: The Graphcore Intelligence Pro-
+cessing Unit (IPU) is designed for efficient execution of fine-
+grained operations across a large number of parallel threads.
+By design, the IPU offers true Multiple Instruction, Multi-
+ple Data (MIMD) parallelism. This unique style of parallel-
+processor design adapts well to fine-grained computations that
+exhibit irregular data-access patterns. Each IPU contains 1,472
+tiles, containing 1 core and 624KiB of SRAM memory. A sin-
+gle core can only access the memory in its own tile. Intra-IPU
+communication relies on a powerful, low-latency interconnect
+called IPU exchange. For inter-IPU communications, each
+chip contains 10 so-called IPU links. The IPU compute
+units, called Accumulating Matrix Product (AMP)
+units, support FP32 arithmetic and are designed to accelerate
+matrix multiplications and convolutions. With respect to the
+programming model, the IPU adopts the Bulk Synchronous
+Parallel (BSP) model [33] through which it organizes its
+compute and data-exchange operations. This abstraction for
+parallel computations consists of multiple sequential super-
+steps. A superstep consists of a local computation phase;
+every process (tile, in the IPU case) operates in isolation
+performing compute only on its local memory, followed by a
+communication phase where each process can exchange values
+needed by other tiles. These activities are concluded with a
+barrier synchronization phase; only when all processes have
+reached the barrier can the next superstep be started. Because
+of this, the IPU can be described as a true BSP machine.
+4) Google TPU [7]: The TPU (version 1) was designed
+as a systolic-array processor for inference, only supporting
+8/16-bit operations. By supporting only matrix-multiply and
+4
+
+TABLE I: Overview of all hardware used in experimental setups
+Device
+On-chip Memory
+Process node
+Transistor count (Bn)
+Base-boost freq. (MHz)
+TDP (W) Software
+AMD 3955WX CPU *
+128 GB DDR4
+7 nm
+19.94
+3900 - 4300
+280
+TF 2.11.0
+GroqChip TSP
+230 MB on-chip
+14 nm
+26.8
+900
+-
+Groq SDK 0.9.1 ***
+Nvidia A100 GPU
+80 GB HBM2e
+7 nm
+54.2
+1275 - 1410
+400
+TF 2.11.0
+Graphcore IPU (GC200) **
+900 MB on-chip
+7 nm
+59.4
+1330
+185
+TF IPU 2.6.3+gc3.0.0
+Google TPUv3
+32 GiB HBM
+16 nm
+(est.) 11
+940
+450
+TF 2.11.0
+*AMD Ryzen Threadripper PRO 3955WX (16-Core) | **Single M2000 in IPU-POD16 (with 4 GC200 chips) | ***TF2ONNX 1.13.0 and ONNX opset 16
+basic nonlinear activation functions, it was unfit for training
+neural networks. Consequentially, an HPC application – for
+example, the one demonstrated in this paper – would also
+not be a suitable fit for this processor. However, with the
+TPUv2, Google shifted their focus towards supporting training
+on their TPU chips. Google added a vector-processing unit
+(VPU) and changed the matrix-multiply units to support the
+FP16 format (FP32, with only a 7-bit mantissa). The VPU
+most likely supports higher precision, as can be deducted from
+results in this work but no confirmation of this is found in the
+public domain. These two major (micro)architectural changes
+made it possible to run a wider range of applications including
+training neural models on the TPU. All are supported through
+the Google XLA compiler taking TensorFlow as input. The
+TPUv3, assessed in this work, is an upgrade in terms of
+functional-unit count, higher memory speed, and optimized
+chip layout, but did not include any fundamental changes.
+A. Performance Predictions
+The IO application has two components that map differently
+onto different types of hardware: i) a part with embarrassingly
+parallel computations for updating local neuron states; and
+ii) a part with SM computations for exchanging membrane
+voltages over gap junctions. Before we proceed to the actual
+experiments, we attempt performance predictions, driven by
+the idiosyncrasies of the different AI-chip architectures.
+Embarrassing parallelism: These are calculations for up-
+dating the state of every single neuron. This boils down to
+elementwise vector operations. The GPU architecture featuring
+one Warp execution per Streaming Multiprocessor or multiple
+Tensor Cores is very well-suited for this type of parallelization.
+The TPU and the GroqChip are both based upon systolic-array
+architectures, both natively supporting Matrix-Multiplication
+but also Vector-Operation operations that can be utilized
+for these calculations. In fact, since neuron updates require
+only
+1-D
+data, the Matrix-Multiplication units (which is
+the focus of these chips) are effectively underutilized in these
+architectures. The IPU, with a large amount of very small
+general-purpose cores, should also do well on parallelizing
+neuron-state calculations, however, its architecture is geared
+towards irregular data-access patterns, which is not essential
+to the particular task. The extra overhead of such advanced
+features, therefore, will not help performance in terms of
+computing this embarrassingly parallel part of the simulation.
+Communication:
+As described previously, gap-junction
+communication employs the gather-scatter operations (essen-
+tially, SM operations) from TensorFlow. For either the GPU or
+the IPU, such operations are handled better due to the different
+execution paths that can be handled within the architecture by
+design. In contrast, the GroqChip and TPU need to handle
+these differently: a naive approach would be to enforce dense-
+matrix operations via one-hot encoding of operands and,
+then, utilizing the matrix-multiplication hardware. In case the
+GroqChip or the TPU happen to use such a strategy, we expect
+that performance will deteriorate very rapidly or memory will
+be depleted with increasing IO-network sizes.
+1) CPU + TensorFlow: For this platform, JIT compila-
+tion through the XLA compiler [34] will be used; it will
+automatically utilize the many threads nowadays available in
+CPUs. We expect decent performance and very accurate results
+because of full FP32 support. Since it is the hardware on which
+brain models are traditionally executed and gives accurate
+results, the CPU will form our baseline. Accelerators should
+outperform this implementation in terms of runtime, especially
+for larger network sizes.
+2) GPU + TensorFlow: The XLA compiler is used, which
+optimizes the graph resulting in a single kernel launch. Among
+others, it does this by “fusing” the calculations. Moreover, this
+fusion keeps intermediate values stored in GPU registers [35].
+The TensorFlow backend for CUDA use Tensor Cores, at
+a loss of FP32 accuracy. However, this only happens when
+explicit matrix-multiplications are requested and not as an
+optimization. So in our case, the compiler will only use float32
+CUDA operations.
+3) IPU + TensorFlow: The IPU architecture is not a perfect
+fit for the embarrassingly parallel part of the computation.
+For the interneuron-communication part, the BSP model is a
+better fit and thus is expected to perform better. However, as
+the topology is given as an unknown parameter to the model,
+the IPU compiler can not be expected to allocate neighboring
+cells on adjacent tiles, resulting in sub-par communication
+performance. Available memory should easily be able to
+handle large problem sizes.
+4) TPUv3 + TensorFlow: The TPU supports FP32 and
+is expected to handle our workload, especially for the un-
+connected case, very well. As Google put much effort into
+TensorFlow support, gather-scatter operations are expected to
+be optimized, to the best of the hardware capabilities. Because
+of FP32 support in the v3 model, we expect correct numerics
+in the output, as well.
+5) CPU/GPU + ONNX: Expectations are the same as for
+CPU/GPU + TensorFlow. We expect the XLA compiler to
+outperform the ONNX runtime slightly for the CPU case
+simply because it can perform whole-program optimization.
+For the GPU, this effect is expected to be much larger and
+the TensorFlow is expected to dominate ONNX as the ability
+5
+
+to fuse kernels will be a big advantage for TensorFlow over
+single-kernel invocations in ONNX. Especially the invocation
+overhead for small GPU kernels will hurt the performance
+of the ONNX-GPU-runtime. TensorRT is also a supported
+backend in ONNX that is expected to outperform the CUDA
+runtime in performance; it will, however, drop precision as the
+backend switches to TF32 numerics.
+6) GroqChip + ONNX: The GroqChip is a new, upcoming
+modified systolic-array processor. Its compiler takes in the
+ONNX graph but is not limited to executing this on an
+operation-per-operation basis as it recompiles the full ONNX
+graph at once. Therefore, it can potentially perform the same
+optimizations as the XLA compiler for the TPU. As the first
+version of the architecture, current compiler development is
+still exploring ways to map non-standard ML-operations to
+the hardware. Besides, the GroqChip VXM is not capable of
+doing all operations in IEEE FP32 arithmetic. Because of this,
+it can be expected to perform slightly better than the TPU at
+the cost of reduced accuracy.
+V. EXPERIMENTAL SETUP
+A. Benchmarking Parameters
+Each platform is benchmarked for performance on a set
+problem (i.e., network) size as well as for its performance
+scalability by simulating the IO network for small population
+sizes in the range [43, 53, . . . , 203] and, again, for larger sizes
+in the range [303, 403, . . . , 1003], where the third power is
+an artifact of the cubic network-topology generation method.
+These experiments are focusing on four different aspects of
+each AI platform, discussed next.
+1) Unconnected Network: By removing the communica-
+tion step (gap junctions) from the model, we obtain a (bio-
+logically unrealistic) compute-heavy, embarrassingly parallel
+workload. First, we measure the setup time for each AI
+platform, including on-chip buffer allocation, Ahead-Of-Time
+(AOT) compilation or definition of Just-In-Time (JIT)-enabled
+functions. Next, we simulate an IO network for 100ms of
+biological time and take the minimum wall-clock time from
+5 runs (including data-transfer times). For JIT targets, the
+first runtime (if outside the other runtimes’ standard deviation)
+minus follow-up runtimes is taken as the JIT compilation time,
+such that we can compare setup times between AOT and JIT
+targets.
+2) Connected Network: By restoring gap junctions into the
+IO network, we assess communication overhead. Runtimes are
+obtained in an identical way as before, yet the expectation here
+is that they are markedly longer than the unconnected case.
+3) Numerical Validation: Measuring performance is our
+main focus, yet this must not come at the cost of functional
+correctness. Here, we simulate connected networks up to 729
+neurons for 10 seconds of biological time and numerically
+compare the various results to the reference CPU output.
+4) Numerical Stress-test: Here, we simulate the IO in a
+more biologically realistic way that is of interest to neurosci-
+entists: We add more variance to the neural parameters and,
+most importantly, a lot of external current inputs (simulating
+other brain regions) that will evoke action potentials (spikes)
+in the IO dynamics. These fast transients will stress-test the
+numerical performance of the AI hardware, especially non-
+IEEE754 targets (Tensor Cores and GroqChip). We perform
+this experiment on the smallest 64-neuron network and then
+compare for numerical accuracy against the CPU.
+Benchmarking
+is
+implemented
+in
+a
+publicly
+avail-
+able
+and
+modular,
+extensible
+framework,
+downloadable
+from GitLab https://gitlab.com/neurocomputing-lab/Inferior
+OliveEMC/ioperf.
+The
+main
+benchmarking
+script
+auto-
+discovers available hardware, runs the appropriate benchmarks
+and records results. Used software versions are also shown in
+Tab. I.
+B. Hardware-specific Optimizations
+While our original goal was not to write platform-specific
+code, we found that by default some of the AI platforms did
+not perform very well. For example, most platforms defaulted
+to copying over the entire parameters arrays for each kernel
+invocation, which was not needed for this mostly constant data.
+For a fair comparison between hardware platforms, we allowed
+optimizations to be applied to hardware-specific code that
+either led to operation fusion across different execution kernels
+or prevented unnecessary device-host data transfers. The exact
+optimizations have been applied in close collaboration with
+Graphcore and Groq for the respective chips, and are as
+follows:
+1) TensorFlow XLA: The TensorFlow graph executor typi-
+cally performs each operation separately when a graph is run
+with a corresponding kernel invocation. A different way to
+run TensorFlow models is made available by XLA, which
+turns a TensorFlow graph into a series of kernels created for
+a particular application. These kernels can take advantage of
+application-specific information for performing optimizations,
+e.g., operation fusion. The CPU, GPU, and TPU are the
+three available backends for the XLA compiler. For the IO
+application, a TensorFlow wrapper function was implemented
+that fuses up to 40 timesteps together for each call in order to
+fully exploit the XLA compiler.
+2) ONNX: Except for the GroqChip, all ONNX imple-
+mentations build on top of onnxruntime or onnxruntime-
+gpu. We enable all backend-supported graph-optimizations.
+Explicit IOBindings are used to prevent unneeded host-
+device data copies. Parameters are copied once to the device
+at simulation start. Then, state is allocated twice, with each
+timestep toggling between two buffers, one as the input state
+and the other as the output (next) state. For TensorRT, we
+leave the default behavior of using TF32 enabled, otherwise,
+it will not utilize its Tensor Cores.
+3) Groq: After the compilation of an ONNX graph with
+the Groq Compiler, the binary can be executed directly on the
+GroqChip. A naive approach here would be to invoke this
+binary 40 times for 40 timesteps and move the data back
+and forth continuously since the GroqChip only has SRAM
+which is fully managed at compile time. However, the Groq
+Compiler is able to tie input and output tensors together into
+6
+
+10
+1
+100
+101
+102
+103
+CPU (ONNX)
+CPU (TF)
+10
+1
+100
+101
+102
+103
+GPU (TF)
+IPU (TF)
+102
+103
+104
+105
+106
+10
+1
+100
+101
+102
+103
+TPU (TF)
+102
+103
+104
+105
+106
+GroqChip (ONNX)
+Network size (#neurons)
+Time required to simulate one biological second (s)
+Fig. 2: Runtime performance (lower is better), comparison between CPU baseline, GPU and AI chips. For scale, the mouse (•) and human (▲) Inferior Olive
+are shown as running in realtime in all figures. The CPU is included twice to explain the observed switching behavior of the IPU. On the CPU, while the XLA
+optimizer builds a single-core, connected-network simulation, it builds a multicore, unconnected-network simulation (as observed by load-testing), leading to
+an unexpectedly slow simulation for the latter case. The same behavior can be observed for the IPU, which uses the XLA compiler as well.
+a persistent memory buffer in the on-chip SRAM. Utilizing
+this results still in 40 invocations of the binary but skips the
+continuous I/O between host and accelerator. A more radical
+way to improve the performance is to compile the 40 timesteps
+into a single ONNX graph that can then be converted with the
+Groq Compiler; this method will reduce 40 invocations to a
+single invocation. We implemented all optimizations as long
+as the compiler was able to compile them. The 40 timesteps at
+once quickly ran into compiler errors with growing networks.
+4) Graphcore:
+The IPU has architectural support for
+streaming memory. This means that we can run a single
+program on-chip for the entire simulation that will stream
+out samples every 40 timesteps. The inner, unsampled 1-
+msec 40-timestep loop, is run using ipu.loops.repeat,
+after
+which
+the
+recorded
+voltages
+are
+pushed
+to
+an
+IPUOutfeedQueue with a 200-sample size. This is, then,
+looped once more using ipu.loops.repeat for the re-
+quired amount of milliseconds to simulate and wrapped in
+a TensorFlow JIT function. Furthermore, the fast-math op-
+timization is enabled, 128 IPU tiles are reserved for I/O
+with place_ops_on_io_tiles = True and program
+execution is limited to a single IPU.
+VI. EXPERIMENTAL RESULTS
+With the exception of the reference CPU, for brevity we
+report here either TensorFlow or ONNX results, depending
+on which of the two leads to better performance. Overall
+performance plots are shown in Fig. 2 and will be detailed
+in the next sections. In general, it is found that, for the
+IO application, the ONNX ports are outperformed by their
+TensorFlow counterparts. This is due to the fact that the
+onnx-runtime library currently does not perform as extensive
+optimizations as the XLA compiler. For example, the CUDA
+target translates each compute step into a single predefined
+kernel call. The TensorRT backend performs operator fusion,
+resulting in multiple kernels that chain arithmetic operations.
+Still, the CUDA XLA-backend vastly outperforms both ONNX
+CUDA targets, and as such we removed the corresponding
+findings from the main analysis. Note that the Groq platform
+only supports AOT compilation of ONNX models.
+A. Compilation Time
+Both software stack and hardware influence program setup
+time, as illustrated in Fig. 3 for the largest network (729 cells)
+that could fit in all AI chips. The CPU compiles the fastest
+across the board as we have a direct translation of ONNX
+operations to their CPU-optimized callbacks. The TensorFlow
+(XLA) version, not included in the figure, was much slower
+7
+
+100
+101
+102
+103
+Compile time (s)
+(A). Network type at compile time (n=729)
+Unconnected
+Connected
+CPU
+IPU
+GPU
+TPU
+GroqChip
+10
+2
+10
+1
+100
+101
+102
+Run time (s)
+(B).Connected network size at run time
+n=64
+n=729
+n=125000
+1.0x
+2.3x
+8.7x
+7.7x
+32.9x
+1.0x
+4.3x
+17.4x
+16.4x
+2.6x
+1.0x
+28.6x
+269.5x
+1208.5x
+Fig. 3: (A) Setup (AOT+JIT compile + memory allocation) times for a network
+of 729 neurons in both Unconnected- and Connected-network configurations.
+JIT compile times are extracted from the first run of 5 performance runs
+and added to the initial setup time (if outside one standard deviation). (B)
+Performance and speedup of different AI chips vs. the CPU reference on
+the Connected benchmark, for different network sizes. Sizes were chosen to
+be the smallest (64) and largest connected networks that could fit on the
+GroqChip (729) and the TPUv3 (125,000). The rightmost GroqChip bar is
+absent, corresponding to the model that could not be compiled.
+due to the increased compiler complexity. Both the IPU and
+GPU exhibit similar JIT compilation speeds. The GroqChip’s
+AOT compiler takes significantly longer for this workload due
+to the explicitly concatenated 40 timesteps. The GroqChip
+version with a single timestep per program compiles much
+faster than the Graphcore or A100 versions, but at a small
+performance loss.
+B. Runtime Performance
+1) Unconnected Network (embarrassingly parallel): For
+unconnected cells, neural dynamics are expressed only using
+vectorized operations. As predicted, this fits the compute
+paradigm of the GPU very well. Performance scales linearly
+with problem size (horizontal line), showing that the GPU
+cores are underutilized for all simulated network sizes.
+The TPU and GroqChip, as systolic-array-based processors,
+were expected to be a poorer architectural fit because large
+parts of the chip would be left unused. Still, the focus on
+efficient vector operations could result in speedups. We can
+indeed observe this in Fig. 2, although in different ways. The
+TPU, similar to the GPU, flatlines across all problem sizes,
+although being 4.1x slower. Consequently, memory capacity
+is not a problem for the TPU but performance capping in raw
+single-cell computations due to architectural design choices.
+In contrast, the GroqChip starts out 2.0x faster than the GPU,
+quickly loses this edge and, between 103 and 104 cells, starts
+to hit its memory-capacity limits, degrading performance with
+higher problem sizes. Networks of more than 640, 000 cells
+simply do not fit on the chip. The GroqView analyzer confirms
+that the problem is core-to-core-memory communication and
+that most dedicated cores are not used most of the time.
+The IPU was expected to perform well given its large core
+count but the very homogeneous compute load proved a poor
+fit for its MIMD design, leading to large under-utilization
+of the chip. With respect to real-time performance, only the
+GPU followed by the GroqChip (ignoring memory issues) and
+marginally the TPU makes the 1-sec cut.
+2) Connected Network (high communication overhead): As
+predicted, communication patterns induced by a small number
+of gap junctions lead to a large performance reduction of 4.6x
+for small networks on the GPU. For higher problem sizes,
+performance drops at a growing rate, with a 141x degradation
+for networks of 106 cells. The AI chips fare much better here,
+most of which initially shows a less than 20% reduction in
+performance against their unconnected counterparts.
+As an exception, the GroqChip’s connected-network sim-
+ulation runs 2.5x slower than the unconnected version; even
+so, it outperforms the GPU on very small, connected neural
+networks by a 3.7x speedup. However, the GroqChip (as
+expected) converts the SM communication into a dense-
+matrix multiplication, making the best out of its deterministic-
+execution hardware. This quickly leads to prohibitively large
+matrix multiplications and, beyond 729 cells, the scheduler
+is unable to allocate the necessary instructions. In effect, the
+GroqChip loses its edge over the GPU for larger networks.
+In contrast, the TPU shows nearly identical behavior to
+the unconnected case and its performance does still not scale
+with problem size. This changes around networks larger than
+105, where the JIT compiler seems to run into performance
+problems. Here, we observed large random fluctuations in
+performance that either led to approx. 1-sec or very long more
+than 400-sec run-times over the 5 repeated runs. We expect
+that these originate from memory limits of the TPU and had
+to stop benchmarking due to impractically large run times.
+However, we could not determine the true source of variation.
+The IPU, severely underutilized for the unconnected case,
+sees in fact a performance improvement when we increase
+the communication overhead in small networks. While coun-
+terintuitive, this is actually the same effect we see on the
+XLA-based CPU backend. Here, we see that gap junctions
+force the simulation to become single core, which actually
+becomes faster than the parallel, multi-core, unconnected case,
+due to the lower synchronization overhead. Around 104 cells,
+this behavior changes, gap-junction communication becomes a
+fixed overhead on top of normal simulation. At a certain point,
+this growth becomes exponential and the largest simulated
+network does not fit on a single IPU anymore.
+C. Numerical Validation
+While all AI chips outperform the CPU baseline, it is wise
+to explore also any potential decrease in numerical accuracy
+of the different runs with respect to that of the same CPU.
+Here, we compare 1-msec sampled cell somatic voltages of an
+extended, 10-sec simulation for a 729-cell, connected network
+8
+
+IPU
+GPU
+TPU
+GroqChip
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Abs. difference to CPU reference (mV)
+Left
+boxp.
+Right
+boxp.
+Vgroqchip
+Vcpu
+Fig. 4: Numerical-accuracy validation (lower is better). Box plots show
+deviations from CPU baseline, as recorded over two 1-sec timespans, one
+at the start (left) and one at the end (right) of the 10-sec numerical-validation
+simulation. The GroqChip result, showing the largest deviation, is plotted in
+the upper left corner together with the two recording spans .
+(the largest population supported by all platforms); results are
+shown in the box plots of Fig. 4.
+As expected, platforms supporting IEEE754 floating-point
+numerics (IPU, GPU, TPU) show accurate reproduction of
+voltage traces. The IPU, even with fast-math enabled, is the
+most faithful to the CPU baseline. The GPU and TPU exhibit
+increasingly large deviations but still fall within limits explain-
+able by floating-point instruction reordering. The GroqChip,
+while supporting FP32 number storage, implements certain
+operations at lower precision including exponent calculation.
+This is visible by a quite large mV -order deviation from the
+CPU baseline, for a process that happens at the 10 − 100mV -
+scales. This voltage difference mostly stems from a slowly
+accrued phase difference for the oscillating cells. TensorRT
+(not shown in this plot) is by default using Nvidia’s TF32, for
+which accuracy was found similar to that of the GroqChip.
+D. Numerical Stress-test
+The numerical stress test increases neuronal variation and
+adds external inputs that lead the neurons to spike. These
+fast transients can not be simulated using FP16 precision, but
+reduced-accuracy FP32 operations as used in Tensor Cores
+or GroqChip are still untested. Once more, we compare the
+deviation of the somatic-voltage traces of the various AI chips
+against the CPU baseline.
+Again, the platforms with native FP32 support show the
+lowest deviation: For the IPU this is 0.087mV , for the
+GPU this is 0.135mV and for the TPU this is a 0.672mV
+maximum absolute difference from the CPU baseline. These
+moderate, mV -order differences can be explained by small
+spike-time differences which due to the large neuronal-spike
+sizes quickly lead to large voltage discrepancies. Importantly,
+all simulations run stably; i.e., do not cause this chaotic IO-
+model simulator to crash. The GroqChip simulation initially
+starts out the same as in the numerical-validation test, but as
+soon as input perturbations are applied, it becomes unstable
+and settles on voltage deviation at a measured maximum
+of 8.51 × 1036mV , unacceptable for scientific applications.
+Notably, the error stabilizes at this point and does not explode
+to infinity or NaN values, as observed with FP16 simulations.
+To regain numerical stability, we tried lowering the time-
+stepping constant ∆t 10-fold and 100-fold for the GroqChip
+simulation, but this did not lead to results more closely in
+range with the CPU ones.
+VII. DISCUSSION
+As this work has shown, utilizing AI platforms for executing
+highly biologically plausible SNN workloads is made exceed-
+ingly user-friendly when using a ML-library like TensorFlow.
+Arguably, even better performances could be obtained by
+coding via the various hardware SDKs (Software Development
+Kits), but it is unrealistic to expect computational scientists
+to learn the low-level details of all hardware options made
+available to them these days.
+As shown, the added benefits from JIT compilation make
+a hand-coded CUDA implementation perform on par with the
+XLA-compiled TensorFlow version while, at the same time,
+allowing one to move easily to a new piece of hardware
+when this is released. We expect that, in the future, more
+classical HPC workloads will see ML-library, that is, tensor-
+based implementations.
+For promising upcoming accelerators like those by Graph-
+core and Groq, we believe that future speedups will chiefly
+come from software and compiler upgrades, as current SDKs
+are mostly optimized for ML workloads. For instance, gather-
+scatter operations on the GroqChip do not have to be im-
+plemented as dense-matrix operations, memory can be better
+utilized, and better support for iterative programs must also
+be introduced. The TPU which is architecturally similar to the
+GroqChip, clearly performs gather-scatter operations in a more
+efficient way than encoding indexing as one-hot vectors.
+Speed-ups could be gained by effective use of mixed
+precision on the IPU or reduced accuracy FP32 operations
+using Tensor Cores or GroqChip. For the IPU, this would
+constitute a separate numerical sensitivity analysis to find out
+which parts of the compute graph can be lowered to (stochastic
+rounded) FP16. As shown, the accuracy loss on Tensor Cores
+and GroqChip does in its current form not allow for brain
+simulation, but possible these could be put to use by switching
+the integration scheme or other numerical optimizations.
+Finally, this work has steered clear off multi-chip topologies.
+All discussed architectures do support specifically developed,
+low-latency, chip-to-chip hardware and assorted communica-
+tion protocols. In many ways, such coherent communication
+is a bigger and more timely challenge than acceleration speed
+itself, which would deliver massive benefits for large-scale
+SNN simulation (or training). However, tapping into those
+platform-specific interfaces requires SDK-specific coding of
+the IO application; relying on TensorFlow or ONNX frame-
+works will, generally, not work. Careful and platform-specific
+coding is necessary, which we leave as future work.
+VIII. CONCLUSION
+In this work, we built the first ML-library-based, effi-
+cient implementation of a large-scale, conductance-level brain
+9
+
+model, the Inferior Olive (IO). Subsequently, we benchmarked
+the performance of simulating this model on a 16-core AMD
+Ryzen Threadripper PRO 3955WX CPU, an Nvidia A100
+GPU, and different AI chips (Graphcore IPU M2000, Gro-
+qChip and Google TPU v3). We found that all accelerators
+provide significant speedups over the CPU implementation.
+For this specific problem, the GPU and TPU seem most
+fit for simulation, with the TPU setting a new record for
+real-time IO simulation. For small networks, the GroqChip
+outperforms the other accelerators, but large networks could
+not fit in the on-chip instruction memory. More generally, we
+hypothesize that modern ML-libraries possess the semantic
+power to model classical problems in scientific computing.
+These, then, map extremely well to ML-driven, novel AI-chip
+architectures, which apart from large performance benefits,
+also benefit from reduced development times. For example, the
+version of our IO application running on the TPU outperforms
+the handwritten and hand-optimized CUDA implementation by
+a large factor, at a fraction of the development cost. The exact
+hardware trade-off will vary on an application-by-application
+basis, and hardware selection also benefits significantly from
+the hardware-agnostic model description.
+ACKNOWLEDGEMENT
+This research would not have been possible without access
+to dedicated hardware: The RTX6000 was gifted from the
+NVIDIA Hardware Grant Program, Google provided free
+cloud credits for TPU access and Graphcore provided access
+to POD16 machines through Paperspace and Gcore cloud
+via its Academia program. Furthermore, we’d like to thank
+Graphcore employees for helping with optimizing the IPU
+code and Dr. Mario Negrello for neuroscientific insights.
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+SURFACES OF MINIMUM CURVATURE VARIATION
+LUIS A. CAFFARELLI, PABLO RA´UL STINGA, AND HERN´AN VIVAS
+Abstract. We establish the analytical theory of surfaces of minimum curvature variation.
+We construct classical, G2 continuous surfaces, as well as weak solutions in the context of
+geometric measure theory.
+1. Introduction
+Computer-aided design (CAD) and computer-aided manufacturing (CAM) are widely pop-
+ular techniques whose basic feature is the use of computer software to create or modify shapes
+in such a way that some aspects of the design process, such as quality of the object or produc-
+tivity of the process, are optimized, see, for example, [9]. Their origins can be traced back
+to the 1950s and 60s and their development have been continuous since then. Nowadays,
+CAD/CAM are used in contexts as varied as engineering, particularly in automotive, ship-
+building and aerospace industries; architectural design; and computer animation for creation
+of special effects in movies, among many other applications.
+Within this realm, of particular interest are geometric problems in computer-aided geomet-
+ric design (CAGD). The goal of CAGD is the creation of complex smoothly shaped models
+and surfaces with specified geometric constraints. The resulting surfaces have to accurately
+reflect these specifications and be free of unwanted wrinkles, bulges and ripples. In many
+instances, the aim is to create fair surfaces that are aesthetically pleasing to the eye. As it
+turns out, many of these problems can be approached via a variational principle, that is, by
+looking for a surface that minimizes an appropriate functional or fairness energy subject to
+adequate geometric boundary conditions, see [10].
+The most commonly used fairness energy functionals can be split into two groups: physical-
+based or geometric-based. The first group roughly corresponds to interpreting the surface as
+an ideal elastic membrane or plate and minimize energies such as
+´
+|∇u|2 dx or
+´
+|∆u|2 dx.
+The second group aims at minimizing energies that relate to geometric invariants of the
+surface such as the area or curvature, see [11] and the references therein. In 1992, Moreton
+and S´equin proposed in [8] a numerical algorithm for the creation of 2-dimensional fair
+surfaces M as minimizers of the energy functional
+ˆ
+M
+��dκ1
+de1
+�2
++
+�dκ2
+de2
+�2�
+dA.
+Here e1 and e2 are the principal directions corresponding to the principal curvatures κ1 and
+κ2 of M and dA is the differential of surface area.
+It is a key aspect in CAGD to be able to construct fair surfaces that preserve several degrees
+of geometric continuity. This is particularly important at the boundary of the domains where
+2010 Mathematics Subject Classification. Primary: 35B65, 49Q10, 53A10.
+Secondary: 49Q20, 65D17,
+68U07.
+Key words and phrases. Curvature variation, computer-aided design, prescribed mean curvature, regularity.
+Research partially supported by NSF grant 1500871 (USA), Simons Foundation grant 580911 (USA), and
+Agencia Nacional de Promoci´on Cient´ıfica y Tecnol´ogica under grant PICT 2019-3530 (Argentina).
+1
+arXiv:2301.00082v1  [math.DG]  31 Dec 2022
+
+2
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+the surfaces meet. The notions of geometric continuity are referred to as G0 continuity, where
+two surfaces meet in a continuous fashion, without jumps; G1 continuity, where the tangent
+planes of the surfaces meet with continuity; and G2 continuity, where the curvatures meet
+with continuity. These are not the same as the classical notions of C0, C1 and C2 continuities,
+as those require some specific combination of the derivatives of the solutions to be continuous
+up to the boundary. In particular, G2 continuity turns out to be crucial in applications such
+as the streamlined surfaces of aircrafts, ships and cars, and this was the main motivation
+for the numerical study in [8].
+In [11], a numerical finite difference method is proposed
+to construct surfaces that would enjoy G2 continuity as steady states of a sixth order flow
+derived from the Euler–Lagrange equation of the energy functional
+ˆ
+M
+|∇H|2 dA
+where H is the mean curvature of M. Numerical evidence of G2 continuity is observed in [11],
+while G1 continuity is expected according to [8]. To the best of our knowledge, the analytical
+theory of surfaces of minimum curvature variation in general is missing. Furthermore, no
+proof of G2 continuity is available thus far.
+The aim of this paper is to fill these gaps and to develop the analytical foundation from
+the PDE perspective of the theory of surfaces of minimum curvature variation. We give
+two constructions of surfaces: classical solutions that are G2-continuous, and weak solutions
+through geometric measure theory methods.
+Therefore, we consider the minimization problem
+(1.1)
+min
+M
+1
+2
+ˆ
+M
+|∇MH|2 dA
+where M ranges over all n−dimensional manifolds in Rn+1, n ≥ 1, with prescribed bound-
+ary, H is the mean curvature of M and dA is the differential of surface area. Notice that
+(1.1) minimizes the (quadratic) variation of the mean curvature of M so that surfaces with
+constant mean curvature such as planes, circles and cylinders are minimizers.
+If M is the graph of a function defined on a bounded domain Ω ⊂ Rn, that is,
+M = {(x, u(x)) : x ∈ Ω}
+for some u : Ω → R, then the values of u at ∂Ω prescribe the boundary ∂M of M. For a
+point x0 ∈ Ω, the tangent plane to M at (x0, u(x0)) and its upward pointing unit normal are
+P(x) = u(x0) + ∇u(x0) · (x − x0)
+and
+ν(x0) =
+(−∇u(x0), 1)
+(1 + |∇u(x0)|2)1/2 ,
+respectively. The mean curvature H of M at a point is defined as the average of the n
+principal curvatures of M at that point. In the coordinates given by u, it takes the form
+H ≡ H(u) = 1
+n div
+�
+∇u
+(1 + |∇u|2)1/2
+�
+.
+If we set
+D(u) := (1 + |∇u|2)1/2
+we then have that dA = D(u) dx.
+Let f be a function in C1(Ω × R). The tangential gradient of f on M is obtained by
+projecting the gradient of f in Rn+1 onto the plane orthogonal to ν:
+∇Mf = ∇Rn+1f − (ν · ∇Rn+1f)ν
+on M.
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+3
+Clearly, ν · ∇Mf = 0 and
+(1.2)
+|∇Mf|2 = |∇Rn+1f|2 − |ν · ∇Rn+1f|2.
+Furthermore, ∇Mf depends only on the values of f on M, see, for instance, [6, Section 16.1].
+To compute ∇MH we extend H as a function of (x, xn+1) ∈ Ω × R by making it constant
+in xn+1: H(x, xn+1) ≡ H(x). This is enough to compute ∇MH because the resulting value
+is independent of the extension. Since ∇Rn+1H = (∇H, Hxn+1) = (∇H, 0), by (1.2) we get
+|∇MH|2 = |(∇H, 0)|2 −
+����(−∇u · ∇H)(−∇u, 1)
+D(u)2
+����
+2
+= |∇H|2 −
+����
+∇u · ∇H
+D(u)
+����
+2
+.
+With this formula the energy in (1.1) becomes
+(1.3)
+E[M] = 1
+2
+ˆ
+Ω
+�
+|∇H|2 −
+����
+∇u · ∇H
+D(u)
+����
+2�
+D(u) dx.
+We will call this the geometric energy. It follows from the Cauchy–Schwartz inequality that
+|∇H|2
+D(u)2 ≤ |∇MH|2 ≤ |∇H|2.
+Therefore, we will also study the (larger) simplified energy functional
+(1.4)
+E[H, u] := 1
+2
+ˆ
+Ω
+|∇H|2D(u) dx.
+In Section 2 we consider (1.4) and show how to construct smooth solutions that satisfy the
+prescribed mean curvature equation for a curvature of minimum variation. Section 3 shows
+how to modify the argument to construct solutions of the geometric energy functional (1.3).
+Finally, in Section 4 we provide a weak formulation of the problem and prove existence of
+minimizers in the context of geometric measure theory.
+2. Existence of G2 surfaces for the simplified energy
+In this section we work with the simplified energy functional (1.4). Let Ω ⊂ Rn be a
+bounded domain such that ∂Ω ∈ C3,α for some 0 < α < 1 fixed. We assume that we are
+given prescribed boundary values g ∈ C3,α(Ω) for u and h ∈ C1,α(Ω) for H on ∂Ω.
+We address the following problem: given Ω and the boundary datum g, find a surface
+given by the graph of a function u such that its mean curvature H is a minimizer of (1.4)
+among all functions with prescribed boundary values h ∈ C1,α(Ω).
+We will use Schauder’s fixed point theorem:
+Theorem 2.1 (see [6, Corollary 11.2]). Let G be a closed convex set in a Banach space B
+and let T be a continuous mapping of G into itself such that the image T(G) is precompact.
+Then T has a fixed point.
+Consider the Banach space B = C1,α(Ω) and its subset
+G :=
+�
+v ∈ C1,α(Ω) : v = g on ∂Ω
+�
+.
+Observe that G is nonempty because g ∈ C3,α(Ω). By classical global Schauder estimates,
+we see that another example of function in G is the harmonic extension v of g inside of Ω:
+�
+∆v = 0
+in Ω
+v = g
+on ∂Ω.
+
+4
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+It is clear that G is convex and closed.
+For any v ∈ G, we define the functional
+(2.1)
+E[H, v] := 1
+2
+ˆ
+Ω
+|∇H|2D(v) dx.
+The map T : G → G is constructed in a 2-step process.
+Step 1. Given any v ∈ G, we find the unique minimizer H ∈ W 1,2(Ω) to (2.1) such that
+H − h ∈ W 1,2
+0 (Ω). This can be done because the coefficient D(v) satisfies
+1 ≤ D(v) ≤ (1 + ∥∇v∥2
+L∞(Ω))1/2 < ∞,
+so that (2.1) is a coercive functional. Then H is the unique weak solution to
+�
+div(D(v)∇H) = 0
+in Ω
+H = h
+on ∂Ω.
+Since v ∈ C1,α(Ω), the coefficient D(v) ∈ C0,α(Ω). Thus, by global Schauder estimates (see
+[6, Section 8.11]),
+(2.2)
+∥H∥C1,α(Ω) ≤ Cn[∂Ω]C1,α∥D(v)∥C0,α(Ω)∥h∥C1,α(∂Ω)
+where Cn > 0 is a constant that depends only on dimension n.
+Step 2. Given H ∈ C1,α(Ω) from Step 1, we find the solution u to the prescribed mean
+curvature equation
+(2.3)
+�
+�
+�
+div
+� ∇u
+D(u)
+�
+= nH
+in Ω
+u = g
+on ∂Ω.
+For this, we use the following result (where we use nH instead of H in [7]).
+Theorem 2.2 ([7, Theorem 3.4.1] and its proof). Let 0 < α < 1 and Ω ⊂ Rn be a bounded
+domain with C3,α boundary. Suppose that H ∈ C1,α(Ω) satisfies
+(2.4)
+∥H∥Ln(Ω) <
+�ˆ
+Rn(1 + |p|2)− n+2
+2
+dp
+�1/n
+and, for any y ∈ ∂Ω,
+(2.5)
+|H(y)| ≤ H∂Ω(y),
+where H∂Ω is the mean curvature of ∂Ω corresponding to the inner unit normal vector to ∂Ω.
+Then for any g ∈ C3,α(Ω) there exists a unique solution u ∈ C3,α(Ω) to (2.3). In particular,
+there exists a constant C∗ > 0, depending only on n, α, ∥H∥Ln(Ω), ∥H∥C1(Ω), ∥g∥C2,α(Ω) and
+Ω, such that
+(2.6)
+∥u∥C2,α(Ω) ≤ C∗.
+The constant in the right-hand side of (2.4) can be simplified. Recall the definition of the
+Beta function and its relation with the Gamma function: for x, y > 0,
+B(x, y) :=
+ˆ ∞
+0
+tx−1
+(1 + t)x+y dt = Γ(x)Γ(y)
+Γ(x + y) .
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+5
+We have that Γ(1) = 1 and xΓ(x) = Γ(x + 1), for all x > 0. By passing to polar coordinates
+p = rθ, for r > 0 and θ ∈ Sn−1, performing the change of variables t = r2 which makes
+2dr/r = dt/t, and using that |Sn−1| = n|B1|, we get
+ˆ
+Rn(1 + |p|2)− n+2
+2 dp = |Sn−1|
+ˆ ∞
+0
+rn
+(1 + r2)
+n+2
+2
+dr
+r
+= n|B1|
+2
+ˆ ∞
+0
+tn/2
+(1 + t)
+n+2
+2
+dt
+t
+= n|B1|
+2
+B(n/2, 1) = n|B1|
+2
+Γ(n/2)
+Γ(n/2 + 1) = |B1|.
+Therefore, (2.4) reads
+(2.7)
+∥H∥Ln(Ω) < |B1|1/n.
+Now (2.5) and (2.7) impose further restrictions on the boundary values h of H. Condition
+(2.5) is natural to assume and cannot be avoided (see, for example, the nonexistence results [6,
+Theorem 16.11] and [7, Theorem 3.4.5]). Therefore, we assume that h ∈ C1,α(Ω) additionally
+satisfies
+(2.8)
+|h(y)| ≤ H∂Ω(y)
+for all y ∈ ∂Ω.
+On the other hand, by the maximum principle (see [6, Section 8.1]), we can estimate
+(2.9)
+ˆ
+Ω
+|H|n dx ≤ |Ω|
+�
+max
+∂Ω |h|
+�n
+.
+or
+∥H∥Ln(Ω) ≤ |Ω|1/n max
+∂Ω |h|.
+Thus, in order to ensure (2.7), we further assume that
+(2.10)
+max
+∂Ω |h| <
+�|B1|
+|Ω|
+�1/n
+.
+From the computer science point of view, this means that for large boundary curvatures h
+the domain Ω for the reconstruction should be sufficiently small.
+Therefore, under the additional assumptions (2.8) and (2.10), we can apply Theorem 2.2
+and find the unique solution u ∈ C3,α(Ω) to (2.3). This completes Step 2.
+Using these two steps, we define T : G → G by T(v) = u. In order to apply Theorem 2.1,
+we need to verify that
+(1) T is continuous, and
+(2) T(G) is precompact.
+Let us begin with (1). Fix v1 ∈ G. We need to show that given any ε > 0 there exists
+δ = δ(ε, v1) > 0 such that for any v2 ∈ G satisfying ∥v1 − v2∥C1,α(Ω) < δ we have ∥u1 −
+u2∥C1,α(Ω) < ε, where uj = Tvj, for j = 1, 2.
+Let Hj denote the minimizer of E[·, vj],
+j = 1, 2, as constructed in Step 1. Then the difference H = H1 − H2 is the unique weak
+solution to
+�
+div(D(v1)∇H) = div
+�
+(D(v2) − D(v1))∇H2
+�
+in Ω
+H = 0
+on ∂Ω.
+
+6
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+By global Schauder estimates (see [6, Section 8.11]),
+(2.11)
+∥H∥C1,α(Ω) ≤ Cn[∂Ω]C1,α∥D(v1)∥C0,α(Ω)∥(D(v2) − D(v1))∇H2∥C0,α(Ω)
+≤ C(n, α, Ω, v1, ∇H2)∥v1 − v2∥C1,α(Ω)
+=: C1∥v1 − v2∥C1,α(Ω).
+Let us now estimate the difference u = u1 − u2 ∈ C3,α(Ω). Since
+�
+�
+�
+div
+� ∇ui
+D(ui)
+�
+= nHi
+in Ω, for i = 1, 2
+u1 = u2 = g
+on ∂Ω
+we find that
+�
+�
+�
+div
+� ∇u1
+D(u1) − ∇u2
+D(u2)
+�
+= nH
+in Ω
+u = 0
+on ∂Ω.
+In order to apply global Schauder estimates one more time we need to find an equation for
+u. Set
+F(p) :=
+p
+�
+1 + |p|2
+p ∈ Rn.
+Then F is a smooth, bounded vector field with entries Fi(p) =
+pi
+√
+1+|p|2 , for i = 1, . . . , n. Note
+that, for j = 1, . . . , n,
+∂jFi(p) =
+�
+�
+�
+1
+√
+1+|p|2 −
+p2
+i
+(1+|p|2)3/2
+if i = j
+−
+pipj
+(1+|p|2)3/2
+if i ̸= j
+=
+δij
+D(p) − pipj
+D(p)3 .
+In particular,
+(2.12)
+∂jFi(p) = ∂iFj(p)
+so that ∇F is a symmetric matrix. It is clear that ∇F is bounded. To see that it is locally
+strictly elliptic, observe that, for any ξ ∈ Rn, by the Cauchy–Schwartz inequality,
+n
+�
+i,j=1
+∂jFi(p)ξiξj =
+n
+�
+i,j=1
+�δijξiξj
+D(p) − pipjξiξj
+D(p)3
+�
+≥ |ξ|2
+�
+1
+D(p) −
+|p|2
+D(p)3
+�
+=
+|ξ|2
+D(p)3 ≥ θ(R)|ξ|2
+for all |p| < R, where θ(R) → 0 as R → ∞. Furthermore, we can write
+Fi(∇u1) − Fi(∇u2) =
+ˆ 1
+0
+d
+dtFi(t∇u1 + (1 − t)∇u2) dt
+=
+ˆ 1
+0
+∇Fi(t∇u1 + (1 − t)∇u2) · ∇(u1 − u2) dt
+so that
+F(∇u1) − F(∇u2) = A(x)∇u
+with
+Aij(x) =
+ˆ 1
+0
+∂jFi(t∇u1 + (1 − t)∇u2) dt.
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+7
+The matrix A is symmetric thanks to (2.12), as well as bounded. Recall that ∇F is locally
+strictly elliptic. Now, u1 ∈ C3,α(Ω) is fixed. By (2.6), the C2,α(Ω) norm of u2 is uniformly
+controlled by the C1(Ω) norm of H2, which in turn is uniformly close to the C1(Ω) norm of
+the initially fixed H1. These facts imply that that A(x) is strictly elliptic. Moreover, we have
+the following technical lemma.
+Lemma 2.3. Let U, V : Ω → Rn, U, V ∈ C0,α(Ω) and let ψ : Rn → R be a smooth function
+such that
+∥ψ∥L∞(Rn) + ∥∇ψ∥L∞(Rn) < ∞.
+Define
+φ(x) :=
+ˆ 1
+0
+ψ(tU(x) + (1 − t)V (x)) dt
+for every x ∈ Ω.
+Then φ ∈ C0,α(Ω), with
+∥φ∥C0,α(Ω) ≤ ∥ψ∥L∞(Rn) + ∥∇ψ∥L∞(Rn)
+�
+[U]Cα(Ω) + [V ]Cα(Ω)
+�
+.
+Proof. The boundedness of ψ implies that φ is bounded with ∥φ∥L∞(Ω) ≤ ∥ψ∥L∞(Rn). To
+bound the H¨older seminorm of φ, let x, y ∈ Ω. Then
+|φ(x) − φ(y)| =
+����
+ˆ 1
+0
+�
+ψ(tU(x) + (1 − t)V (x)) − ψ(tU(y) + (1 − t)V (y))
+�
+dt
+����
+≤ ∥∇ψ∥L∞(Rn)
+ˆ 1
+0
+|t(U(x) − U(y)) + (1 − t)(V (x) − V (y))| dt
+≤ ∥∇ψ∥L∞(Rn) (|U(x) − U(y)| + |V (x) − V (y)|)
+≤ ∥∇ψ∥L∞(Rn)
+�
+[U]Cα(Ω) + [V ]Cα(Ω)
+�
+|x − y|α.
+□
+Lemma 2.3 gives the H¨older continuity of A(x). Indeed, let ψ be any of the entries of the
+gradient matrix of F:
+(∇F(p))ij =
+δij
+D(p) − pipj
+D(p)3
+for i, j = 1, . . . , n,
+which are smooth and bounded, so ∥ψ∥L∞(Rn) ≤ M1, where M1 is independent of i and j.
+For any k = 1, . . . , n, we have
+∂k(∇F(p))ij = −δijpk + δikpj + δjkpi
+D(p)3
++ pipjpk
+D(p)5
+and these are all bounded. Therefore, ∥∇ψ∥L∞(Rn) ≤ M2, where M2 > 0 is independent of i
+and j. By setting U = ∇u1 and V = ∇u2 in Lemma 2.3, we get
+∥A∥C0,α(Ω) ≤ M1 + M2
+�
+[∇u1]Cα(Ω) + ([∇u2]Cα(Ω)
+�
+≤ M3
+with M3 > 0 a constant depending only on n, α, ∥H1∥Ln(Ω), ∥H1∥C1(Ω), ∥g∥C2,α(Ω), and
+Ω, see (2.6). Observe that all these quantities are independent of u2 if v2 is close to v1 in
+C1,α(Ω). In summary, we have found that u is a solution to
+�
+div(A(x)∇u) = nH
+in Ω
+u = 0
+on ∂Ω
+and so, by Schauder estimates,
+(2.13)
+∥u∥C1,α(Ω) ≤ Cn[∂Ω]C1,αM3∥H∥C1,α(Ω) =: C2∥H∥C1,α(Ω).
+
+8
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+Therefore, by collecting estimates (2.11) and (2.13), and recalling that u = u1 − u2 = Tv1 −
+Tv2, and H = H1 − H2, we obtain
+∥Tv1 − Tv2∥C1,α(Ω) ≤ C1C2∥v1 − v2∥C1,α(Ω).
+If we choose δ = ε/(C1C2) then we see that T is continuous, as desired.
+Let us now turn to (2), which will follow from a priori estimates for prescribed mean
+curvature equations. Let {vk}k≥1 be a sequence in G such that
+sup
+k≥1
+∥vk∥C1,α(Ω) ≤ N1 < ∞
+and consider the corresponding solutions Hk ∈ C1,α(Ω) found in Step 1. Set uk = Tvk. By
+(2.6),
+∥uk∥C2,α(Ω) ≤ Ck
+where Ck > 0 is a constant depending only on n, α, ∥Hk∥Ln(Ω), ∥Hk∥C1(Ω), ∥h∥C2,α(Ω), and
+Ω. Since all Hk have the same boundary values h, by (2.9), we get that
+sup
+k≥1
+∥Hk∥Ln(Ω) = N2 < ∞.
+Furthermore, from the C1,α estimate in (2.2),
+sup
+k≥1
+∥Hk∥C1(Ω) ≤ Cn[∂Ω]C1,α∥h∥C1,α(∂Ω) sup
+k≥1
+∥D(vk)∥C0,α(Ω) = N3 < ∞.
+Consequently,
+sup
+k≥1
+∥uk∥C2,α(Ω) ≤ sup
+k≥1
+Ck = N4 < ∞.
+By the Arzel`a–Ascoli compact embedding C2,α(Ω) ⊂⊂ C1,α(Ω), there exist a subsequence
+{ukj}j≥1 of {uk}k≥1 and u ∈ G such that ukj → u in C1,α(Ω). We conclude that T(G) is
+precompact and (2) is proved.
+Thus, by Theorem 2.1, there exists u ∈ G such that Tu = u. We have proved the following:
+Theorem 2.4 (Existence for the simplified energy). Let Ω ⊂ Rn be a bounded domain with
+C3,α boundary ∂Ω, for some 0 < α < 1. Fix g ∈ C3,α(Ω). Let h ∈ C1,α(Ω) such that
+(2.14)
+|h(y)| ≤ H∂Ω(y)
+for all y ∈ ∂Ω,
+where H∂Ω is the mean curvature of ∂Ω corresponding to the inner unit normal vector to ∂Ω,
+and
+(2.15)
+max
+∂Ω |h| <
+�|B1|
+|Ω|
+�1/n
+.
+Then there exist u ∈ C3,α(Ω) and H ∈ C1,α(Ω) such that H minimizes the energy
+1
+2
+ˆ
+Ω
+|∇H|2D(u) dx
+among all H ∈ W 1,2(Ω) such that H − h ∈ W 1,2
+0 (Ω), or, equivalently, H is the unique weak
+solution to
+�
+div(D(u)∇H) = 0
+in Ω
+H = h
+on ∂Ω,
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+9
+and, in addition, H is the mean curvature of the graph of u with prescribed values on ∂Ω,
+that is,
+�
+�
+�
+1
+n div
+� ∇u
+D(u)
+�
+= H
+in Ω
+u = g
+on ∂Ω.
+Remark 2.5 (Nonexistence of solutions). The conditions imposed on the curvature at the
+boundary datum h in Theorem 2.4 come from restrictions already present when one seeks for
+solutions of the prescribed mean curvature equation. Indeed, the divergence form equation for
+H is uniformly elliptic when u is, say, Lipschitz continuous and therefore is always solvable.
+On the other hand, if condition (2.14) is not satisfied, that is,
+|h(y0)| > H∂Ω(y0)
+for some y0 ∈ ∂Ω
+and h ≥ 0 (or h ≤ 0) on ∂Ω then H ≥ 0 (or H ≤ 0) in Ω and we have that for any ε > 0
+there exists g ∈ C∞(Ω) with |g| < ε such that the prescribed mean curvature equation with
+curvature H and boundary values h is not solvable (see [7, Theorem 3.4.5] or [6, Corollary
+14.13]) and hence neither is the minimum curvature variation system.
+On the other hand, a necessary condition for existence of solutions of the prescribed mean
+curvature equation is
+(2.16)
+����
+ˆ
+Ω
+Hη dx
+���� ≤ 1 − ε0
+n
+ˆ
+Ω
+|∇η| dx
+for all η ∈ C1
+0(Ω) and with
+1 − ε0 = sup
+Ω
+|∇u|
+�
+1 + |∇u|2 ,
+see [6, eq. (16.60)]. This condition implies ∥H∥Ln(Ω) < |B1|1/n, which is the structural con-
+dition on H that motivates (2.15). The requirement in Theorem 2.4 could be thus weakened,
+but (2.16) is the least requirement under which existence for the prescribed mean curvature
+equation can be obtained and hence also for the system at hand.
+3. Existence of G2 surfaces for the geometric energy
+In this section we discuss how the technique we developed in the previous section can be
+applied to the geometric energy functional
+E[M] = 1
+2
+ˆ
+Ω
+�
+|∇H|2 −
+����
+∇u · ∇H
+D(u)
+����
+2�
+D(u) dx.
+Let Ω, α, h and g be as in Section 2. Fix v ∈ C1,α(Ω) such that v = g on ∂Ω. Consider the
+energy
+Ev[H] := 1
+2
+ˆ
+Ω
+�
+|∇H|2 −
+����
+∇v · ∇H
+D(v)
+����
+2�
+D(v) dx =
+ˆ
+Ω
+L(∇H) dx
+where the smooth Lagrangian L is given by
+L(p) = 1
+2
+�
+|p|2 −
+����
+∇v · p
+D(v)
+����
+2�
+D(v)
+for p ∈ Rn.
+Then L is coercive, as
+L(p) ≥ 1
+2
+�
+|p|2 − |∇v|2|p|2
+D(v)2
+�
+D(v) = 1
+2
+�
+D(v) − |∇v|2
+D(v)
+�
+|p|2 =
+1
+2D(v)|p|2.
+
+10
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+To prove that L is convex, first observe that, for i = 1, . . . , n,
+Lpi(p) =
+�
+pi − (∇v · p)
+D(v)2 vxi
+�
+D(v) =
+n
+�
+j=1
+�
+δijD(v) − vxivxj
+D(v)
+�
+pj
+and, for i, j = 1, . . . , n,
+Lpipj(p) = δijD(v) − vxivxj
+D(v) .
+Then, for any ξ ∈ Rn,
+Lpipj(p)ξiξj = D(v)|ξ|2 − (∇v · ξ)2
+D(v)
+≥
+�
+D(v) − |∇v|2
+D(v)
+�
+|ξ|2 =
+1
+D(v)|ξ|2.
+Thus, D2
+pL is a positive definite matrix, and L is uniformly convex. It follows that there
+exists a unique minimizer H ∈ W 1,2(Ω) of the energy Ev[H] such that H − h ∈ W 1,2
+0 (Ω). In
+particular, H is the unique weak solution to
+�
+�
+�
+�
+�
+n
+�
+i=1
+(Lpi(∇H))xi = 0
+in Ω
+H = h
+on ∂Ω.
+Since
+Lpi(∇H) =
+n
+�
+j=1
+�
+δijD(v) − vxivxj
+D(v)
+�
+Hxj
+we find that H is the unique weak solution to the linear problem
+�
+div(a(x)∇H) = 0
+in Ω
+H = h
+on ∂Ω
+where
+aij(x) = δijD(v) − vxivxj
+D(v) = Lpipj.
+Observe that
+|aij(x)| ≤ C
+�
+D(v) + |∇v|2
+D(v)
+�
+≤ C(D(v) + |∇v|) ≤ C(n, ∥∇v∥L∞(Ω)).
+We have already seen that aij(x) is uniformly elliptic. Moreover, if v ∈ C1,α(Ω) then aij(x) ∈
+C0,α(Ω). Hence, H ∈ C1,α(Ω), with
+∥H∥C1,α(Ω) ≤ Cn[∂Ω]C1,α∥v∥C1,α(Ω)∥h∥C1,α(∂Ω).
+If h satisfies (2.8) and (2.10) then we can apply Theorem 2.2 and find the unique solution
+u ∈ C3,α(Ω) to (2.3). From here on we can continue with the fixed point arguments we did
+in Section 2 to conclude the following result.
+Theorem 3.1 (Existence for the geometric functional). Let Ω ⊂ Rn be a bounded domain
+with C3,α boundary ∂Ω, for some 0 < α < 1. Fix g ∈ C3,α(Ω). Let h ∈ C1,α(Ω) such that
+|h(y)| ≤ H∂Ω(y)
+for all y ∈ ∂Ω,
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+11
+where H∂Ω is the mean curvature of ∂Ω corresponding to the inner unit normal vector to ∂Ω,
+and
+max
+∂Ω |h| <
+�|B1|
+|Ω|
+�1/n
+.
+Then there exist u ∈ C3,α(Ω) and H ∈ C1,α(Ω) such that H minimizes the energy
+1
+2
+ˆ
+Ω
+�
+|∇H|2 −
+����
+∇u · ∇H
+D(u)
+����
+2�
+D(u) dx
+among all H ∈ W 1,2(Ω) such that H − h ∈ W 1,2
+0 (Ω), or, equivalently, H is the unique weak
+solution to
+�
+div(a(x)∇H) = 0
+in Ω
+H = h
+on ∂Ω,
+where
+aij(x) = δijD(u) − uxiuxj
+D(u)
+and, in addition, H is the mean curvature of the graph of u with prescribed values on ∂Ω,
+that is,
+�
+�
+�
+1
+n div
+� ∇u
+D(u)
+�
+= H
+in Ω
+u = g
+on ∂Ω.
+4. Weak solutions
+In this section we develop the weak formulation of the minimum curvature variation prob-
+lem in the context of geometric measure theory.
+Given a Lipschitz bounded domain Ω, we denote by BV(Ω) the space of functions of
+bounded variation in Ω. We start by recalling that u ∈ BV(Ω) is a generalized solution
+to the prescribed mean curvature equation with (weak) mean curvature H ∈ L1(Ω) and
+boundary value g ∈ L1(∂Ω) if
+(WPMC)
+J [u] =
+min
+v∈BV(Ω) J [v]
+where
+J [v] :=
+ˆ
+Ω
+D(v) +
+ˆ
+Ω
+nHv dx +
+ˆ
+∂Ω
+|v − g| dS
+and
+(4.1)
+ˆ
+Ω
+D(v) := sup
+� ˆ
+Ω
+�
+v
+n
+�
+i=1
+∂xiφi + φn+1
+�
+dx : φi ∈ C1
+c (Ω),
+n+1
+�
+i=1
+φ2
+i ≤ 1
+�
+.
+Note that
+´
+Ω
+�
+1 + |∇u|2 dx does not make usual sense a priori for a function of bounded
+variation and so (4.1) is indeed a definition. Furthermore, this definition is consistent in the
+sense that for v ∈ W 1,1(Ω) we have
+ˆ
+Ω
+D(v) =
+ˆ
+Ω
+�
+1 + |∇v|2 dx,
+see the proof of Lemma 4.1.
+
+12
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+In [3, Theorem 1.1], Giaquinta proved that if H is a measurable function then (WPMC)
+is solvable in BV(Ω) if and only if there exists ε0 > 0 such that, for every measurable subset
+A ⊂ Ω,
+(4.2)
+����
+ˆ
+A
+H dx
+���� ≤ (1 − ε0) 1
+nP(∂A)
+where P(∂A) denotes the perimeter of A. Clearly, (4.2) is significant only when A is a set of
+finite perimeter (or Caccioppoli set).
+We need a generalized measure of surface area. In that regard, we recall that the dis-
+tributional gradient of u ∈ BV(Ω) is a vector valued Radon measure whose total variation
+is identified with |∇u|. This is again consistent in the sense that if u ∈ W 1,1(Ω) then the
+total variation equals
+´
+Ω |∇u| dx (see [2, Chapter 5] for this and other properties of the space
+BV(Ω) used hereafter). In general, for an open set U ⊂⊂ Ω the variation measure of ∇u
+over U is given by
+|∇u|(U) = sup
+�ˆ
+U
+u div φ dx : φ ∈ C1
+c (U; Rn), |φ| ≤ 1
+�
+and, for an arbitrary set V ⊂ Ω,
+|∇u|(V ) = inf
+�
+|∇u|(U) : V ⊂ U and U is open
+�
+.
+Taking this into account, and in analogy with (4.1), we define for the area measure by
+(4.3)
+D(u)(U) = sup
+� ˆ
+Ω
+�
+u
+n
+�
+i=1
+∂xiφi + φn+1
+�
+dx : φi ∈ C1
+c (U),
+n+1
+�
+i=1
+φ2
+i ≤ 1
+�
+for any U ⊂⊂ Ω open and, for an arbitrary set V ⊂ Ω,
+D(u)(V ) = inf {D(u)(U) : V ⊂ U and U is open} .
+Although (4.3) could be defined, in principle, for functions in L1(Ω), it is easy to check that
+(4.3) is finite if and only if u ∈ BV(Ω).
+Similarly as for the variation measure, D(u) is a Radon measure, namely, a locally finite,
+Borel regular measure in Rn (to prove that it is locally finite, see the ideas in [5, eq. (14.2)]).
+The following observation will be useful.
+Lemma 4.1. Let U ⊂ Ω be a Borel set. Then
+(4.4)
+|U| ≤ D(u)(U).
+Proof. Due to the Borel regularity of both D(u) and the Lebesugue measure it suffices to
+prove (4.4) for open sets. Let U ⊂ Rn be open.
+First, we note that
+(4.5)
+D(u)(U) =
+ˆ
+U
+�
+1 + |∇u|2 dx
+for any u ∈ C1(Ω).
+Indeed, an integration by parts yields
+ˆ
+Ω
+�
+u
+n
+�
+i=1
+∂xiφi + φn+1
+�
+dx =
+ˆ
+U
+(−∇u, 1) · Φ dx
+where Φ = (φ1, . . . , φn, φn+1). Then the Cauchy-Schwartz inequality in Rn+1 and the condi-
+tion |Φ| ≤ 1 give
+D(u)(U) ≤
+ˆ
+U
+�
+1 + |∇u|2 dx.
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+13
+On the other hand,
+�
+1 + |∇u|2 ∈ L1(U) and so there exists a sequence Φj = (φj
+1, . . . , φj
+n, φj
+n+1)
+with φj
+i ∈ C1
+c (U), j ≥ 1, that converges in L1(U) and almost everywhere to
+(−∇u,1)
+√
+1+|∇u|2 . Fur-
+thermore,
+(−∇u,1)
+√
+1+|∇u|2 is a unit vector so we may assume that �n+1
+i=1 (φj
+i)2 ≤ 1. Since
+��Φj · (−∇u, 1)
+�� ≤ |Φj|
+�
+1 + |∇u|2 ≤
+�
+1 + |∇u|2 ∈ L1(U)
+we can use the dominated convergence theorem to get
+lim
+j→∞
+ˆ
+Ω
+�
+u
+n
+�
+i=1
+∂xiφj
+i + φj
+n+1
+�
+dx = lim
+j→∞
+ˆ
+U
+(−∇u, 1) · Φj dx =
+ˆ
+U
+�
+1 + |∇u|2 dx
+and the supremum is achieved. Thus (4.5) holds.
+Second, we have that u ∈ BV(U) and there exists {uk}k≥1 ⊂ BV(U) ∩ C∞(U) such that
+uk → u in L1(Ω) and
+(4.6)
+lim
+k→∞ D(uk)(U) = D(u)(U),
+see [2, Theorem 5.3]. Since, by (4.5), the conclusion (4.4) is trivial for C1 functions, we have
+|U| ≤ lim
+k→∞ D(uk)(U) = D(u)(U)
+as desired.
+□
+From now on, we fix a bounded, C1,1 domain Ω. We consider the minimization problem
+min
+(u,H)∈A I[u, H]
+where
+(4.7)
+I[u, H] :=
+ˆ
+Ω
+|∇H|2 dD(u)
+and dD(u) stands for the area measure defined in (4.3). The admissible set A is defined as
+follows. Let h ∈ W 2,2(Ω) ∩ Lip(∂Ω) satisfying
+(4.8)
+|h(y)| ≤ n − 1
+n
+Λ(y), y ∈ ∂Ω,
+and
+max
+∂Ω |h| ≤ (1 − ε0)
+�|B1|
+|Ω|
+�1/n
+,
+where Λ(y) is the weak mean curvature of ∂Ω at y ∈ ∂Ω and
+(4.9)
+n − 1
+n
+< ε0 < 1.
+Define
+(4.10)
+A :=
+� (u, H) ∈ BV(Ω) × (Ln(Ω) ∩ W 2,2(Ω)) : u solves (WPMC)
+and ∥H∥Ln(Ω) + ∥H∥W 2,2(Ω) ≤ C0, H = h on ∂Ω
+�
+with C0 > 0 is to be appropriately chosen. The equality H = h is understood in the sense of
+traces.
+Remark 4.2. The condition H ∈ W 2,2(Ω) is certainly natural for applications to the design
+of fair G2-continuous surfaces in CAD/CAM/CAGD. Indeed, in dimensions n = 1, 2, 3, the
+Sobolev embedding gives that the curvature H is H¨older continuous.
+The main result of this section is the following:
+Theorem 4.3 (Existence of weak solutions). Let I be defined by (4.7), h ∈ W 2,2(Ω)∩Lip(∂Ω)
+satisfying (4.8) and ε0 ∈ (0, 1) satisfying (4.9). Then the set of admissible functions A in
+(4.10) is nonempty and there exists a minimizer of I within the class A.
+
+14
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+To prove Theorem 4.3 we recall the notion and properties of Γ−convergence in our context,
+referring the reader to [1] for an introduction to the topic.
+Let Jk, k ≥ 1, and J∞ be
+functionals defined on the common space BV(Ω) and taking values in [−∞, ∞]. The sequence
+{Jk}k≥1 is said to Γ−converge to J∞ if the following two conditions hold:
+(a) For every v ∈ BV(Ω) and every sequence {vk}k≥1 ⊂ BV(Ω) such that vk → v in BV(Ω)
+it holds
+lim inf
+k→∞ Jk(vk) ≥ J∞(v).
+(b) For every v ∈ BV(Ω) there exists a sequence {vk}k≥1 ⊂ BV(Ω) such that vk → v in
+BV(Ω) for which
+lim sup
+k→∞
+Jk(vk) ≤ J∞(v).
+We will use the following result.
+Theorem 4.4 (see [1, Theorem 1.21]). Let (X, d) be a metric space and let {fk}k≥1 be an
+equi-mildly coercive sequence of functions on X that Γ−converges to f∞. Then, there exits
+min
+X f∞ = lim
+k→∞ inf
+X fk.
+Moreover, if {xk}k≥1 ⊂ X is a precompact sequence such that
+lim
+k→∞ fk(xk) = lim
+k→∞ inf
+X fk
+then every limit of {xk}k≥1 is a minimum point for f∞.
+Here f is said to be mildly coercive if there exists a nonempty compact set K ⊂ X such
+that infK f = infX f, and equi-mild coercivity means that the set K is the same for the whole
+sequence {fk}k≥1.
+Proof of Theorem 4.3. The proof is divided into 4 steps.
+Step 1. A ̸= ∅. We can extend h to Ω by solving
+�
+∆H = 0
+in Ω
+H = h
+on ∂Ω.
+By classical elliptic regularity, H ∈ W 2,2(Ω) and
+∥H∥W 2,2(Ω) ≤ C0
+where C0 = C0(∂Ω, ∥h∥L∞(∂Ω)) > 0. Moreover, by the H¨older and isoperimetric inequalities,
+����
+ˆ
+A
+H dx
+���� ≤ ∥H∥Ln(Ω)|A|
+n−1
+n
+≤ ∥H∥Ln(Ω)
+P(∂A)
+n|B1|1/n .
+By the maximum principle and (4.8) we have
+∥H∥Ln(Ω) ≤ |Ω|1/n max
+∂Ω |h| ≤ (1 − ε0)|B1|1/n,
+where we make C0 larger if needed. Therefore,
+����
+ˆ
+A
+H dx
+���� ≤ (1 − ε0)
+n
+P(∂A)
+and (WPMC) is solvable for this H. Let u ∈ BV(Ω) be the corresponding minimizer of J .
+We have that A ̸= ∅. We further point out that
+´
+Ω D(u) < ∞ and H ∈ Lip(Ω) so that
+ˆ
+Ω
+|∇H|2 dD(u) ≤ ∥∇H∥2
+L∞(Ω)D(u)(Ω) < ∞.
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+15
+In particular,
+0 ≤
+inf
+(u,H)∈A I[u, H] < ∞.
+Step 2. Construction of a minimizer. Let {(uk, Hk)}k≥1 ⊂ A be a minimizing sequence:
+m :=
+inf
+(u,H)∈A I[u, H] = lim
+k→∞ I[uk, Hk].
+To get a convergent subsequence of {uk}k≥1 we show its uniform boundedness in BV(Ω) and
+use that BV(Ω) embedds compactly in L1(Ω). Since every uk is a minimizer of the functional
+Jk defined by
+(4.11)
+Jk[v] :=
+ˆ
+Ω
+D(v) +
+ˆ
+Ω
+nHkv dx +
+ˆ
+∂Ω
+|v − g| dS
+we have that, for any u0 ∈ BV(Ω),
+ˆ
+Ω
+D(uk) +
+ˆ
+Ω
+nHkuk dx +
+ˆ
+∂Ω
+|uk − g| dS ≤
+ˆ
+Ω
+D(u0) +
+ˆ
+Ω
+nHku0 dx +
+ˆ
+∂Ω
+|u0 − g| dS
+from where
+(4.12)
+ˆ
+Ω
+D(uk) +
+ˆ
+Ω
+nHkuk dx ≤ C +
+ˆ
+Ω
+nHku0 dx
+for C > 0 independent of k. Reasoning as in [3, eq. (1.4)] we have that
+(4.13)
+ˆ
+Ω
+Hkuk dx ≥ −(1 − ε0)
+ˆ
+Ω
+|∇uk| − C.
+Furthermore, BV(Ω) ⊂ L
+n
+n−1 (Ω) so (4.13) and the H¨older inequality in (4.12) give
+ˆ
+Ω
+D(uk) ≤ −n
+ˆ
+Ω
+Hkuk dx + C +
+ˆ
+Ω
+nHku0 dx
+≤ n(1 − ε0)
+ˆ
+Ω
+|∇uk| + n∥Hk∥Ln(Ω)∥u0∥L
+n
+n−1 (Ω) + C
+for a new constant C > 0 that is independent of k. Moreover, the uniform bound on the
+Ln(Ω) norm of {Hk}k≥1 (they all belong to A) gives
+ˆ
+Ω
+|∇uk| ≤ n(1 − ε0)
+ˆ
+Ω
+|∇uk| + nC0∥u0∥L
+n
+n−1 (Ω) + C.
+Thus, after rearranging terms and recalling (4.9),
+ˆ
+Ω
+|∇uk| ≤
+1
+(1 − n(1 − ε0))
+�
+nC0∥u0∥L
+n
+n−1 (Ω) + C
+�
+.
+Hence, by compactness in BV(Ω), there exists a subsequence of {uk}k≥1, still denoted by the
+same indexes, and u∞ ∈ BV(Ω) such that
+uk → u∞ in L1(Ω) as k → ∞, and
+|∇u∞|(Ω) ≤ lim inf
+k→∞ |∇uk|.
+Note that we also have
+(4.14)
+D(u∞) ≤ lim inf
+k→∞ D(uk).
+By Poincar´e’s inequality and the Rellich–Kondrachov compactness theorem, there exist a
+subsequence of {Hk}k≥1, still denoted by the same indexes, and H∞ ∈ W 2,2(Ω) such that
+(4.15)
+∇Hk → ∇H∞ in L2(Ω), as k → ∞.
+
+16
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+Further, due to the uniform bound on ∥Hk∥Ln(Ω), we may assume that Hk converges weakly
+in Ln(Ω) to H∞. Finally, the weak convergence ensures that
+∥H∞∥Ln(Ω) + ∥H∞∥W 2,2(Ω) ≤ C0.
+Step 3. (u∞, H∞) ∈ A. For this step we use Γ−convergence. Recall the functionals Jk
+defined in (4.11) (for the subsequence Hk we found in Step 2) and define J∞ analogously.
+We want to show that u∞ is a solution of (WPMC), namely, that u∞ is a minimizer of J∞
+over BV(Ω). Let us show that {Jk}k≥1 Γ−converges to J∞. A first remark is that it is
+enough to prove the Γ−convergence of
+�
+Jk(v) :=
+ˆ
+Ω
+vHk dx
+to
+�
+J∞(v) :=
+ˆ
+Ω
+vH∞ dx
+since the other two terms do not depend on k and can be considered as continuous pertur-
+bations of Jk, see [1, Remark 1.7]. To prove the liminf inequality (a), let {vk}k≥1 ⊂ BV(Ω)
+and v ∈ BV(Ω) such that vk → v in BV(Ω). We write
+ˆ
+Ω
+vkHk dx −
+ˆ
+Ω
+vH∞ dx = Ik + IIk + IIIk
+with
+Ik =
+ˆ
+Ω
+(vk − v)H∞ dx
+IIk =
+ˆ
+Ω
+(vk − v)(Hk − H∞) dx
+IIIk =
+ˆ
+Ω
+v(Hk − H∞) dx.
+By lower semicontinuity [4, Proposition 2.1],
+lim inf
+k→∞ Ik ≥ 0
+Next, we bound
+|IIk| ≤ ∥vk − v∥L
+n
+n−1 (Ω)
+�
+∥Hk∥Ln(Ω) + ∥H∞∥Ln(Ω)
+�
+.
+Since vk converge to v in BV(Ω), by the isoperimetric embedding, the convergence also holds
+in L
+n
+n−1 (Ω). This and the uniform bound of Hk in Ln(Ω) give
+lim
+k→∞ IIk = 0.
+Finally, limk→∞ IIIk = 0 by the weak convergence of Hk to H∞ in Ln(Ω). As for the limsup
+inequality (b), given any v ∈ BV(Ω), consider the constant sequence vk = v for all k ≥ 1 and
+notice that, using the weak convergence of Hk to H∞ in Ln(Ω), we have that
+lim
+k→∞
+�
+Jk(vk) = �
+J∞(v).
+Hence, {Jk}k≥1 converges to J∞ in the Γ sense. Therefore, we can apply Theorem 4.4 with
+X = BV(Ω), fk = Jk, f∞ = J∞ and {xk}k≥1 and x∞ given by {uk}k≥1 and u∞, respectively
+(note that the sequence {Jk}k is equi-mildly coercive), to conclude that u∞ is a minimizer
+of J∞. We have thus shown that (u∞, H∞) ∈ A.
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+17
+Step 4. (u∞, H∞) is a minimizer. Recall that L(p) = 1
+2|p|2, p ∈ Rn, is convex, that is,
+1
+2|p|2 ≥ 1
+2|p0|2 + p0 · (p − p0)
+for every p, p0 ∈ Rn. Then we can write
+1
+2
+ˆ
+Ω
+|∇Hk|2 dD(uk) ≥ 1
+2
+ˆ
+Ω
+|∇H∞|2 dD(uk)
++
+ˆ
+Ω
+∇H∞ · (∇Hk − ∇H∞) dD(uk).
+As k → ∞, the left hand side of this inequality converges to m. As for the right hand side,
+(4.6) implies that
+lim inf
+k→∞
+1
+2
+ˆ
+Ω
+|∇H∞|2 dD(uk) ≥ 1
+2
+ˆ
+Ω
+|∇H∞|2 dD(u∞).
+It remains to analyze the second term on the right hand side. For this, notice that Lemma
+4.1 implies that dD(uk) is absolutely continuous with respect to the Lebesgue measure, see
+(4.4). This and H¨older’s inequality give
+����
+ˆ
+Ω
+∇H∞ · (∇Hk − ∇H∞) dD(uk)
+���� ≤
+ˆ
+Ω
+|∇H∞||∇Hk − ∇H∞| dD(uk)
+≤ C
+ˆ
+Ω
+|∇H∞||∇Hk − ∇H∞| dx
+≤ C∥∇H∞∥L2(Ω)∥∇Hk − ∇H∞∥L2(Ω).
+In view of (4.15), this term goes to 0 as k → ∞. We have shown that
+m ≥
+ˆ
+Ω
+|∇H∞|2 dD(u∞).
+Since (u∞, H∞) ∈ A equality must be attained and (u∞, H∞) is a minimizer, as desired.
+□
+References
+[1] A. Braides, Γ-convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications 22,
+Oxford University Press, Oxford, 2002.
+[2] L. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition, Text-
+books in Mathematics, CRC Press, Boca Raton, FL, 2015.
+[3] M. Giaquinta, On the Dirichlet problem for surfaces of prescribed mean curvature, Manuscripta Math.
+12 (1974), 73–86.
+[4] E. Giusti, Boundary value problems for non-parametric surfaces of prescribed mean curvature, Ann.
+Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), 501–548.
+[5] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics 80,
+Birkh¨auser Verlag, Basel, 1984.
+[6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in
+Mathematics, Springer-Verlag, Berlin, 2001.
+[7] Q. Han, Nonlinear Elliptic Equations of the Second Order, Graduate Studies in Mathematics 171, Amer-
+ican Mathematical Society, Providence, RI, 2016.
+[8] H. P. Moreton and C. H. S´equin Functional optimization for fair surface design, ACM SIGGRAPH
+Computer Graphics 2 (1992), 167-176.
+[9] K. L. Narayan, K. M. Rao and M. M. M. Sacar, Computer Aided Design and Manufacturing, PHI Learning
+Pvt. Ltd., 2008.
+[10] W. Welch and A. Witkin, Variational surface modeling, ACM SIGGRAPH computer graphics 2 (1992),
+157-166.
+
+18
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+[11] G. Xu and Q. Zhang, Minimal mean-curvature-variation surfaces and their applications in surface mod-
+eling, in: International Conference on Geometric Modeling and Processing, 357–370, Springer, Berlin,
+Heidelberg, 2006.
+Department of Mathematics, The University of Texas at Austin, 2515 Speedway, Austin, TX
+78712, United States of America
+Email address: [email protected]
+Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA 50011, United
+States of America
+Email address: [email protected]
+Centro Marplatense de Investigaciones Matem´aticas/CONICET, Dean Funes 3350, 7600 Mar
+del Plata, Argentina
+Email address: [email protected]
+

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+arXiv:2301.00581v1  [cs.IT]  2 Jan 2023
+1
+Bent Partitions, Vectorial Dual-Bent Functions and Partial Difference Sets†
+Jiaxin Wang, Fang-Wei Fu, Yadi Wei
+Abstract
+Bent partitions of V (p)
+n
+are quite powerful in constructing bent functions, vectorial bent functions and
+generalized bent functions, where V (p)
+n
+is an n-dimensional vector space over Fp, n is an even positive
+integer and p is a prime. It is known that partial spreads is a class of bent partitions. In [4], [18], two
+classes of bent partitions whose forms are similar to partial spreads were presented. In [3], more bent
+partitions Γ1, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2 were presented from (pre)semifields, including the bent partitions given
+in [4], [18]. In this paper, we investigate the relations between bent partitions and vectorial dual-bent
+functions. For any prime p, we show that one can generate certain bent partitions (called bent partitions
+satisfying Condition C) from certain vectorial dual-bent functions (called vectorial dual-bent functions
+satisfying Condition A). In particular, when p is an odd prime, we show that bent partitions satisfying
+Condition C one-to-one correspond to vectorial dual-bent functions satisfying Condition A. We give an
+alternative proof that Γ1, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2 are bent partitions in terms of vectorial dual-bent functions.
+We present a secondary construction of vectorial dual-bent functions, which can be used to generate
+more bent partitions. We show that any ternary weakly regular bent function f : V (3)
+n
+→ F3 (n even) of
+2-form can generate a bent partition. When such f is weakly regular but not regular, the generated bent
+partition by f is not coming from a normal bent partition, which answers an open problem proposed in
+[4]. We give a sufficient condition on constructing partial difference sets from bent partitions, and when
+p is an odd prime, we provide a characterization of bent partitions satisfying Condition C in terms of
+partial difference sets.
+Index Terms
+Bent partitions; bent functions; vectorial bent functions; vectorial dual-bent functions; semifields;
+partial difference sets
+Jiaxin Wang, Fang-Wei Fu and Yadi Wei are with Chern Institute of Mathematics and LPMC, Nankai University, Tianjin
+300071, China, Emails: [email protected], [email protected], [email protected].
+†This research is supported by the National Key Research and Development Program of China (Grant Nos. 2018YFA0704703
+and 2022YFA1005001), the National Natural Science Foundation of China (Grant Nos. 12141108, 61971243, 12226336), the
+Natural Science Foundation of Tianjin (20JCZDJC00610), the Fundamental Research Funds for the Central Universities of China
+(Nankai University), and the Nankai Zhide Foundation.
+January 3, 2023
+DRAFT
+
+2
+I. INTRODUCTION
+Boolean bent functions were introduced by Rothaus [21] and were generalized to p-ary bent
+functions by Kumar, Scholtz and Welch [15], where p is an arbitrary prime. Due to applications
+of p-ary bent functions in cryptography, coding theory, sequence and combinatorics, they have
+been extensively studied. We refer to surveys [5], [17] and a book [19] on p-ary bent functions
+and their generalizations such as vectorial bent functions and generalized bent functions.
+In [10], C¸ es¸melio˘glu et al. introduced vectorial dual-bent functions, which is a special class
+of vectorial bent functions. In [7], [8], [22], vectorial dual-bent functions were used to construct
+partial difference sets. In particular, Wang and Fu [22] showed that for certain vectorial dual-bent
+functions F : V (p)
+n
+→ V (p)
+s
+(where V (p)
+n
+is an n-dimensional vector space over the prime field
+Fp), the preimage set of any subset of V (p)
+s
+for F forms a partial difference set.
+Very recently, bent partitions of V (p)
+n
+were introduced [4], [18], which are quite powerful in
+constructing bent functions, vectorial bent functions and generalized bent functions. The well-
+known partial spreads is a class of bent partitions. In [18], Meidl and Pirsic for the first time
+presented two classes of bent partitions for p = 2 different from partial spreads. In [4], Anbar and
+Meidl generalized the contributions in [18] to the case of p being odd and gave the corresponding
+two classes of bent partitions for odd p. In [3], Anbar, Kalaycı and Meidl presented more bent
+partitions Γ1, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2 from (pre)semifields, including the bent partitions given in [4],
+[18]. In [2], Anbar, Kalaycı and Meidl showed that any union of elements in the bent partition
+given in [4], [18] forms a partial difference set. In terms of constructing partial difference sets,
+certain vectorial dual-bent functions and certain bent partitions seem to play the same role.
+Therefore, it is interesting to investigate the relations between vectorial dual-bent functions
+and bent partitions. In this paper, we show that by using certain vectorial dual-bent functions
+(called vectorial dual-bent functions satisfying Condition A), we can construct bent partitions
+of V (p)
+n
+with certain properties (called bent partitions satisfying Condition C) for any prime
+p. Particularly, when p is an odd prime, we prove that bent partitions of V (p)
+n
+with Condition
+C one-to-one correspond to vectorial dual-bent functions satisfying Condition A. In terms of
+vectorial dual-bent functions, we provide an alternative proof that Γ1, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2 given
+in [3] are bent partitions. We provide a secondary construction of vectorial dual-bent functions,
+which can be used to generate more bent partitions. We prove that any ternary weakly regular
+bent function f : V (3)
+n
+→ F3 (n even) of 2-form can generate a bent partition. In the special case
+January 3, 2023
+DRAFT
+
+3
+that f is weakly regular but not regular, the generated bent partition by f is not coming from
+a normal bent partition, which answers an open problem proposed in [4]. By using vectorial
+dual-bent functions as the link between bent partitions and partial difference sets, we give a
+sufficient condition on constructing partial difference sets from bent partitions. When p is an
+odd prime, we provide a characterization of bent partitions satisfying Condition C in terms of
+partial difference sets.
+The rest of the paper is organized as follows. In Section II, we state some needed results on
+vectorial dual-bent functions and bent partitions. In Section III, we present relations between
+certain bent partitions and certain vectorial dual-bent functions. In Section IV, we give a sec-
+ondary construction of vectorial dual-bent functions, which can be used to generate more bent
+partitions. In Section V, we present relations between certain bent partitions and certain partial
+difference sets. In Section VI, we make a conclusion.
+II. PRELIMINARIES
+In this section, we state some basic results on vectorial dual-bent functions and bent partitions.
+First, we fix some notations used throughout this paper.
+• p is a prime.
+• ζp = e
+2π√−1
+p
+is a complex primitive p-th root of unity. Note that ζ2 = −1.
+• Fpn is the finite field with pn elements.
+• Fn
+p is the vector space of the n-tuples over Fp.
+• V (p)
+n
+is an n-dimensional vector space over Fp.
+• ⟨·⟩n denotes a (non-degenerate) inner product of V (p)
+n . In this paper, when V (p)
+n
+= Fpn,
+let ⟨a, b⟩n = Trn
+1(ab), where a, b ∈ Fpn, Trn
+k(·) denotes the trace function from Fpn to
+Fpk, k | n; when V (p)
+n
+= Fn
+p, let ⟨a, b⟩n = a · b = �n
+i=1 aibi, where a = (a1, . . . , an), b =
+(b1, . . . , bn) ∈ Fn
+p; when V (p)
+n
+= V (p)
+n1 ×· · ·×V (p)
+nm (n = �m
+i=1 ni), let ⟨a, b⟩n = �m
+i=1⟨ai, bi⟩ni,
+where a = (a1, . . . , am), b = (b1, . . . , bm) ∈ V (p)
+n .
+• For any set A ⊆ V (p)
+n
+and u ∈ V (p)
+n , let χu(A) = �
+x∈A χu(x), where χu denotes the
+character χu(x) = ζ⟨u,x⟩n
+p
+.
+A. Vectorial dual-bent functions
+A function F : V (p)
+n
+→ V (p)
+s
+is called a vectorial p-ary function, or simply p-ary function
+when s = 1. The Walsh transform of a p-ary function f : V (p)
+n
+→ Fp is the complex valued
+January 3, 2023
+DRAFT
+
+4
+function defined by
+Wf(a) =
+�
+x∈V (p)
+n
+ζf(x)−⟨a,x⟩n
+p
+, a ∈ V (p)
+n .
+(1)
+A p-ary function f : V (p)
+n
+→ Fp is called bent if |Wf(a)| = p
+n
+2 for all a ∈ V (p)
+n . Note that
+when f is a Boolean bent function, that is, p = 2, then n must be even since in this case, Wf is
+an integer valued function. A vectorial p-ary function F : V (p)
+n
+→ V (p)
+s
+is called vectorial bent
+if all component functions Fc : V (p)
+n
+→ Fp, c ∈ V (p)
+s
+\{0} defined as Fc(x) = ⟨c, F(x)⟩s are bent.
+It is known that if F : V (p)
+n
+→ V (p)
+s
+is vectorial bent, then s ≤ n
+2 if p = 2, and s ≤ n if p is
+an odd prime. If f : V (p)
+n
+→ Fp is bent, then so are cf, c ∈ F∗
+p, that is, any p-ary bent function
+is vectorial bent. For F : V (p)
+n
+→ V (p)
+s
+, the vectorial bentness of F is independent of the inner
+products of V (p)
+n
+and V (p)
+s
+. The Walsh transform of a p-ary bent function f : V (p)
+n
+→ Fp satisfies
+that for any a ∈ V (p)
+n , when p = 2, we have
+Wf(a) = 2
+n
+2 (−1)f∗(a),
+(2)
+and when p is an odd prime, we have
+Wf(a) =
+
+
+
+±p
+n
+2 ζf∗(a)
+p
+if p ≡ 1 (mod 4) or n is even,
+±
+√
+−1p
+n
+2 ζf∗(a)
+p
+if p ≡ 3 (mod 4) and n is odd,
+(3)
+where f ∗ is a function from V (p)
+n
+to Fp, called the dual of f. A p-ary bent function f : V (p)
+n
+→ Fp
+is called weakly regular if Wf(a) = εfp
+n
+2 ζf∗(a)
+p
+, where εf is a constant independent of a,
+otherwise f is called non-weakly regular. In particular, if εf = 1, f is called regular. The (non-
+)weakly regularity of f is independent of the inner product of V (p)
+n
+and if f is weakly regular,
+εf is independent of the inner product of V (p)
+n . By (2), all Boolean bent functions are regular.
+If f is a p-ary weakly regular bent function, then the dual f ∗ of f is also weakly regular bent
+with (f ∗)∗(x) = f(��x) (see [9]).
+In 2018, C¸ es¸melio˘glu et al. [10] introduced vectorial dual-bent functions.
+Definition 1. A vectorial p-ary bent function F : V (p)
+n
+→ V (p)
+s
+is called vectorial dual-bent
+if there exists a vectorial bent function G : V (p)
+n
+→ V (p)
+s
+such that (Fc)∗ = Gσ(c) for any
+c ∈ V (p)
+s
+\{0}, where (Fc)∗ is the dual of the component function ⟨c, F(x)⟩s and σ is some
+permutation over V (p)
+s
+\{0}. The vectorial bent function G is called a vectorial dual of F and
+denoted by F ∗.
+January 3, 2023
+DRAFT
+
+5
+It is known in [10] that the property of being vectorial dual-bent is independent of the inner
+products of V (p)
+n
+and V (p)
+s
+. Note that for a vectorial dual-bent function, its vectorial dual is not
+unique since being vectorial bent and vectorial dual-bent for a function is a property of the
+vector space consisting of all component functions (see Remark 1 of [10]). For example, if a
+p-ary function f (seen as a vectorial function into V (p)
+1
+, p odd) is vectorial dual-bent under any
+fixed inner product, then its dual f ∗ is unique, but its vectorial dual is not unique since for any
+c ∈ F∗
+p, cf ∗ is a vectorial dual of f. A p-ary function f : V (p)
+n
+→ Fp is called an l-form if
+f(ax) = alf(x) for any a ∈ F∗
+p and x ∈ V (p)
+n , where 1 ≤ l ≤ p − 1 is an integer. By the results
+in [7], [22], we have the following proposition.
+Proposition 1 ( [7], [22]). A p-ary function f with f(0) = 0 is a weakly regular vectorial dual-
+bent function if and only if f is a weakly regular bent function of l-form with gcd(l−1, p−1) = 1.
+In particular, a p-ary function f is a weakly regular vectorial dual-bent function with (cf)∗ = cf ∗
+for any c ∈ F∗
+p if and only if f is a weakly regular bent function of (p − 1)-form.
+In the rest of this subsection, we recall an important class of p-ary bent functions, called
+Maiorana-McFarland bent functions.
+• Let f : Fpn × Fpn → Fp be defined as
+f(x, y) = Trn
+1(αxπ(y)) + g(y),
+where α ∈ F∗
+pn, π is a permutation over Fpn and g : Fpn → Fp is an arbitrary function.
+Then f is bent and is called a Maiorana-McFarland bent function. The dual f ∗ of f is
+f ∗(x, y) = Trn
+1(−π−1(α−1x)y) + g(π−1(α−1x)).
+(4)
+All Maiorana-McFarland bent functions are regular (see [15]).
+B. Bent partitions
+Very recently, the concept of bent partitions of V (p)
+n
+were introduced [4], [18].
+Definition 2. Let n be an even positive integer, K be a positive integer divisible by p.
+• Let Γ = {A1, . . . , AK} be a partition of V (p)
+n . Assume that every function f for which every
+i ∈ Fp has exactly K
+p of sets Aj in Γ in its preimage, is a p-ary bent function. Then Γ is
+called a bent partition of V (p)
+n
+of depth K and every such bent function f is called a bent
+function constructed from bent partition Γ.
+January 3, 2023
+DRAFT
+
+6
+• Let Γ = {U, A1, . . . , AK} be a partition of V (p)
+n . Assume that every function f with the
+following properties is bent:
+(1) Every c ∈ Fp has exactly K
+p of the sets A1, . . . , AK in its preimage set;
+(2) f(x) = c0 for all x ∈ U and some fixed c0 ∈ Fp.
+Then we call Γ a normal bent partition of V (p)
+n
+of depth K.
+Bent partitions are very powerful in constructing bent functions, vectorial bent function and
+generalized bent functions. In this paper, we focus on the relations between bent partitions and
+vectorial bent functions.
+Proposition 2 ( [4]). Let Γ = {A1, . . . , Aps} be a bent partition of V (p)
+n . Then every function
+F : V (p)
+n
+→ V (p)
+s
+such that every element i ∈ V (p)
+s
+has the elements of exactly one of the sets
+Aj, 1 ≤ j ≤ ps, in its preimage, is a vectorial bent function.
+It is known that partial spreads is a class of bent partitions (for instance see Section 2 of [4]). In
+[4], [18], two classes of explicit bent partitions different from partial spreads were presented. In
+[3], bent partitions Γ1, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2 were presented from certain (pre)semifields, including
+the bent partitions given in [4], [18]. We will recall bent partitions Γ1, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2 given
+in [3]. First, we need to introduce some basic knowledge on (pre)semifields.
+Definition 3. Let ◦ be a binary operation defined on (V (p)
+n , +) such that
+(i) x ◦ y = 0 implies x = 0 or y = 0,
+(ii) (x + y) ◦ z = (x ◦ z) + (y ◦ z), (z ◦ (x + y) = (z ◦ x) + (z ◦ y), respectively),
+for all x, y, z ∈ V (p)
+n . Then (V (p)
+n , +, ◦) is called a right (left, respectively) prequasifield. If
+(V (p)
+n , +, ◦) is a right and a left prequasifield, then it is called a presemifield. If (V (p)
+n , +, ◦) is
+a presemifield for which there is an element e ̸= 0 such that e ◦ x = x ◦ e = x for all x ∈ V (p)
+n ,
+then it is called a semifield.
+Let P = (Fpn, +, ◦) be a presemifield. Then one can obtain presemifields P • = (Fpn, +, •)
+and P ⋆ = (Fpn, +, ⋆) from P, where • and ⋆ are given by
+x • y = y ◦ x for all x, y ∈ Fpn,
+and
+Trn
+1(z(x ◦ y)) = Trn
+1(x(z ⋆ y)) for all x, y, z ∈ Fpn,
+January 3, 2023
+DRAFT
+
+7
+respectively. The presemifield P ⋆ is called the dual of P. Let s be a positive divisor of n.
+If x ◦ (cy) = c(x ◦ y) holds for any x, y ∈ Fpn, c ∈ Fps, then P is called right Fps-linear.
+Each presemifield P = (Fpn, +, ◦) can induce a semifield P ′ = (Fpn, +, ∗) via the following
+transformation: choose any α ∈ F∗
+pn and give ∗ by
+(x ◦ α) ∗ (α ◦ y) = x ◦ y for all x, y ∈ Fpn.
+By Lemma 2 of [3], if P is right Fps-linear, then P ′ is also right Fps-linear. The finite field Fpn
+is a right Fps-linear semifield (that is, ◦ is the field multiplication). For more right Fps-linear
+(pre)semifields, see Section 3 of [3].
+Now we recall bent partitions Γ1, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2 given in [3].
+• Let n, s be positive integers satisfying s | n and gcd(pn−1, ps+p−1) = 1. Set u = ps+p−1,
+and let d be an integer with du ≡ 1 mod (pn − 1). Let P = (Fpn, +, ◦) be a (pre)semifield
+such that its dual P ⋆ = (Fpn, +, ⋆) is right Fps-linear. For given x ∈ Fpn, if x = 0, then let
+ηx = 0, and if x ̸= 0, then let ηx be given by x ⋆ η−1
+x
+= 1.
+• Define
+Ut = {(x, t ◦ xu) : x ∈ F∗
+pn} if t ∈ Fpn, and U = {(0, y) : y ∈ Fpn}.
+Let i0 ∈ Fps be an arbitrary element. Define
+Γ1 = {Ai, i ∈ Fps},
+(5)
+where Ai = �
+t∈Fpn:Trn
+s (t)=i Ut if i ̸= i0, Ai0 = �
+t∈Fpn:Trn
+s (t)=i0 Ut
+� U.
+• Define
+U•
+t = {(x, xu ◦ t) : x ∈ F∗
+pn} if t ∈ Fpn, and U = {(0, y) : y ∈ Fpn}.
+Let i0 ∈ Fps be an arbitrary element. Define
+Γ•
+1 = {A•
+i , i ∈ Fps},
+(6)
+where A•
+i = �
+t∈Fpn:Trn
+s (t)=i U•
+t if i ̸= i0, A•
+i0 = �
+t∈Fpn:Trn
+s (t)=i0 U•
+t
+� U.
+• Define
+Vt = {(t ◦ xd, x) : x ∈ F∗
+pn} if t ∈ Fpn, and V = {(x, 0) : x ∈ Fpn}.
+Let i0 ∈ Fps be an arbitrary element. Define
+Γ2 = {Bi, i ∈ Fps},
+(7)
+January 3, 2023
+DRAFT
+
+8
+where Bi = �
+t∈Fpn:Trn
+s (t)=i Vi if i ̸= i0, Bi0 = �
+t∈Fpn:Trn
+s (t)=i0 Vi
+� V .
+• Define
+V •
+t = {(xd ◦ t, x) : x ∈ F∗
+pn} if t ∈ Fpn, and V = {(x, 0) : x ∈ Fpn}.
+Let i0 ∈ Fps be an arbitrary element. Define
+Γ•
+2 = {B•
+i , i ∈ Fps},
+(8)
+where B•
+i = �
+t∈Fpn:Trn
+s (t)=i V •
+i if i ̸= i0, Bi0 = �
+t∈Fpn:Trn
+s (t)=i0 V •
+i
+� V .
+• Define
+Xt = {(tηd
+x, x) : x ∈ F∗
+pn} if t ∈ Fpn, and X = {(x, 0) : x ∈ Fpn}.
+Let i0 ∈ Fps be an arbitrary element. Define
+Θ1 = {Si, i ∈ Fps},
+(9)
+where Si = �
+t∈Fpn:Trn
+s (t)=i Xt if i ̸= i0, Si0 = �
+t∈Fpn:Trn
+s (t)=i0 Xt
+� X.
+• Define
+Yt = {(x, tηu
+x) : x ∈ F∗
+pn} if t ∈ Fpn, and Y = {(0, y) : y ∈ Fpn}.
+Let i0 ∈ Fps be an arbitrary element. Define
+Θ2 = {Ti, i ∈ Fps},
+(10)
+where Ti = �
+t∈Fpn:Trn
+s (t)=i Yt if i ̸= i0, Ti0 = �
+t∈Fpn:Trn
+s (t)=i0 Yt
+� Y .
+Remark 1. In the finite field case, that is, ◦ and ⋆ are the field multiplication, then Γ1 = Γ•
+1 =
+Θ2, Γ2 = Γ•
+2 = Θ1, which reduces to the two classes bent partitions given in [4], [18].
+Remark 2. In fact, for the parameter u in the bent partitions Γ1, Γ•
+1, Γ2, Γ•
+2, Θ1, Θ2, one can
+consider the more general form u ≡ pj mod (ps − 1) by the proofs in [3].
+III. RELATIONS BETWEEN CERTAIN BENT PARTITIONS AND CERTAIN VECTORIAL
+DUAL-BENT FUNCTIONS
+Throughout this section, we consider bent partitions and vectorial dual-bent functions satisfy-
+ing the following conditions, respectively.
+Condition C: Let n be an even positive integer, s be a positive integer with s ≤
+n
+2. Let
+Γ = {Ai, i ∈ V (p)
+s
+} be a bent partition of V (p)
+n
+which satisfies that F∗
+pAi = Ai for all i ∈ V (p)
+s
+January 3, 2023
+DRAFT
+
+9
+and all bent functions f constructed from Γ are regular (that is, εf = 1) or weakly regular but
+not regular (that is, εf = −1). We denote by ε = εf for all bent functions f constructed from Γ.
+Condition A: Let n be an even positive integer, s be a positive integer with s ≤
+n
+2. Let
+F : V (p)
+n
+→ V (p)
+s
+be a vectorial dual-bent function with (Fc)∗ = (F ∗)c, c ∈ V (p)
+s
+\{0} for a
+vectorial dual F ∗ of F and all component functions being regular or weakly regular but not
+regular, that is, εFc, c ∈ V (p)
+s
+\{0} are all the same. We denote by ε = εFc for all c ∈ V (p)
+s
+\{0}.
+It is easy to see that the known bent partitions, including partial spreads and Γi, Γ•
+i , Θi, i = 1, 2
+defined by (5)-(10), all satisfy F∗
+pAi = Ai, i ∈ V (p)
+s
+. By the results in [3], [11], [14], all bent
+functions constructed from partial spreads and Γi, Γ•
+i , Θi, i = 1, 2 are regular. Thus, the known
+bent partitions all satisfy Condition C with ε = 1. Moreover, when p = 2, it is easy to see that
+Condition C is trivial for any bent partition of V (2)
+n
+of depth powers of 2. In this section, we
+present relations between bent partitions satisfying Condition C and vectorial dual-bent functions
+satisfying Condition A. First, we need a lemma.
+Lemma 1. Let n be an even positive integer, s be a positive integer with s ≤ n
+2, and F : V (p)
+n
+→
+V (p)
+s
+. Then the following two statements are equivalent.
+(1) F is a vectorial dual-bent function satisfying Condition A.
+(2) There exist pairwise disjoint sets Wi ⊆ V (p)
+n , i ∈ V (p)
+s
+with �
+i∈V (p)
+s
+Wi = V (p)
+n
+and a
+constant ε ∈ {±1} (ε = 1 if p = 2) such that for any nonempty set I ⊆ V (p)
+s
+,
+χu(DF,I) = pn−sδ{0}(u)|I| + εp
+n
+2 −s(psδWI(u) − |I|), u ∈ V (p)
+n ,
+(11)
+where DF,I = {x ∈ V (p)
+n
+: F(x) ∈ I}, WI = �
+i∈I Wi, and for any set S, δS denotes the
+indicator function of S.
+Proof. By Proposition 3 of [22] (Note that although Proposition 3 of [22] only considers the
+case of p being odd, p = 2 also holds), for any u ∈ V (p)
+n , i ∈ V (p)
+s
+we have
+χu(DF,i) = pn−sδ{0}(u) + p−s
+�
+c∈V (p)
+s
+\{0}
+WFc(−u)ζ−⟨c,i⟩s
+p
+,
+(12)
+where DF,i = {x ∈ V (p)
+n
+: F(x) = i}.
+January 3, 2023
+DRAFT
+
+10
+(1) ⇒ (2): If F is a vectorial dual-bent function satisfying Condition A (Note that if p = 2,
+then ε = 1 since all Boolean bent functions are regular), then
+χu(DF,i) = pn−sδ{0}(u) + εp
+n
+2 −s
+�
+c∈V (p)
+s
+\{0}
+ζ(Fc)∗(−u)−⟨c,i⟩s
+p
+= pn−sδ{0}(u) + εp
+n
+2 −s
+�
+c∈V (p)
+s
+\{0}
+ζ(F ∗)c(−u)−⟨c,i⟩s
+p
+= pn−sδ{0}(u) + εp
+n
+2 −s
+�
+c∈V (p)
+s
+\{0}
+ζ⟨c,F ∗(−u)−i⟩s
+p
+= pn−sδ{0}(u) + εp
+n
+2 −s(psδ{0}(F ∗(−u) − i) − 1).
+(13)
+Define Wi = {x ∈ V (p)
+n
+: F ∗(−x) = i}, i ∈ V (p)
+s
+. Then Wi
+� Wj = ∅ for any i ̸= j and
+�
+i∈V (p)
+s
+Wi = V (p)
+n . By (13), for any nonempty set I ⊆ V (p)
+s
+and u ∈ V (p)
+n
+we have
+χu(DF,I) =
+�
+i∈I
+χu(DF,i)
+=
+�
+i∈I
+pn−sδ{0}(u) + εp
+n
+2 −s(psδWi(u) − 1)
+= pn−sδ{0}(u)|I| + εp
+n
+2 −s(psδWI(u) − |I|).
+(2) ⇒ (1): By the assumption on Wi, i ∈ V (p)
+s
+, we have that for any x ∈ V (p)
+n , there exists a
+unique i ∈ V (p)
+s
+such that x ∈ Wi. Define G : V (p)
+n
+→ V (p)
+s
+by
+G(x) = i if − x ∈ Wi.
+By the definition of G, for any u ∈ V (p)
+n , i ∈ V (p)
+s
+we have
+χu(DF,i) = pn−sδ{0}(u) + εp
+n
+2 −s(psδ{0}(G(−u) − i) − 1).
+(14)
+For any c ∈ V (p)
+s
+\{0},
+WFc(−u) =
+�
+x∈V (p)
+n
+ζ⟨c,F (x)⟩s+⟨u,x⟩n
+p
+=
+�
+i∈V (p)
+s
+�
+x∈V (p)
+n
+:F (x)=i
+ζ⟨c,i⟩s+⟨u,x⟩n
+p
+January 3, 2023
+DRAFT
+
+11
+=
+�
+i∈V (p)
+s
+ζ⟨c,i⟩s
+p
+χu(DF,i)
+=
+�
+i∈V (p)
+s
+\{G(−u)}
+ζ⟨c,i⟩s
+p
+(pn−sδ{0}(u) − εp
+n
+2 −s) + (pn−sδ{0}(u) + εp
+n
+2 −s(ps − 1))ζ⟨c,G(−u)⟩s
+p
+= (pn−sδ{0}(u) − εp
+n
+2 −s)
+�
+i∈V (p)
+s
+ζ⟨c,i⟩s
+p
++ εp
+n
+2 ζGc(−u)
+p
+= εp
+n
+2 ζGc(−u)
+p
+.
+(15)
+By (15) and the assumption that ε = 1 if p = 2, F is a vectorial bent function with εFc = ε
+and (Fc)∗ = Gc for any c ∈ V (p)
+s
+\{0}. Since Fc is a weakly regular bent function, we have that
+Gc = (Fc)∗ is also weakly regular bent and G is vectorial bent. Thus, F is vectorial dual-bent
+with εFc = ε and (Fc)∗ = (F ∗)c for any c ∈ V (p)
+s
+\{0}, where F ∗ = G, that is, F satisfies
+Condition A.
+Based on Lemma 1, we have the following theorem.
+Theorem 1. Let F : V (p)
+n
+→ V (p)
+s
+be a vectorial dual-bent function satisfying Condition A.
+Define
+Ai = DF,i, i ∈ V (p)
+s
+,
+where DF,i = {x ∈ V (p)
+n
+: F(x) = i}. Then Γ = {Ai, i ∈ V (p)
+s
+} is a bent partition satisfying
+Condition C.
+Proof. By Lemma 1 and its proof, for any i ∈ V (p)
+s
+and u ∈ V (p)
+n ,
+χu(Ai) = χu(DF,i) = pn−sδ{0}(u) + εp
+n
+2 −s(psδ{0}(F ∗(−u) − i) − 1),
+where ε = 1 if p = 2 since all Boolean bent functions are regular. For any union S of ps−1 sets
+of {Ai : i ∈ V (p)
+s
+}, we have
+χu(S) =
+
+
+
+pn−1δ{0}(u) + εp
+n
+2 −1(p − 1), if AF ∗(−u) ⊆ S,
+pn−1δ{0}(u) − εp
+n
+2 −1, if AF ∗(−u) ⊈ S.
+(16)
+Let f be an arbitrary function such that for every j ∈ Fp, there are exactly ps−1 sets Ai in Γ in
+its preimage. Define g(u) = f(AF ∗(−u)). Note that g is a p-ary function from V (p)
+n
+to Fp. Then
+by (16), we have
+χu(Df,j) =
+
+
+
+pn−1δ{0}(u) + εp
+n
+2 −1(p − 1), if j = g(u),
+pn−1δ{0}(u) − εp
+n
+2 −1, if j ̸= g(u).
+(17)
+January 3, 2023
+DRAFT
+
+12
+By (17), for any u ∈ V (p)
+n
+we have
+Wf(−u) =
+�
+x∈V (p)
+n
+ζf(x)+⟨u,x⟩n
+p
+=
+�
+j∈Fp
+ζj
+p
+�
+x∈V (p)
+n
+:f(x)=j
+ζ⟨u,x⟩n
+p
+=
+�
+j∈Fp
+ζj
+pχu(Df,j)
+=
+�
+j∈Fp\{g(u)}
+ζj
+p(pn−1δ{0}(u) − εp
+n
+2 −1) + ζg(u)
+p
+(pn−1δ{0}(u) + εp
+n
+2 −1(p − 1))
+= (pn−1δ{0}(u) − εp
+n
+2 −1)
+�
+j∈Fp
+ζj
+p + εp
+n
+2 ζg(u)
+p
+= εp
+n
+2 ζg(u)
+p
+.
+(18)
+By (18) and ε = 1 if p = 2, f is a weakly regular bent function with εf = ε and f ∗(x) = g(−x).
+Let
+Wj = {u ∈ V (p)
+n
+: g(u) = j}, j ∈ Fp,
+then Wj, j ∈ Fp are pairwise disjoint and �
+j���Fp Wj = V (p)
+n . By (17), for any u ∈ V (p)
+n
+and
+nonempty set J ⊆ Fp we have
+χu(Df,J) = pn−1δ{0}(u)|J| + εp
+n
+2 −1(pδWJ(u) − |J|),
+(19)
+where Df,J = {x ∈ V (p)
+n
+: f(x) ∈ J}, WJ = �
+j∈J Wj. By (19) and Lemma 1, f is vectorial dual-
+bent with (cf)∗ = c(βf ∗), c ∈ F∗
+p for some β ∈ F∗
+p (since all vectorial duals of f are cf ∗, c ∈ F∗
+p).
+Let c = 1, we obtain β = 1, that is, f is vectorial dual-bent with (cf)∗ = cf ∗, c ∈ F∗
+p. By
+Proposition 1, f is a (p − 1)-form. In particular, Fc is a (p − 1)-form for any c ∈ F∗
+ps, which
+yields that F(αx) = F(x) for any α ∈ F∗
+p and F∗
+pAi = Ai, i ∈ V (p)
+s
+. Hence, Γ is a bent partition
+satisfying Condition C.
+By Theorem 1, we have the following corollary.
+Corollary 1. Let n be an even positive integer. Let f : V (p)
+n
+→ Fp be a weakly regular bent
+function of (p − 1)-form, then {Df,j, j ∈ Fp} is a bent partition of V (p)
+n , where Df,j = {x ∈
+V (p)
+n
+: f(x) = j}.
+Proof. By Proposition 1, f is a weakly regular vectorial dual-bent function with (cf)∗ = cf ∗.
+Since n is even, εcf = εf for all c ∈ F∗
+p (see Theorem 1 of [6]). Then by Theorem 1, the
+conclusion holds.
+January 3, 2023
+DRAFT
+
+13
+A bent partition Γ = {A1, . . . , AK} of depth K is called coming from a normal bent partition
+if there is U ⊆ Ai for some i such that {U, A1, . . . , Ai−1, Ai\U, Ai+1, . . . , AK} is a normal bent
+partition. In [4], there is an open problem: Do bent partitions exist which are not coming from
+a normal bent partition of depth K > 2? In the following, we provide a positive answer for
+this open problem. By the definition of l-form, a ternary function f is a 2-form if and only if
+f(x) = f(−x). Let n be an even positive integer. If f : V (3)
+n
+→ F3 with f(x) = f(−x) is a
+ternary weakly regular but not regular bent function (that is, εf = −1), then by Corollary 1,
+{Df,0, Df,1, Df,2} is a bent partition of depth 3. There exist such ternary bent functions f, for
+instance see [7], [17]:
+•
+f(x) = Trn
+1(αx2), x ∈ F3n,
+(20)
+where n is even, α ∈ F∗
+3n is a square element if 4 | n, and α ∈ F∗
+3n is a non-square element
+if 4 ∤ n;
+•
+f(x) = Trn
+1(ax
+3n−1
+4
++3m+1), x ∈ F3n,
+(21)
+where n = 2m, m odd, a = α
+3m+1
+4
+for a primitive element α of F3n;
+•
+f(x) = Trn
+1(α(x33k+32k−3k+1 + x2)), x ∈ F3n,
+(22)
+where n = 4k for an arbitrary positive integer k, α ∈ F∗
+32k;
+•
+f(x, y, z) = (g(x) − h(x))z2 + yz + g(x), (x, y, z) ∈ F3n × F3 × F3,
+(23)
+where n is even, g and h are distinct bent functions constructed by (20) or (22) if 4 | n, g
+and h are distinct bent functions constructed by (20) or (21) if 4 ∤ n.
+For any ternary weakly regular but not regular bent function f : V (3)
+n
+→ F3 (n even) with
+f(x) = f(−x), the corresponding bent partition {Df,0, Df,1, Df,2} is not coming from a normal
+bent partition by Theorem 4 (i) of [4], which provides a positive answer for the above open
+problem proposed in [4]. We first recall Theorem 4 (i) of [4] and then give an example to
+illustrate this fact.
+Lemma 2 ( [4]). Let Γ = {U, A1, . . . , AK} be a normal bent partition of V (p)
+n . Then |U| = p
+n
+2
+and |Aj| = pn−p
+n
+2
+K
+, 1 ≤ j ≤ K.
+January 3, 2023
+DRAFT
+
+14
+Example 1. Let f : F34 → F3 be defined by f(x) = Tr4
+1(x2). Then f is ternary weakly regular
+bent with f(x) = f(−x) and εf = −1. By Corollary 1, {Df,0, Df,1, Df,2} is a bent partition.
+By the result of Nyberg [20], for any weakly regular p-ary bent function g : V (p)
+n
+→ Fp with
+n even, we have {|Dg,i|, i ∈ Fp} = {pn−1 + εfp
+n
+2 −1(p − 1), pn−1 − εfp
+n
+2 −1}. For our example,
+|Df,0| = 21, |Df,1| = |Df,2| = 30. By Lemma 2, it is easy to see that {Df,0, Df,1, Df,2} can not
+be from a normal bent partition.
+In the following, based on Theorem 1, we give an alternative proof that Γi, Γ•
+i , Θi, i = 1, 2
+defined by (5)-(10) given in [3] are bent partitions.
+Let s, n be positive integers with s | n, u be an integer with u ≡ pj0 mod (ps − 1) for some
+0 ≤ j0 ≤ s − 1 and gcd(u, pn − 1) = 1, and let d be an integer with du ≡ 1 mod (pn − 1). Let
+P = (Fpn, +, ◦) be a (pre)semifield such that its dual P ⋆ = (Fpn, +, ⋆) is right Fps-linear. For
+given x ∈ Fpn, if x = 0, then let ηx = 0, and if x ̸= 0, then let ηx be given by x ⋆ η−1
+x
+= 1 (For
+convention we set η−1
+0
+= ηpn−2
+0
+= 0). For any α ∈ F∗
+pn and i0 ∈ Fps, define
+F(x, y) = Trn
+s (αa(x, y)) + i0(1 − xpn−1), (x, y) ∈ Fpn × Fpn,
+(24)
+where for given (x, y), if x = 0, then a(x, y) = 0, and if x ̸= 0, then a(x, y) is given by
+a(x, y) ◦ xu = y, and
+F •(x, y) = Trn
+s (αa•(x, y)) + i0(1 − xpn−1), (x, y) ∈ Fpn × Fpn,
+(25)
+where for given (x, y), if x = 0, then a•(x, y) = 0, and if x ̸= 0, then a•(x, y) is given by
+xu ◦ a•(x, y) = y, and
+G(x, y) = Trn
+s (αb(x, y)) + i0(1 − ypn−1), (x, y) ∈ Fpn × Fpn,
+(26)
+where for given (x, y), if y = 0, then b(x, y) = 0, and if y ̸= 0, then b(x, y) is given by
+b(x, y) ◦ yd = x, and
+G•(x, y) = Trn
+s (αb•(x, y)) + i0(1 − ypn−1), (x, y) ∈ Fpn × Fpn,
+(27)
+where for given (x, y), if y = 0, then b•(x, y) = 0, and if y ̸= 0, then b•(x, y) is given by
+yd ◦ b•(x, y) = x, and
+M(x, y) = Trn
+s (αη−u
+x y) + i0(1 − xpn−1), (x, y) ∈ Fpn × Fpn,
+(28)
+and
+N(x, y) = Trn
+s (αxη−d
+y ) + i0(1 − ypn−1), (x, y) ∈ Fpn × Fpn.
+(29)
+January 3, 2023
+DRAFT
+
+15
+Proposition 3. Let F, F •, G, G•, M, N be defined as above. Then they are all vectorial dual-bent
+functions satisfying Condition A with ε = 1.
+Proof. We only prove the result for F and M since the proofs for F •, G, G• are similar to the
+proof for F, and the proof for N is similar to the proof for M.
+• For F:
+For any c ∈ F∗
+ps, we have
+Fc(x, y) = Trn
+1(cαa(x, y)) + Trs
+1(ci0)(1 − xpn−1).
+For any c ∈ F∗
+ps and (w, v) ∈ Fpn × Fpn, we have
+WFcu(w, v) =
+�
+x∈F∗
+pn
+�
+y∈Fpn
+ζTrn
+1 (cuαa(x,y))−Trn
+1 (wx+vy)
+p
++ ζTrs
+1(cui0)
+p
+�
+y∈Fpn
+ζ−Trn
+1 (vy)
+p
+=
+�
+x∈Fpn
+�
+y∈Fpn
+ζTrn
+1 (cuαa(x,y))−Trn
+1 (wx+vy)
+p
++ pn(ζTrs
+1(cui0)
+p
+− 1)δ{0}(v)
+= Wh(w, v) + pn(ζTrs
+1(cui0)
+p
+− 1)δ{0}(v),
+where h(x, y) = Trn
+1(cuαa(x, y)). For given x ∈ Fpn, if x = 0, then let λx = 0, and if
+x ̸= 0, then let λx be given by x ⋆ λ−1
+x
+= α (For convention we set λ−1
+0
+= λpn−2
+0
+= 0). Define
+ρ(x) = λ−d
+x . Then ρ is a permutation over Fpn. For any x ∈ F∗
+pn, set z = ρ−1(c−1x). Then
+λ−d
+z
+= ρ(z) = c−1x. By du ≡ 1 mod (pn − 1), we have λ−1
+z
+= c−uxu. Since z ̸= 0 and P ⋆ is
+right Fps-linear, we have α = z ⋆ λ−1
+z
+= z ⋆ (c−uxu) = c−u(z ⋆ xu), that is, ρ−1(c−1x) ⋆ xu =
+αcu for any x ̸= 0. Thus, when x ̸= 0, Trn
+1(cuαa(x, y)) = Trn
+1(a(x, y)(ρ−1(c−1x) ⋆ xu)) =
+Trn
+1(ρ−1(c−1x)(a(x, y) ◦ xu)) = Trn
+1(ρ−1(c−1x)y). When x = 0, by a(0, y) = ρ−1(0) = 0, we
+have Trn
+1(cuαa(x, y)) = Trn
+1(ρ−1(c−1x)y) = 0. Hence, h(x, y) = Trn
+1(ρ−1(c−1x)y), which is a
+Maiorana-McFarland bent function and by (4),
+Wh(w, v) = pnζ−Trn
+1 (cwρ(v))
+p
+.
+Therefore, for any c ∈ F∗
+ps,
+WFcu(w, v) = pn(ζ−Trn
+1 (cwρ(v))
+p
++ (ζTrs
+1(cui0)
+p
+− 1)δ{0}(v))
+= pnζ−Trn
+1 (cwρ(v))+Trs
+1(cui0)(1−vpn−1)
+p
+.
+(30)
+By (30) and ud ≡ 1 mod (pn − 1), we have that for any c ∈ F∗
+ps, Fc is a regular bent function
+with
+(Fc)∗(x, y) = −Trn
+1(cdxρ(y)) + Trs
+1(ci0)(1 − ypn−1)
+= −Trn
+1(cdpj0(xρ(y))pj0) + Trs
+1(ci0)(1 − ypn−1).
+January 3, 2023
+DRAFT
+
+16
+Since u ≡ pj0 mod (ps − 1) and du ≡ 1 mod (pn − 1), we have d ≡ ps−j0 mod (ps − 1) and
+thus (cd)pj0 = c for any c ∈ F∗
+ps. Therefore, F is a vectorial bent function with εFc = 1 and
+(Fc)∗ = Hc for all c ∈ F∗
+ps, where
+H(x, y) = −Trn
+s ((xρ(y))pj0) + i0(1 − ypn−1) = −(Trn
+s (xρ(y)))pj0 + i0(1 − ypn−1).
+Since Fc is regular bent, we have that (Fc)∗ = Hc is also regular bent and H is vectorial bent.
+Thus, F is vectorial dual-bent with εFc = 1 and (Fc)∗ = (F ∗)c for all c ∈ F∗
+ps, where F ∗ = H,
+that is, F satisfies Condition A.
+• For M:
+For any c ∈ F∗
+ps,
+Mc(x, y) = Trn
+1(cαη−u
+x y) + Trs
+1(ci0)(1 − xpn−1).
+Similar to the discussion for F, for any c ∈ F∗
+ps and (w, v) ∈ Fpn × Fpn we have
+WMc(w, v) = Wg(w, v) + pn(ζTrs
+1(ci0)
+p
+− 1)δ{0}(v),
+where g(x, y) = Trn
+1(cαη−u
+x y). Let π(x) = η−u
+x , then π is a permutation over Fpn. Since g is a
+Maiorana-McFarland bent function, then by (4),
+Wg(w, v) = pnζ−Trn
+1 (wπ−1(c−1α−1v))
+p
+.
+For any given y ∈ F∗
+pn, set π−1(c−1α−1y) = z. Then c−1α−1y = π(z) = η−u
+z . By ud ≡
+1 mod (pn − 1), we have η−1
+z
+= c−dα−dyd. Since z ̸= 0 and P ⋆ is right Fps-linear, we have
+1 = z ⋆ η−1
+z
+= z ⋆ (c−dα−dyd) = c−d(z ⋆ α−dyd), that is, π−1(c−1α−1y) ⋆ α−dyd = cd. For given
+(x, y) ∈ Fpn × Fpn, if y = 0, then let r(x, y) = 0, and if y ̸= 0, then let r(x, y) be given by
+r(x, y)◦α−dyd = x. When v ̸= 0, we have Trn
+1(wπ−1(c−1α−1v)) = Trn
+1(π−1(c−1α−1v)(r(w, v)◦
+α−dvd)) = Trn
+1(r(w, v)(π−1(c−1α−1v) ⋆ α−dvd)) = Trn
+1(cdr(w, v)) = Trn
+1(c(r(w, v))pj0). When
+v = 0, since π−1(0) = 0 and r(w, 0) = 0, we have Trn
+1(wπ−1(c−1α−1v)) = Trn
+1(c(r(w, v))pj0) =
+0. Thus, −Trn
+1(wπ−1(c−1α−1v)) = −Trn
+1(c(r(w, v))pj0) and
+WMc(w, v) = pn(ζ−Trn
+1 (c(r(w,v))pj0 )
+p
++ (ζTrs
+1(ci0)
+p
+− 1)δ{0}(v))
+= pnζ−Trn
+1 (c(r(w,v))pj0 )+Trs
+1(ci0)(1−vpn−1)
+p
+,
+which implies that M is a vectorial dual-bent function with εMc = 1 and (Mc)∗ = (M∗)c for all
+c ∈ F∗
+ps, where
+M∗(x, y) = −Trn
+s ((r(x, y))pj0) + i0(1 − ypn−1),
+January 3, 2023
+DRAFT
+
+17
+that is, M satisfies Condition A.
+By Theorem 1 and Proposition 3, we have that {DF,i, i ∈ Fps}, {DF ��,i, i ∈ Fps}, {DG,i, i ∈
+Fps}, {DG•,i, i ∈ Fps}, {DM,i, i ∈ Fps} and {DN,i, i ∈ Fps} are bent partitions. It is easy to
+verify that
+DF,i =
+
+
+
+
+
+
+
+
+
+
+
+�
+t∈Fpn:T rn
+s (αt)=i
+Ut, if i ̸= i0,
+�
+t∈Fpn:T rn
+s (αt)=i0
+Ut
+�
+U, if i = i0,
+, DF •,i =
+
+
+
+
+
+
+
+
+
+
+
+�
+t∈Fpn:T rn
+s (αt)=i
+U •
+t , if i ̸= i0,
+�
+t∈Fpn:T rn
+s (αt)=i0
+U •
+t
+�
+U, if i = i0,
+,
+DG,i =
+
+
+
+
+
+
+
+
+
+
+
+�
+t∈Fpn:T rn
+s (αt)=i
+Vt, if i ̸= i0,
+�
+t∈Fpn:T rn
+s (αt)=i0
+Vt
+�
+V, if i = i0,
+, DG•,i =
+
+
+
+
+
+
+
+
+
+
+
+�
+t∈Fpn:T rn
+s (αt)=i
+V •
+t , if i ̸= i0,
+�
+t∈Fpn:T rn
+s (αt)=i0
+V •
+t
+�
+V, if i = i0,
+,
+DM,i =
+
+
+
+
+
+
+
+
+
+
+
+�
+t∈Fpn:T rn
+s (αt)=i
+Xt, if i ̸= i0,
+�
+t∈Fpn:T rn
+s (αt)=i0
+Xt
+�
+X, if i = i0,
+, DN,i =
+
+
+
+
+
+
+
+
+
+
+
+�
+t∈Fpn:T rn
+s (αt)=i
+Yt, if i ̸= i0,
+�
+t∈Fpn:T rn
+s (αt)=i0
+Yt
+�
+Y, if i = i0,
+where
+Ut = {(x, t ◦ xu) : x ∈ F∗
+pn} if t ∈ Fpn, and U = {(0, y) : y ∈ Fpn},
+U•
+t = {(x, xu ◦ t) : x ∈ F∗
+pn} if t ∈ Fpn, and U = {(0, y) : y ∈ Fpn},
+Vt = {(t ◦ xd, x) : x ∈ F∗
+pn} if t ∈ Fpn, and V = {(x, 0) : x ∈ Fpn},
+V •
+t = {(xd ◦ t, x) : x ∈ F∗
+pn} if t ∈ Fpn, and V = {(x, 0) : x ∈ Fpn},
+Xt = {(tηd
+x, x) : x ∈ F∗
+pn} if t ∈ Fpn, and X = {(x, 0) : x ∈ Fpn},
+Yt = {(x, tηu
+x) : x ∈ F∗
+pn} if t ∈ Fpn, and Y = {(0, y) : y ∈ Fpn}.
+For the above bent partitions from vectorial dual-bent functions F, F •, G, G•, M, N, by set-
+ting α = 1, u = ps + p − 1 with gcd(u, pn − 1) = 1, then we can obtain bent partitions
+Γ1, Γ•
+1, Γ2, Γ•
+2, Θ1, Θ2 defined by (5)-(10) respectively. Thus by the above analysis, we provide
+an alternative derivation that Γ1, Γ•
+1, Γ2, Γ•
+2, Θ1, Θ2 are bent partitions.
+When p is an odd prime, we show that the converse of Theorem 1 also holds.
+Theorem 2. Let p be an odd prime. Let Γ = {Ai, i ∈ V (p)
+s
+} be a bent partition of V (p)
+n
+satisfying
+Condition C. Define F : V (p)
+n
+→ V (p)
+s
+by
+F(x) = i if x ∈ Ai.
+Then F is a vectorial dual-bent function satisfying Condition A.
+January 3, 2023
+DRAFT
+
+18
+Proof. Since F∗
+pAi = Ai for any i ∈ V (p)
+s
+, all bent functions constructed from Γ are (p−1)-form.
+When s = 1, the conclusion follows from Proposition 1. In the following, we consider the case
+of s ≥ 2.
+Let f be an arbitrary bent function constructed from Γ. By Lemma 3.4 of [13], for any
+u ∈ V (p)
+n
+and j ∈ Fp we have
+χu(Df,j) =
+
+
+
+pn−1δ{0}(u) + εp
+n
+2 −1(p − 1), if f ∗(u) = j,
+pn−1δ{0}(u) − εp
+n
+2 −1, if f ∗(u) ̸= j,
+(31)
+where Df,j = {x ∈ V (p)
+n
+: f(x) = j}, j ∈ Fp. For any fixed u ∈ V (p)
+n , since
+{χu(Df,j), j ∈ Fp} = {pn−1δ{0}(u) + εp
+n
+2 −1(p − 1), pn−1δ{0}(u) − εp
+n
+2 −1}
+for any bent function f constructed from Γ, we have that for any fixed u ∈ V (p)
+n , there exists a
+unique G(u) ∈ V (p)
+s
+such that χu(Ai), i ̸= G(u) are all the same and χu(Ai) ̸= χu(AG(u)), i ̸=
+G(u). Note that G is a function from V (p)
+n
+to V (p)
+s
+. Moreover by (31), for any fixed u ∈ V (p)
+n
+we have
+χu(Ai) =
+
+
+
+pn−sδ{0}(u) + εp
+n
+2 −s(ps − 1), if i = G(u),
+pn−sδ{0}(u) − εp
+n
+2 −s, if i ̸= G(u).
+(32)
+Define
+Wi = {u ∈ V (p)
+n
+: G(u) = i}, i ∈ V (p)
+s
+.
+Then obviously Wi, i ∈ V (p)
+s
+are pairwise disjoint and �
+i∈V (p)
+s
+Wi = V (p)
+n . By (32), for any
+u ∈ V (p)
+n
+and nonempty set I ⊆ V (p)
+s
+we have
+χu(DF,I) =
+�
+i∈I
+χu(Ai) = pn−sδ{0}(u)|I| + εp
+n
+2 −s(psδWI(u) − |I|),
+(33)
+where DF,I = {x ∈ V (p)
+n
+: F(x) ∈ I}, WI = �
+i∈I Wi. By (33) and Lemma 1, the conclusion
+holds.
+When p is an odd prime, from Theorems 1 and 2 we obtain a characterization of bent partitions
+satisfying Condition C in terms of vectorial dual-bent functions.
+Theorem 3. Let p be an odd prime. Let Γ = {Ai, i ∈ V (p)
+s
+} be a partition of V (p)
+n , where n is
+even, s ≤ n
+2. Define F : V (p)
+n
+→ V (p)
+s
+as
+F(x) = i if x ∈ Ai.
+Then Γ is a bent partition satisfying Condition C if and only if F is a vectorial dual-bent function
+satisfying Condition A.
+January 3, 2023
+DRAFT
+
+19
+IV. CONSTRUCTING BENT PARTITIONS FROM VECTORIAL DUAL-BENT FUNCTIONS
+In this section, we construct bent partitions from vectorial dual-bent functions.
+The following theorem provides a secondary construction of vectorial dual-bent functions,
+which can be used to generate more bent partitions.
+Theorem 4. Let n, m, s be positive integers for which n is even and s ≤ n
+2, s | m, s ̸= m. For
+any i ∈ Fps, let F(i; x) : V (p)
+n
+→ Fps be a vectorial dual-bent function with ((F(i; x))c)∗ =
+((F(i; x))∗)c and ε(F (i;x))c = ε for any c ∈ F∗
+ps, where (F(i; x))∗ is a vectorial dual of F(i; x)
+and ε ∈ {±1} is a constant independent of i, c. Let α, β ∈ Fpm be linearly independent over Fps.
+Let R be a permutation over Fpm with R(0) = 0 and T : Fps → Fps be an arbitrary function.
+Define H : V (p)
+n
+× Fpm × Fpm → Fps as
+H(x, y1, y2) = F(T rm
+s (αR(y1ypm−2
+2
+)); x) + T rm
+s (βR(y1ypm−2
+2
+)) + T (T rm
+s (αR(y1ypm−2
+2
+))).
+Then H is a vectorial dual-bent function satisfying Condition A and Γ = {Ai, i ∈ Fps} is a bent
+partition satisfying Condition C, where Ai = {(x, y1, y2) ∈ V (p)
+n
+×Fpm ×Fpm : H(x, y1, y2) = i}.
+Proof. Denote
+d(y) = Trm
+s (βR(y1ypm−2
+2
+)), e(y) = Trm
+s ((β − α)R(y1ypm−2
+2
+)), y = (y1, y2) ∈ Fpm × Fpm.
+For any c ∈ F∗
+ps and (a, b) = (a, b1, b2) ∈ V (p)
+n
+× Fpm × Fpm, we have
+WHc(a, b)
+=
+�
+x∈V (p)
+n
+�
+y=(y1,y2)∈Fpm×Fpm
+ζT rs
+1(cF (d(y)−e(y);x))+T rs
+1(cd(y))+T rs
+1(cT (d(y)−e(y)))
+p
+ζ−⟨a,x⟩n−T rm
+1 (b1y1+b2y2)
+p
+=
+�
+i∈Fps
+�
+y=(y1,y2)∈Fpm×Fpm:d(y)−e(y)=i
+�
+x∈V (p)
+n
+ζT rs
+1(cF (i;x))+T rs
+1(cd(y))+T rs
+1(cT (i))
+p
+ζ−⟨a,x⟩n−T rm
+1 (b1y1+b2y2)
+p
+= p−s �
+i∈Fps
+W(F (i;x))c(a)ζT rs
+1(cT (i))
+p
+�
+y=(y1,y2)∈Fpm×Fpm
+ζT rs
+1(cd(y))−T rm
+1 (b1y1+b2y2)
+p
+�
+j∈Fps
+ζT rs
+1(cj(i−(d(y)−e(y))))
+p
+= p−s �
+i∈Fps
+W(F (i;x))c(a)ζT rs
+1(cT (i))
+p
+�
+j∈Fps
+ζT rs
+1(ijc)
+p
+�
+y=(y1,y2)∈Fpm×Fpm
+ζT rs
+1(c((1−j)d(y)+je(y)))−T rm
+1 (b1y1+b2y2)
+p
+.
+By Theorem 3 of [10], for any j ∈ Fps, J(j; y) = (1−j)d(y)+je(y) is a partial spread vectorial
+dual-bent function with ε(J(j;y))c = 1 and ((J(j; y))c)∗ = ((1−j)d∗(y)+je∗(y))c for any c ∈ F∗
+ps,
+January 3, 2023
+DRAFT
+
+20
+where d∗(y) = Trm
+s (βR(−ypm−2
+1
+y2)), e∗(y) = Trm
+s ((β − α)R(−ypm−2
+1
+y2)). Therefore,
+WHc(a, b)
+= pm−s �
+i∈Fps
+W(F (i;x))c(a)ζT rs
+1(cT (i))
+p
+�
+j∈Fps
+ζT rs
+1(ijc)
+p
+ζT rs
+1(c((1−j)d∗(b)+je∗(b)))
+p
+= pm−sζT rs
+1(cd∗(b))
+p
+�
+i∈Fps
+W(F (i;x))c(a)ζT rs
+1(cT (i))
+p
+�
+j∈Fps
+ζT rs
+1(cj(i−(d∗(b)−e∗(b))))
+p
+= pmζT rs
+1(cd∗(b))
+p
+W(F (d∗(b)−e∗(b);x))c(a)ζT rs
+1(cT (d∗(b)−e∗(b)))
+p
+= εp
+n
+2 +mζ
+((F (T rm
+s (αR(−bpm−2
+1
+b2));x))c)∗(a)+T rs
+1(cT rm
+s (βR(−bpm−2
+1
+b2)))+T rs
+1(cT (T rm
+s (αR(−bpm−2
+1
+b2))))
+p
+= εp
+n
+2 +mζ((F (T rm
+s (αR(−bpm−2
+1
+b2));x))∗)c(a)+T rs
+1(cT rm
+s (βR(−bpm−2
+1
+b2)))+T rs
+1(cT (T rm
+s (αR(−bpm−2
+1
+b2))))
+p
+.
+(34)
+Note that ε = 1 if p = 2 since all Boolean bent functions are regular. By (34), H is a vectorial
+bent function with (Hc)∗ = Gc and εHc = ε for any c ∈ F∗
+ps, where
+G(a, b1, b2) = (F(T rm
+s (αR(−bpm−2
+1
+b2)); x))∗(a) + T rm
+s (βR(−bpm−2
+1
+b2)) + T (T rm
+s (αR(−bpm−2
+1
+b2))).
+Since Hc is weakly regular bent, we have that Gc = (Hc)∗ is also weakly regular bent and G is
+vectorial bent. Thus, H is vectorial dual-bent with (Hc)∗ = (H∗)c and εHc = ε for any c ∈ F∗
+ps,
+where H∗ = G, that is, H satisfies Condition A. By Theorem 1, the partition Γ generated from
+H is a bent partition satisfying Condition C.
+The following explicit construction of bent partitions is an immediate result of Proposition 3
+and Theorem 4.
+Theorem 5. Let n, m, s be positive integers with s | n, s | m, s ̸= m, and ui, i ∈ Fps be
+integers for which for any i ∈ Fps, ui ≡ pji mod (ps − 1) for some 0 ≤ ji ≤ s − 1 and
+gcd(ui, pn − 1) = 1. For any i ∈ Fps, let di be an integer with uidi ≡ 1 mod (pn − 1), and
+Pi = (Fpn, +, ◦i) be a (pre)semifield for which its dual P ⋆
+i is right Fps-linear. For any i ∈ Fps,
+let F(i; x1, x2) : Fpn × Fpn → Fps be an arbitrary vectorial dual-bent function constructed by
+Proposition 3 with u = ui, d = di, P = Pi. Let α, β ∈ Fpm be linearly independent over Fps, R
+be a permutation over Fpm with R(0) = 0 and T : Fps → Fps be an arbitrary function. Define
+H : Fpn × Fpn × Fpm × Fpm → Fps as
+H(x1, x2, y1, y2) = F(T rm
+s (αR(y1ypm−2
+2
+)); x1, x2) + T rm
+s (βR(y1ypm−2
+2
+)) + T (T rm
+s (αR(y1ypm−2
+2
+))).
+Then
+Γ = {Ai, i ∈ Fps}
+is a bent partition satisfying Condition C, where
+Ai = {(x1, x2, y1, y2) ∈ Fpn × Fpn × Fpm × Fpm : H(x1, x2, y1, y2) = i}.
+January 3, 2023
+DRAFT
+
+21
+Remark 3. With the same notation as in Theorem 4. Note that in Theorem 4, by setting vectorial
+dual-bent functions H constructed by Theorem 5 as building blocks (that is, as F(i; x)), we can
+obtain more explicit vectorial dual-bent functions which can generate more bent partitions by
+Theorem 4.
+We give an example by using Theorem 5.
+Example 2. Let p = 3, s = 4, n = m = 8. Let α be a primitive element of F38 and β = 1, R be
+the identity map and T = 0. For any i ∈ F34, let
+F(i; x1, x2) =
+
+
+
+Tr8
+4(x−89
+1
+x2), if i ∈ F∗
+34,
+Tr8
+4(x1x−83
+2
+), if i = 0.
+Then
+H(x1, x2, y2, y2) = (Tr8
+4(αy1y6559
+2
+))80(Tr8
+4(x−89
+1
+x2 − x1x−83
+2
+)) + Tr8
+4(x1x−83
+2
++ y1y6559
+2
+),
+and Γ = {DH,i, i ∈ F34} is a bent partition satisfying Condition C, where DH,i = {(x1, x2, y1, y2) ∈
+(F38)4 : H(x1, x2, y1, y2) = i}.
+V. RELATIONS BETWEEN BENT PARTITIONS AND PARTIAL DIFFERENCE SETS
+In this section, by taking vectorial dual-bent functions as the link between bent partitions and
+partial difference sets, we give a sufficient condition on constructing partial difference sets from
+bent partitions. When p is an odd prime, we characterize bent partitions satisfying Condition C
+in terms of partial difference sets.
+Definition 4. Let (G, +) be a finite abelian group of order v and D be a subset of G with k
+elements. Then D is called a (v, k, λ, µ) partial difference set of G, if the expressions d1 − d2,
+for d1 and d2 in D with d1 ̸= d2, represent each nonzero element in D exactly λ times, and
+represent each nonzero element in G \ D exactly µ times. When λ = µ, then D is called a
+(v, k, λ) difference set.
+Note that if D is a partial difference set of G with −D = D, then so are D∪{0}, D \ {0}, G \ D
+(see [16]). There is an important tool to characterize partial difference sets in terms of characters.
+January 3, 2023
+DRAFT
+
+22
+Lemma 3 ( [16]). Let G be an abelian group of order v. Suppose that D is a subset of G with
+k elements which satisfies −D = D and 0 /∈ D. Then D is a (v, k, λ, µ) partial difference set
+if and only if for each non-principal character χ of G,
+χ(D) = β ±
+√
+∆
+2
+,
+where χ(D) = �
+x∈D χ(x), β = λ − µ, γ = k − µ, ∆ = β2 + 4γ.
+When p is an odd prime or s ≥ 2, we give the value distribution of vectorial dual-bent
+functions satisfying Condition A.
+Proposition 4. Let F : V (p)
+n
+→ V (p)
+s
+be a vectorial dual-bent function satisfying Condition A,
+where p is odd or s ≥ 2. Then
+|DF,F (0)| = pn−s + εp
+n
+2 −s(ps − 1), |DF,i| = pn−s − εp
+n
+2 −s if i ̸= F(0).
+Proof. Note that if f is a weakly regular p-ary bent function, then for any a ∈ Fp, f − a is
+a weakly regular bent function with (f − a)∗ = f ∗ − a and εf−a = εf. Since F is a vectorial
+dual-bent function with (Fc)∗ = (F ∗)c, c ∈ V (p)
+s
+\{0}, we have that F(x) − F(0) is a vectorial
+bent function and for any c ∈ V (p)
+s
+\{0},
+((F − F(0))c)∗ = (Fc)∗ − ⟨c, F(0)⟩s = (F ∗)c − ⟨c, F(0)⟩s = (F ∗ − F(0))c,
+which implies that F(x) − F(0) is a vectorial dual-bent function with ((F − F(0))c)∗ = (F ∗ −
+F(0))c and ε(F −F (0))c = ε for any c ∈ V (p)
+s
+\{0}. By the proof of Theorem 1, F(ax) = F(x) for
+any a ∈ F∗
+p and thus F(x) = F(−x). By Corollary 1 of [22] (Note that although Corollary 1 of
+[22] only considers the case of p being odd, the conclusion of Corollary 1 of [22] also holds
+for p = 2, s ≥ 2), we have
+|DF −F (0),0| = pn−s + εp
+n
+2 −s(ps − 1), |DF −F (0),i| = pn−s − εp
+n
+2 −s if i ̸= 0,
+that is,
+|DF,F (0)| = pn−s + εp
+n
+2 −s(ps − 1), |DF,i| = pn−s − εp
+n
+2 −s if i ̸= F(0).
+In the following, we give a characterization of vectorial dual-bent functions satisfying Con-
+dition A in terms of partial difference sets.
+January 3, 2023
+DRAFT
+
+23
+Theorem 6. Let n be an even positive integer, s be a positive integer with s ≤ n
+2, and F :
+V (p)
+n
+→ V (p)
+s
+. The following two statements are equivalent.
+(1) F is a vectorial dual-bent function satisfying Condition A.
+(2) When p = 2, s = 1, then the support supp(F) of F defined as supp(F) = {x ∈ V (2)
+n
+:
+F(x) = 1} is a (2n, 2n−1 ± 2
+n
+2 −1, 2n−2 ± 2
+n
+2 −1) difference set, and when p is odd or s ≥ 2,
+then for any nonempty set I ⊆ V (p)
+s
+, DF,I\{0} is a (pn, k, λ, µ) partial difference set for which
+−DF,I = DF,I and if F(0) ∈ I, then
+k = pn−s|I| + εp
+n
+2 −s(ps − |I|) − 1,
+λ = pn−2s|I|2 + εp
+n
+2 −s(ps − |I|) − 2,
+µ = pn−2s|I|2 + εp
+n
+2 −s|I|,
+(35)
+and if F(0) /∈ I, then
+k = pn−s|I| − εp
+n
+2 −s|I|,
+λ = pn−2s|I|2 + εp
+n
+2 −s(ps − 3|I|),
+µ = pn−2s|I|2 − εp
+n
+2 −s|I|,
+(36)
+where DF,I = {x ∈ V (p)
+n
+: F(x) ∈ I} and ε ∈ {±1} is a constant (ε = 1 if p = 2).
+Proof. It is easy to see that a Boolean function F is a vectorial dual-bent function satisfying
+Condition A if and only if F is bent, that is, Condition A is trivial for any Boolean bent function.
+By the well-known result that a Boolean function F : V (2)
+n
+→ F2 is bent if and only if its support
+supp(F) = {x ∈ V (2)
+n
+: F(x) = 1} is a (2n, 2n−1 ± 2
+n
+2 −1, 2n−2 ± 2
+n
+2 −1) difference set (see [11]),
+the conclusion obviously holds for p = 2, s = 1. In the following, we prove the conclusion for
+p being odd or s ≥ 2.
+(1) ⇒ (2): By the proof of Theorem 1, F(−x) = F(x), that is, −DF,I = DF,I. For any
+u ∈ V (p)
+n \{0}, with the same argument as in the proof of Theorem 2 of [22],
+χu(DF,I) =
+
+
+
+εp
+n
+2 − εp
+n
+2 −s|I|, if F ∗(−u) ∈ I,
+−εp
+n
+2 −s|I|, if F ∗(−u) /∈ I.
+where ε = 1 if p = 2 since all Boolean bent functions are regular.
+If F(0) ∈ I, then |DF,I\{0}| = |DF,I|−1 and χu(DF,I\{0}) = χu(DF,I)−1. By Proposition 4,
+|DF,I\{0}| = (|I|−1)(pn−s−εp
+n
+2 −s)+(pn−s+εp
+n
+2 −s(ps−1)−1) = pn−s|I|+εp
+n
+2 −s(ps−|I|)−1.
+By Lemma 3, DF,I\{0} is a (pn, k, λ, µ) partial difference set, where k, λ, µ are given in (35).
+January 3, 2023
+DRAFT
+
+24
+If F(0) /∈ I, then |DF,I\{0}| = |DF,I| and χu(DF,I\{0}) = χu(DF,I). By Proposition 4,
+|DF,I\{0}| = |I|(pn−s − εp
+n
+2 −s). By Lemma 3, DF,I\{0} is a (pn, k, λ, µ) partial difference set,
+where k, λ, µ are given in (36).
+(2) ⇒ (1): By Lemma 3, for any u ∈ V (p)
+n
+and nonempty set I ⊆ V (p)
+s
+we have
+χu(DF,I) = pn−sδ{0}(u)|I| + εp
+n
+2 − εp
+n
+2 −s|I| or χu(DF,I) = pn−sδ{0}(u)|I| − εp
+n
+2 −s|I|.
+(37)
+For any i ∈ V (p)
+s
+, define Wi = {u ∈ V (p)
+n
+: χu(DF,i) = pn−sδ{0}(u) + εp
+n
+2 − εp
+n
+2 −s}, where
+DF,i = {x ∈ V (p)
+n
+: F(x) = i}. We claim that Wi
+� Wi′ = ∅ for any i ̸= i′ and �
+i∈V (p)
+s
+Wi = V (p)
+n .
+Indeed, if there exist i ̸= i′ such that Wi
+� Wi′ ̸= ∅, that is, there exists u ∈ V (p)
+n
+such that
+χu(DF,i) = χu(DF,i′) = pn−sδ{0}(u) + εp
+n
+2 − εp
+n
+2 −s, then χu(DF,i
+� DF,i′) = 2pn−sδ{0}(u) +
+2εp
+n
+2 − 2εp
+n
+2 −s, which contradicts with (37). Thus, Wi
+� Wi′ = ∅ for any i ̸= i′. If there exists
+u ∈ V (p)
+n
+such that u /∈ Wi for any i ∈ V (p)
+s
+, that is, χu(DF,i) = pn−sδ{0}(u) − εp
+n
+2 −s for
+any i ∈ V (p)
+s
+, then χu(V (p)
+n ) = �
+i∈V (p)
+s
+χu(DF,i) = pnδ{0}(u) − εp
+n
+2 , which contradicts with
+χu(V (p)
+n ) = �
+x∈V (p)
+n
+ζ⟨u,x⟩n
+p
+= pnδ{0}(u). Thus, �
+i∈V (p)
+s
+Wi = V (p)
+n . By the above analysis, we
+have
+χu(DF,I) = pn−sδ{0}(u)|I| + εp
+n
+2 −s(psδWI(u) − |I|),
+(38)
+where WI = �
+i∈I Wi. By (38) and Lemma 1, F is a vectorial dual-bent function satisfying
+Condition A.
+The following theorem provides a sufficient condition on constructing partial difference sets
+from bent partitions.
+Theorem 7. Let n be an even positive integer and s be a positive integer with s ≤ n
+2. Assume
+that Γ = {Ai, i ∈ V (p)
+s
+} is a bent partition of V (p)
+n
+for which the function F : V (p)
+n
+→ V (p)
+s
+defined by
+F(x) = i if x ∈ Ai
+is a vectorial dual-bent function satisfying Condition A. Then when p = 2, s = 1, A0 and A1 are
+(2n, 2n−1 ± 2
+n
+2 −1, 2n−2 ± 2
+n
+2 −1) difference set and (2n, 2n−1 ∓ 2
+n
+2 −1, 2n−2 ∓ 2
+n
+2 −1) difference set,
+respectively, and when p is odd or s ≥ 2, for any nonempty set I ⊆ V (p)
+s
+, AI\{0} = �
+i∈I Ai\{0}
+is a (pn, k, λ, µ) partial difference set, where (k, λ, µ) are given in (35) if 0 ∈ AI and (k, λ, µ)
+are given in (36) if 0 /∈ AI.
+January 3, 2023
+DRAFT
+
+25
+Proof. Note that if D is a (v, k, λ) difference set of a finite abelian group G, then G\D is a
+(v, v − k, v − 2k + λ) difference set of G (for instance see [12]). Then the result follows from
+Theorem 6.
+Remark 4. By Proposition 3, the bent partition Γ1 (resp. Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2) satisfies the
+condition in Theorem 7. By Theorem 7, any union of sets from Γ1 (resp, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2)
+forms a partial difference set. Thus, the results given in Corollary 15 of [1] on constructing
+partial difference sets from Γ1 (resp. Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2) (which includes the results given in
+Theorem 2 of [2] on constructing partial difference sets from Γ1, resp. Γ2, in the finite field)
+can also be illustrated by our results.
+Since the bent partitions constructed in Theorem 5 satisfy the condition in Theorem 7, we
+have the following corollary from Theorem 7.
+Corollary 2. Let Γ = {Ai, i ∈ Fps} be a bent partition constructed by Theorem 5. Then when
+p = 2, s = 1, A0 and A1 are (2n, 2n−1 ± 2
+n
+2 −1, 2n−2 ± 2
+n
+2 −1) difference set and (2n, 2n−1 ∓
+2
+n
+2 −1, 2n−2 ∓ 2
+n
+2 −1) difference set, respectively, and when p is odd or s ≥ 2, for any nonempty
+set I ⊆ Fps, AI\{0} = �
+i∈I Ai\{0} is a (pn, k, λ, µ) partial difference set, where (k, λ, µ) are
+given in (35) with ε = 1 if 0 ∈ AI and (k, λ, µ) are given in (36) with ε = 1 if 0 /∈ AI.
+We give an example by Corollary 2.
+Example 3. Let Γ = {DH,i, i ∈ F34} be the bent partition constructed in Example 2. By Corol-
+lary 2, DH,i is a (1853020188851841, 22876791923520, 282470988879, 282429005040) partial difference
+set for any i ∈ F∗
+34, DH,0\{0} is a (1853020188851841, 22876834970240, 282472051759, 282430067922)
+partial difference set, (DH,0
+� DH,1)\{0} is a (1853020188851841, 45753626893760, 1129760129761,
+1129719208806) partial difference set.
+When p is an odd prime, we immediately obtain the following characterization of bent
+partitions of V (p)
+n
+satisfying Condition C from Theorems 3 and 6.
+Theorem 8. Let p be an odd prime. Let Γ = {Ai, i ∈ V (p)
+s
+} be a partition of V (p)
+n , where n is
+even and s ≤ n
+2. Then the following two statements are equivalent.
+(1) Γ is a bent partition satisfying Condition C.
+(2) For any nonempty set I ⊆ V (p)
+s
+, AI\{0} = �
+i∈I Ai\{0} is a (pn, k, λ, µ) partial difference
+January 3, 2023
+DRAFT
+
+26
+set with −AI = AI, where (k, λ, µ) are given in (35) if 0 ∈ AI and (k, λ, µ) are given in (36)
+if 0 /∈ AI.
+VI. CONCLUSION
+In this paper, we investigated relations between bent partitions and vectorial dual-bent functions
+(Theorems 1, 2, 3) and gave some new constructions of bent partitions satisfying Condition C
+(Corollary 1, Theorems 4 and 5). We illustrated that for any ternary weakly regular bent function
+f : V (3)
+n
+→ F3 (n even) with f(x) = f(−x) and εf = −1, the generated bent partition by f
+is not coming from a normal bent partition (see Example 1), which answers an open problem
+proposed in [4]. By taking vectorial dual-bent functions as the link between bent partitions and
+partial difference sets, we give a sufficient condition on constructing partial difference sets from
+bent partitions (Theorem 7). When p is an odd prime, we characterized bent partitions satisfying
+Condition C in terms of partial difference sets (Theorem 8).
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+

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@@ -0,0 +1,792 @@
+Draft version January 13, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+The implications of large binding energies of massive stripped core collapse supernova progenitors on
+the explosion mechanism
+Dmitry Shishkin1 and Noam Soker1
+1Department of Physics, Technion, Haifa, 3200003, Israel; [email protected]; [email protected]
+(Dated: January 2023)
+ABSTRACT
+We examine the binding energy of massive stripped-envelope core collapse supernova (SECCSN)
+progenitors with the stellar evolution code mesa, and find that only the jittering jets explosion mech-
+anism can account for explosions where carbon-oxygen cores with masses of ≳ 20M⊙ collapse to leave
+a neutron star (NS) remnant. We calculate the binding energy at core collapse under the assumption
+that the remnant is a NS. Namely, stellar gas above mass coordinate of ≃ 1.5−2.5M⊙ is ejected in the
+explosion. We find that the typical binding energy of the ejecta of stripped-envelope progenitors with
+carbon-oxygen core masses of MCO ≳ 20M⊙ is Ebind ≳ 2 × 1051 erg. Since only jet-driven explosion
+mechanisms can supply such high energies, we conclude that jets must explode such cores. We apply
+our results to SN 2020qlb, which is a SECCSN with a claimed core mass of ≃ 30−50M⊙, and conclude
+that the jittering jets explosion mechanism best account for such an explosion that leaves a NS.
+Keywords: stars: jets – stars: massive – supernovae: general – supernovae: individual: 2020qlb
+1. INTRODUCTION
+The binding energy of a core collapse supernova
+(CCSN) progenitor plays a crucial role in determining
+the explosion outcome, like explosion energy and rem-
+nant mass. Typical explosion energies are estimated to
+be in the range of Eexp ≃ 1050 − 1052 erg (e.g., Yang
+& Chevalier 2015; Utrobin et al. 2015; Gal-Yam 2019;
+Burrows & Vartanyan 2021a). The binding energy of
+the most massive pre-collapse cores have similar or even
+larger values than these typical explosion energies (e.g.,
+Pejcha & Thompson 2015; Bruenn et al. 2016; Chan
+et al. 2020; Wang et al. 2022; Burrows et al. 2020). The
+explosion mechanism should both overcome the binding
+energy and account for the explosion energy (radiation
++ final kinetic energy of the ejecta).
+Two theoretical explosion mechanisms of non-rotating
+(or slowly rotating) pre-collapse cores utilize the grav-
+itational energy of the collapsing core to power CC-
+SNe. These are the delayed-neutrino explosion mech-
+anism (e.g., Bethe & Wilson 1985; Ertl et al. 2016; Bur-
+rows et al. 2020; Bruenn et al. 2020; Bollig et al. 2021;
+Burrows & Vartanyan 2021b; Zha et al. 2023) and the
+jittering-jets explosion mechanism (e.g., Soker 2010; Pa-
+pish & Soker 2011; Gilkis & Soker 2015; Soker 2019,
+2022a). Studies show that the maximum energy that the
+delayed neutrino mechanism can supply to overcome the
+binding energy of the ejecta is ≃ 2 × 1051 erg, resulting
+in maximum explosion energies (after removing binding
+energy) of Eexp ≃ 2 × 1051 erg (e.g., Fryer et al. 2012;
+Ertl et al. 2016; Sukhbold et al. 2016; Gogilashvili et al.
+2021). In addition, the delayed neutrino mechanism has
+problems in producing the observed amount of 56Ni, in
+particular in stripped-envelope CCSNe (SECCSNe; for
+a recent study see Sawada & Suwa 2023)
+The jittering jets explosion mechanism, on the other
+hand, can account for much larger explosion energies
+(e.g., Gilkis et al. 2016; Soker 2022b).
+The mag-
+neto rotational explosion mechanism that works only
+for rapidly rotating pre-collapse cores can also account
+for large explosion energies (e.g., LeBlanc & Wilson
+1970; Khokhlov et al. 1999; L´opez-C´amara et al. 2013;
+Wheeler et al. 2015; Bromberg & Tchekhovskoy 2016;
+Kuroda et al. 2020; Gottlieb et al. 2022c,a; Fujibayashi
+et al. 2022; Powell et al. 2022) because the newly born
+neutron star (NS) launches fixed-axis jets. Whether the
+explosion is by jittering jets (most CCSNe according to
+that model; e.g., Soker 2022b) or by fixed-axis jets, the
+jets operate in a negative feedback cycle, i.e., the jet
+feedback explosion mechanism (e.g., Soker 2016a). As
+well, the jets can influence the direction of later jets (e.g.,
+Papish & Soker 2014; Gottlieb et al. 2022b). Even if the
+stochastically accreted mass has sub-Keplerian angular
+arXiv:2301.05144v1  [astro-ph.HE]  12 Jan 2023
+
+2
+momentum, it might still form an accretion belt that
+can launch jets (e.g., Schreier & Soker 2016; Garain &
+Kim 2022).
+The above discussion implies that CCSNe with large
+kinetic energies of Eexp ≳ 2 × 1051, e.g., SN 2020jfo
+(Ailawadhi et al. 2022 – Eexp
+=
+2.9 × 1051 erg);
+SN 2020qlb (West et al. 2022 – Eexp = 20 × 1051 erg);
+SN 2012au (Pandey et al. 2021 – Eexp = 4.8 − 5.4 ×
+1051 erg) require jets to drive the explosion.
+In this study we examine the interesting case of the
+hydrogen-poor and super-energetic CCSN SN 2020qlb.
+West et al. (2022) estimate the kinetic energy of
+SN 2020qlb as ≃ 20 × 1051 erg and suggest that a mag-
+netar supplies the large amount of energy of the ejecta.
+They also provide fitting parameters for the magnetar
+model (e.g., Maeda et al. 2007; Kasen & Bildsten 2010;
+Woosley 2010; Metzger et al. 2015; Nicholl et al. 2017;
+Gomez et al. 2022) and estimate the progenitor pre-
+explosion mass, namely, about the ejecta plus the rem-
+nant mass, to be Mej + Mrem ≃ few × 10M⊙.
+It seems nonetheless, that there are two reasons why
+the explosion of SN 2020qlb must be driven by jets and
+not by the delayed neutrino mechanism. The first one is
+that the formation of an energetic magnetar must be ac-
+companied by the launching of even more energetic jets
+at the explosion itself and possibly after the explosion
+as well (e.g. Soker 2016b, 2017; Soker & Gilkis 2017;
+Shankar et al. 2021; Soker 2022c,b). The second rea-
+son is the new finding of the present study, where by a
+stellar evolutionary code (section 2) we show (section 3)
+that the binding energies of such massive cores are much
+above what the delayed neutrino mechanism can supply
+(Janka 2012; Soker 2022b). In section 4 we summarize
+our results and discuss their implications in the context
+of the jet feedback explosion mechanism.
+2. NUMERICAL SCHEME
+We use the stellar evolution code mesa (Paxton et al.
+2010, 2013, 2015, 2018, 2019) to simulate the struc-
+ture of 52 CCSN progenitor models, all with initial
+metalicity of z = 0.02.
+We base our numerical rou-
+tine on the ‘20M pre ms to core collapse’ example from
+mesa r22.05.1, but simulate the evolution using ver-
+sion r15140. Starting from the zero age main sequence
+(ZAMS), we evolve the star until center He depletion
+(when helium abundance in the center is either ≃ 1%
+or ≃ 5%), at which point we numerically remove the
+hydrogen-rich envelope, leaving only the He core. We
+then evolve the star until core-collapse. We introduce
+several changes to the example routine to both adhere
+to our requirements, i.e., having at least 21 isotopes
+and having sufficiently high resolution (Shishkin & Soker
+2021), and to fit mesa version r15140. We expand on
+the numerical scheme in appendix A.
+By varying the ZAMS stellar mass, the wind parame-
+ters, and the exact time when we numerically remove
+the envelope we obtain stripped-envelope CCSN pro-
+genitors, i.e., hydrogen-poor stellar progenitors, with
+varying mass of a carbon-oxygen (CO) core, MCO ≃
+4 − 35M⊙. This mass range of SECCSNe corresponds
+to ZAMS stellar masses range of ≃ 20 − 82M⊙. In some
+cases the core contains several solar masses of helium,
+while in other cases the core has a much lower helium
+mass, depending on the above parameters.
+Because the inner part of the core collapses to form
+a NS or a black hole (BH) remnant, only the binding
+energy of the outer core is relevant to our study. We
+calculate the binding energy of the outer core, Ebind(r),
+by integrating over the sum of the internal energy and
+gravitational energy from the surface down to the mass
+coordinate that separates the ejecta and the final rem-
+nant m = Min, i.e., the inner boundary of the ejecta. We
+calculate the binding energy of the ejecta for two values
+of this mass coordinate Min = 1.5 , 2.5M⊙, because we
+consider cases where the remnant is a NS.
+We refer to the latest evolution point we simulate
+in the stellar evolution as collapse. This point of col-
+lapse must adhere to an iron core more massive than
+MFe > 1.5, where we assume collapse is imminent. This
+value of iron core mass is more or less when the iron
+core mass reaches its maximum value (as at the onset
+of collapse the iron disintegrates). About half of simu-
+lations reach infall velocities at the edge of the iron core
+of vfe,infall > 100 km s−1. Other simulations encounter
+numerical difficulties and we had to terminate them at
+somewhat earlier times.
+3. RESULTS
+3.1. Binding Energy towards Collapse
+As the stellar core evolves towards collapse and nu-
+cleosynthesis of heavier elements takes place, the core
+becomes denser and the binding energy of the inner lay-
+ers of the star increases. We demonstrate this for one
+stellar model of carbon-oxygen core mass (mainly oxy-
+gen) of MCO = 13.2M⊙ in Fig 1 where we present the
+star at three times: at center oxygen depletion, at cen-
+ter silicon depletion, and at collapse. We present the
+composition of the main isotopes by lines with differ-
+ent colors and the binding energy Ebind(m) by the black
+lines. Here Ebind(m) is the binding energy (gravitational
++ internal) of the envelope laying above mass coordi-
+nate m. Relevant to this study is the binding energy
+of the ejecta, which is the mass above mass coordinate
+m = Min ≃ 1.5, 2.5M⊙. The mass Min is the baryonic
+
+3
+Figure 1. Abundances and the binding energy as function
+of mass coordinate at three times for a SECCSN (hydrogen-
+poor) progenitor model with carbon-oxygen core mass of
+M collapse
+CO
+= 13.2M⊙, which corresponds to a ZAMS mass of
+MZAMS ≈ 40M⊙. The colored step-like lines are the abun-
+dances according to the inset in the lower panel. The black
+and smoothly varying lines represent Ebind(m), which is the
+binding energy of the envelope laying above mass coordi-
+nate m. The three panels present these quantities at three
+different times: center oxygen depletion (upper panel, bind-
+ing energy by the black solid-line), center silicon depletion
+(middle panel; binding energy by the black dashed-line), core
+collapse (lower panel; binding energy by black dotted-line).
+Note that the middle panel contains the binding energy at
+the three times to allow for comparison. The two vertical
+lines mark the mass coordinates m = Min = 1.5M⊙ and
+m = Min = 2.5M⊙. Helium that appears only at collapse
+results from disintegration of iron.
+mass of the NS remnant (the corresponding final gravi-
+tational masses will be ≃ 1.35, 2.1M⊙). We mark these
+two masses by the vertical lines. We see that at collapse
+the binding energy Ebind(Min) is larger than at earlier
+times.
+In Fig.
+2 we present composition and binding en-
+ergy for a model with a much more massive core of
+MCO = 27M⊙.
+There are two qualitative differences
+between this model and the one we present in Fig. 1.
+The first qualitative difference is that the binding en-
+Figure 2. Similar to Fig. 1 but for a more massive core of
+M collapse
+CO
+= 26.5M⊙, which corresponds to a ZAMS mass of
+MZAMS ≈ 65M⊙. Note that the left vertical axis is scaled
+differently than in Fig. 1.
+ergy at collapse is somewhat smaller than at the earlier
+time that we present in the figure. The explanation to
+the decreasing binding energy shortly before collapse is
+that the envelope expands starting from deep in the oxy-
+gen burning shell and outwards. We find (by drawing
+the density profiles) that moving from the upper to the
+middle panel of Fig. 2 the density from m ≃ 10M⊙ and
+outward decreases, reducing the binding energy. This
+mass coordinate is deep inside the shell where oxygen
+(teal line) burns to S+Si (yellow line).
+The second qualitative difference comes from the much
+higher binding energy of the ejecta of the descendant
+CCSN of the more massive model, i.e., Ebind(Min) ≳ 2×
+1051 erg. The implication is that we do not expect that
+the neutrino driven explosion mechanism can account
+for explosions of such cores. We argue that jets explode
+these cores. We leave the discussion of this point, as
+well as our view that jets also explode cores with lower
+binding energy, to section 4, where we also refer to the
+claim of a very massive core of SN 2020qlb (West et al.
+2022).
+We first find the range of such high-binding-
+energy cores.
+
+0
+3
+OCoreDepletion
+2
+91
+erg
+0
+SiCoreDepletion
+BEatO
+dep.
+BEatSi
+dep
+BEatCollapse
+2
+0
+3
+Collapse
+2
+Helium
+Carbon
+Oxygen
+S+Si
+"Fe Group"
+2
+0
+2
+4
+6
+8
+10
+12Abundance6
+0
+4
+OCoreDepletion
+2
+6
+0
+Si Core Depletion
+BEatOont
+. dep.
+- BEat Si
+dep
+"BEatCollapse
+2
+0
+4
+Collapse
+2
+Helium
+Oxygen
+"Fe Group'
+Carbon
+S+Si
+2
+0
+4
+8
+12
+16
+20
+24Abundance4
+3.2. High-binding-energy cores
+We search for the mass range of cores that have bind-
+ing energies at collapse of Ebind(Min) ≳ 2×1051 erg. We
+present the results in Fig. 3. We present the binding en-
+ergy for an inner ejecta mass coordinate of Min = 2.5M⊙
+(upper panel) and Min = 1.5M⊙ (lower panel).
+We
+focus on the binding energy of these two mass coordi-
+nates Min = 1.5 − 2.5M⊙ as the iron core masses at col-
+lapse falls within this mass range. The horizontal line
+at 2 × 1051 erg is the approximate energy above which
+we do not expect that neutrino heating by itself can ex-
+plode the core. In appendix B we provide linear fits to
+the binding energy at collapse as function of CO core
+mass for these two mass coordinates Min = 1.5, 2.5M⊙
+(table B.1).
+We simulated 52 stripped-hydrogen envelope cases. In
+27 cases the cores reach collapse as we present by the
+red circles in the figure. In 25 cases cases the numeri-
+cal code encountered difficulties and we had to stop the
+simulation before reaching collapse. In these cases we
+extrapolate from an evolutionary time before collapse
+to the collapse time as we explain in appendix B (red
+stars in the figure). We also include 35 in the figure the
+binding energies of models with hydrogen-rich envelope
+that we take from Shishkin & Soker (2022), as we mark
+by open purple circles.
+From Fig. 3 (with a more rigorous derivation in ap-
+pendix B) we draw our conclusion that in cases where
+the inner mass of Min = 2.5M⊙ of the core collapses
+to form a NS, the delayed neutrino mechanism cannot
+explode cores with masses of MCO ≳ 15M⊙ (or maybe
+rarely do so). For Min = 1.5M⊙ we find this limit to be
+MCO ≳ 13M⊙.
+In Fig. 4 we present a more detailed binding energy
+profile of the pre-collapse stripped-envelope models that
+we simulated. When we take the lowest binding energy
+of the inner core at the onset of collapse we find the limit
+of core mass that the neutrino-driven explosion cannot
+account for to be MCO > 20M⊙. Namely, still a large
+range.
+In table B.1 we provide the linear fit parameters for
+the binding energy at collapse for the edge of the iron
+core (gray squares in the figure) and the binding energy
+curve break (black circles). The binding energy curve
+break is point we refer to as separating the inner core
+from the outer core, and is the point of lowest binding
+energy for most simulated cases that reached collapse.
+4. DISCUSSION AND SUMMARY
+We simulated the evolution of 52 massive SECCSN
+progenitor models corresponding to ZAMS masses of
+20 ≲ MZAMS ≲ 82. We removed the entire hydrogen-
+rich envelope, and calculated the binding energy just
+before core collapse.
+The final core mass depends on
+the ZAMS mass and on the mass loss parameter (ap-
+pendix C). We present the structure of the pre-collapse
+progenitor for two cases in Figs. 1 and 2. We find that
+to a fare accuracy we can linearly fit the binding energy
+of these stripped-envelope progenitors to the CO core
+mass MCO (Fig. 3 and table B.1).
+We present our main results in Fig.
+3.
+In those
+figures the horizontal gray line represents the approx-
+imate maximum energy that the neutrino-driven mech-
+anism can supply, Emax
+ν
+= 2 × 1051 erg. We find that
+the binding energy calculated at Min = 1.5M⊙ and
+Min = 2.5M⊙, of progenitors with a carbon-oxygen core
+mass of MCO ≳ 13M⊙ and MCO ≳ 15M⊙, respectively,
+are larger than Emax
+ν
+. Namely,
+Emax
+ν
+≲
+�
+�
+�
+Ebind,1.5
+for
+MCO ≳ 13M⊙
+Ebind,2.5
+for
+MCO ≳ 15M⊙.
+(1)
+The main conclusion is that the delayed neutrino ex-
+plosion mechanism cannot explode stars with a core
+mass of MCO ≳ 13 − 15M⊙. The jittering jets explosion
+mechanism, on the other hand, has no limiting explosion
+energy in these ranges as it is fueled by accretion onto
+the compact remnant (e.g., (Gilkis et al. 2016; Soker &
+Gilkis 2017)).
+Let us apply our results to a specific SECCSN. In a
+recent paper West et al. (2022) deduce that SN 2020qlb
+had an explosion energy of ≃ 20×1051 erg and estimate
+the progenitor pre-explosion mass, ejecta plus remnant
+mass, to be Mej + Mrem ≃ 30 − 50M⊙. According to
+our results the binding energy alone of such cores is
+Ebind > 3 × 1051 erg. We therefore conclude that jets
+must have exploded SN 2020qlb. Jets can also supply
+the kinetic energy of the ejecta. Namely, jet-driven ex-
+plosions might make the magnetar powering less criti-
+cal or not needed at all (although a magnetar might be
+present). Most likely the explosion is via jittering jets.
+The reason is that an explosion driven by a fixed-axis
+jets, like if the core is rapidly rotating, will not expel
+mass from the equatorial region, which it turn is ac-
+creted by the newly formed central object. Therefore,
+the final mass of the remnant will be large and the rem-
+nant will be a BH (see discussion in Soker 2022b).
+On a large scope, our study adds to the growing evi-
+dence pointing to the major roles that jets play in the
+explosion, as well as pre-explosion and post-explosion,
+of CCSNe (for a recent review see Soker 2022b).
+ACKNOWLEDGMENTS
+This research was supported by a grant from the Israel
+Science Foundation (769/20).
+
+5
+Figure 3. The binding energies of the simulated models as a function of the carbon-oxygen core mass nearing collapse (vertical
+axis). The panels show the final binding energy at two mass coordinates: Min = 2.5M⊙ (top) and Min = 1.5M⊙ (bottom). The
+red data points (filled circles and stars at the outer panels) are stripped-envelope (SE) models (SECCSNe), whilst purple empty-
+circles data points are models from Shishkin & Soker (2022) that have hydrogen rich envelopes. Red stars are extrapolated
+data points, as explained in appendix B. The horizontal line at 2 × 1051 erg denotes the binding energy above which we do not
+expect the neutrino delayed explosion mechanism to explode the core.
+Data availability
+The data underlying this article will be shared upon
+reasonable request to the corresponding author.
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+9
+APPENDIX
+A. NUMERICAL PRESCRIPTION DETAILS
+Our numerical scheme files (‘inlists’) are a modified version of the ‘20 pre ms to cc’ mesa version r22.05.1 ‘test suite’
+example. We adapted this example to run on mesa version r15140 and incorporated certain parameters (e.g., over-
+shooting and mesh resolution) according to our previous works (Shishkin & Soker 2021; Shishkin & Soker 2022). The
+full ’inlists’ that we used are available online1. Here we mention some of the important parameters.
+In a similar fashion to our previous works which focus on the convective profile of the inner layers of massive stars
+(Shishkin & Soker 2021; Shishkin & Soker 2022), we use the exponential overshooting prescription (Herwig 2000)
+with symmetrical (both ‘bottom’ and ‘above’) and uniform (all burning regions) settings and f = 0.01, f0 = 0.004
+parameters.
+We chose the Henyey scheme (Henyey et al. 1965) for mixing length theory (MLT, B¨ohm-Vitense 1958) with
+αMLT = 1.5.
+We also enable the Ledoux criterion (Ledoux 1947) and set thermohaline option to ‘Kippenhahn’
+with ‘thermohaline coeff = 1’ alongside ‘alpha semiconvection = 0.01.
+We make use of the ’Dutch’ wind loss scheme (e.g., Vink et al. 2001; Nugis & Lamers 2000), and vary the scaling
+factor (along with the initial mass) to achieve different core masses.
+For the nuclear network we use the 22 isotopes of ‘approx21 cr60 plus co56’ (e.g., Timmes 1999), aimed at
+stellar evolution up to collapse.
+This network includes hydrogen, He3 and He4 up to the heavier isotopes of
+Fe52 , Fe54 , Fe56 , Co56 , Ni56 , Cr60.
+We scale mesh refinement gradually up to ‘max dq’ values of 1d − 4 at the later stages (from the default value of
+1d − 2) to properly resolve the fine burning features close to core collapse.
+The mesa equation of state (EOS) is a blend of the OPAL (Rogers & Nayfonov 2002), SCVH (Saumon et al. 1995),
+FreeEOS (Irwin 2004), HELM (Timmes & Swesty 2000), PC (Potekhin & Chabrier 2010), and Skye (Jermyn et al.
+2021) EOSs. Nuclear reaction rates are from JINA REACLIB (Cyburt et al. 2010), NACRE (Angulo et al. 1999) and
+additional tabulated weak reaction rates Fuller et al. (1985); Oda et al. (1994); Langanke & Mart´ınez-Pinedo (2000).
+Screening is included via the prescription of Chugunov et al. (2007). Thermal neutrino loss rates are from Itoh et al.
+(1996). Radiative opacities are primarily from OPAL (Iglesias & Rogers 1993, 1996), with low-temperature data from
+Ferguson et al. (2005) and the high-temperature, Compton-scattering dominated regime by Poutanen (2017). Electron
+conduction opacities are from Cassisi et al. (2007) and Blouin et al. (2020).
+B. BINDING ENERGY ESTIMATION
+Because of numerical difficulties of stripped-envelope progenitors (specifically some steep gradients) some simulations
+did not reach the phase of core collapse, although they did reach oxygen depletion and/or silicon depletion at the center.
+Time steps became much too short and we had to terminate the simulations before core collapse. In these cases we
+estimated the binding energy at collapse (red-stars in Fig. 3) by extrapolating the binding energy during earlier phases
+using linear fits.
+We made linear fits to the binding energies as function of the CO core masses at three evolutionary phases: oxygen
+depletion, silicon depletion, and core collapse. In Fig. B.1 we present these three fittings by blue, orange, and red
+lines, respectively, for Min = 2.5M⊙ (upper panel) and Min = 1.5M⊙ (lower panel). From these three lines we can find
+the ratio of the binding energy at core collapse to the binding energy at oxygen depletion and to the binding energy
+at silicon depletion. In cases where we did not reach core collapse we use this ratio at the given CO core mass to
+calculate the expected binding energy at core collapse. We mark these energies by red-stars in Fig. B.1 and use them
+in Fig. 3. Error bars attached to the red stars signify the 1σ intervals of the this extrapolation procedure. We note
+that the CO core mass does not change much after oxygen depletion in the non-extended helium phase. The average
+difference between the CO core mass at central oxygen depletion and at core collapse is ∆mcore
+CO = 0.06 ± 0.33M⊙.
+1 Zenodo: Modified inlists to reproduce the models. Also included
+a full simulation list and the simulated models at different time
+points.
+
+10
+Figure B.1. The binding energy of the envelope above mass coordinate Min = 2.5M⊙ (upper panel) and Min = 1.5M⊙ (lower
+panel) as a function of the final carbon-oxygen core mass. The blue circles are at central oxygen depletion (5% oxygen in the
+center), the orange circles are at silicon depletion (5% silicon in the center), and red circles are at core collapse. The three
+respective lines are the linear fit to the points. Red stars are the extrapolated values for the binding energy at collapse based
+on available earlier data points (oxygen depletion or silicon depletion) for the cases that did not reach collapse (see text).
+We fit the binding energy Ebind versus the CO core mass MCO by a linear fit Ebind = aMCO + b. In Table B.1 we
+list the values of the two coefficients for the six lines (three stage for two values of the mass that collapses to form the
+NS). We also list (last column) the number of data points that were used at each fitting.
+
+11
+aMCO + b
+Fit1.5M⊙
+Fit2.5M⊙
+FitBEbreak
+FitFecore
+No. of points
+Collapse
+a [1051erg/M⊙]
+0.122 ± 0.024
+0.137 ± 0.024
+0.117 ± 0.021
+0.114 ± 0.02
+27
+b [1051erg]
+0.537 ± 0.46
+−0.053 ± 0.444
+0.165 ± 0.423
+0.442 ± 0.406
+Sicntr depletion
+a [1051erg/M⊙]
+0.132 ± 0.017
+0.145 ± 0.02
+−−
+−−
+43
+b [1051erg]
+0.159 ± 0.38
+−0.099 ± 0.445
+−−
+−−
+Ocntr depletion
+a [1051erg/M⊙]
+0.15 ± 0.017
+0.175 ± 0.02
+−−
+−−
+47
+b [1051erg]
+0.002 ± 0.394
+−0.447 ± 0.479
+−−
+−−
+Table B.1. The linear fits to the lines in Fig. B.1 and the number of data points for each of the simulation groups: at collapse
+(second row), at center silicon depletion (third row) and center oxygen depletion (bottom row). The third and fourth columns
+are the fits to the binding energy Ebind,1.5, Ebind,2.5 at mass coordinates Min = 1.5M⊙ and Min = 2.5M⊙, respectively. In the
+fifth column we present the linear fit to the variation of the binding energy at the black dots in Fig. 4 with the CO core mass.
+In the fifth column we present the linear fit to the variation of the binding energy at the edge of the iron core (gray squares in
+Fig. 4) with the CO core mass. Linear fits are in units of energy Ebind [1051erg] to CO core mass MCO [M⊙]. Values and errors
+(2σ) are in accordance with Fig. B.1.
+
+12
+C. SIMULATIONS LIST
+In Fig. C.2 we present the simulations that we conducted in a three-parameters space. As input we show the dutch
+wind scaling factor in mesa in the range of 0.5 < αDutch,wind < 1 and the zero age main sequence (ZAMS) mass in the
+range of 20M⊙ < MZAMS < 82M⊙. As an output we present the final CO core mass (mainly oxygen mass) in units of
+solar mass according to the color bar.
+Figure C.2. Wind (Dutch scheme) scaling factors (vertical axis) and the ZAMS masses of the cases (horizontal axis) that we
+simulated, and the final CO core mass (by color bar). We denote with a black X the cases where we extended the He burning
+to a later stage before removing the hydrogen envelope.
+

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