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+Efficiently unquenching QCD+QED at O(𝜶)
+Tim Harris,𝑎,∗ Vera Gülpers,𝑎 Antonin Portelli𝑎 and James Richings𝑎,𝑏
+𝑎School of Physics and Astronomy, University of Edinburgh,
+Edinburgh EH9 3FD, United Kingdom
+𝑏EPCC, University of Edinburgh,
+EH8 9BT, Edinburgh, United Kingdom
+E-mail: tharris@ed.ac.uk
+We outline a strategy to efficiently include the electromagnetic interactions of the sea quarks
+in QCD+QED. When computing iso-spin breaking corrections to hadronic quantities at leading
+order in the electromagnetic coupling, the sea-quark charges result in quark-line disconnected
+diagrams which are challenging to compute precisely. An analysis of the variance of stochastic
+estimators for the relevant traces of quark propagators helps us to improve the situation for certain
+flavour combinations and space-time decompositions. We present preliminary numerical results
+for the variances of the corresponding contributions using an ensemble of 𝑁f = 2 + 1 domain-wall
+fermions generated by the RBC/UKQCD collaboration.
+The 39th International Symposium on Lattice Field Theory (Lattice2022),
+8-13 August, 2022
+Bonn, Germany
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.03995v1  [hep-lat]  10 Jan 2023
+
+Efficiently unquenching QCD+QED at O(𝛼)
+Tim Harris
+1.
+Introduction
+Several lattice QCD predictions which form important input for precision tests of the Standard
+Model have uncertainties at or below the 1% level, for example the HVP contribution to (𝑔 − 2)𝜇,
+𝑓𝐾/ 𝑓𝜋, 𝑔A or the Wilson flow scale √𝑡0 to name a few [1, 2].
+However, to further improve
+such predictions, QCD with iso-spin symmetry is not a sufficiently accurate effective description
+of the low-energy dynamics and QED, which contributes one source of iso-spin breaking due to
+the different up- and down-quark electric charges, must be included. Recent efforts have been
+successful at including iso-spin breaking corrections, and some of which fully account for the
+effects of the sea-quark electric charges [3, 4, 5, 6, 7].
+Nevertheless, many computations of
+iso-spin breaking effects still neglect to incorporate these dynamical effects in an approximation
+known as electroquenching. As the FLAG report notes in Section 3.1.2 [2], computations using the
+electroquenched approximation might feature an uncontrolled systematic error.
+In this work we aim to include the effects of the electric charge of the sea quarks in the
+perturbative method known as the RM123 approach. This amounts to computing at least two
+additional Wick contractions.
+In order to sum the vertices in the resulting diagrams over the
+lattice volume, some approximations must be used which often introduce additional fluctuations,
+for example due to the auxiliary fields of a stochastic estimator. Here we investigate some simple
+decompositions which may avoid large contributions to the variance, so that sufficiently precise
+results can be obtained to systematically include all sources of iso-spin breaking without incurring
+a large computational cost.
+2.
+Sea-quark effects in the RM123 method
+Due to the smallness of the fine-structure constant 𝛼 ∼ 1/137 and the renormalized light-
+quark mass difference (𝑚R
+u − 𝑚R
+d )/Λ ∼ 1%, it is natural to expand physical observables (i.e. in
+QCD+QED) in these parameters to compute iso-spin breaking corrections, as was first outlined in
+Refs. [8, 9]. In the resulting expansion of an observable 𝑂
+⟨𝑂⟩ = ⟨𝑂⟩
+���
+𝑒=0 + 1
+2𝑒2� 𝜕
+𝜕𝑒
+𝜕
+𝜕𝑒 ⟨𝑂⟩
+�
+𝑒=0 + . . .
+(1)
+the leading corrections in the electric charge 𝑒 =
+√
+4𝜋𝛼 are parameterized in terms of the correlation
+function
+𝜕
+𝜕𝑒
+𝜕
+𝜕𝑒 ⟨𝑂⟩ = (−i)2
+∫
+d4𝑥
+∫
+d4𝑦 ⟨𝐽𝜇(𝑥)𝐴𝜇(𝑥)𝐽𝜈(𝑦)𝐴𝜈(𝑦)𝑂⟩c
+(2)
+where the electromagnetic current for u, d, s quark flavours is defined
+𝐽𝜇 =
+∑︁
+𝑓 =u,d,s
+𝑄 𝑓 ¯𝜓 𝑓 𝛾𝜇𝜓 𝑓 ,
+𝑄u = 2
+3,
+𝑄d = 𝑄s = −1
+3.
+(3)
+By choosing the expansion point to be a theory with 𝛼 = 0 and iso-spin symmetry 𝑚u = 𝑚d,
+only correlation functions in the 𝑁f = 2 + 1 theory need to be evaluated, which we denote with
+𝑒 = 0 in Eq. (1). The precise definition of such a theory using an additional set of renormalization
+conditions is necessary to fix the meaning of the leading-order term on the right-hand side (and
+2
+
+Efficiently unquenching QCD+QED at O(𝛼)
+Tim Harris
+𝑊1
+𝑂
+𝑊2
+𝑂
+𝑊3
+𝑂
+𝑊4
+𝑂
+Figure 1: Wick contractions which appear at leading order in the expansion of a hadronic observable 𝑂
+in the electromagnetic coupling. Each closed fermion line has contributions from all of the quark flavours
+u, d, s, . . . with the appropriate charge factors.
+conversely the iso-spin breaking corrections themselves). Otherwise the predictions of QCD+QED
+are unambiguously defined, up to its intrinsic accuracy, by fixing 𝑁f quark masses and the QCD
+coupling as the electric coupling does not renormalize at this order. In the above, the ellipsis stands
+for the mass counterterms which are needed to make physical predictions due to the contribution to
+the quark self-energy induced by QED.
+After integrating out the fermion and photon fields, the resulting Wick contractions 𝑊𝑖 are
+shown in Fig. 1, which contribute to the derivative with respect to the electric charge through the
+connected correlation function
+𝜕
+𝜕𝑒
+𝜕
+𝜕𝑒 ⟨𝑂⟩ =
+4
+∑︁
+𝑖=1
+⟨𝑂𝑊𝑖⟩c.
+(4)
+The first two subdiagrams, which arise soley from the electric charges of the sea quarks, can be
+expressed in terms of a convolution with the photon propagator (in some fixed gauge) 𝐺 𝜇𝜈(𝑥) =
+⟨𝐴𝜇(𝑥)𝐴𝜈(0)⟩
+𝑊1,2 = −𝑎8 ∑︁
+𝑥,𝑦
+𝐻𝜇𝜈
+1,2(𝑥, 𝑦)𝐺 𝜇𝜈(𝑥 − 𝑦),
+(5)
+where 𝐻1,2 are the traces of quark propagators 𝑆 𝑓 (𝑥, 𝑦) = ⟨𝜓 𝑓 (𝑥) ¯𝜓 𝑓 (𝑦)⟩
+𝐻𝜇𝜈
+1 (𝑥, 𝑦) =
+∑︁
+𝑓 ,𝑔
+𝑄 𝑓 𝑄𝑔 tr{𝛾𝜇𝑆 𝑓 (𝑥, 𝑥)} tr{𝛾𝜈𝑆𝑔(𝑦, 𝑦)},
+(6)
+𝐻𝜇𝜈
+2 (𝑥, 𝑦) = −
+∑︁
+𝑓
+𝑄2
+𝑓 tr{𝛾𝜇𝑆 𝑓 (𝑥, 𝑦)𝛾𝜈𝑆 𝑓 (𝑦, 𝑥)}.
+(7)
+These two diagrams are the main subject of these proceedings, and the techniques advocated for
+the first can be effectively reused for the third diagram, 𝑊3. In the following sections we introduce
+stochastic estimators only for the quark lines and compute the subdiagrams by convoluting with the
+exact photon propagator which avoids introducing additional stochastic fields for the U(1) gauge
+potential. The final diagram 𝑊4, which only contributes if the observable 𝑂 depends explicitly
+on the (charged) fermion fields, is the only one surviving the electroquenched approximation, and,
+can in most cases be computed efficiently provided that the leading-order diagram is already under
+control.
+3
+
+Efficiently unquenching QCD+QED at O(𝛼)
+Tim Harris
+We note that the variance of the contributions to the connected correlation functions on the
+r.h.s. of Eq. (4) crudely factorizes
+𝜎2
+𝑂𝑊1,2 ≈ ⟨𝑂⟩2
+c ⟨𝑊1,2⟩2
+c + ⟨𝑂𝑊1,2⟩c
+(8)
+≈ 𝜎2
+𝑂𝜎2
+𝑊1,2,
+(9)
+where in the first line we have made the Gaussian approximation, and in the second line we have
+assumed that the fluctuations are much larger than the signal ⟨𝑂𝑊1,2⟩c. Thus, in the following
+sections we will analyse the variance of individual subdiagrams 𝑊1,2 in order to gain a rough
+insight into the fluctuations of the total correction, in a similar fashion to the analysis of Ref. [10].
+In that case, however, the correction to the factorization of the variance is exponentially suppressed
+in the separation between the vertices of the subdiagrams.
+3.
+Quark-line disconnected subdiagram 𝑊1
+We begin by noting that the hadronic part of the diagram factorizes into two traces,
+𝐻𝜇𝜈
+1 (𝑥, 𝑦) = 𝑇𝜇(𝑥)𝑇𝜈(𝑦),
+(10)
+each of which, with the current defined in Eq. (3) and in the 𝑁f = 2 + 1 theory with iso-spin
+symmetry, is the difference of the light- and strange-quark propagators
+𝑇𝜇(𝑥) = 1
+3 tr{𝛾𝜇[𝑆ud(𝑥, 𝑥) − 𝑆s(𝑥, 𝑥)]}.
+(11)
+It is convenient to rewrite this difference as a product [10]
+𝑆ud − 𝑆s = (𝑚s − 𝑚ud)𝑆ud𝑆s
+(12)
+which makes the explicit suppression of 𝑇𝜇 in the SU(3)-symmetry breaking parameter 𝑚s − 𝑚ud
+explicit. This additionally results in a suppression of the variance of 𝑊1 by (𝑚s − 𝑚ud)4. This
+suppression results in a cancellation of a quartic short-distance divergence in the variance of the
+contribution of each individual flavour to 𝑊1, explaining this favourable flavour combination.
+While the identity in Eq. (12) is easily derived for Wilson-type fermions, here we sketch that
+it holds exactly for the domain-wall fermion valence propagator 𝑆 𝑓 = ˜𝐷−1
+𝑓 which (approximately)
+satisfies the Ginsparg-Wilson relation [11]. Recalling the definition of ˜𝐷 𝑓 in terms of the 5D
+Wilson matrix 𝐷5, 𝑓 (see Ref. [12] for unexplained notation)
+˜𝐷−1
+𝑓 = (P−1𝐷−1
+5, 𝑓 𝑅5P)11,
+(13)
+where the matrix indices indicate the coordinate in the fifth dimension, the result is obtained
+immediately from
+˜𝐷−1
+ud − ˜𝐷−1
+s
+= (𝑚s − 𝑚ud)(P𝐷−1
+5,ud𝑅5𝐷−1
+5,s𝑅5)11
+(14)
+by noting that the following matrix projects on the physical boundary
+(𝑅5)·· = (𝑅5P)·1(P−1)1·.
+(15)
+4
+
+Efficiently unquenching QCD+QED at O(𝛼)
+Tim Harris
+𝐿/𝑎
+𝑇/𝑎
+𝑚 𝜋
+𝑚 𝜋𝐿
+𝑎
+𝑁cfg
+24
+64
+340 MeV
+4.9
+0.12 fm
+50
+Table 1: The parameters of the C1 ensemble of 𝑁f = 2 + 1 Shamir domain-wall fermions used in the
+numerical experiments in this work, see Ref. [17] for details.
+The preceding identity is easily demonstrated using the explicit representations
+𝑅5 =
+���
+�
+𝑃+
+𝑃−
+���
+�
+,
+P−1 =
+������
+�
+𝑃−
+𝑃+
+𝑃+
+...
+...
+𝑃+
+𝑃−
+������
+�
+,
+(16)
+where 𝑃± = 1 ± 𝛾5.
+Using the identity for the difference, there are two independent estimators for the trace
+Θ𝜇(𝑥) = 1
+3 (𝑚s − 𝑚ud) 1
+𝑁s
+𝑁s
+∑︁
+𝑖=1
+𝜂†
+𝑖 (𝑥)𝛾𝜇{𝑆ud𝑆s𝜂𝑖}(𝑥),
+(17)
+T𝜇(𝑥) = 1
+3 (𝑚s − 𝑚ud) 1
+𝑁s
+𝑁s
+∑︁
+𝑖=1
+{𝜂†
+𝑖 𝑆s}(𝑥)𝛾𝜇{𝑆ud𝜂𝑖}(𝑥),
+(18)
+where the auxiliary quark fields 𝜂𝑖(𝑥) have zero mean and finite variance.
+The properties of
+both estimators were investigated in detail in Ref. [10], where it was shown that the contribution
+to the variance from the auxiliary fields for the second split-even estimator was in the region of
+a factor O(100) smaller than the first standard estimator, which translates into the same factor
+reduction in the cost. The split-even estimator has since been used extensively for disconnected
+current correlators [13, 14, 15], while in the context of the twisted-mass Wilson formulation similar
+one-end trick estimators have often been employed for differences of twisted-mass propagators [16].
+In this work we propose an estimator for the first diagram 𝑊1 using
+W1 ≈
+�
+𝑎4 ∑︁
+𝑥
+T𝜇(𝑥)
+� �
+𝑎4 ∑︁
+𝑦
+T𝜈(𝑦)𝐺 𝜇𝜈(𝑥 − 𝑦)
+�
+(19)
+where independent estimators are used for the two traces to avoid incurring a bias with a finite
+sample size. The convolution in the second parentheses can be efficiently computed using the
+Fast Fourier Transform (FFT). With a minor modification, an estimator using all possible unbiased
+combinations of samples can be written at the cost of performing O(𝑁s) FFTs.
+The standard
+estimator is obtained by replacing both occurances of T𝜇 with Θ𝜇 in Eq. (19).
+We performed an analysis of the variance for the standard and split-even estimators for W1
+using the domain-wall ensemble generated by the RBC/UKQCD collaboration whose parameters
+are listed in Tab. 1. The photon propagator is computed in the QED𝐿 formulation [18] in the
+Feynman gauge. The results for the variances, which are dimensionless numbers, are shown in
+Fig. 2. In addition, we plot the variance for the contribution of a single flavour Wu
+1 using the
+5
+
+Efficiently unquenching QCD+QED at O(𝛼)
+Tim Harris
+10−4
+10−3
+10−2
+10−1
+100
+101
+102
+103
+104
+105
+106
+107
+108
+109
+1
+10
+100
+1000
+σ2
+Ns
+Wu
+1
+Wuds
+1
+(standard)
+Wuds
+1
+(split-even)
+1/N 2
+s
+Figure 2: Left: Comparison of the variance versus the number of sources for the 𝑊1 quark-line disconnected
+diagram, using a single flavour (red squares), the standard estimator for u, d, s flavours (blue circles) and the
+split-even estimator (green triangles). The dashed line shows 1/𝑁2
+s scaling. In this figure, the (local) currents
+are not renormalized and the charge factors are not included.
+standard estimators for the traces. We note that all the variances are dominated by the fluctuations
+of the auxiliary fields for small 𝑁s, and in particular scale like 1/𝑁2
+s in that region.
+As expected, the standard estimator including the light-quark and strange-quark contributions
+(blue circles) is suppressed with respect to the contribution of a single flavour (red squares).
+Furthermore, the variance of the split-even estimator (green triangles) is reduced by a factor of 104
+with respect to the standard one (blue circles). This reduction is commensurate with the reduction
+in the variance observed for the disconnected contribution to the current correlator [10], which
+suggests the same mechanisms are present here. For 𝑁s ∼ 100, the variance is independent of
+the number of auxiliary field samples which indicates that it is dominated by the fluctuations of
+the gauge field. In this case no further variance reduction is possible for a fixed number of gauge
+configurations. Finally we note that the convolution of the second parentheses of Eq. (19) can be
+simply inserted sequentially in any of the diagrams of type 𝑊3.
+4.
+Quark-line connected subdiagram 𝑊2
+In contrast to the quark-line disconnected subdiagram, there is no cancellation in the variance
+in the connected subdiagram 𝑊2 between the light and strange-quark contributions. In this case,
+power counting suggests that the variance diverges with the lattice spacing like 𝑎−4 as 𝑎 → 0 and is
+expected to be dominated by short-distance contributions. Translation averaging should therefore
+be very effective and one way to implement it is to use an all-to-all estimator [19] for the quark
+propagator
+S 𝑓 (𝑥, 𝑥 + 𝑟) = 1
+𝑁s
+𝑁s
+∑︁
+𝑖=1
+{𝑆 𝑓 𝜂𝑖}(𝑥)𝜂†
+𝑖 (𝑥 + 𝑟),
+(20)
+6
+
+Efficiently unquenching QCD+QED at O(𝛼)
+Tim Harris
+10−7
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+102
+103
+104
+105
+106
+107
+108
+109
+0
+2
+4
+6
+8
+10
+12
+σ2
+|r|/a
+H2(r)G(r)
+H2, Ns = 1
+¯H2, NX = 1
+H2, Ns = ∞
+10−2
+10−1
+100
+101
+102
+103
+104
+105
+106
+1
+10
+100
+1000
+σ2
+Ninv
+W2, R/a = 4
+�
+r≤R H2G
+�
+r>R ¯H2G
+R = 0, Ns = ∞
+Figure 3: Left: the variance for the stochastic estimator (red squares) and point source estimator (blue
+circles) for the minimum number of inversions required, for the contribution with fixed separation between
+the currents |𝑟|. The green triangle indicates the gauge variance for the point 𝑟 = 0. Right: the variance for
+the short-distance (red squares) and long-distance (blue circles) for the choice 𝑅/𝑎 = 4, versus the number
+of inversions. The green band indicates the gauge variance for the contribution from 𝑟 = 0 only. The dashed
+lines indicate the expected leading 𝑁−2
+inv and 𝑁−1
+inv scaling for the short- and long-distance components.
+using independent fields for each propagator in the trace
+H 𝜇𝜈
+2
+(𝑟) = 𝑎4 ∑︁
+𝑥
+∑︁
+𝑓
+𝑄2
+𝑓 tr{𝛾𝜇S 𝑓 (𝑥, 𝑥 + 𝑟)𝛾𝜈S 𝑓 (𝑥 + 𝑟, 𝑥)}.
+(21)
+As written, the estimator is feasible to compute for a small number of separations 𝑟 between the
+vertices and, although it introduces a (mild) signal-to-noise ratio problem at large 𝑟, should be
+efficient at small |𝑟| ≤ 𝑅 given the leading extra contribution vanishes like 𝑁−2
+s , c.f. Sec. 3.
+For the remainder |𝑟| > 𝑅, we propose using 𝑁𝑋 randomly selected point sources 𝑋𝑛 [20]
+¯𝐻𝜇𝜈
+2 (𝑟) = 𝐿3𝑇
+𝑁𝑋
+𝑁𝑋
+∑︁
+𝑛=1
+𝐻𝜇𝜈
+2 (𝑋𝑛, 𝑋𝑛 + 𝑟)
+(22)
+so that the total is split between short- and long-distance contributions
+W2 = 𝑎4 ∑︁
+|𝑟 |≤𝑅
+H2(𝑟)𝐺 𝜇𝜈(𝑟) + 𝑎4 ∑︁
+𝑟>𝑅
+¯𝐻𝜇𝜈
+2 (𝑟)𝐺 𝜇𝜈(𝑟),
+(23)
+using the efficient stochastic estimator for the noisy short-distance contribution. Ref. [21] introduced
+an importance sampling based on current separations for higher-point correlation functions, whereas
+in this case we make the separation based on the expected contributions to the variance. This
+approach avoids completely factorizing the trace which would require either O(𝑉) contractions or
+O(𝑁2
+s ) FFTs to include the photon line which we deemed unfeasible.
+In Fig. 3 (left) we illustrate the variance of each of the terms in Eq. (23) for the sum over a
+fixed separation |𝑟| between the currents, for the case 𝑁s = 𝑁𝑋 = 1. As expected, the variance
+from the contribution around |𝑟| ∼ 0 dominates both the stochastic (red squares) and point source
+estimator (blue circles), and we observe the mild signal-to-noise ratio problem in the stochastic
+7
+
+Efficiently unquenching QCD+QED at O(𝛼)
+Tim Harris
+estimator. The green triangle denotes the gauge variance for the case 𝑟 = 0, which is approximately
+suppressed by (𝐿3𝑇)/𝑎4 compared to 𝑁𝑋 = 1 indicating translation averaging is very effective for
+the short-distance contribution. In the right-hand panel, we see variance of the short- and long-
+distance contributions with the choice 𝑅/𝑎 = 4 as a function of the number of inversions (where
+𝑁𝑋 = 1 corresponds to 12 inversions). The variance is dominated by the short-distance contribution
+(red squares) which however scales favourably like 𝑁−2
+inv, while the long-distance contribution (blue
+circles) which scales only like 𝑁−1
+inv is much suppressed. Deviations from the former scaling indicate
+that the gauge variance may be reached with just 𝑁inv ∼ 1000, which although is larger than required
+for 𝑊1 is still achievable with modern computational resources, and universal for all observables.
+5.
+Conclusions
+In this work we have examined the Wick contractions which arise due to the charge of the
+sea quarks in the RM123 method. Such diagrams contribute, in principle, even to observables
+constructed from neutral fields and are therefore ubiquitous in the computation of iso-spin breaking
+corrections. We have proposed stochastic estimators for the quark lines in such diagrams which
+completely avoids the need to sample the Maxwell action stochastically, thus eliminating one
+additional source of variance. As for the case of disconnected contributions to current correlators,
+we have shown it is beneficial to consider certain flavour combinations which have greatly suppressed
+fluctuations. We have shown that the split-even estimators generalize also to domain-wall fermions
+and perform well compared with naïve estimators. Thus the frequency-splitting strategy of Ref. [10]
+should generalize appropriately for this fermion formulation. In the second topology, however, there
+is no cancellation of the short-distance effects in the variance by considering multiple flavours.
+In this case, we propose decomposing the diagram into a short-distance part to be estimated
+stochastically and a long-distance part estimated using position-space sampling. The variance is
+reduced sufficiently so that the gauge variance can be reached with a reasonable computational cost.
+Given their short-distance nature, these estimators should also succeed with smaller quark masses,
+and furthermore as the diagrams are universal to all iso-spin breaking corrections we anticipate
+that these simple decompositions ought to be beneficial in large-scale simulations. In particular we
+are developing these methods for refinements of our computations of iso-spin breaking corrections
+within the RBC/UKQCD collaboration, for example to meson (leptonic) decay rates [22, 23].
+Acknowledgments
+We use the open-source and free software Grid as the data parallel C++ library
+for the lattice computations [24]. The authors warmly thank the members of the RBC/UKQCD
+collaboration for valuable discussions and the use of ensembles of gauge configurations. T.H., A.P.
+and V.G. are supported in part by UK STFC 1039 grant ST/P000630/1. A.P. and V.G. received
+funding from the European Research Council (ERC) under the European Union’s Horizon 2020
+research and innovation programme under grant agreement No 757646 and A.P. additionally under
+grant agreement No 813942. This work used the DiRAC Extreme Scaling service at the University
+of Edinburgh, operated by the Edinburgh Parallel Computing Centre on behalf of the STFC DiRAC
+HPC Facility (www.dirac.ac.uk). This equipment was funded by BEIS capital funding via STFC
+capital grant ST/R00238X/1 and STFC DiRAC Operations grant ST/R001006/1. DiRAC is part of
+the National e-Infrastructure.
+8
+
+Efficiently unquenching QCD+QED at O(𝛼)
+Tim Harris
+References
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+9
+

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf,len=370
+page_content='Efficiently unquenching QCD+QED at O(𝜶)  Tim Harris,𝑎,∗ Vera Gülpers,𝑎 Antonin Portelli𝑎 and James Richings𝑎,𝑏  𝑎School of Physics and Astronomy, University of Edinburgh,  Edinburgh EH9 3FD, United Kingdom  𝑏EPCC, University of Edinburgh,  EH8 9BT, Edinburgh, United Kingdom  E-mail: tharris@ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='uk  We outline a strategy to efficiently include the electromagnetic interactions of the sea quarks  in QCD+QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' When computing iso-spin breaking corrections to hadronic quantities at leading  order in the electromagnetic coupling, the sea-quark charges result in quark-line disconnected  diagrams which are challenging to compute precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' An analysis of the variance of stochastic  estimators for the relevant traces of quark propagators helps us to improve the situation for certain  flavour combinations and space-time decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' We present preliminary numerical results  for the variances of the corresponding contributions using an ensemble of 𝑁f = 2 + 1 domain-wall  fermions generated by the RBC/UKQCD collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  The 39th International Symposium on Lattice Field Theory (Lattice2022),  8-13 August, 2022  Bonn, Germany  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='it/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='03995v1  [hep-lat]  10 Jan 2023  Efficiently unquenching QCD+QED at O(𝛼)  Tim Harris  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  Introduction  Several lattice QCD predictions which form important input for precision tests of the Standard  Model have uncertainties at or below the 1% level, for example the HVP contribution to (𝑔 − 2)𝜇,  𝑓𝐾/ 𝑓𝜋, 𝑔A or the Wilson flow scale √𝑡0 to name a few [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  However, to further improve  such predictions, QCD with iso-spin symmetry is not a sufficiently accurate effective description  of the low-energy dynamics and QED, which contributes one source of iso-spin breaking due to  the different up- and down-quark electric charges, must be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Recent efforts have been  successful at including iso-spin breaking corrections, and some of which fully account for the  effects of the sea-quark electric charges [3, 4, 5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  Nevertheless, many computations of  iso-spin breaking effects still neglect to incorporate these dynamical effects in an approximation  known as electroquenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' As the FLAG report notes in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='2 [2], computations using the  electroquenched approximation might feature an uncontrolled systematic error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  In this work we aim to include the effects of the electric charge of the sea quarks in the  perturbative method known as the RM123 approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' This amounts to computing at least two  additional Wick contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  In order to sum the vertices in the resulting diagrams over the  lattice volume, some approximations must be used which often introduce additional fluctuations,  for example due to the auxiliary fields of a stochastic estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Here we investigate some simple  decompositions which may avoid large contributions to the variance, so that sufficiently precise  results can be obtained to systematically include all sources of iso-spin breaking without incurring  a large computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  Sea-quark effects in the RM123 method  Due to the smallness of the fine-structure constant 𝛼 ∼ 1/137 and the renormalized light-  quark mass difference (𝑚R  u − 𝑚R  d )/Λ ∼ 1%, it is natural to expand physical observables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' in  QCD+QED) in these parameters to compute iso-spin breaking corrections, as was first outlined in  Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' In the resulting expansion of an observable 𝑂  ⟨𝑂⟩ = ⟨𝑂⟩  ���  𝑒=0 + 1  2𝑒2� 𝜕  𝜕𝑒  𝜕  𝜕𝑒 ⟨𝑂⟩  �  𝑒=0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  (1)  the leading corrections in the electric charge 𝑒 =  √  4𝜋𝛼 are parameterized in terms of the correlation  function  𝜕  𝜕𝑒  𝜕  𝜕𝑒 ⟨𝑂⟩ = (−i)2  ∫  d4𝑥  ∫  d4𝑦 ⟨𝐽𝜇(𝑥)𝐴𝜇(𝑥)𝐽𝜈(𝑦)𝐴𝜈(𝑦)𝑂⟩c  (2)  where the electromagnetic current for u, d, s quark flavours is defined  𝐽𝜇 =  ∑︁  𝑓 =u,d,s  𝑄 𝑓 ¯𝜓 𝑓 𝛾𝜇𝜓 𝑓 ,  𝑄u = 2  3,  𝑄d = 𝑄s = −1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  (3)  By choosing the expansion point to be a theory with 𝛼 = 0 and iso-spin symmetry 𝑚u = 𝑚d,  only correlation functions in the 𝑁f = 2 + 1 theory need to be evaluated, which we denote with  𝑒 = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' The precise definition of such a theory using an additional set of renormalization  conditions is necessary to fix the meaning of the leading-order term on the right-hand side (and  2  Efficiently unquenching QCD+QED at O(𝛼)  Tim Harris  𝑊1  𝑂  𝑊2  𝑂  𝑊3  𝑂  𝑊4  𝑂  Figure 1: Wick contractions which appear at leading order in the expansion of a hadronic observable 𝑂  in the electromagnetic coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Each closed fermion line has contributions from all of the quark flavours  u, d, s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' with the appropriate charge factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  conversely the iso-spin breaking corrections themselves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Otherwise the predictions of QCD+QED  are unambiguously defined, up to its intrinsic accuracy, by fixing 𝑁f quark masses and the QCD  coupling as the electric coupling does not renormalize at this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' In the above, the ellipsis stands  for the mass counterterms which are needed to make physical predictions due to the contribution to  the quark self-energy induced by QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  After integrating out the fermion and photon fields, the resulting Wick contractions 𝑊𝑖 are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' 1, which contribute to the derivative with respect to the electric charge through the  connected correlation function  𝜕  𝜕𝑒  𝜕  𝜕𝑒 ⟨𝑂⟩ =  4  ∑︁  𝑖=1  ⟨𝑂𝑊𝑖⟩c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  (4)  The first two subdiagrams, which arise soley from the electric charges of the sea quarks, can be  expressed in terms of a convolution with the photon propagator (in some fixed gauge) 𝐺 𝜇𝜈(𝑥) =  ⟨𝐴𝜇(𝑥)𝐴𝜈(0)⟩  𝑊1,2 = −𝑎8 ∑︁  𝑥,𝑦  𝐻𝜇𝜈  1,2(𝑥, 𝑦)𝐺 𝜇𝜈(𝑥 − 𝑦),  (5)  where 𝐻1,2 are the traces of quark propagators 𝑆 𝑓 (𝑥, 𝑦) = ⟨𝜓 𝑓 (𝑥) ¯𝜓 𝑓 (𝑦)⟩  𝐻𝜇𝜈  1 (𝑥, 𝑦) =  ∑︁  𝑓 ,𝑔  𝑄 𝑓 𝑄𝑔 tr{𝛾𝜇𝑆 𝑓 (𝑥, 𝑥)} tr{𝛾𝜈𝑆𝑔(𝑦, 𝑦)},  (6)  𝐻𝜇𝜈  2 (𝑥, 𝑦) = −  ∑︁  𝑓  𝑄2  𝑓 tr{𝛾𝜇𝑆 𝑓 (𝑥, 𝑦)𝛾𝜈𝑆 𝑓 (𝑦, 𝑥)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  (7)  These two diagrams are the main subject of these proceedings, and the techniques advocated for  the first can be effectively reused for the third diagram, 𝑊3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' In the following sections we introduce  stochastic estimators only for the quark lines and compute the subdiagrams by convoluting with the  exact photon propagator which avoids introducing additional stochastic fields for the U(1) gauge  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' The final diagram 𝑊4, which only contributes if the observable 𝑂 depends explicitly  on the (charged) fermion fields, is the only one surviving the electroquenched approximation, and,  can in most cases be computed efficiently provided that the leading-order diagram is already under  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  3  Efficiently unquenching QCD+QED at O(𝛼)  Tim Harris  We note that the variance of the contributions to the connected correlation functions on the  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' (4) crudely factorizes  𝜎2  𝑂𝑊1,2 ≈ ⟨𝑂⟩2  c ⟨𝑊1,2⟩2  c + ⟨𝑂𝑊1,2⟩c  (8)  ≈ 𝜎2  𝑂𝜎2  𝑊1,2,  (9)  where in the first line we have made the Gaussian approximation, and in the second line we have  assumed that the fluctuations are much larger than the signal ⟨𝑂𝑊1,2⟩c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Thus, in the following  sections we will analyse the variance of individual subdiagrams 𝑊1,2 in order to gain a rough  insight into the fluctuations of the total correction, in a similar fashion to the analysis of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  In that case, however, the correction to the factorization of the variance is exponentially suppressed  in the separation between the vertices of the subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  Quark-line disconnected subdiagram 𝑊1  We begin by noting that the hadronic part of the diagram factorizes into two traces,  𝐻𝜇𝜈  1 (𝑥, 𝑦) = 𝑇𝜇(𝑥)𝑇𝜈(𝑦),  (10)  each of which, with the current defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' (3) and in the 𝑁f = 2 + 1 theory with iso-spin  symmetry, is the difference of the light- and strange-quark propagators  𝑇𝜇(𝑥) = 1  3 tr{𝛾𝜇[𝑆ud(𝑥, 𝑥) − 𝑆s(𝑥, 𝑥)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  (11)  It is convenient to rewrite this difference as a product [10]  𝑆ud − 𝑆s = (𝑚s − 𝑚ud)𝑆ud𝑆s  (12)  which makes the explicit suppression of 𝑇𝜇 in the SU(3)-symmetry breaking parameter 𝑚s − 𝑚ud  explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' This additionally results in a suppression of the variance of 𝑊1 by (𝑚s − 𝑚ud)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' This  suppression results in a cancellation of a quartic short-distance divergence in the variance of the  contribution of each individual flavour to 𝑊1, explaining this favourable flavour combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  While the identity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' (12) is easily derived for Wilson-type fermions, here we sketch that  it holds exactly for the domain-wall fermion valence propagator 𝑆 𝑓 = ˜𝐷−1  𝑓 which (approximately)  satisfies the Ginsparg-Wilson relation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Recalling the definition of ˜𝐷 𝑓 in terms of the 5D  Wilson matrix 𝐷5, 𝑓 (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' [12] for unexplained notation)  ˜𝐷−1  𝑓 = (P−1𝐷−1  5, 𝑓 𝑅5P)11,  (13)  where the matrix indices indicate the coordinate in the fifth dimension, the result is obtained  immediately from  ˜𝐷−1  ud − ˜𝐷−1  s  = (𝑚s − 𝑚ud)(P𝐷−1  5,ud𝑅5𝐷−1  5,s𝑅5)11  (14)  by noting that the following matrix projects on the physical boundary  (𝑅5)·· = (𝑅5P)·1(P−1)1·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  (15)  4  Efficiently unquenching QCD+QED at O(𝛼)  Tim Harris  𝐿/𝑎  𝑇/𝑎  𝑚 𝜋  𝑚 𝜋𝐿  𝑎  𝑁cfg  24  64  340 MeV  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='12 fm  50  Table 1: The parameters of the C1 ensemble of 𝑁f = 2 + 1 Shamir domain-wall fermions used in the  numerical experiments in this work, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' [17] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  The preceding identity is easily demonstrated using the explicit representations  𝑅5 =  ���  �  𝑃+  𝑃−  ���  �  ,  P−1 =  ������  �  𝑃−  𝑃+  𝑃+  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  𝑃+  𝑃−  ������  �  ,  (16)  where 𝑃± = 1 ± 𝛾5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  Using the identity for the difference, there are two independent estimators for the trace  Θ𝜇(𝑥) = 1  3 (𝑚s − 𝑚ud) 1  𝑁s  𝑁s  ∑︁  𝑖=1  𝜂†  𝑖 (𝑥)𝛾𝜇{𝑆ud𝑆s𝜂𝑖}(𝑥),  (17)  T𝜇(𝑥) = 1  3 (𝑚s − 𝑚ud) 1  𝑁s  𝑁s  ∑︁  𝑖=1  {𝜂†  𝑖 𝑆s}(𝑥)𝛾𝜇{𝑆ud𝜂𝑖}(𝑥),  (18)  where the auxiliary quark fields 𝜂𝑖(𝑥) have zero mean and finite variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  The properties of  both estimators were investigated in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' [10], where it was shown that the contribution  to the variance from the auxiliary fields for the second split-even estimator was in the region of  a factor O(100) smaller than the first standard estimator, which translates into the same factor  reduction in the cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' The split-even estimator has since been used extensively for disconnected  current correlators [13, 14, 15], while in the context of the twisted-mass Wilson formulation similar  one-end trick estimators have often been employed for differences of twisted-mass propagators [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  In this work we propose an estimator for the first diagram 𝑊1 using  W1 ≈  �  𝑎4 ∑︁  𝑥  T𝜇(𝑥)  � �  𝑎4 ∑︁  𝑦  T𝜈(𝑦)𝐺 𝜇𝜈(𝑥 − 𝑦)  �  (19)  where independent estimators are used for the two traces to avoid incurring a bias with a finite  sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' The convolution in the second parentheses can be efficiently computed using the  Fast Fourier Transform (FFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' With a minor modification, an estimator using all possible unbiased  combinations of samples can be written at the cost of performing O(𝑁s) FFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  The standard  estimator is obtained by replacing both occurances of T𝜇 with Θ𝜇 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  We performed an analysis of the variance for the standard and split-even estimators for W1  using the domain-wall ensemble generated by the RBC/UKQCD collaboration whose parameters  are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' The photon propagator is computed in the QED𝐿 formulation [18] in the  Feynman gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' The results for the variances, which are dimensionless numbers, are shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' we plot the variance for the contribution of a single flavour Wu  1 using the  5  Efficiently unquenching QCD+QED at O(𝛼)  Tim Harris  10−4  10−3  10−2  10−1  100  101  102  103  104  105  106  107  108  109  1  10  100  1000  σ2  Ns  Wu  1  Wuds  1  (standard)  Wuds  1  (split-even)  1/N 2  s  Figure 2: Left: Comparison of the variance versus the number of sources for the 𝑊1 quark-line disconnected  diagram,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' using a single flavour (red squares),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' the standard estimator for u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' s flavours (blue circles) and the  split-even estimator (green triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' The dashed line shows 1/𝑁2  s scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' In this figure, the (local) currents  are not renormalized and the charge factors are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  standard estimators for the traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' We note that all the variances are dominated by the fluctuations  of the auxiliary fields for small 𝑁s, and in particular scale like 1/𝑁2  s in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  As expected, the standard estimator including the light-quark and strange-quark contributions  (blue circles) is suppressed with respect to the contribution of a single flavour (red squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  Furthermore, the variance of the split-even estimator (green triangles) is reduced by a factor of 104  with respect to the standard one (blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' This reduction is commensurate with the reduction  in the variance observed for the disconnected contribution to the current correlator [10], which  suggests the same mechanisms are present here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' For 𝑁s ∼ 100, the variance is independent of  the number of auxiliary field samples which indicates that it is dominated by the fluctuations of  the gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' In this case no further variance reduction is possible for a fixed number of gauge  configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Finally we note that the convolution of the second parentheses of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' (19) can be  simply inserted sequentially in any of the diagrams of type 𝑊3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  Quark-line connected subdiagram 𝑊2  In contrast to the quark-line disconnected subdiagram, there is no cancellation in the variance  in the connected subdiagram 𝑊2 between the light and strange-quark contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' In this case,  power counting suggests that the variance diverges with the lattice spacing like 𝑎−4 as 𝑎 → 0 and is  expected to be dominated by short-distance contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Translation averaging should therefore  be very effective and one way to implement it is to use an all-to-all estimator [19] for the quark  propagator  S 𝑓 (𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' 𝑥 + 𝑟) = 1  𝑁s  𝑁s  ∑︁  𝑖=1  {𝑆 𝑓 𝜂𝑖}(𝑥)𝜂†  𝑖 (𝑥 + 𝑟),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  (20)  6  Efficiently unquenching QCD+QED at O(𝛼)  Tim Harris  10−7  10−6  10−5  10−4  10−3  10−2  10−1  100  101  102  103  104  105  106  107  108  109  0  2  4  6  8  10  12  σ2  |r|/a  H2(r)G(r)  H2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Ns = 1  ¯H2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' NX = 1  H2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Ns = ∞  10−2  10−1  100  101  102  103  104  105  106  1  10  100  1000  σ2  Ninv  W2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' R/a = 4  �  r≤R H2G  �  r>R ¯H2G  R = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Ns = ∞  Figure 3: Left: the variance for the stochastic estimator (red squares) and point source estimator (blue  circles) for the minimum number of inversions required,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' for the contribution with fixed separation between  the currents |𝑟|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' The green triangle indicates the gauge variance for the point 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Right: the variance for  the short-distance (red squares) and long-distance (blue circles) for the choice 𝑅/𝑎 = 4, versus the number  of inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' The green band indicates the gauge variance for the contribution from 𝑟 = 0 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' The dashed  lines indicate the expected leading 𝑁−2  inv and 𝑁−1  inv scaling for the short- and long-distance components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  using independent fields for each propagator in the trace  H 𝜇𝜈  2  (𝑟) = 𝑎4 ∑︁  𝑥  ∑︁  𝑓  𝑄2  𝑓 tr{𝛾𝜇S 𝑓 (𝑥, 𝑥 + 𝑟)𝛾𝜈S 𝑓 (𝑥 + 𝑟, 𝑥)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  (21)  As written, the estimator is feasible to compute for a small number of separations 𝑟 between the  vertices and, although it introduces a (mild) signal-to-noise ratio problem at large 𝑟, should be  efficient at small |𝑟| ≤ 𝑅 given the leading extra contribution vanishes like 𝑁−2  s , c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  For the remainder |𝑟| > 𝑅, we propose using 𝑁𝑋 randomly selected point sources 𝑋𝑛 [20]  ¯𝐻𝜇𝜈  2 (𝑟) = 𝐿3𝑇  𝑁𝑋  𝑁𝑋  ∑︁  𝑛=1  𝐻𝜇𝜈  2 (𝑋𝑛, 𝑋𝑛 + 𝑟)  (22)  so that the total is split between short- and long-distance contributions  W2 = 𝑎4 ∑︁  |𝑟 |≤𝑅  H2(𝑟)𝐺 𝜇𝜈(𝑟) + 𝑎4 ∑︁  𝑟>𝑅  ¯𝐻𝜇𝜈  2 (𝑟)𝐺 𝜇𝜈(𝑟),  (23)  using the efficient stochastic estimator for the noisy short-distance contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' [21] introduced  an importance sampling based on current separations for higher-point correlation functions, whereas  in this case we make the separation based on the expected contributions to the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' This  approach avoids completely factorizing the trace which would require either O(𝑉) contractions or  O(𝑁2  s ) FFTs to include the photon line which we deemed unfeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' 3 (left) we illustrate the variance of each of the terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' (23) for the sum over a  fixed separation |𝑟| between the currents, for the case 𝑁s = 𝑁𝑋 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' As expected, the variance  from the contribution around |𝑟| ∼ 0 dominates both the stochastic (red squares) and point source  estimator (blue circles), and we observe the mild signal-to-noise ratio problem in the stochastic  7  Efficiently unquenching QCD+QED at O(𝛼)  Tim Harris  estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' The green triangle denotes the gauge variance for the case 𝑟 = 0, which is approximately  suppressed by (𝐿3𝑇)/𝑎4 compared to 𝑁𝑋 = 1 indicating translation averaging is very effective for  the short-distance contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' In the right-hand panel, we see variance of the short- and long-  distance contributions with the choice 𝑅/𝑎 = 4 as a function of the number of inversions (where  𝑁𝑋 = 1 corresponds to 12 inversions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' The variance is dominated by the short-distance contribution  (red squares) which however scales favourably like 𝑁−2  inv, while the long-distance contribution (blue  circles) which scales only like 𝑁−1  inv is much suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Deviations from the former scaling indicate  that the gauge variance may be reached with just 𝑁inv ∼ 1000, which although is larger than required  for 𝑊1 is still achievable with modern computational resources, and universal for all observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  Conclusions  In this work we have examined the Wick contractions which arise due to the charge of the  sea quarks in the RM123 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Such diagrams contribute, in principle, even to observables  constructed from neutral fields and are therefore ubiquitous in the computation of iso-spin breaking  corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' We have proposed stochastic estimators for the quark lines in such diagrams which  completely avoids the need to sample the Maxwell action stochastically, thus eliminating one  additional source of variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' As for the case of disconnected contributions to current correlators,  we have shown it is beneficial to consider certain flavour combinations which have greatly suppressed  fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' We have shown that the split-even estimators generalize also to domain-wall fermions  and perform well compared with naïve estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' Thus the frequency-splitting strategy of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' [10]  should generalize appropriately for this fermion formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' In the second topology, however, there  is no cancellation of the short-distance effects in the variance by considering multiple flavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  In this case, we propose decomposing the diagram into a short-distance part to be estimated  stochastically and a long-distance part estimated using position-space sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' The variance is  reduced sufficiently so that the gauge variance can be reached with a reasonable computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  Given their short-distance nature, these estimators should also succeed with smaller quark masses,  and furthermore as the diagrams are universal to all iso-spin breaking corrections we anticipate  that these simple decompositions ought to be beneficial in large-scale simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' In particular we  are developing these methods for refinements of our computations of iso-spin breaking corrections  within the RBC/UKQCD collaboration, for example to meson (leptonic) decay rates [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  Acknowledgments  We use the open-source and free software Grid as the data parallel C++ library  for the lattice computations [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' The authors warmly thank the members of the RBC/UKQCD  collaboration for valuable discussions and the use of ensembles of gauge configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='  and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' are supported in part by UK STFC 1039 grant ST/P000630/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' received  funding from the European Research Council (ERC) under the European Union’s Horizon 2020  research and innovation programme under grant agreement No 757646 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' additionally under  grant agreement No 813942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' This work used the DiRAC Extreme Scaling service at the University  of Edinburgh, operated by the Edinburgh Parallel Computing Centre on behalf of the STFC DiRAC  HPC Facility (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' This equipment was funded by BEIS capital funding via STFC  capital grant ST/R00238X/1 and STFC DiRAC Operations grant ST/R001006/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
+page_content=' DiRAC is part of  the National e-Infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'}
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+What Decreases Editing Capability?
+Domain-Specific Hybrid Refinement for Improved GAN Inversion
+Pu Cao1,2
+Lu Yang1
+Dongxu Liu1
+Zhiwei Liu3
+Shan Li1
+Qing Song1*
+1Beijing University of Posts and Telecommunications
+2Metavatar
+3Institute of Automation Chinese Academy of Sciences
+{caopu, soeaver, ldx, ls1995, priv}@bupt.edu.cn zhiwei.liu@nlpr.ia.ac.cn
+Abstract
+Recently, inversion methods have focused on additional
+high-rate information in the generator (e.g., weights or
+intermediate features) to refine inversion and editing results
+from embedded latent codes.
+Although these techniques
+gain reasonable improvement in reconstruction,
+they
+decrease editing capability, especially on complex images
+(e.g., containing occlusions, detailed backgrounds, and
+artifacts). A vital crux is refining inversion results, avoiding
+editing capability degradation.
+To tackle this problem,
+we introduce Domain-Specific Hybrid Refinement (DHR),
+which draws on the advantages and disadvantages of two
+mainstream refinement techniques to maintain editing
+ability with fidelity improvement.
+Specifically, we first
+propose Domain-Specific Segmentation to segment images
+into two parts: in-domain and out-of-domain parts. The
+refinement process aims to maintain the editability for
+in-domain areas and improve two domains’ fidelity.
+We
+refine these two parts by weight modulation and feature
+modulation, which we call Hybrid Modulation Refinement.
+Our proposed method is compatible with all latent code
+embedding methods.
+Extension experiments demonstrate
+that our approach achieves state-of-the-art in real image
+inversion and editing.
+Code is available at https:
+//github.com/caopulan/Domain-Specific_
+Hybrid_Refinement_Inversion.
+1. Introduction
+Generative Adversarial Networks (GANs) have shown
+promising results in image generation. Synthetic images
+are photorealistic with high resolution and are difficult to
+distinguish from real images [24, 27, 28, 26, 61]. Mean-
+while, image manipulation and controllable generation are
+deeply explored thanks to their highly semantic latent space.
+Moreover, GANs can represent a high-quality image prior
+*corresponding author.
+Figure 1. Inversion and editing results of our method. We pre-
+serve image details, including background and occlusion, in both
+inversion and manipulation processes.
+to improving various tasks, such as face parsing [59, 58,
+62, 60, 57, 56], style transfer [33, 63], face super-resolution
+[53].
+Inversion is built to convert real images into GANs’ la-
+tent space. The inverted latent codes are required to recon-
+struct given images by pretrained generator, which also em-
+arXiv:2301.12141v1  [cs.CV]  28 Jan 2023
+
+TO
+indul
+Inversion
+ASA
+TO
+Smile
+SAANN
+LUTO
+Young
+TO
+Exposure
+LTO
+Lipstick
+SAANNbeds semantic information to edit or apply in other GAN-
+based tasks. Two types of methods generally reach image
+embedding. One is training an image encoder to convert
+given images to latent codes [49, 41], while another is min-
+imizing the discrepancy between given images and recon-
+structed images to optimize initial latent codes iteratively
+[28]. This process attains the corresponding latent codes
+to reconstruct or edit the images. However, latent codes are
+low bit-rate [52], and high-rate details of images may not be
+reconstructed faithfully. Hence, many works focus more on
+refining results by additional high-rate information, e.g., in-
+termediate features [36], generator weight [44, 5], recently.
+As reconstruction performance increases by refinement
+with high-rate information, editing capability is inevitably
+decreased, especially on images containing complex parts.
+This phenomenon is due to the destruction of pretrained
+GAN prior.
+High-rate information needs drastic change
+to reconstruct complex parts.
+We demonstrate this phe-
+nomenon in Figure 3. Meanwhile, complex images pre-
+vail in the natural world. For example, face accessories,
+hats, occlusions, and complex backgrounds usually appear
+in face photos.
+As there are two mainstream refinement methods, they
+show different manipulation impacts. One is weight modu-
+lation, in which the generator’s weight is tuned [44, 12] or
+predicted [5] by given images. Another is feature modula-
+tion [52, 36], in which the input image would also invert to
+feature space by encoder or optimization. Generally, weight
+modulation can maintain editing capacity better, while fea-
+ture modulation breaks it since high-rate level editing is re-
+quired [38].
+Based on the above illustration, we further explore the
+idea of ”divide and conquer.” Specifically, we divide the
+image into in-domain and out-of-domain parts. In-domain
+parts imply areas close to generators’ output distribution
+and are desired to perform well on both inversion and edit-
+ing.
+Correspondingly, out-of-domain parts are segments
+challenging to inverse or edit and desired to reconstruct
+faithfully. Hence, we introduce a hybrid method to han-
+dle them. We refine in-domain parts by tuning generator
+weight since it can maintain editing capability. For out-of-
+domain parts, we straightforwardly invert them by interme-
+diate features to keep spatial image details. Notably, our
+hybrid refinement method first analyzes and combines fea-
+ture and weight modulation for improved GAN inversion,
+and achieves extraordinary results as shown in Figure 1.
+Extensive experiments are presented to demonstrate the
+effects of our Domain-Specific Hybrid Refinement.
+We
+achieve state-of-the-art and gain significant improvement in
+both fidelity and editability. The key contributions of this
+work are summarized as follows:
+• We analyze the reasons for editing capability degrada-
+tion in the refinement process. Based on our analysis,
+we introduce in-domain and out-of-domain and pro-
+pose Domain-Specific Segmentation to segment im-
+ages into these two parts for better inversion.
+• We propose Hybrid Modulation Refinement to im-
+prove inversion results of in-domain and out-of-
+domain parts. We conduct weight modulation on in-
+domain part and feature modulation on out-of-domain
+part, which can preserve editing capability when refin-
+ing the image details.
+• We conduct extensive experiments and user studies to
+demonstrate the effects of our method. We reach ex-
+traordinary performance on real-world image inversion
+and editing and achieve state-of-the-art.
+2. Related Work
+2.1. GAN Inversion
+GAN inversion aims to embed real-world images into a
+pretrained generator’s latent space, which can be used to
+reconstruct and edit input images. Generally, methods can
+be divided into two stages.
+The first stage aims to attain low-rate latent codes, usu-
+ally in Z/W/W + spaces. The latent codes are gained by
+an encoder or optimization process. Training an encoder
+[49, 43, 54, 17, 8, 41] to predict latent codes is efficient for
+inference and is easier to get better trade-offs between fi-
+delity and manipulation [49, 41]. Optimizing initial latent
+codes by reconstruction discrepancy gains better fidelity.
+However, it may cost several minutes per image [28, 8, 1, 2]
+and decreases editability during per-image tuning. Due to
+low-rate characteristics, latent codes can only reconstruct
+coarse information and drop the details from original im-
+ages. Meanwhile, there is a trade-off between fidelity and
+editability, and many methods introduce additional regular-
+ization modules (e.g., latent code discriminator [49] and la-
+tent space alignment [41]) to address it.
+In the second stage, reconstruction and manipulation re-
+sults from latent codes are refined by high-rate information.
+Refinement methods are mainly divided into weight mod-
+ulation and feature modulation. Weight modulation meth-
+ods predict or finetune generator weight to improve fidelity.
+Some methods use hypernet [18] to predict weight offsets.
+The others tune generator by given images, which attain bet-
+ter fidelity but cost much time. Another branch further in-
+vert images to latent feature, which we call feature modula-
+tion. HFGI [52] proposes a distortion consultation approach
+for high-fidelity reconstruction. SAM [36] segments images
+into various parts and inverts them into different intermedi-
+ate layers by predicting ”invertibility.” All of them only use
+one of feature and weight modulation to refine results and
+suffer editing capability degradation. In this work, we com-
+
+generator
+manifold
+latent space
+weight
+modulation
+𝐺(𝑤; 𝜃∗)
+𝐺(𝑤)
+𝑤
+generator
+manifold
+latent space
+𝐺(𝑤, 𝑓)
+𝐺(𝑤)
+𝑤
+image space
+image space
+feature
+modulation
+generator
+manifold
+latent space
+feature
+modulation
+𝐺(𝑤, 𝑓; 𝜃∗)
+𝐺(𝑤)
+𝑤
+image space
+weight
+modulation
+𝐺(𝑤)
+𝐺(𝑤; 𝜃∗)
+𝐺(𝑤; 𝜃∗)
+𝐺(𝑤, 𝑓; 𝜃∗)
+(a) Weight Modulation
+(b) Feature Modulation
+(c) Hybrid Modulation
+Figure 2. Comparison of different refinement mechanisms. We suppose that the refinement results are similar to the given images in
+all pipelines. The first row shows two previous mainstream refinement mechanisms, weight, and feature modulation. Weight modulation
+changes the generator manifold and feature modulation introduces spatial high-rate information to recover image details. The bottom
+part demonstrates our hybrid refinement method, combining these two modulation mechanisms to retain editing capability. We tune the
+generator on invertible and editable areas, which causes lower manifold deviation, and the result is shown on G(w; θ∗). To reconstruct
+faithfully, we use feature modulation on the other area and attain G(w, f; θ∗).
+bine these two aspects by their pros and cons to reach more
+promising results.
+2.2. GAN-based Manipulation
+GANs’ latent spaces encode highly rich semantic infor-
+mation, which develop the GAN-based manipulation task.
+It aims to edit given images by changing latent codes in
+certain directions.
+Many works propose multiple meth-
+ods to find semantic editing directions in latent spaces.
+Some methods obtain the edit vectors of the correspond-
+ing attributes by means of supervision with the help of
+attribute-labeled datasets [10, 15, 47, 45]. And others ex-
+plore the latent space by unsupervised [19, 46, 50, 51] or
+self-supervised ways [23, 39] to find more semantic direc-
+tions way.
+3. Method
+3.1. Preliminaries
+Inversion is built to bridge real-world images and GANs’
+latent space.
+As latent codes are low-rate, which limits
+their reconstruction performance, much research has re-
+cently focused on additional high-rate information in gen-
+eration process, which we call refinement methods. They
+can be mainly divided into two categories: weight modula-
+tion and feature modulation. We first formulate them and
+analyze the causes of editing capacity degradation.
+Formulation.
+We denote the original generation process
+as X = G(w), where G is the generator, w is latent code
+which can represent each latent space (e.g., Z/W/W +).
+We use encoded latent codes as w in the refinement process.
+Weight modulation methods predict [5] or optimize [28]
+θ by minimizing reconstruction error, and are denoted as
+X
+= G(w; θ∗).
+And feature modulation methods in-
+vert images into the intermediate feature, which follows
+X = G(w, f). Defining L as the distance of images, we
+can illustrate these two refinement processes as follow:
+θ∗ = arg min
+θ
+L(x, G(w; θ))
+(1)
+f ∗ = arg min
+f
+L(x, G(w, f))
+(2)
+Impacts on editing capability. Weight and feature modu-
+lation impacts image manipulation in different aspects. The
+
+Easy Sample
+Easy Sample
+Hard Sample, Occlusion
+Hard Sample, Artifact
+Figure 3. Impacts on editing capability of weight modulation.
+We show the input images, inversion results, and two editing re-
+sults (smile and age) from PTI [44]. For those easy samples, edit-
+ing results are reasonable. However, editing capability degrades
+significantly on hard samples.
+schematics are shown in Figure 2. Since the feature modu-
+lation mechanism fixes the intermediate feature distribution
+at one of layers, the effects of edit vectors applied to previ-
+ous layers cannot edit the features of latter layers. Although
+many existing works make efforts to maintain the editing ef-
+fects, including training with adaptive distortion alignment
+[25, 52], their solutions still sacrifice fidelity or editing re-
+sults [36].
+Meanwhile, weight modulation shows promising editing
+performance but also gains unreasonable results on com-
+plex images, as shown in Figure 3. Editing results are more
+reasonable on easy samples than on hard samples.
+The
+main reason for editing capacity degradation is the signif-
+icant weight deviation caused by refining complex images,
+which we show in Figure 2(a). To reconstruct given images,
+the weight modulation mechanism may change the genera-
+tor manifold much. Therefore, the highly semantic charac-
+teristic of the pretrained generator is broken, which would
+decrease the editing capability.
+In conclusion, the critical problem of refinement meth-
+ods is how to decrease weight deviation with reconstruc-
+tion improvement. We next propose our method to answer
+this question.
+3.2. Overview
+In this work, we conduct Domain-Specific Hybrid
+Refinement (DHR) to deal with real-world image inversion
+and the pipline is shown in Figure 4. Based on the above
+analysis, we explore the idea of ”divide and conquer.”
+We first propose the concepts of in-domain and out-of-
+domain. In-domain implies areas that have a similar distri-
+bution with the generator’s output space and are easy to in-
+vert, while out-of-domain areas misalign with output space
+and are difficult to invert. For example, in face domain,
+in-domain areas mainly consist of face and hair, while out-
+of-domain areas consist of occlusions, backgrounds, and ar-
+tifacts. Meanwhile, in-domain areas are more editable, e.g.,
+smile, lipstick, and eyes openness.
+Hence, we propose a hybrid refinement method, which
+segments images into in-domain and out-of-domain areas
+and applies weight and feature modulation to improve fi-
+delity and preserve editing capability. Our framework is
+shown in Figure 4, which consists of three components.
+The image Embedding module aims to embed images
+into latent codes, which we use an off-the-shelf encoder
+(i.e., e4e [49] and LSAP [41]). Given input images X, the
+encoder predicts its W + space latent codes, which we de-
+note as w = E(X), where E is an encoder.
+Domain-Specific Segmentation predicts a binary mask
+which indicates in-domain and out-of-domain areas:
+m = S(X)
+where m ∈ {0, 1}h×w. It segments images into two parts,
+which will be used for refinement.
+In Hybrid Modulation Refinement, weight modulation is
+applied to in-domain areas to recover image details in both
+inversion and editing results. Thanks to the low reconstruc-
+tion discrepancy of in-domain part, weight deviation would
+not be large, and editing capacity would be preserved. For
+out-of-domain parts, we use feature modulation to refine
+them spatially and not to edit them. Hence, those hard-to-
+invert parts would not influence editing ability. We mod-
+ulate weight θ and feature f by minimizing reconstruction
+error in in-domain and out-of-domain part, respectively:
+Xrec = G(w, f, m; θ)
+(3)
+The difference with vanilla weight and feature modula-
+tion can be seen in Figure 2. Based on hybrid ways, genera-
+tor manifold would not change a lot, which highly maintains
+the editing capability of the original GAN.
+3.3. Domain-specific Segmentation
+The first challenge is segmenting images into in-domain
+and out-of-domain at the pixel-level.
+An end-to-end
+domain-specific segmentation model is required for a large,
+manually annotated dataset. Although previous work [36]
+
+omFEOTURUUSH
+FEATORLNSTPTOAIKO
+2,2
+OTomHybrid Modulation Refinement
+Domain-Specific Segmentation
+θ∗
+Segmentation
+𝒘"
+𝒇∗
+Input
+Encoder
+Image Embedding
+Figure 4. Overview of our Domain-Specific Hybrid Refinement framework. We use an off-the-shelf image encoding mechanism and
+introduce Domain-Specific Segmentation and Hybrid Modulation Refinement. The former segments the input images with two domains:
+in-domain and out-of-domain. They are refined by weight modulation and feature modulation in the latter method.
+Figure 5. Illustration of Domain-Specific Segmentation module. We use a parsing model and superpixel algorithm with coarse opti-
+mization to segment input images into in-domain (white areas) and out-of-domain (black areas) parts.
+trains an invertibility prediction model by self-supervision,
+results are inaccurate in some complex areas, which we il-
+lustrate in our ablation study. In this work, we propose a
+Domain-Specific Segmentation module, combining parsing
+and superpixel modules to generate domain segments. Our
+module is robust for real images and does not require data
+annotation. The pipeline is shown in Figure 5.
+The parsing model categorizes face components [69] like
+eye, mouth, and background. For parsing results mp, we
+manually set some categories as out-of-domain and the oth-
+ers as in-domain. However, it is not robust on some complex
+images, as shown in Figure 5. The parsing result represents
+a coarse mask, where complicated paradigms are not seg-
+mented well. Therefore, we introduce a superpixel module
+with coarse optimization to improve the segmentation re-
+sults.
+We use a superpixel algorithm [3] for image partition-
+ing, shown in the middle route of Figure 5. This step finely
+segments images to distinguish each area. We denote each
+partition as {mi
+s}S
+i=1. Categorizing each partition into in-
+domain and out-of-domain without manual annotation is a
+crucial challenge. We first apply a coarse optimization in W
+space, where latent codes initialed by mean values are only
+optimized by a few steps. Since in-domain are those easy-
+to-invert areas, the coarse inversion result Xcoarse could
+reconstruct in-domain areas. We calculate the perceptual
+loss L between the coarse reconstruction image and the in-
+put image, as shown at the bottom of Figure 5. White area
+
+Parsing Result
+Domain Segment
+om:
+Face
+Parsing
+loma
+ome
+om
+Superpixel
+K
+Input X
+Superpixel Result
+om
+out-of-domain
+Coarse
+Optimization
+in-domain
+Xcoarse
+LPIPS(X, Xcoarse)m
+KPTOAIKO
+2,2
+OTmeans higher loss value, while black area means lower loss
+value. As can be seen, the loss of the occlusion area is sig-
+nificantly higher than the face area. We calculate the aver-
+age loss of each partition as follows:
+vi = L ⊙ mi
+s
+||mis||
+and the result {mi
+s, vi}S
+i=1 is visualized. Then we binarize
+superpixel results by adaptive threshold τ and attain ms.
+Finally, domain-specific segmentation results are com-
+bined by parsing results and superpixel results:
+m = mp × ms
+Our Domain-Specific Segmentation module can gain fine
+segmentation results without data annotation.
+3.4. Hybrid Modulation Refinement
+Figure 6. Illustration of Hybrid Modulation Refinement mod-
+ule. We refine in-domain areas and out-of-domain areas by weight
+and feature modulation, respectively. Black lines indicate forward
+flow, and orange and blue lines represent gradients.
+To faithfully recover image details and maintain editing
+capability from original GAN, we introduce a Hybrid Mod-
+ulation Refinement module. It consists of two mainstream
+refinement aspects: weight modulation and feature modu-
+lation. Weight modulation aims to minimize in-domain re-
+construction error by tuning the generator’s parameters. In
+contrast, feature modulation is applied to out-of-domain ar-
+eas by optimizing an intermediate feature of the generator.
+The forward and backward processes are shown in Figure 6.
+For lth layer of total k stages in generator, the original
+generator’s feature is denoted as fl = Gl(w; θ), and an ad-
+ditional modulated feature is marked as f, which is initialed
+by fl. Fixing the latent codes, the original feature fl is only
+relevant to θ. Given segmentation result m, we formulate
+the forward process as follows:
+f ′ = fl ⊙ m + f ⊙ (1 − m)
+(4)
+Then f ′ represents the output of the first l layers and gener-
+ates the final images, which follow Eq 3.
+For backward, we use mean square error L2 and percep-
+tual loss Llpips as objectives in the refinement process. To
+make weight and feature focus on the corresponding areas,
+we update them in a parallel optimization process. Calcu-
+lating the reconstruction errors, we backward loss with seg-
+mentation result m:
+L = L2 + λLlpips
+(5)
+∇f = ∂
+∂f [L ⊙ (1 − m)
+||1 − m||
+]
+(6)
+∇θ = ∂
+∂θ[L ⊙ (m)
+||m||
+]
+(7)
+where λ is a hyper-parameter. The parallel optimization
+mechanism constrains the impact from different domains.
+Based on Domain-Specific Segmentation and Hybrid
+Modulation Refinement, we segment images into in-domain
+and out-of-domain areas and refine them by weight and fea-
+ture modulation, which significantly improve fidelity with
+editing capability remaining.
+4. Experiments
+4.1. Experimental Settings
+Datasets.
+We evaluate all methods on the CelebA-HQ
+[24, 35] test set (2,824 images). Encoders and the generator
+are trained on FFHQ [27] (70,000 images).
+Baselines. We compare our model with previous state-of-
+the-art refinement methods, i.e., ReStyle [4], HFGI [52],
+SAM [36], and PTI [44].
+We use pSp [43], e4e [49]
+and LSAP [41] as encoders. Moreover, the performance
+of encoder-based methods is also reported. All of model
+weights of encoders and generators come from their official
+release.
+Metrics.
+We evaluate all methods in two respects: inver-
+sion and editing. For inversion ability, we conduct MSE,
+LPIPS [67], and identity similarity calculated by a face
+recognition model [22]. MSE straightforwardly measures
+the image distortion, and LPIPS evalutaes the visual dis-
+crepancy. Identity similarity further compares the identity
+consistency during inversion. Moreover, we perform user
+studies to evaluate the perceptual performance of inversion
+and editing.
+4.2. Main Results
+Quantitative results. We first evaluate the reconstruction
+ability. Qualitative results are reported in Table 1. We com-
+pare our method with four previous refinement methods and
+employ two encoders. We conduct experiments with en-
+coders and original Wpivot for PTI. As one can see, DHR
+achieves the best performance on all metrics. Employed
+with e4e, it gains 0.0036 MSE, which is 7.5% e4e, 8.3%
+
+om
+Loss
+gradient of weight modulation
+gradient of feature modulationFigure 7. Inversion and editing results. The second column is the segmentation results of in-domain and out-of-domain areas. Our method
+restores almost all image details.
+ReStyle, 17% HFGI, 25% SAM, and 48% PTI. For LPIPS
+and identity similarity, it demonstrates similar superiority
+and surpasses other methods by a large margin. DHR with
+LSAP achieves the best performance of all metrics.
+Qualitative results.
+We illustrate the inversion and edit-
+ing results of DHR in Figure 7.
+The second column is
+the results from the Domain-Specific Segmentation module,
+and the third is our inversion results. Some dedicated ar-
+eas are categorized into out-of-domain domain (black area).
+We edit them by InterFaceGAN and GANSpace, using four
+editing directions, i.e., smile, young, exposure, and lipstick.
+All image details are preserved in both inversion and editing
+results, such as hairstyle (the first row), earrings (the fifth
+row), and hats. Our results are faithful and photorealistic.
+We further conduct a qualitative comparison with other
+methods, shown in Figure 8. Although inversion results are
+
+Input
+Inversion
+Smile
+Young
+Exposure
+LipstickFigure 8. Comparisons with previous methods. We compare the inversion and editing results with PTI [4], HFGI [52], and SAM [36].
+Although HFGI and SAM reach reasonable inversion results, image distortion and details loss occur in editing results. Our method attains
+the best fidelity and editing performance. Image details are reserved in both phases, and our results are the most natural.
+Method
+Encoder
+MSE ↓
+LPIPS ↓
+Similarity ↑
+ReStyle [4]
+pSp
+0.0276
+0.1298
+0.5816
+e4e
+0.0429
+0.1904
+0.5062
+HFGI [52]
+e4e
+0.0210
+0.1172
+0.6816
+LSAP
+0.0210
+0.0945
+0.7405
+SAM [36]
+e4e
+0.0143
+0.1104
+0.5568
+LSAP
+0.0117
+0.0939
+0.6184
+PTI [44]
+e4e
+0.0074
+0.0750
+0.8633
+LSAP
+0.0067
+0.0666
+0.8696
+Wpivot
+0.0084
+0.0845
+0.8402
+DHR (ours)
+e4e
+0.0036
+0.0455
+0.8704
+LSAP
+0.0035
+0.0436
+0.8780
+Encoder-Only
+pSp [43]
+0.0351
+0.1628
+0.5591
+e4e [49]
+0.0475
+0.1991
+0.4966
+LSAP [41]
+0.0397
+0.1766
+0.5305
+Table 1. Fidelity results on face domain. We compare DHR to
+three previous refinement methods with two powerful encoders.
+The results of these encoders are also presented at the bottom.
+reasonable, editing capability degradation occurs in base-
+lines, e.g., decorations and occlusion blur.
+4.3. User Study
+We conduct user studies to demonstrate the performance
+of inversion and editing.
+Results are shown in Table 2.
+We randomly select 50 different images and invert and edit
+them by HFGI [52], SAM [36], PTI [44], and our method.
+We then ask three users to make a preference for each pair of
+images. A higher value implies users prefer our results. As
+can be seen, our results are highly preferred by users, which
+Method
+Inversion
+Editing
+Smile
+Young
+Exposure
+Lipstick
+ReStyle [4]
+94%
+100%
+100%
+90%
+94%
+HFGI [52]
+94%
+84%
+86%
+100%
+96%
+SAM [36]
+100%
+92%
+94%
+100%
+100%
+PTI [44]
+96%
+100%
+100%
+84%
+88%
+Table 2. User Study. We conduct user studies on inversion and
+editing tasks. The values in the table indicate the percentage of
+images where users prefer our results. Results show our method is
+more faithful and photorealistic.
+are most all above 90%, compared to previous state-of-the-
+art methods. It illustrates that our method decreases image
+distortion and attains better photorealism of reconstruction
+and manipulation results.
+5. Conclusion
+In this work, we propose Domain-Specific Hybrid Re-
+finement to improve GAN inversion and editing capabil-
+ity. Specifically, we analyze the causes of editing ability
+degradation in refinement process and introduce ”divide and
+conquer” to address it. Our method consists of Domain-
+Specific Segmentation and Hybrid Modulation Refinement,
+which segments images into in-domain and out-of-domain
+parts and refines them by weight and feature modulation,
+respectively. Our method attains promising results in both
+inversion and editing with considerable improvement.
+
+Young
+Lipstick
+Exposure
+50
+Smile
+Input
+Inversion
+Inversion
+edit
+Inversion
+edit
+Inversion
+edit
+PTI
+HFGI
+SAM
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+Conference on Computer Vision and Pattern Recognition,
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+Dongdong Chen, Yangyu Huang, Lu Yuan, Dong Chen,
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+arXiv:2112.03109, 2021.
+

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+INRADIUS OF RANDOM LEMNISCATES
+MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN
+ABSTRACT. A classically studied geometric property associated to a complex polynomial p is the
+inradius (the radius of the largest inscribed disk) of its (filled) lemniscate Λ := {z ∈ C : |p(z)| < 1}.
+In this paper, we study the lemniscate inradius when the defining polynomial p is random, namely,
+with the zeros of p sampled independently from a compactly supported probability measure µ. If
+the negative set of the logarithmic potential Uµ generated by µ is non-empty, then the inradius is
+bounded from below by a positive constant with overwhelming probability. Moreover, the inradius
+has a determinstic limit if the negative set of Uµ additionally contains the support of µ.
+On the other hand, when the zeros are sampled independently and uniformly from the unit circle,
+then the inradius converges in distribution to a random variable taking values in (0, 1/2).
+We also consider the characteristic polynomial of a Ginibre random matrix whose lemniscate we
+show is close to the unit disk with overwhelming probability.
+1. INTRODUCTION
+Let p(z) be a polynomial of degree n and Λ be its (filled) lemniscate defined by Λ = {z : |p(z)| <
+1}. Denote by ρ(Λ) the inradius of Λ. By definition, this is the radius of the largest disk that is
+completely contained in Λ. In this paper, we study the inradius of random lemniscates for various
+models of random polynomials.
+The lemniscate {z : |zn − 1| < 1} has an inradius asymptotically proportional to 1/n. In 1958, P.
+Erd¨os, F. Herzog, and G. Piranian posed a number of problems [10] on geometric properties of
+polynomial lemniscates. Concerning the inradius, they asked [10, Problem 3] whether the rate of
+decay in the example {|zn − 1| = 1} is extremal, that is, whether there exists a positive constant C
+such that for any monic polynomial of degree n, all of whose roots lie in the closed unit disk, the
+inradius ρ of its lemniscate Λ satisfies ρ ≥ C
+n . This question remains open. C. Pommerenke [33]
+showed in this context that the inradius satisfies the lower bound ρ ≥
+1
+2e n2 .
+Our results, which we state below in Sec. 1.4 of the Introduction, show within probabilistic set-
+tings that the typical lemniscate admits a much better lower bound on its inradius. Namely, if the
+zeros of p are sampled independently from a compactly supported measure µ whose logarithmic
+potential has non-empty negative set, then the inradius of Λ is bounded below by a positive con-
+stant with overwhelming probability, see Theorem 1.1 below. Let us provide some insight on this
+result and explain why the logarithmic potential of µ plays an important role. First, the lemniscate
+Λ can alternatively be described as the sublevel set { 1
+n log |p(z)| < 0} of the discrete logarith-
+mic potential 1
+n log |p(z)| =
+1
+n
+� log |z − zk| where zk are the zeros of p(z). For fixed z the sum
+1
+n
+� log |z − zk| is a Monte-Carlo approximation for the integral defining the logarithmic potential
+Uµ(z) of µ, and, in particular, it converges pointwise, by the law of large numbers, to Uµ(z). With
+the use of large deviation estimates, we can further conclude that each z in the negative set Ω− of
+Uµ is in Λ with overwhelming probability. The property of holding with overwhelming probabil-
+ity survives (by way of a union bound) when taking an intersection of polynomially many such
+events. This fact, together with a suitable uniform estimate for the derivative p′(z) (for which we
+can use a Bernstein-type inequality), allows for a standard epsilon-net argument showing that an
+1
+arXiv:2301.13424v1  [math.PR]  31 Jan 2023
+
+2
+MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN
+arbitrary compact subset of Ω− is contained in Λ with overwhelming probability. Since Ω− is as-
+sumed nonempty, this leads to the desired lower bound on the inradius, see the proof of Theorem
+1.1 in Section 3 for details.
+Under an additional assumption that the negative set Ω− of the logarithmic potential of µ contains
+the support of µ, the inradius converges to the inradius of Ω− almost surely, see Corollary 1.2; in
+particular, the inradius has a deterministic limit.
+On the other hand, for certain measures µ, the inradius does not have a deterministic limit and
+rather converges in distribution to a nondegenerate random variable, see Theorem 1.5 addressing
+the case when µ is uniform measure on the unit circle. We also consider the lemniscate associated
+to the characteristic polynomial of a random matrix sampled from the Ginibre ensemble, and we
+show that the inradius is close to unity (in fact the whole lemniscate is close to the unit disk) with
+overwhelming probability, see Theorem 1.6.
+See Section 1.4 below for precise statements of these results along with some additional results
+giving further insight on the geometry of Λ.
+1.1. Previous results on random lemniscates. The current paper fits into a series of recent stud-
+ies investigating the geometry and topology of random lemniscates. Let us summarize previous
+results in this direction. We note that the lemniscates studied in the results cited below, in contrast
+to the filled lemniscates of the current paper, are level sets (as opposed to sublevel sets).
+Partly motivated to provide a probabilistic counterpart to the Erd¨os lemniscate problem on the
+extremal length of lemniscates [10], [5], [11], [12], the second and third authors in [23] studied the
+arclength and topology of a random polynomial lemniscate in the plane. When the polynomial has
+i.i.d. Gaussian coefficients, it is shown in [23] that the average length of its lemniscate approaches
+a constant. They also showed that with high probability the length is bounded by a function with
+arbitrarily slow rate of growth, which means that the length of a lemniscate typically satisfies a
+much better estimate than the extremal case. It is also shown in [23] that the number of connected
+components of the lemniscate is asymptotically n (the degree of the defining polynomial) with
+high probability, and there is at least some fixed positive probability of the existence of a “giant
+component”, that is, a component having at least some fixed positive length. Of relevance to the
+focus of the current paper, we note that the proof of the existence of the giant component in [23]
+shows that for a fixed 0 < r < 1, there is a positive probability that the inradius ρ of the lemniscate
+satisfies the lower bound ρ > r.
+Inspired by Catanese and Paluszny’s topological classification [7] of generic polynomials (in terms
+of the graph of the modulus of the polynomial with equivalence up to diffeomorphism of the do-
+main and range), in [9] the second author with M. Epstein and B. Hanin studied the so-called
+lemniscate tree associated to a random polynomial of degree n. The lemniscate tree of a poly-
+nomial p is a labelled, increasing, binary, nonplane tree that encodes the nesting structure of the
+singular components of the level sets of the modulus |p(z)|. When the zeros of p are i.i.d. sam-
+pled uniformly at random according to a probability density that is bounded with respect to Haar
+measure on the Riemann sphere, it is shown in [9] that the number of branches (nodes with two
+children) in the induced lemniscate tree is o(n) with high probability, whereas a lemniscate tree
+sampled uniformly at random from the combinatorial class has asymptotically
+�
+1 − 2
+π
+�
+n many
+branches on average.
+In [21], partly motivated by a known result [11], [43]) stating that the maximal length of a rational
+lemniscate on the Riemann sphere is 2πn, the second author with A. Lerario studied the geometry
+of a random rational lemniscate and showed that the average length on the Riemann sphere is
+
+INRADIUS OF RANDOM LEMNISCATES
+3
+asymptotically π2
+2
+√n. Topological properties (the number of components and their nesting struc-
+ture) were also considered in [21], where the number of connected components was shown to
+be asymptotically bounded above and below by positive constants times n. Z. Kabluchko and I.
+Wigman subsequently established an asymptotic limit law for the number of connected compo-
+nents in [16] by adapting a method of F. Nazarov and M. Sodin [28] using an integral geometry
+sandwich and ergodic theory applied to a translation-invariant ensemble of planar meromorphic
+lemniscates obtained as a scaling limit of the rational lemniscate ensemble.
+1.2. Motivation for the study of lemniscates. The study of lemniscates has a long and rich history
+with a wide variety of applications. The problem of computing the length of Bernoulli’s lemnis-
+cate played a role in the early study of elliptic integrals [1]. Hilbert’s lemniscate theorem and its
+generalizations [26] show that lemniscates can be used to approximate rather arbitrary domains,
+and this density property contributes to the importance of lemniscates in many of the applications
+mentioned below. In some settings, sequences of approximating lemniscates arise naturally for ex-
+ample in holomorphic dynamics [25, p. 159], where it is simple to construct a nested sequence of
+“Mandelbrot lemniscates” that converges to the Madelbrot set. In the classical inverse problem of
+logarithmic potential theory—to recover the shape of a two-dimensional object with uniform mass
+density from the logarithmic potential it generates outside itself—uniqueness has been shown to
+hold for lemniscate domains [37]. This is perhaps surprising in light of Hilbert’s lemniscate the-
+orem and the fact that the inverse potential problem generally suffers from non-uniqueness [40].
+Since lemniscates are real algebraic curves with useful connections to complex analysis, they have
+frequently received special attention in studies of real algebraic curves, for instance in the study of
+the topology of real algebraic curves [7], [2]. Leminscates such as the Arnoldi lemniscate appear
+in applications in numerical analysis [39]. Lemniscates have seen applications in two-dimensional
+shape compression, where the “fingerprint” of a shape constructed from conformal welding sim-
+plifies to a particularly convenient form—namely the nth root of a Blaschke product—in the case
+the two-dimensional shape is assumed to be a degree-n lemniscate [8], [44], [35]. Lemniscates have
+appeared in studies of moving boundary problems of fluid dynamics [18], [24], [19]. In the study
+of planar harmonic mappings, rational lemniscates arise as the critical sets of harmonic polyno-
+mials [17], [22] as well as critical sets of lensing maps arising in the theory of gravitational lensing
+[30, Sec. 15.2.2]. Lemniscates also have appeared prominently in the theory and application of
+conformal mapping [3], [15], [13]. See also the recent survey [36] which elaborates on some of the
+more recent of the above mentioned lines of research.
+1.3. Definitions and Notation. Throughout the paper, µ will denote a Borel probability measure
+with compact support S ⊂ C. The logarithmic potential of µ is defined by
+Uµ(z) =
+�
+S
+log |z − w|dµ(w).
+It is well known that Uµ is a subharmonic function in the plane, and harmonic in C \ S. For such
+µ, we denote the associated negative and positive sets of its potential by
+Ω− = {z ∈ C : Uµ(z) < 0}, Ω+ = {z ∈ C : Uµ(z) > 0}.
+It is easy to see that Ω− is a (possibly empty) bounded open set.
+Assumptions on the measure. Let µ be a Borel probability measure with compact support S ⊂ C.
+We define the following progressively stronger conditions on µ.
+(A) For each compact K ⊂ C,
+C(K) = sup
+z∈K
+�
+S
+(log |z − w|)2 dµ(w) < ∞.
+
+4
+MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN
+(B) There is some C < ∞ and ε > 0 such that for all z ∈ C and all r ≤ 1, we have
+µ (B(z, r)) ≤
+C
+(log(1/r))2+ε .
+(C) There exists δ > 0 such that
+sup
+z∈C
+�
+S
+dµ(w)
+|z − w|δ < ∞.
+(D) There is some C < ∞ and ε > 0 such that for all z ∈ C and all r > 0, we have
+µ (B(z, r)) ≤ Crε.
+1.4. Main results. In all theorems (except Theorem 1.6), we have the following setting:
+Setting: µ is a compactly supported probability measure on C with support S. The random
+variables Xi are i.i.d. from the distribution µ. We consider the random polynomial pn(z) :=
+(z − X1) . . . (z − Xn) and its lemniscate Λn := {z
+: |pn(z)| < 1}. We write ρn = ρ(Λn) for the
+inradius of Λn.
+Throughout the paper, w.o.p. means with overwhelming probability, i.e., with probability at least
+1 − e−cn for some c > 0.
+The theorems below concern the random lemniscate Λn. Observe that Λn consists of all z for which
+log |pn(z)| < 0, or what is the same,
+1
+n
+n
+�
+k=1
+log |z − Xk| < 0.
+By the law of large numbers, the quantity on the left converges to Uµ(z) pointwise. Hence we
+may expect the asymptotic behaviour of Λn to be described in terms of Uµ and its positive and
+negative sets Ω+, Ω−. The first three theorems make this precise under different conditions on the
+underlying measure µ.
+Theorem 1.1. Assume that µ satisfies assumption (A). Suppose that Ω− ̸= ∅ and let ρ = ρ(Ω−). Fix
+compact sets K ⊂ Ω−, and L ⊂ Ω+ \ S. Then for all large n,
+K ⊂ Λn,
+w.o.p.,
+and
+L ⊂ Λc
+n
+w.o.p.
+In particular, if ρn denotes the inradius of Λn, then
+ρn ≥ a
+w.o.p.,
+∀a ∈ (0, ρ)
+Corollary 1.2. In the setting of Theorem 1.1, lim inf ρn ≥ ρ a.s. Further, if S ⊆ Ω−, then ρn → ρ a.s.
+Ideally, we would have liked to say that a compact set L ⊆ Ω+ is contained inside Λc
+n w.o.p.
+However, this is clearly not true if some of the roots fall inside L. Making the stronger assumption
+(D) on the measure and further assuming that Uµ is bounded below by a positive number on L,
+we show that L is almost entirely contained in Λc
+n.
+Theorem 1.3. Let µ satisfy assumption (D). Let L be a compact subset of {Uµ ≥ m} for some m > 0.
+Then there exists c0 > 0 such that
+Λn ∩ L ⊂
+n�
+k=1
+B(Xk, e−c0n),
+w.o.p.
+
+INRADIUS OF RANDOM LEMNISCATES
+5
+In particular, if Uµ ≥ m everywhere, then the whole lemniscate is small. It suffices to assume that
+Uµ ≥ m on the support of µ, by the minimum principle for potentials (Theorem 3.1.4 in [34]).
+Corollary 1.4. Suppose µ satisfies assumption (D) and Uµ ≥ m on S. Then there is a c0 > 0 such that
+Λn ⊂ �n
+k=1 B(Xk, e−c0n) and ρn ≤ ne−c0n w.o.p.
+A class of examples illustrating Theorem 1.1 and Theorem 1.3 is given at the end of the section.
+What happens when the potential Uµ vanishes on a non-empty open set? In this case log |pn| has
+zero mean, and is (approximately) equally likely to be positive or negative. Because of this, one
+may expect that the randomness in Λn and ρn persists in the limit and we can at best hope for a
+convergence in distribution. The particular case when µ is uniform on the unit circle is dealt with
+in the following theorem.
+Theorem 1.5. Let µ be the uniform probability measure on S1, the unit circle in the complex plane. Then,
+ρn
+d→ ρ for some random variable ρ taking values in (0, 1
+2). Further, P{ρ < ε} > 0 and P{ρ > 1
+2 − ε} > 0
+for every ε > 0.
+As shown in the proof of Theorem 1.5, the random function log |pn(z)| converges, after appropri-
+ate normalization, almost surely to a nondegenerate Gaussian random function on D, and this
+convergence underlies the limiting random inradius ρ. We note that similar methods can be used
+to study other measures µ for which Uµ vanishes on non-empty open set (such as other instances
+where µ is the equilibrium measure of a region with unit capacity), however the case of the uni-
+form measure on the circle is rather special, as the resulting random function log |pn(z)| as well as
+its limiting Gaussian random function has a deterministic zero at the origin (which is responsible
+for the limiting inradius taking values only up to half the radius of D.
+Another setting where one can rely on convergence of the defining function log |pn(z)| is in the
+case when the polynomial pn has i.i.d. Gaussian coefficients. Actually, the convergence in this
+case is more transparent (and does not require additional tools such as Skorokhod’s Theorem) as
+pn can already be viewed as the truncation of a power series with i.i.d. coefficients. This case has
+a similar outcome as in Theorem 1.5, except the value 1/2 is replaced by 1 due to the absence of a
+deterministic zero.
+One can ask for results analogous to Theorems 1.1 and 1.3 when the zeros are dependent random
+variables. A natural class of examples are determinantal point processes. We consider one special
+case here.
+The Ginibre ensemble is a random set of n points in C with joint density proportional to
+e− �n
+k=1 |λk|2 �
+j<k
+|λj − λk|2.
+(1)
+This arises in random matrix theory, as the distribution of eigenvalues of an n × n random matrix
+whose entries are i.i.d. standard complex Gaussian. After scaling by √n, the empirical distribu-
+tions
+(2)
+µn = 1
+n
+n
+�
+j=1
+δ λj
+√n
+converge to the uniform measure on D. Hence, we may expect the lemniscate of the corresponding
+polynomial to be similar to the case when the roots are sampled independently and uniformly
+from D.
+
+6
+MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN
+Theorem 1.6. Let λ1, . . . , λn have joint density given by (1) and let Xj =
+λj
+√n. Let Λn be the unit
+lemniscate of the random polynomial pn(z) = �n
+k=1(z − Xk). Given r ∈ (0, 1) and s ∈ (1, ∞), we have
+for large n,
+Dr ⊆ Λn ⊆ Ds,
+w.o.p.
+Example 1.7. Let µr be the normalized area measure on the disk rD and suppose the roots are
+sampled from µr. It is easy to check that µr satisfies assumptions (A) − (D). We claim that
+(3)
+Uµr(z) =
+�
+|z|2−r2
+2r2
++ log r
+if |z| < r,
+log |z|
+if |z| ≥ r.
+Therefore, Ω− = rcD where
+rc =
+�
+�
+�
+�
+�
+1
+if r ≤ 1,
+r√1 − 2 log r
+if 1 ≤ r ≤ √e,
+0
+if r ≥ √e.
+Hence Theorem 1.1 implies that when r < √e, any disk Ds with radius s < rc is contained in Λn
+with overwhelming probability as n → ∞. When r ≤ 1, Corollary 1.2 implies that Λn is almost
+the same as Ω− = D. For r > √e, Theorem 1.3 applies to show that Λn is contained in a union of
+very small disks.
+Let us carry out the computations to verify (3). By rescaling, it is clear that Uµr(z) = log r +
+Uµ1(z/r), hence it suffices to consider r = 1.
+Uµ1(z) = 1
+π
+�
+D
+log |z − w|dA(w).
+For |z| ≥ 1, the integrand is harmonic with respect to w ∈ D, hence Uµ(z) = log |z| by the mean-
+value theorem. For |z| < 1, we separate the integral over the two regions where |w| < |z| and
+|w| > |z|. Harmonicity of w �→ log |z − w| on {|w| < |z|} and the mean-value property gives
+�
+|w|<|z|
+log |z − w|dA(w) = π|z|2 log |z|.
+We switch to polar coordinates w = reiθ for the second integral.
+� 1
+|z|
+� 2π
+0
+log |z − reiθ|rdθdr =
+� 1
+|z|
+� 2π
+0
+log |ze−iθ − r|dθ
+�
+��
+�
+2π log r
+rdr
+=
+� 1
+|z|
+2πr log rdr
+= 2π
+�1
+4 − |z|2
+2 log |z| + |z|2
+4
+�
+,
+where we have again used the mean value property (this time over a circle) for harmonic functions
+to compute the inside integral in the first line above. Combining these integrals over the two
+regions and dividing by π we arrive at (3).
+
+INRADIUS OF RANDOM LEMNISCATES
+7
+FIGURE 1. Lemniscates of degree n = 30, 40, 400, 15 with zeros sampled uniformly
+from the disks of radii 0.5, 1, 1.5, 1.7 (order: from top-left to bottom-right). The
+dotted circle has radius rc.
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+FIGURE 2. Lemniscates of degree n = 20, 30, 40 with zeros sampled uniformly
+from the unit circle. A unit circle is also plotted for reference in each case.
+Outline of the paper. We review some preliminary results in Section 2 that serve as tools in the
+proofs the results stated above. We prove Theorem 1.1 and Corollary 1.2 in Section 3, and we
+prove Theorem 1.3 and Corollary 1.4 in Section 4. The proof of Theorem 1.5, concerning uniform
+measure on the circle, is presented in Section 5, and Theorem 1.6, related to the Ginibre ensemble,
+is proved in Section 6.
+2. PRELIMINARY RESULTS
+We start with two preparatory lemmas which we use repeatedly in the proofs of our theorems.
+Lemma 2.1. Let µ be a Borel probability measure with compact support S ⊂ C satisfying Assumption
+(A). Whenever K is a non-empty compact subset of Ω− or a compact subset of Ω+ with K ∩ S = ∅, there
+
+1.0
+1.0
+0.5
+0.5
+0.0
+0.0
+-0.5
+0.5
+-1.0
+1.0 
+1.0
+0.5
+0.0
+0.5
+1.0
+1.0
+0.5
+0.0
+0.5
+1.0
+1.0
+1.0F
+0.5
+0.5
+0.0
+0.0F
+-0.5
+0.5 
+..
+1.0
+1.0 
+1.0
+0.5
+0.0
+0.5
+1.0
+1.0
+0.5
+0.0
+0.5
+1.08
+MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+FIGURE 3. Lemniscates of degree n = 20, 30, 40 with i.i.d. Gaussian coefficients
+plotted together with a unit circle for reference.
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+FIGURE 4. Lemniscates of degree n = 20, 30, 40 generated by the characteristic
+polynomial of a Ginibre matrix together with a unit circle plotted for reference.
+exists a constant c(K) > 0 such that
+inf
+z∈K
+�
+S
+(log |z − w|)2 dµ(w) ≥ c(K).
+Proof. Let µ satisfy assumption (A), and let K ̸= ∅ be compact. Then, by the Cauchy-Schwarz
+inequality we have for all z ∈ K
+�
+S
+(log |z − w|)2 dµ(w) ≥
+��
+S
+|log |z − w|| dµ(w)
+�2
+≥ |Uµ(z)|2
+Thus in order to prove the lemma, it suffices to show that |Uµ(z)|2 is bounded away from zero for
+z ∈ K, whenever K ⊂ Ω−, or K ⊂ Ω+ and K ∩ S = ∅.
+Suppose first that K ⊂ Ω− is compact. Since subharmonic functions are upper semi-continuous
+and hence attain a maximum on any compact set, there exists c1(K) > 0 such that Uµ(z) ≤
+−c1(K), for all z ∈ K. Hence |Uµ(z)|2 ≥ c1(K)2, for z ∈ K. In the other case, let K ⊂ Ω+ be
+compact and disjoint from the support S of µ. Notice then that Uµ(z) is positive and harmonic
+on K. An application of Harnack’s inequality now gives the existence of the required constant
+(depending only on K). This concludes the proof of the lemma.
+□
+The second lemma is based on a net argument which allows us to control the size of the modulus
+of a polynomial by its values at the points of the net.
+
+INRADIUS OF RANDOM LEMNISCATES
+9
+Lemma 2.2. Let G be a bounded Jordan domain with rectifiable boundary. Let p(z) be a polynomial of
+degree n. Then, there exists a constant C = C(G) > 0, and points w1, w2...wCn2 ∈ ∂G such that
+(4)
+∥p∥∂G ≤ 2
+max
+1≤k≤Cn2 |p(wk)|
+Proof. The key to the proof is a Bernstein-type inequality (see [32, Thm. 1])
+(5)
+|p′(z)| ≤ C1n2M,
+where M := ∥p∥∂G, and C1 is a constant that depends only on G. With this estimate in hand, the
+proof reduces to the following argument that is well-known but which we nevertheless present in
+detail for the reader’s convenience. Let ℓ = ℓ(∂G) denote the length of ∂G. Let N be a positive
+integer to be specified later. Divide ∂G into N pieces of equal length, with w0, w1..., wN denoting
+the points of subdivision. Let z0 ∈ ∂G be such that M = ∥p∥∂G = |p(z0)|. If z0 is one of the wj,
+then the estimate (4) clearly holds. If that is not the case, then z0 lies |z0 −wj| ≤ ℓ
+N , for some j with
+0 ≤ j ≤ N. We can now write
+(6)
+M − |p(wj)| ≤ |p(z0) − p(wj)| =
+�����
+� z0
+wj
+p′(t)dt
+����� ≤ C1n2M ℓ
+N .
+Here we have used the Bernstein-type inequality (5) to estimate the size of |p′|. If we now choose
+N = 2ℓC1n2, then the estimate (6) becomes
+M − |p(wj)| ≤ M
+2
+which concludes the proof of the lemma.
+□
+We will also need the following concentration inequality (see Section 2.7 of [6]). This result, re-
+ferred to as “Bennett’s inequality”, is similar to the well-known Hoeffding inequality, but note
+that, instead of being bounded, the random variables are merely assumed to be bounded from
+above.
+Theorem 2.3 (Bennett’s inequality). Let X1, X2, ..., Xn be independent random variables with finite
+variance such that Xi ≤ b for some b > 0 almost surely for all i ≤ n. Let
+S =
+n
+�
+i=1
+(Xi − E(Xi))
+and ν = �n
+i=1 E(X2
+i ). Then for any t > 0,
+P(S > t) ≤ exp
+�−ν
+b2 h
+�bt
+ν
+��
+,
+where h(u) = (1 + u) log(1 + u) − u for u > 0.
+3. PROOFS OF THEOREM 1.1 AND COROLLARY 1.2
+Proof of Theorem 1.1. We divide the proof into two steps.
+Step 1: Compact subsets of Ω− lie in Λn
+By our hypothesis Ω− ̸= ∅. Let K ⊂ Ω− be compact. We wish to show that K ⊂ Λn w.o.p. We
+may assume without loss of generality that K = G for some bounded Jordan domain G with
+
+10
+MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN
+rectifiable boundary, since any connected compact is contained such a domain. Recall that Λn =
+{z : log |pn(z)| < 0}. Writing
+log |pn(z)| =
+n
+�
+k=1
+log |z − Xk|
+as a sum of i.i.d. random variables, for z ∈ Ω− we will use a concentration inequality to show that
+log |pn(z)| is negative with overwhelming probability. We then use lemma 2.2 to get a uniform
+estimate on K and finish the proof.
+Fix z0 ∈ K. For k = 1, 2, ..., n, define Yk = log |z0 − Xk|, and let
+Z :=
+n
+�
+k=1
+(Yk − EYk).
+Notice since z0 ∈ Ω−
+EYk = Uµ(z0) < 0
+and by the assumption in the statement of the theorem, we also have
+σ2
+z0 := EY 2
+k =
+�
+S
+(log |z0 − u|)2dµ(u) < ∞
+Now applying Theorem 2.3 to our problem with b ≥ supz∈K,w∈S log(|z|+|w|), ν = nσ2
+z0, we obtain
+P{log |pn(z0)| > − log(2)} = P{log |pn(z0)| − nUµ(z0) > −nUµ(z0) − log(2)}
+= P{Z > −nUµ(z0) − log(2)}
+≤ exp
+�
+−nσ2
+z0
+b2 h
+� −b
+σ2z0
+Uµ(z0) − b log(2)
+nσ2z0
+��
+.
+Since subharmonic functions are upper semi-continuous and hence attain a maximum on any
+compact set, we have, Uµ(z) ≤ −M for all z ∈ K and some M > 0. Also, by Lemma 2.1, 0 <
+c1(K) ≤ σ2
+z ≤ c2(K) < ∞, for all z ∈ K. This bound together with the fact that h is an increasing
+function can now be used in the above estimate to get
+(7)
+P{log |pn(z0)| > − log(2)} ≤ exp
+�
+−nσ2
+z0
+b
+h
+� −b
+σ2z0
+Uµ(z0) − b log(2)
+nσ2z0
+��
+≤ exp (−cn)
+for some constant c = c(K) > 0 depending only on K. Using lemma 2.2 in combination with a
+union bound and the estimate (7), we obtain
+P{log ∥pn∥K < 0} ≥ P{
+max
+1≤k≤Cn2 log |pn(wk,n)| + log(2) < 0}
+= 1 − P{
+max
+1≤k≤Cn2 log |pn(wk,n)| > − log(2)}
+= 1 − P
+�
+�
+Cn2
+�
+k=1
+{log |pn(wk,n)| > − log(2)}
+�
+�
+≥ 1 − Cn2 exp (−cn)
+where in the last inequality we used (7). This proves that K ⊂ Λn w.o.p. and concludes the proof
+of the first part.
+
+INRADIUS OF RANDOM LEMNISCATES
+11
+Step 2: Compact subsets L of Ω+ \ S are in Λc
+n.
+Without loss of generality, we may assume that L is a closed disc in Ω+ \ S. Since S is a compact
+set disjoint from L, there exists δ > 0 such that the distance d(L, S) = δ. Notice that for all z ∈ L,
+we have − log |z − Xi| ≤ − log δ. Now fix z0 ∈ L. An application of Bennett’s inequality to the
+random variables − log |z0 − Xi| yields,
+(8)
+P (− log |pn(z0)| + nUµ(z0) ≥ nUµ(z0) − 1) ≤ exp
+�
+−nσ2
+z0
+b
+h
+� b
+σ2z0
+Uµ(z0) −
+b
+nσ2z0
+)
+��
+.
+The quantities b, h have an analogous meaning as in Step 1. By Lemma 2.1, σ2
+z is bounded below,
+and by assumption it is also bounded above, by some positive constants depending only on L.
+Furthermore, Lemma 2.1 shows that Uµ(z) ≥ c(L) > 0 for all z ∈ L. Making use of all this in (8),
+we can now estimate
+P (log |pn(z0)| > 1) = P (log |pn(z0)| − nUµ(z0) > −nUµ(z0) + 1)
+= 1 − P (log |pn(z0)| − nUµ(z0) ≤ −nUµ(z0) + 1)
+= 1 − P (− log |pn(z0)| + nUµ(z0) ≥ nUµ(z0) − 1)
+≥ 1 − exp
+�
+−nσ2
+z0
+b
+h
+�bUµ(z0)
+σ2z0
+−
+b
+nσ2z0
+��
+≥ 1 − exp (−C0(L)n) .
+(9)
+This estimate shows that individual points of L are in Λc
+n with overwhelming probability. To finish
+the proof, we once again use a net argument to show that L ⊂ Λc
+n w.o.p. We first observe that if
+z, w ∈ L, and X is one of the Xk’s, the mean value theorem gives
+| log |z − X| − log |w − X|| ≤ |z − w|
+δ
+,
+where we have used that d(L, S) = δ > 0 (and that L is a disk). The triangle inequality then yields
+(10)
+| log |pn(z)| − log |pn(w)|| ≤ n|z − w|
+δ
+,
+for z, w ∈ L.
+Choose a net of n2 equally spaced points w1, w2, ...wn2 on ∂L, and note that any point on ∂L is
+within C1/n2 of some point in the net, where C1 is a constant depending on the radius of L. From
+(10) we have that
+(11)
+| log |pn(z)| − log |pn(w)|| ≤ C2
+n ,
+for z, w ∈ L with |z − w| ≤ C1
+n2 ,
+where C2 = C1/δ is a constant.
+We are now ready to show that for large n, infz∈L log |pn(z)| > 0 w.o.p. Indeed, note that the point
+on ∂L where the infimum of log |pn| is attained must be within C1/n2 of some point in the net
+{w1, w2..., wn2}. Then by (11),
+P
+�
+inf
+L log |pn(z)| > 0
+�
+≥ P
+�
+�
+n2
+�
+k=1
+{log |pn(wk)| > 1}
+�
+� .
+
+12
+MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN
+Therefore, we obtain
+P
+�
+inf
+L log |pn(z)| > 0
+�
+≥ P
+�
+�
+n2
+�
+k=1
+{log |pn(wk)| > 1}
+�
+�
+= 1 −
+n2
+�
+k=1
+P (log |pn(wk)| ≤ 1)
+≥ 1 − n2 exp(−C0 n).
+by the pointwise estimate (9). This concludes the proof of the theorem.
+□
+Proof of Corollary 1.2. We assume that the measure µ is as in Theorem 1.1. Let ρn = ρ(Λn) be the
+inradius of the lemniscate of pn and let ρ = ρ(Ω−) be the inradius of Ω−. By Theorem 1.1, we
+immediately get lim inf ρn ≥ ρ.
+Let S be the support of µ. As S ∩ Ω+ = ∅, Theorem 1.3 shows that if m > 0 then Λn ∩ {Uµ ≥ m} is
+contained in a union of at most n circles each of radius e−cn. Writing ρn(m) for ρ(Λn ∩ {Uµ < m})
+and ρ(m) for ρ({Uµ < m}), it is then clear that ρn ≤ ρn(m)+2ne−cn ≤ ρ(m)+2ne−cn and therefore,
+first letting n → ∞ and then letting m ↓ 0 we see that
+lim sup
+n→∞ ρn ≤ lim
+m↓0 ρ(m).
+As Uµ is continuous on C\S, it follows that for any ε > 0 there is m > 0 such that {Uµ < m} ⊆ Ω−
+ε ,
+the ε enlargement of Ω−. Hence, with ρ′(ε) := ρ(Ω−
+ε ), we have
+lim sup
+n→∞ ρn ≤ lim
+ε↓0 ρ′(ε).
+Under the additional assumption that S ⊆ Ω−, we show that ρ′(ε) ↓ ρ as ε ↓ 0 and that completes
+the proof that lim sup ρn ≤ ρ. That ρ′(ε) ↓ ρ requires a proof as inradius is not continuous under
+decreasing limits of sets. For example, the inradius of the slit disk D \ [0, 1) is 1/2 but any ε-
+enlargement of it has inradius 1.
+As Uµ is harmonic on C \ S and S ⊆ Ω− and Uµ(z) ∼ log |z| near ∞, the level set {Uµ = 0} is
+a compact set comprised of curves that are real analytic except for a discrete set of points (the
+critical points of Uµ are zeros of locally defined analytic functions). It also separates S from ∞.
+Thus, {Uµ < 0} can be written as a union of Jordan domains, and there are at most finitely many
+components that have inradius more than any given number.
+Pick a component V of Ω− that attains the inradius ρ. The boundary of V can have a finite number
+of critical points of Uµ. Locally around any such critical point, Uµ is the real part of a holomorphic
+function that looks like czp for some p, and hence Uµ = 0 is like a system of equi-angular lines
+with angle π/p between successive rays. In particular, there are no cusps. What this shows is that
+V satisfies the following “external ball condition”: There is a δ0 > 0 and B < ∞, such that for any
+δ < δ0 and each w ∈ ∂V , there is a
+(12)
+w′ ∈ C \ V such that |w′ − w| = δ and |w′ − z| ≥ δ/B for all z ∈ Ω−.
+Now suppose D(z, r) ⊆ Ω−
+ε . If ε < δ0/B, we claim that D(z, r−2Bε) ⊆ Ω−, which of course proves
+that ρ ≥ ρ′(ε) − Bε, completing the proof. If the claim was not true, then we could find w ∈ ∂V
+such that |w − z| ≤ r − 2Bε. Find w′ as in (12) with δ = Bε. Then w′ ̸∈ Ωδ/B = Ωε but
+|w′ − z| ≤ |w′ − w| + |w − z| ≤ δ
+B + r − 2Bε < r.
+This is a contradiction as w′ ∈ D(z, r) ⊆ Ωε.
+□
+
+INRADIUS OF RANDOM LEMNISCATES
+13
+4. PROOF OF THEOREM 1.3 AND COROLLARY 1.4
+A standard net argument can be used to prove the theorem. But we would like to first present a
+proof of Corollary 1.4 by a different method, which may be of independent interest. At the end of
+the section, we outline the net argument to prove Theorem 1.3.
+We will need the following lemma in the proof of Corollary 1.4.
+Lemma 4.1. Under the assumptions of Corollary 1.4, there exists c1 > 0 such that
+P
+�
+log |p′
+n(X1)| ≤ m
+2 (n − 1)
+�
+≤ e−c1n.
+First we prove the corollary assuming the above Lemma.
+Proof of Corollary 1.4. Let Gi be the connected component of Λn containing Xi. Then by Bernstein’s
+inequality we have
+(13)
+|p′
+n(Xi)| ≤ C
+n2
+diam(Gi)∥pn∥∂Gi = C
+n2
+diam(Gi).
+By Lemma 4.1 we have
+|p′
+n(Xi)| ≥ exp
+�m
+2 (n − 1)
+�
+,
+w.o.p.
+and we conclude from (13) that
+(14)
+diam(Gi) ≤ Cn2 exp
+�
+−m
+2 (n − 1)
+�
+,
+w.o.p.
+The event Λn ⊂ �n
+k=1 Drn(Xk) occurs if diam(Gi) < rn for each i = 1, 2, ..., n. Using (14) and a
+union bound, all these events occur with overwhelming probability if we choose rn = exp{−c0n}
+for a suitable c0.
+□
+It remains to prove Lemma 4.1.
+Proof of Lemma 4.1. We have
+P
+�
+log |p′
+n(X1)| ≤ m
+2 (n − 1)
+�
+=
+�
+S
+P
+�
+log |p′
+n(X1)| ≤ m
+2 (n − 1)
+��X1 = z
+�
+dµ(z)
+(15)
+=
+�
+S
+P
+� n
+�
+k=2
+log |z − Xk| ≤ m
+2 (n − 1)
+�
+�
+��
+�
+(∗)
+dµ(z).
+Let us rewrite the integrand (∗) as
+(∗) = P
+�
+Z ≥
+�
+Uµ(z) − m
+2
+�
+(n − 1)
+�
+,
+Z = (n − 1)Uµ(z) −
+n
+�
+k=2
+log |z − Xk|.
+Then we have (with θ to be chosen below)
+(∗) = P
+�
+eθZ ≥ eθ(n−1)(Uµ−m/2)�
+(since Uµ ≥ m)
+≤ P
+�
+eθZ ≥ eθ(n−1)(m/2)�
+≤ e−θ(n−1)(m/2)EeθZ.
+
+14
+MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN
+Let Zk = − log |z − Xk| + Uµ(z) so that Z = Z2 + . . . + Zn. As Xi are i.i.d., so are Zi and we have
+EeθZ =
+�
+EeθZ2�n−1
+.
+We claim that there exist τ < ∞ and θ0 > 0 (not depending on z ∈ S) such that
+E[eθZ2] ≤ eτθ2
+for |θ| < θ0.
+(16)
+Assuming this, the proof can be completed as follows:
+(∗) ≤ e−θ(n−1)m/2e(n−1)τθ2
+= e− 1
+4 mθ(n−1)
+(17)
+provided we choose θ < m
+4τ . Using this in (15) we obtain
+P
+�
+log |p′
+n(X1)| ≤ m
+2 (n − 1)
+�
+≤ e− 1
+4 mθ(n−1),
+which implies the statement in the lemma.
+It remains to prove (16). Assumption (D) in definition A yields that for z ∈ S,
+P{Z1 > t} = P{|z − X1| ≤ eUµ(z)−t}
+≤ Ceε(M−t)
+where M = supz∈S Uµ(z). On the other hand, P{Z1 < −t} = 0 for large t, hence by choosing a
+smaller ε if necessary, we have the bound
+P{|Z1| > t} ≤ 2e−εt.
+A random variable satisfying the above tail bound is said to be sub-exponential (see Section 2.7
+in [41]). It is well-known (see the implication (a)
+=⇒
+(e) of Proposition 2.7.1 in [41]) that if a
+sub-exponential random variable has zero mean, then (16) holds.
+□
+Now we outline the argument for the proof of Theorem 1.3
+Proof of Theorem 1.3. The same argument (basically that − log |z − X1| + Uµ(z) has sub-exponential
+distribution) that led to (17) shows that there exists θ > 0
+P{log |pn(z)| < 1
+2mn} ≤ e−θn
+(18)
+for any z ∈ L. Let rn = e− θ
+4 n. Then, if z ∈ L \ �n
+k=1 B(Xk, rn), we have
+|∇ log |pn(z)|| =
+��
+n
+�
+k=1
+1
+z − Xk
+�� ≤
+n
+rn
+.
+Therefore, if z ∈ L \ �n
+k=1 B(Xk, (1 + m
+4 )rn), then combining the bound on the gradient with (18),
+we get
+P
+�
+inf
+B(z, 1
+4 mrn)
+log |pn| ≥ 1
+4mn
+�
+≥ 1 − e−θn.
+Assuming without loss of generality that m ≤ 1, we may choose a net of C/r2
+n points in L such that
+every of point of L\�n
+k=1 B(Xk, 2rn) is within distance mrn/4 of one of the points of the net. Then,
+log |pn| > 1
+4mn everywhere on L\�n
+k=1 B(Xk, 2rn), with probability at least 1− C
+r2n e−θn ≥ 1−Ce− θ
+2 n,
+by our choice of rn.
+□
+
+INRADIUS OF RANDOM LEMNISCATES
+15
+5. PROOF OF THEOREM 1.5
+First we claim that Λn ⊆ (1 + ε)D w.o.p. for any ε > 0. Deterministically, Λn ⊆ 2D, since µ is
+supported on S1. Further, Uµ(z) = log+ |z|, hence L = {z : 1 + ε ≤ |z| ≤ 2} is a compact subset
+of Ω+. By Theorem 1.1 or Theorem 1.3, we see that L ∩ Λn = ∅ w.o.p. proving that Λn ⊆ (1 + ε)D
+w.o.p.
+Thus, it suffices to consider Λn ∩ D. Consider
+gn(z) =
+1
+√n
+n
+�
+k=1
+log |z − Xk|
+for z ∈ D. As Xk are uniform on S1, it follows that E[log |z − X1|] = 0. Let
+K(z, w) = E[(log |z − X1|)(log |w − X1|)] = 1
+2π
+� 2π
+0
+log |z − eiθ| log |w − eiθ| dθ.
+Hence E[gn(z)] = 0 and E[gn(z)gn(w)] = K(z, w).
+Let g be the (real-valued) Gaussian process on D with expectation E[g(z)] = 0 and covariance
+function E[g(z)g(w)] = K(z, w). Then by the central limit theorem, it follows that
+(gn(z1), . . . , gn(zk)) d→ (g(z1), . . . , g(zk))
+for any z1, . . . , zk ∈ D. We observe that gn(0) = 0 and claim that suprD |∇gn| is tight, for any
+r < 1. By a well-known criterion for tightness of measures (on the space C(D) endowed with
+the topology of uniform convergence on compacts), this proves that gn → g in distribution, as
+processes (see Theorem 7.2 in [4]).
+To prove the tightness of suprD |∇gn|, fix r < s < 1 and note that ∇gn(z) is essentially the same as
+Fn(z) =
+1
+√n
+�n
+k=1
+1
+z−Xk which is holomorphic on D. By Cauchy’s integral formula, for |z| < r,
+|Fn(z)|2 =
+�� 1
+2π
+� 2π
+0
+Fn(seiθ)
+z − seiθ iseiθdθ
+��2
+≤
+� 1
+2π
+� 2π
+0
+|Fn(seiθ)|2dθ
+� � 1
+2π
+� 2π
+0
+1
+|z − seiθ|2 dθ
+�
+≤
+1
+(s − r)2
+1
+2π
+� 2π
+0
+|Fn(seiθ)|2dθ.
+The bound does not depend on z, hence taking expectations,
+E[(sup
+rD
+|Fn|)2] ≤
+1
+(s − r)2
+1
+2π
+� 2π
+0
+E
+�
+|Fn(seiθ)|2�
+dθ
+≤
+1
+(s − r)2
+1
+2π
+� 2π
+0
+E
+�
+1
+|seiθ − X1|2
+�
+dθ
+≤
+1
+(s − r)2(1 − s)2 .
+The boundedness in L2 implies tightness of the distributions of Fn, as claimed.
+In order to formulate a precise statement on almost sure convergence it is necessary to construct
+gn and g on a single probability space. One way to accomplish that is by the Skorokhod represen-
+tation theorem (see Theorem 6.7 in [4]) from which it follows that gn and g can be constructed on
+one probability space so that gn → g uniformly on compacta, a.s. Hence, the proof of Theorem 1.5
+will be complete if we prove the following lemma.
+
+16
+MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN
+Lemma 5.1. Let fn, f; D → R be smooth functions such that {f = 0} ∩ {∇f = 0} = ∅. Suppose fn → f
+uniformly on compact sets of D. Then, ρ({fn < 0}) → ρ({f < 0}).
+Indeed, applying this to gn, g, we see that ρ(Λn ∩ D) → ρ({g < 0}) almost surely. On the other
+hand, for any ε > 0, Theorem 1.3 shows that Λn ∩ ((1 + ε)D)c is contained in a union of n disks of
+radius e−cn, w.o.p. Putting these together, ρ(Λn) → ρ({g < 0}) a.s. and hence in distribution. This
+completes the proof of the convergence claim in Theorem 1.5.
+Proof of Lemma 5.1. For any U ⊆ D, it is clear that ρ(U) − ε ≤ ρ(U ∩ (1 − ε)D) ≤ ρ(U). Applying
+this to U = {fn < 0} and U = {f < 0}, we see that to show that ρ({fn < 0}) → ρ({f < 0}), it is
+sufficient to show that ρ({fn < 0}∩(1−ε)D) → ρ({f < 0}∩(1−ε)D) for every ε > 0. On (1−ε)D,
+the convergence is uniform, hence for any δ > 0, we have {f < −δ} ⊆ {fn < 0} ⊆ {f < δ} for
+sufficiently large n. It remains to show that δ �→ ρ({f < δ}) is continuous at δ = 0.
+First we show that ρ({f < −δ}) ↑ ρ({f < 0}) as δ ↓ 0. If B(z, r) ⊆ {f < 0}, then for any ε > 0,
+the maximum of f on B(z, r − ε) is some −δ < 0. Hence ρ({f ≤ −δ}) ≥ r − ε proving that
+ρ({f < −δ}) ↑ ρ({f < 0}).
+Next we show that ρ({f ≤ δn}) ↓ ρ({f ≤ 0}) for some δn ↓ 0. Let rn = ρ({f ≤
+1
+n}) and find
+zn such that B(zn, rn) ⊆ {f ≤ 1
+n}. Let rn ↓ r0 and zn → z0 without loss of generality. Then if
+w ∈ B(z0, r0), then w ∈ B(zn, rn) for large enough n, hence f(w) ≤ 1
+n for large n. Thus f ≤ 0 on
+B(z0, r0) showing that ρ({f ≤ 0}) ≥ lim
+δ↓0 ρ({f ≤ δ}).
+From the assumption that {f = 0} ∩ {∇f = 0} = ∅, we claim that ρ({f ≤ 0}) = ρ({f < 0}).
+Indeed, if B(z, r) ⊆ {f ≤ 0}, then in fact B(z, r) ⊆ {f < 0}. Otherwise, we would get w ∈ B(z, r)
+with f(w) = 0 which implies that w is a local maximum of f and hence ∇f(w) = 0.
+This proves the continuity of δ �→ ρ({f < δ}) at δ = 0, and hence the lemma.
+□
+This completes the proof of the first part that ρn = ρ({gn < 0}) converges in distribution to
+ρ = ρ({g < 0}). To show that P({ρ < ε}) > 0, it suffices to show that g > 0 on (1−ε)D∩{| Im z| > ε}
+with positive probability. To show that P({ρ >
+1
+2 − ε}) > 0, it suffices to show that g < 0 in
+(1 − ε)D ∩ {| Im z| > ε} with positive probability. We do this in two steps.
+(1) There exist u0 : D → R, harmonic with u0(0) = 0 such that u0 < 0 on (1−ε)D∩{| Im z| > ε}.
+This is known, see either the proof of Theorem 6.1 of [27] or take log |p| of the polynomial
+p constructed in Lemma 5 of Wagner [42].
+(2) For any u : D → R, harmonic with u(0) = 0 and any r < 1 and ε > 0, we claim that
+∥g −u∥sup(rD) < ε with positive probability. Applying this to u0 and −u0 from the previous
+step show that ρ > 1
+2 − ε with positive probability and ρ < ε with positive probability.
+To this end, we observe that the process g can be represented as
+g(z) = Re
+∞
+�
+k=1
+2
+kakzk
+where ak are i.i.d. standard complex Gaussian random variables. The covariance of g
+defined as above is
+E[g(z)g(w)] =
+�
+k≥1
+1
+k2 (zk ¯wk + wk¯zk)
+
+INRADIUS OF RANDOM LEMNISCATES
+17
+which can be checked to match with the integral expression for K(z, w) given earlier. Given
+any harmonic u : D → R with u(0) = 0, write it as
+u(z) = Re
+�
+k≥1
+ckzk
+and choose N such that
+∥
+�
+k>N
+ckzk∥sup(rD) < ε.
+If both the events
+AN =
+�
+∥
+�
+k>N
+ak
+k zk∥sup(rD) < ε
+�
+,
+BN =
+�
+|2ak
+k
+− ck| < ε
+N for 1 ≤ k ≤ N
+�
+occur, then |g − u| < 3ε on rD. As AN and BN are independent and have positive proba-
+bility, we also have P(AN ∩ BN) > 0.
+6. PROOF OF THEOREM 1.6
+The idea of the proof proceeds along earlier lines: first we fix t > 0 and show that log |pn(z)| is
+negative w.o.p. for a fixed z lying on |z| = 1 − t. It then follows from a net argument that the
+whole circle (and hence the disk) is contained in Λn w.o.p.
+Let t ∈ (0,
+1
+100) and fix z with |z| = 1 − t. Taking logarithms, we have as before that
+log |pn(z)| =
+n
+�
+k=1
+log |z − Xk|,
+except now the roots are no longer i.i.d. Define Ft : C → R by
+Ft(w) =
+�
+�
+�
+�
+�
+log 1
+t ,
+|z − w| ≥ 1
+t
+log |z − w|,
+t < |z − w| < 1
+t
+log t,
+|z − w| ≤ t.
+Next, we write
+log |pn(z)| =
+n
+�
+k=1
+Ft(Xk) +
+�
+k:|z−Xk|≥ 1
+t
+�
+log |z − Xk| − log 1
+t
+�
++
+�
+k:|z−Xk|≤t
+(log |z − Xk| − log t)
+=: L1 + L2 + L3.
+Since the term L3 is negative, we have
+(19)
+P
+�
+log |pn(z)| ≥ − t
+4n
+�
+≤ P
+�
+L1 + L2 ≥ − t
+4n
+�
+We claim that the right hand side of (19) decays exponentially. For that we will need the following
+Proposition 6.1. Fix t > 0. There exist constants ct, c2 > 0 such that for all large n, we have
+P
+�
+L1 ≥ − t
+2n
+�
+≤ 5 exp(−ctn),
+P(L2 ≥ t
+4n) ≤ n exp(−c2n).
+
+18
+MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN
+Assume the Proposition is true for now. Then, it is easy to see that the right hand side of (19) goes
+to 0 exponentially with n. Indeed,
+P
+�
+L1 + L2 ≥ − t
+4n
+�
+= P
+�
+L1 + L2 ≥ − t
+4n, L2 < t
+4n
+�
++ P
+�
+L1 + L2 ≥ − t
+4n, L2 ≥ t
+4n
+�
+≤ P
+�
+L1 ≥ − t
+2n
+�
++ P
+�
+L2 ≥ t
+4n
+�
+≤ 5 exp(−ctn) + n exp(−c2n).
+which establishes the claim. We now proceed with the proof of Proposition 6.1.
+Proof of Proposition 6.1. Step 1: Estimate on L2
+Let Nt = |{k : |z −Xk| ≥ 1
+t }|. If L2 ≥ t
+4n, then we must have Nt ≥ 1, which has probability at most
+e−cn for some c > 0. To see this, let us recall the following fact about eigenvalues of the Ginibre
+ensemble.
+Lemma 6.2 (Kostlan [20], [14]). Let λj be the eigenvalues (indexed in order of increasing modulus) of a
+Ginbre random matrix (un-normalized). Then,
+{|λ1|2, |λ2|2, ..., |λn|2} ∼ {Y1, Y2, ..., Yn},
+where Yj is a sum of j i.i.d. Exp(1) random variables.
+Now for the proof of the claim. Since |z| < 1 and t ∈ (0,
+1
+100), |z − Xk| ≥ 1
+t implies for instance that
+|Xk| > 99. Therefore, by elementary steps and applying Lemma 6.2, we obtain
+P(Nt ≥ 1)
+≤ P(maxk |Xk| ≥ 99)
+= P(maxk |Xk|2 ≥ 992)
+= P(maxk |λk|2 > 992n)
+= P(maxk Yk > 992n),
+where we have used Xj = λj
+√n in going from the second to third line above. Then a union bound
+and a Cramer-Chernoff estimate gives
+P(max
+k
+Yk > 992n)
+≤ nP(Yn > 992n)
+≤ n exp(−c2n),
+and combining this with the above estimate we obtain
+P(Nt ≥ 1) ≤ n exp(−c2n),
+as desired.
+Step 2: Estimate on L1
+The desired estimate is equivalent to
+(26)
+P
+�
+L1 − E(L1) ≥ − t
+2n − E(L1)
+�
+≤ 5 exp(−ctn).
+As preparation towards this, observe that 1
+nE(L1) = E
+��
+Ftdµn
+�
+, where µn is the empirical spec-
+tral measure defined in (2). By the circular law of random matrices [38], almost surely µn and
+
+INRADIUS OF RANDOM LEMNISCATES
+19
+its expectation both converge to the uniform measure on the unit disk. As a result, taking into
+account that Ft is a bounded continuous function, we obtain
+(27)
+lim
+n→∞
+1
+nE(L1) = 1
+π
+�
+D
+Ftdm = |z|2 − 1
+2
++ t2
+2 ,
+where the second equality in (27) follows from a computation similar to the one in Example 1.7.
+Using |z| = 1 − t, the quantity on the right reduces to −t + t2. Hence, for large n, we have
+E(L1) ≤ − 3
+4tn and hence, if the event in (26) holds, then
+L1 − E(L1) ≥ t
+4n.
+Thus, our immediate goal is reduced to showing that the probability of the above event is at
+most 5 exp(−ctn) for an appropriate constant ct. We invoke the following result of Pemantle and
+Peres [29, Thm. 3.2].
+Theorem 6.3. Given a determinantal point process with n < ∞ points and f a Lipschitz-1 function on
+finite counting measures, for any a > 0 we have
+P (|f − E(f)| ≥ a) ≤ 5 exp
+�
+−
+a2
+16(a + 2n)
+�
+.
+To say that f is Lipschitz-1 on the space of finite counting measures means that
+���f
+�k+1
+�
+i=1
+δxi
+�
+− f
+� k
+�
+i=1
+δxi
+� ��� ≤ 1
+for any k ≥ 0 and any points x1, . . . , xk.
+In our case, as we have recalled, {X1, X2, ..., Xn} is a determinantal point process with exactly n
+points. Moreover, L1 is Lipschitz with Lipschitz constant ∥Ft∥sup = log 1
+t . Applying Theorem 6.3
+to L1/ log(1/t), we see that
+P
+�
+L1 − E(L1) ≥ t
+4n
+�
+= 5 exp
+�
+−
+t2n2
+256(log(1/t))2(
+tn
+4 log(1/t) + 2n)
+�
+≤ 5 exp{−ctn}
+where we may take ct = ct2/ log(1/t)2 for a large constant c. This completes the proof of the
+proposition.
+□
+Now that we have proved the pointwise estimate, the net argument from Lemma 6 can be used
+to show that the whole circle |z| = 1 − t lies in the lemniscate w.o.p. The maximum principle then
+shows that the corresponding disk lies in the lemniscate w.o.p. This concludes the proof that Λn
+contains Dr w.o.p.
+We next prove that Λn ⊆ Ds w.o.p. for s > 1. Fix 1 < s′ < s and let δ = s − s′ and ε = 1
+2 log s. We
+present the proof in four steps.
+(1) |λj|
+√n < s′ for all j, w.o.p., i.e., with probability at least 1−e−cn. To see this, invoke Lemma 6.2
+to see that the complementary event has probability less than ne−c(s′)n by the same reason-
+ing used in (25), noting that 992 may be replaced by any constant greater than 1.
+
+20
+MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN
+(2) Fix z with |z| = s and let fz,δ(w) = log
+�
+min{max{|z − w|, δ}, 1
+δ}
+�
+, a bounded continuous
+function. Then by [31] (Theorem 9),
+P
+�
+�
+�
+�� 1
+n
+n
+�
+j=1
+fz,δ(λj/√n) −
+�
+D
+fz,δ(w)dm(w)
+π
+�� > ε
+�
+�
+� ≤ e−cε,δn2.
+(3) On the event in (1), fz,δ(λj/√n) = log |z − |λj
+√n| for all j and all |z| = s. Also, fz,δ(w) =
+log |z − w| for all w ∈ D. Hence, w.o.p.
+P
+��� 1
+n log |pn(z)| − log s
+�� > ε
+�
+≤ e−cε,δn2 + e−cn.
+Hence, 1
+n log |pn(z)| > 1
+2ε w.o.p. by the choice of ε = 1
+2 log s.
+(4) Let m = 100
+εδ and let z1, . . . , zm be equispaced points on ∂Ds. Then w.o.p. infj≤m 1
+n log |pn(zj)| >
+1
+2ε by the previous step. On the event in (1), ∥∇ 1
+n log |pn(z)|∥ ≤ 1
+δ, hence
+inf
+|z|=s
+1
+n log |pn(z)| > 0
+w.o.p. On this event Λn ⊆ Ds.
+This concludes the proof of Theorem 1.6.
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+

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+Deepfake CAPTCHA: A Method for Preventing Fake Calls
+Lior Yasur,∗ Guy Frankovits,∗ Fred M. Grabovski, Yisroel Mirsky
+{lioryasu,guyfrank,freddie}@post.bgu.ac.il,yisroel@bgu.ac.il
+Ben-Gurion University of the Negev
+Israel
+ABSTRACT
+Deep learning technology has made it possible to generate realistic
+content of specific individuals. These ‘deepfakes’ can now be gen-
+erated in real-time which enables attackers to impersonate people
+over audio and video calls. Moreover, some methods only need a
+few images or seconds of audio to steal an identity. Existing de-
+fenses perform passive analysis to detect fake content. However,
+with the rapid progress of deepfake quality, this may be a losing
+game.
+In this paper, we propose D-CAPTCHA: an active defense against
+real-time deepfakes. The approach is to force the adversary into
+the spotlight by challenging the deepfake model to generate con-
+tent which exceeds its capabilities. By doing so, passive detection
+becomes easier since the content will be distorted. In contrast to
+existing CAPTCHAs, we challenge the AI’s ability to create content
+as opposed to its ability to classify content. In this work we focus
+on real-time audio deepfakes and present preliminary results on
+video.
+In our evaluation we found that D-CAPTCHA outperforms state-
+of-the-art audio deepfake detectors with an accuracy of 91-100%
+depending on the challenge (compared to 71% without challenges).
+We also performed a study on 41 volunteers to understand how
+threatening current real-time deepfake attacks are. We found that
+the majority of the volunteers could not tell the difference between
+real and fake audio.
+KEYWORDS
+Deepfake, deep fake, voice cloning, impersonation, CAPTCHA,
+deep learning, fake calls, social engineering, security
+ACM Reference Format:
+Lior Yasur,∗ Guy Frankovits,∗ Fred M. Grabovski, Yisroel Mirsky. 2023.
+Deepfake CAPTCHA: A Method for Preventing Fake Calls. In Proceedings
+of ACM Conference (Conference’17). ACM, New York, NY, USA, 15 pages.
+https://doi.org/10.1145/nnnnnnn.nnnnnnn
+1
+INTRODUCTION
+A deepfake is any media, generated by a deep neural network,
+which is authentic from a human being’s perspective [40]. Since
+the emergence of deepfakes in 2017, the technology has improved
+∗These authors have equal contribution
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for components of this work owned by others than ACM
+must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
+to post on servers or to redistribute to lists, requires prior specific permission and/or a
+fee. Request permissions from permissions@acm.org.
+Conference’17, July 2017, Washington, DC, USA
+© 2023 Association for Computing Machinery.
+ACM ISBN 978-1-4503-XXXX-X/18/06...$15.00
+https://doi.org/10.1145/nnnnnnn.nnnnnnn
+Attacker
+Victim
+Deepfake
+real
+fake
+validate
+Figure 1: Overview of the proposed defense: the victim re-
+quests the caller to perform a task which is challenging for
+a deepfake model to perform. If the response is distorted or
+does not contain the task, then the caller is likely a deepfake.
+in terms of quality and has been adopted in a variety of applications.
+For example, deepfake technology is used to enhance productiv-
+ity [47], education [33] and provide entertainment [10]. However,
+the same technology has been used for unethical and malicious
+purposes as well. For example, with a deepfake, anyone can imper-
+sonate a target identity by reenacting the target’s face and/or voice.
+This ability has enabled threat actors to perform defamation, black-
+mail, misinformation, and social engineering attacks on companies
+and individuals around the world [58]. For example, since 2017,
+the technology has been used to ‘swap’ the identity of individuals
+into explicit videos for unethical [20] and malicious [35] reasons.
+More recently, in March 2022 during the Russian-Ukraine conflict,
+a deepfake video was circulated depicting the prime minister of
+Ukraine telling his troops to give up and stop fighting [60].
+1.1
+Real-time Deepfakes (RT-DF)
+Deepfake technology has improved over the last few years in terms
+of efficiency. This has enabled attackers to create real-time deep-
+fakes (RT-DF)a
+With an RT-DF, an attacker can impersonate people over voice
+and video calls. The danger of this emerging threat is that (1) the
+attack vector is not expected, (2) familiarity can be mistaken as
+authenticity and (3) the quality of RT-DFs is constantly improving.
+aExamples of RT-DF tools: https://github.com/iperov/DeepFaceLive
+https://github.com/alievk/avatarify-python
+https://samsunglabs.github.io/MegaPortraits/
+https://www.respeecher.com/
+arXiv:2301.03064v1  [cs.CR]  8 Jan 2023
+
+Conference’17, July 2017, Washington, DC, USA
+Yasur et al.
+To conceptualize this threat, let’s perform the following thought
+experiment. Imagine someone receives a call from their mother
+who is in trouble and urgently needs a money transfer. The caller
+sounds exactly like her, but the situation seems a bit out of place.
+Under stress and frustration, she hands the phone over to someone
+who sounds like the victim’s father, who confirms the situation.
+Without hesitation, many would transfer the money even though
+they’re technically talking to a stranger. Now consider state-actors
+with considerable amounts of time and resources. They could target
+workers at power plants and other critical infrastructure by posing
+as their administrators. Over a phone call, they could convince the
+worker to change a configuration or reveal confidential informa-
+tion which would lead to a cyber breach or a catastrophic failure.
+Attackers could even pose as military officials or politicians leading
+to a breach of national security.
+These scenarios are plausible because some existing real-time
+frameworks can impersonate an individual’s face or voice using
+very little information. For example, some real-time methods can
+reenact a face with one sample image [16, 50] and some can clone
+a voice with just a few seconds of audio [15, 37]. Using these tech-
+nologies, an attacker would only need to call the source voice for
+a few seconds or scrape the source’s image from the internet to
+perform the attack.
+1.2
+The Emerging Threat of RT-DFs
+Threat actors already understand the utility of RT-DFs. This is
+evident in recent events where RT-DFs have been used to perform
+criminal acts. The first case was discovered in 2019 when a CEO
+was tricked into transferring $243k due to an RT-DF phone call
+[51]. In 2021, senior European MPs participated in Zoom meetings
+with someone masquerading as Russian opposition figures [48]. In
+the same year, cyber criminals pulled off a $35 million bank heist
+involving RT-DF audio calls to a company director, tricking him
+to perform money transfers [12]. In June 2022, the FBI released a
+warning that cyber criminals are using RT-DFs in job interviews in
+order to secure remote work positions and gain insider information.
+Then in August that year, cyber criminals attended Zoom meetings
+masquerading as the CEO of Binance [59].
+1.3
+The Gap in Current Defenses
+Many methods have been proposed for detecting deepfakes [4,
+40]. These methods typically use deep learning models to either
+(1) detect mistakes or artifacts in generated media, or (2) search
+for forensic evidence such as a latent noise patterns (examples of
+these works can be found in section 4). However, there are two
+fundamental problems with existing defenses:
+Longevity. Methods which identify semantic errors or artifacts
+have the assumption that the quality of deepfakes will not
+significantly improve. However, it is clearly evident that the
+quality of deepfakes is improving and at a fast rate [39]. There-
+fore, artifact-based methods have a high potential of becoming
+obsolete within a short time-frame.
+Evasion. Methods which rely on latent noise patterns can be
+evaded by applying a post-processor. For example a deepfake
+can be passed through a low pass filter, undergo compression or
+be given additive noise. Moreover, these processes are common
+in audio and video calls. Therefore, the attacker may not need
+to do anything to remove the forensic evidence in the call.
+1.4
+Real-Time CAPTCHA
+In this paper, we propose Deepfake-CAPTCHA (D-CAPTCHA): a
+system for automatically detecting deepfake calls through challenge
+response analysis. Instead of passively observing call content, we
+actively interact with the caller by requesting that he or she to
+perform a task (the challenge). The task is easy for a human to
+perform but extremely hard for a deepfake model to recreate due to
+limitations in attack practicality and technology. When a deepfake
+tries to perform the task, the resulting content (the response) will
+be severely distorted –making it easier for an anomaly detector,
+classifier, or even the victim to detect. In addition, we propose using
+an identity model and task detection model to mitigate evasion
+tactics. The identity model compares the identity of the caller before
+and during the response to ensure that the caller cannot turn off
+the RT-DF during the task or splice in content from other identities.
+Similarly, the task detection model ensures that the caller has indeed
+performed the task as opposed to doing nothing.
+Existing CAPTCHA systems, such as reCAPTHCA,b challenge
+AI to interpret content. In contrast, we propose a system which chal-
+lenges AI to create content, with additional constraints on realism,
+identity, task (complexity), and time.
+In this work, we focus on audio-based RT-DF attacks (voice
+cloning). We consider audio RT-DFs a more significant threat over
+video RT-DFs because it is easier for an attacker to make a phone
+call than setup a video call with the victim. Also, their occurrences
+in the wild are increasing [5]. Therefore, RT-DF audio calls are
+arguably a bigger threat at this time. However, we note that the
+same D-CAPTCHA system proposed in this paper can be applied
+to video calls as well. In section 9 we present initial results in this
+domain.
+In our evaluation, we collected five state-of-the-art audio RT-DF
+technologies. We performed a panel survey to see what the public
+thinks about their quality and we evaluated the top two models
+on our defense and on others as well. We found that our method
+can significantly enhance the performance of state-of-the-art audio-
+based deepfake detectors.
+1.5
+Contributions
+In summary, our work has the following contributions:
+• We propose the first active defense against RT-DFs. Com-
+pared to existing artifact-base methods, our approach (1)
+provides stronger guarantees of detection than using only
+passive detection and (2) has better longevity because the
+challenges are extensible.
+• We define what a D-CAPTCHA is and what constitutes a
+strong deepfake CAPTCHA: We identify the limitations of
+existing RT-DF systems and propose four constraints a chal-
+lenge must present to a caller. We also present how these
+constraints can be verified in a response both manually and
+automatically. We also provide an initial set of CAPTCHAs
+and analyze their security and usability.
+bhttps://developers.google.com/recaptcha/
+
+Deepfake CAPTCHA: A Method for Preventing Fake Calls
+Conference’17, July 2017, Washington, DC, USA
+• We evaluated the quality of five state-of-the-art RT-DF voice
+cloning models with 41 volunteers. Doing so enables us to
+better understand the current threat which this technology
+poses.
+• We provide thorough evaluations on (1) how well the CAPTCHA
+system performs and (2) how robust it is against an evasive
+adversary.
+2
+BACKGROUND
+In this work, we focus on mitigating the threat of real-time voice
+cloning. Furthermore, we focus on methods that perform speech-
+to-speech voice conversion (VC) [19, 23, 30, 36, 37, 44] as opposed
+to text-to-speech (TTS) methods such as [22].
+Let 𝑡 be a target identity which we’d like to clone, and 𝑎𝑠 be an
+audio clip of identity 𝑠 speaking. Content is the part of speech that
+is independent of a speaker’s vocal anatomy (e.g., words, accent,
+enunciation, and so on). The objective of voice cloning is to perform
+𝑓𝑡 (𝑎𝑠) = 𝑎𝑔 where 𝑎𝑔 is generated audio containing the content of
+𝑎𝑠 in the style of 𝑡. In an attack, 𝑡 is an individual who is familiar
+to the victim, and 𝑠 is the attacker (or a voice actor hired by the
+attacker).
+To convert unbounded audio streams in real-time, audio is pro-
+cessed as a sequence of short audio frames (approximately 10-
+1000ms each). In this way, the 𝑖-th input frame 𝑎(𝑖)
+𝑡
+is converted
+into 𝑎(𝑖)
+𝑔
+within one second. We consider 𝑓𝑡 to be an RT-DF if the
+pipeline can be executed with no more than a 1 second delay from
+the microphone to speaker. In other words, the time it takes for an
+utterance to be recorded, converted, and played back is no longer
+than 1 second. Longer delays may raise the victim’s suspicion. Meth-
+ods which process entire recordings all at once form non-casual
+systems. Therefore, we do not consider them as RT-DF systems
+(e.g., [46]).
+There are various levels of flexibility when it comes to prior
+knowledge of 𝑠 and 𝑡. For instance, not every model can drive 𝑎𝑔
+with content from 𝑠 without prior training on 𝑠. Many of the audio
+RT-DF models can be categorized as follows:
+many-to-many. Are models which require both the source voice
+𝑠 (used in 𝑐) and the target voice 𝑡 to be in 𝑓 ’s training set. Since
+𝑠 is the attacker, the only challenge is collecting samples of 𝑡.
+any-to-many. Are models which can use any source voice to drive
+the content in 𝑥𝑔 without retraining the model.
+any-to-any. Are models which do not need to see the source 𝑠
+or target 𝑡 during training to perform 𝑓𝑡 (𝑐𝑠) = 𝑥𝑔. This makes
+any-to-any models the flexible solution for attackers.
+3
+THREAT MODEL
+There are two ways an adversary can use the RT-DF 𝑓𝑡 maliciously:
+the adversary can (1) call a victim while impersonating 𝑡 or (2) call
+a target and threaten to impersonate him. The call may take place
+over the phone through a virtual meeting (such as over Zoom). We
+refer to these calls as “fake calls”.
+3.1
+Attack Goals
+There are several attack goals which an adversary can achieve using
+a fake call:
+Cyber attacks. Fake calls can be used in social engineering attacks
+(SE). For example, instead of sending spear phishing emails to
+get employees to install malware, the attacker can call a victims
+up as their manager and ask them to do it directly. These SE
+attacks can also be used during an adversary’s reconnaissance
+on an organization to obtain system information and credentials.
+For example, the attacker can call a victim posing as a colleague,
+asking for help to login or claiming that he has "forgotten" some
+information.
+Sabotage. An attacker can impersonate a victim’s supervisor in
+an attempt to have the victim change some settings or config-
+urations in a system. For example, in a chemical processing
+plant, an adversary can use a manager’s voice to tell a worker
+to urgently alter the balance of some process –leading to cata-
+strophic results.
+Espionage. Fake calls can also be used by state agents as a means
+for extracting sensitive and confidential information. For exam-
+ple, an adversary can gain a political advantage by posing as
+a politician’s assistant and a military advantage by posing as
+a military official. Moreover, sensitive documents and source
+code can be leaked in a similar manner if the adversary imper-
+sonates a leading figure who directly asks employees for this
+material. Finally, by impersonating professionals with LinkedIn
+profiles, an adversary can obtain remote job interviews which
+may lead to remote work with a company –ultimately placing
+an insider within the organization [17].
+Scams. An attacker can prey upon people and trick them into giv-
+ing them money. For example, the adversary can impersonate
+a family member of the victim to convince the victim that his
+family is in danger and needs an urgent money transfer. Similar
+schemes can be done on business and banks where the attacker
+convinces the victim to make a money transfer under false
+pretexts [12, 51].
+Blackmail. To coerce a victim to perform an action (pay money,
+reveal information, ...) an attacker can blackmail the victim
+using RT-DF technology. For example, the attacker can speak
+to the victim using the victim’s voice and threaten the victim
+that calls will be made to reporters, friends, colleagues, or a
+spouse as the victim if the blackmail terms are not met (similar
+to a case that happened in Singapore [26]).
+Defamation. An adversary can defame the victim by perform-
+ing embarrassing or unethical acts over calls to the victim’s
+colleagues or reporters while masquerading as the victim.
+Misinformation. An attacker can call reporters and do interviews
+as politicians and other public figures to spread misinformation
+in the media.
+3.2
+Attack Setup
+The flexibility of the attacker depends on the flexibility of the RT-
+DF model. To train the model 𝑓𝑡, the attacker can use one of two
+common approaches:
+Batch Learning. If the attacker uses conventional learning mod-
+els such as [23, 30, 36, 44], then the attacker will need to collect
+a large audio training set of 𝑡 (typically around 20-30 minutes)
+and train 𝑓 on this data. This dataset can be obtained from
+the Internet if 𝑡 is a celebrity (e.g., interviews on YouTube). If
+
+Conference’17, July 2017, Washington, DC, USA
+Yasur et al.
+𝑡 doesn’t have an internet presence, then the dataset may be
+obtained via long phone calls, wiretaps, and secret recordings
+(bugs). These models are usually many-to-many or any-to-
+many.
+Few/Zero-shot Learning. When using methods such as [15, 19,
+37, 55], the attacker only needs a few seconds of 𝑡’s audio. In
+this case, the attacker can make a short phone call to 𝑡 and
+record his/her voice. The adversary may also find short video
+clips on social media or resort to wiretaps and bugs as well.
+These types of models are usually any-to-any.
+We note that most modern RT-DF technologies do not require
+labeled data since they are trained in a self-supervised manner
+[40]. Regarding quality, batch model training methods are typically
+preferred over few-shot or zero-shot methods.
+4
+RELATED WORKS
+Most audio deepfake detection systems (ADDS) use a common
+pipeline to detect deepfake audio: given an audio clip 𝑎, the pipeline
+(1) converts 𝑎 into a stream of one or more audio frames 𝑎(1), ...𝑎(𝑛),
+(2) extracts a feature representation from each frame which sum-
+marizes the frames’ waveforms 𝑥 (1), ...𝑥 (𝑛), and then (3) passes
+the frame(s) through a detector which predicts the likelihood of
+𝑎 being real or fake. The audio features in 𝑥 (𝑖) are either a Short
+Time Fourier Transform (STFT) [6, 61], spectrogram, Mel Frequency
+Cepstral Coefficients (MFCC) [27, 52], or the Constant Q Cepstral
+Coefficients (CQCC) [31, 34] of 𝑎(𝑖). Some methods simply use the
+actual waveform of 𝑎(𝑖) [53, 54].
+With this representation, an ADDS can either use a classifier
+[25, 29, 54] or anomaly detector [3, 28] to identify generated audio.
+A good summary of modern ADDS can be found in [4]. In gen-
+eral, classifiers are trained on labeled audio data consisting of two
+classes: real and deepfake. By providing labeled data, the model
+can automatically identify the relevant features (semantic or latent)
+during training. An intuitive example is the case where a deep-
+fake voice cannot accurately pronounce the letter ‘B’ [1]. In this
+scenario, the model will consider this pattern as a distinguishing
+feature for that deepfake. A disadvantage of classifiers is that they
+follow a closed-world assumption; that all examples of the deepfake
+class are in the training set. This assumption requires that detectors
+be retrained whenever new technologies are released. As for the
+model, some works use classical machine learning models such as
+SVMs and decision trees [11, 27, 32] while the majority use deep
+learning architectures such as DNNs [61, 63], CNNs [13, 38], and
+RNNs [7, 49]. To improve generalization to new deepfakes, some ap-
+proaches try to train on a diverse set of deepfake datasets (e.g., [24]).
+However, even with this strategy, ADDS systems still generalize
+poorly to new audio distributions recorded in new environments
+and to novel deepfake new technologies [42].
+In contrast to classifiers, anomaly detectors are trained on real
+voice data only and flag audio that has abnormal patterns within it.
+One approach for anomaly detection is to use the embeddings from
+a voice recognition model to compare the similarity between real
+and authentic voices [43]. Other approaches use one-class machine
+learning models such as OC-SVMs and statistical models such as
+Gaussian Mixture Models (GMM) [3, 28, 56, 63].
+What’s common with the above defenses is that they are all
+passive defenses. This means that they analyze 𝑎 but they do not
+interact with the caller to reveal the true nature of 𝑎. In contrast,
+our proposed method is active in that it can force 𝑓 to try and create
+content it is not capable of doing. By ‘pressing’ on the limitations
+of 𝑓 , we are causing 𝑓 to generate audio with significantly larger
+artifacts, making it easier for us to detect using classifiers and
+anomaly detection. Our approach also ensures some longevity since
+the attacker cannot easily overcome the limitations our challenges
+pose (further discussed in section 5.1).
+Another advantage of our system compared to others is that we
+know exactly where the anomaly should be in the media stream
+(due to the challenge response nature of the CAPTCHA protocol).
+This means that our system is more efficient since it only needs to
+execute its models over specific segments and not entire streams
+(e.g., in contrast to [9]).
+The work most similar to ours is rtCAPTCHA [57]. In this work
+the authors perform liveliness detection by (1: challenge) asking
+the caller to read out a text CAPTCHA, (2: response) verifying
+that the CAPTCHA was read back correctly, and (3: robustness)
+verifying that the face and voice match an existing user in a database.
+The concept of rtCAPTCHA is that the system assumes that the
+attacker will not be able to generate a response with the target’s
+face and voice in real-time. However, with the advent of RT-DFs,
+this rtCAPTCHA can easily be bypassed since the human attacker
+can read the text CAPTCHA back through 𝑓𝑡. Moreover, our D-
+CAPTCHA defense does not require users to register in advance,
+making the solution widely applicable to many users and scenarios.
+5
+DEEPFAKE CAPTCHAS
+In this section we discuss the limitations of RT-DFs and then use
+these limitations to define how D-CAPTCHAs work.
+5.1
+RT-DF Limitations
+Current RT-DF models can only generate content within the scope
+of the task they were trained on. For example, a model trained
+to reenact 𝑡’s face in a somewhat frontal position or generate 𝑡’s
+voice in a calm speaking tone will not be able to generate other
+content. This is evident in facial reenactment models such as [50]
+and DeepFaceLive. These models have excellent performance in
+creating faces with frontal poses, but they cannot generate the back
+of the target’s head. Similarly, for audio-based RT-DFs, it is hard
+for the model to identify and then produce certain sounds if the
+training data, loss functions, and overall pipeline focuses on the
+perfection of normal speech.
+An ideal RT-DF model would be able to create content of 𝑡
+performing an arbitrary task, where the content is both realistic
+and authentic to 𝑡’s identity. However, RT-DF models are not ideal
+because they are scoped to specific tasks during training. This is
+because doing so enables the model to perfect the identity and
+realism in 𝑥𝑔 when driven by 𝑥𝑠. Therefore, even if out of domain
+tasks can be anticipated, 𝑓𝑡 cannot be trained recreate them all. This
+is due to limitations in technology and practicality:
+5.1.1
+Technology. This set of limitations relates to the fact that
+current technology is not yet capable of creating the ideal RT-DF.
+
+Deepfake CAPTCHA: A Method for Preventing Fake Calls
+Conference’17, July 2017, Washington, DC, USA
+Inference Speed. The rate at which audio frames can be gener-
+ated depends on the efficiency of deepfake generation pipeline
+and the complexity of the model’s architecture. However, in or-
+der to handle a wide variety of different tasks, a model requires
+significantly more parametersc and possibly more complex fea-
+ture extractors in its pipeline. For example, existing RT-DF
+models would need higher resolution STFTs and MFCCs to
+capture a wider band of frequencies.
+Feature Representation. In order to capture certain patterns in
+the input 𝑎���, a model must extract appropriate feature repre-
+sentations from the input waveform. Voice tends to use lower
+frequencies and has a rather consistent spectral envelope com-
+pared to other sounds such as singing and clapping. Existing
+pipelines use compressed features such as MFCCs or STFTs
+with lower sample rates (e.g., 16-24 KHz [4]). To capture a more
+dynamic range of frequencies, higher resolution is needed. How-
+ever, increasing input resolution generally makes it harder for
+a model to converge and increases model complexity.
+Training. To train a model, a loss function must be provided to
+guide the optimization process. Modern RT-DF systems use at
+least two loss functions: one for the realism (adversarial loss)
+and one for preserving the identity of 𝑡 in 𝑎𝑔 (e.g., perceptual
+loss) [40]. If additional tasks are considered, then the model will
+likely need additional loss functions to cover each aspect. How-
+ever, loss functions compete during optimization and therefore
+some aspects will suffer. Furthermore, adding loss functions can
+make it harder for the model to converge. Finally, it’s possible
+that 𝑎𝑠 may contain a mix of voice and other audio (e.g., music
+or some other voice). To work on this audio, the model would
+have to convert the voice component and not the other audio,
+and then mix the two components back together in 𝑥𝑔. To the
+best of our knowledge this is an open problem.
+5.1.2
+Resources. This set of limitations relates to cases where the
+desired result is achievable with existing technology, however it
+may be prohibitively expensive or impractical to obtain it.
+Data Collection. To make a high quality RT-DF of 𝑡, a significant
+amount of audio samples of 𝑡 are required (e.g., [36] requires
+20-30 minutes). However, it is impractical for an attacker to
+obtain audio of 𝑡 performing specific tasks other than talking.
+If quality can be sacrificed, then zero-shot learning could be
+used. However, there is still the challenge of (1) gathering an
+extensive dataset of all possible tasks and (2) training a model
+that can generalize the samples to new identities.
+Knowledge. Creating a system that can handle even a subset of
+arbitrary tasks requires some in-depth knowledge on making
+generative deep learning models. This raises the difficulty bar
+for casual attackers, but not for advanced adversaries.
+Labeling. The process of annotating and labeling large datasets is
+expensive and time consuming. This becomes more apparent
+as the number of classes (tasks) increases.
+cAs a point of reference, StarGAN [36] is a state-of-the-art audio-based RT-DF
+models which has about 53 million parameters. In contrast, models that produce arbi-
+trary content (such as DALL-E 2 and Imagen) use 3.5-4.6 billion parameters. Moreover,
+methods such as stable-diffusion requires multiple passes.
+Assets. The ideal RT-DF model would likely be a complex model
+to handle the arbitrary tasks. Executing such a model in real-
+time would require a powerful GPU. Depending on the model’s
+complexity, the GPU may either be prohibitively expensive or
+simply non-existent.
+5.1.3
+Outlook on RT-DF Limitations. We note that the limitations
+described in this section apply to existing RT-DF systems. Although
+these limitations are hard to overcome, there is no guarantee that
+future RT-DF technologies will have the same limitations. However,
+we expect that some of the limitations, such as data collection and
+training, will still apply to novel systems in the near future.
+Therefore, to gain advantage over the adversary, we suggest
+that defenses should exploit the limitations of RT-DFs whenever
+possible.
+5.2
+D-CAPTCHA
+According to [2], a CAPTCHA is “a cryptographic protocol whose
+underlying hardness assumption is based on an AI problem.” The pro-
+tocol follows the form of a challenge-response procedure between
+server 𝐴 (the server/victim) and client 𝐵 (the client/caller), where
+(1) 𝐴 sends challenge 𝑐 to 𝐵, (2) 𝐵 sends response 𝑟𝑐 on 𝑐 back to 𝐴,
+and (3) 𝐴 verifies whether 𝑟𝑐 resolves challenge 𝑐:
+(1) 𝐴 → 𝐵 : 𝑐
+(2) 𝐵 → 𝐴 : 𝑟𝑐
+(3) 𝐴 : 𝑉 (𝑟𝑐) ∈ {𝑝𝑎𝑠𝑠, 𝑓 𝑎𝑖𝑙}
+For example, the popular reCAPTCHA prevents bots from perform-
+ing automated activities on the web by challenging the client to
+perform a human skill which is hard for software but easy for hu-
+mans (e.g., decoding distorted letters). In contrast, a D-CAPTCHA
+challenges a client by requiring the client to create content with
+the following constraints:
+(1) Realism: The content must be realistic to a human or a
+machine learning model
+(2) Identity: The content must reflect the identity 𝑡
+(3) Task: The content must have 𝑡 performing an arbitrary task
+which is hard to generate
+(4) Time: The content must be generated in real-time
+Creating a response to this challenge where 𝑉 (𝑟𝑐) = 𝑝𝑎𝑠𝑠 is hard
+for existing RT-DF technologies but easy for humans. In our system
+the ‘hardness��� of the CAPTCHA directly relates to the limitations of
+existing RT-DF technology (section 5.1). Moreover, just like modern
+CAPTCHA systems, a D-CAPTCHA system can be easily extended
+to new limitations of RT-DFs over time. This gives our system
+flexibility to defend against future threats.
+5.2.1
+Creating a Challenge. A challenge demonstrates whether
+a caller can or cannot create content with realism, identity, task
+and time constraints. Realism constraints are necessary to ensure
+there are no latent or semantic anomalies in the response. Iden-
+tity constraints are needed to ensure that the attacker isn’t just
+recording him/herself during the challenge. Task constraints are
+required to ensure that the deepfake model tries to operate outside
+the bounds of its abilities. Finally, Time constraints are involved
+to guarantee that the caller is using an RT-DF model since (1) we
+don’t want the caller to switch to an offline model and (2) real-time
+
+Conference’17, July 2017, Washington, DC, USA
+Yasur et al.
+Table 1: Examples of audio-based tasks which can be used as challenges in a D-CAPTCHA. Strong challenges are hard for
+the adversary on all four constraints: realism, identity, complexity and time. The measures in this list are based on existing
+RT-DFs methods. Playback is where the caller must play some provided audio from his/her phone into the microphone.
+Hardness
+Weakness
+Effectiveness
+Task (𝑇)
+Acronym
+Usability
+Realism
+Identity
+Task
+Time
+Evasions
+Naive Attacker
+Advanced Attacker
+Clear Throat
+CT
+•
+•
+◦
+•
+•
+•
+◦
+Hold Musical Note
+HN
+•
+◦
+◦
+•
+•
+•
+•
+Hum Tune
+HT
+•
+•
+•
+•
+•
+•
+•
+Laugh
+L
+◦
+•
+•
+•
+•
+•
+•
+Mimic Speaking Style
+MS
+◦
+•
+•
+◦
+•
+◦
+◦
+Repeat Accent
+R
+◦
+•
+•
+◦
+•
+◦
+◦
+Sing
+S
+•
+•
+•
+•
+•
+•
+•
+Speak with Emotion
+SE
+•
+•
+•
+◦
+•
+•
+•
+Yawn
+Y
+◦
+•
+◦
+•
+•
+•
+•
+Blow Noises
+BN
+•
+•
+−
+•
+•
+bypass
+•
+−
+Blow on Mic
+BM
+◦
+•
+−
+•
+•
+bypass
+•
+−
+Clap
+Cl
+•
+◦
+−
+•
+•
+bypass
+•
+−
+Click Tongue
+Clk
+•
+•
+−
+•
+•
+bypass
+•
+−
+Cough
+Co
+•
+•
+−
+•
+•
+bypass
+•
+−
+Horse Lips
+HL
+◦
+•
+−
+•
+•
+bypass
+•
+−
+Knock
+K
+◦
+◦
+−
+•
+•
+bypass
+•
+−
+Playback Audio
+PA
+−
+•
+−
+•
+•
+bypass
+•
+−
+Raspberry
+R
+•
+•
+−
+•
+•
+bypass
+•
+−
+Sound Effect
+SFX
+•
+•
+−
+•
+•
+bypass
+•
+−
+Touch Mic
+TM
+◦
+•
+−
+•
+•
+bypass
+•
+−
+Type
+T
+◦
+•
+−
+•
+•
+bypass
+•
+−
+Whistle
+W
+−
+•
+−
+•
+•
+bypass
+•
+−
+Talk & Clap
+T&C
+◦
+•
+•
+•
+•
+mix
+•
+−
+Talk & Knock
+T&K
+◦
+•
+•
+•
+•
+mix
+•
+−
+Talk & Playback
+P
+−
+•
+•
+•
+•
+mix
+•
+−
+Talk with Tones
+TT
+•
+•
+•
+•
+•
+mix
+•
+•
+Vary Speed
+VS
+•
+•
+•
+◦
+•
+mix
+•
+•
+Vary Volume
+V
+•
+•
+•
+◦
+•
+mix
+•
+•
+•: high, ◦: medium, −: low
+models are more limited since they can only process frames and
+not entire audio clips.
+The core component of a challenge in our system is the task
+which the caller must perform. Let 𝑇 denote a specific task, such
+that 𝑇 = ℎ𝑢𝑚 might be “hum a specific song.” We define the set of
+all possible challenges for task 𝑇 as 𝐶𝑇 . For example, 𝐶ℎ𝑢𝑚 would
+be all possible requests for different songs to be hummed. To select
+a challenge, (1) random seeds 𝑧0,𝑧1 are generated, (2) 𝑧0 is used
+to select a random task 𝑇 and (3) 𝑧1 is used to select a random
+challenge 𝑐 ∈ 𝐶𝑇 .
+In Table 1 we present some example tasks which can be used in
+D-CAPTCHA challenges. In the table, we assume that the RT-DF
+under test has been trained to have the best performance on one
+task; regular talking. Using observations over five state-of-the-art
+RT-DF models we assess the hardness, weakness, and effectiveness
+of each task as a challenge (see 7.1.1 for details on these five models).
+Under hardness, we express the difficulty of a modern RT-DF in
+successfully creating a deepfake of𝑡 given the respective constraints.
+For weakness, we state how an adversary can evade detection if the
+respective task is chosen. For instance, bypass is where the RT-DF
+is turned off and the attacker speaks directly to our system. The
+other case is mix is where the attacker can mix other audio sources
+into 𝑎𝑔. For example, to evade ‘talk & clap’ the attacker creates
+𝑎′𝑔 = 𝑎𝑔 + 𝑎𝑐𝑙𝑎𝑝 where 𝑎𝑐𝑙𝑎𝑝 is taken from another microphone so
+as not to disrupt the RT-DF (i.e., execute 𝑓𝑡 (𝑎𝑠 +𝑎𝑚)). Finally, in the
+table under effectiveness we consider how effective the challenge is
+given two levels of attackers: naive and advanced. A naive attacker
+is one which (1) will use existing datasets and only a limited number
+of samples of 𝑡 to train 𝑓𝑡 and (2) forwards all audio through 𝑓𝑡
+(e.g., if a library is used as-is from GitHub). An advanced attacker
+is one which will collect a practical number of samples on 𝑡 (e.g.,
+20 minutes) and is able to mix other audio sources into 𝑎𝑔.
+Overall, a strong challenge is a random 𝑐 drawn from a random𝑇
+which is hard for the adversary to perform given all four constraints.
+5.2.2
+Verifying a Challenge. To determine whether𝑉 (𝑟𝑐) = 𝑝𝑎𝑠𝑠 or
+𝑓 𝑎𝑖𝑙, we must verify whether 𝑟𝑐 adheres to the realism, identity,
+Task, and time constraints. All four constraints can be verified
+by a human (a moderator or the victim him/herself). For exam-
+ple, if 𝑐 =“say ’I’m hungry’ with anger” but (1) the audio sounds
+strange/distorted, (2) the voice does not sound like 𝑡, (3) the task
+is not completed, or (4) it takes too long for the caller to respond,
+then this would raise suspicion. However, many users may not trust
+themselves enough or they may give in to social pretexts and ig-
+nore the signs –to avoid rejecting a peer. Therefore, we propose an
+automated way to verify each constraint without prior knowledge
+of 𝑡.
+To verify 𝑟𝑐, we validate each constraint separately:
+Realism Verification (R). If an RT-DF attempts to perform𝑐 then
+𝑟𝑐 will likely contain distortions and artifacts. This is because
+(1) the RT-DF is operating outside of its capabilities or (2) be-
+cause the caller simply is using a poor-quality RT-DF. These
+distortions will make it easier for existing anomaly detectors
+and existing deepfake classifiers to identify the RT-DF. The
+output of R is a score on the range [0, ∞) or [0, 1] indicating
+how unrealistic the content of 𝑟𝑐 is.
+Identity Verification (I). To determine if 𝑟𝑐 has the identity 𝑡,
+we can do as follows: (1) collect a short audio sample 𝑎𝑡 of the
+
+Deepfake CAPTCHA: A Method for Preventing Fake Calls
+Conference’17, July 2017, Washington, DC, USA
+suspicious?
+no
+yes
+challenge 𝑐
++ instructions
+response 𝑟𝑐
+drop call
+evidence (𝑐, 𝑟𝑐)
+RT-DF Call
+(3) Response Verification 𝑉 𝑟𝑐
+Victim
+Attacker
+Deepfake
+Realism
+Time
+Task
+𝒯 𝑑 < 𝜙1
+Get voice 
+sample 𝑎𝑡
+from caller
+Select 
+challenge
+seed  𝑧1
+(2) Challenge Creation c ∈ 𝐶𝑇
+ℛ 𝑟𝑐 < 𝜙2
+𝒞 𝑟𝑐, 𝑐 < 𝜙4
+true
+true
+true
+(1) Call 
+Forwarding
+Select task
+𝑇
+𝑎𝑡
+acknowledged?
+true
+true
+Ask victim if 
+accept call from 𝑡
+given 𝑎𝑡? 
+Call connected/resumed
+seed  𝑧0
+Identity
+ℐ 𝑟𝑐, 𝑎𝑡 < 𝜙3
+Figure 2: An overview of the proposed D-CAPTCHA system: (1) Calls are forwarded to the system using a blacklist, whitelist,
+policy or the victim’s intuition, (2) a random D-CAPTCHA 𝑐 with accompanying instructions is generated and send to the
+caller as a challenge, (3) the response 𝑟𝑐 is verified against the four constraints (time, realism, identity, task) and if all four pass
+then the call is connected/resumed. Otherwise, the call is dropped and evidence is provided to the victim.
+caller prior to the challenge and have the victim acknowledge
+the identity, and (2) use zero-shot voice recognition model to
+verify that the identity in 𝑎𝑡 and 𝑟𝑐 are the same. The reason we
+have the victim acknowledged 𝑡 in 𝑎𝑡 is to prevent the attacker
+from switching the identity after the challenge. Alternatively,
+interaction with the victim can be avoided if continuous voice
+verification is used on the caller. However, doing so would be
+expensive. The output of I is a similarity score between 𝑎𝑡 and
+𝑟𝑐.
+Task Verification (C). There are two cases where 𝑟𝑐 would not
+contain the requested task: (1) the model failed to generate
+the content and (2) the attacker is trying to evade generating
+artifacts by performing another task or nothing at all. To en-
+sure that 𝑟𝑐 contains the task, we can use a machine learning
+classifier. The output of C is the probability that 𝑟𝑐 does not
+contain the task.
+Time Verification (T). The time constraint can be verified by en-
+suring that the first frame of 𝑟𝑐 is received within roughly 1
+second after of the challenge’s start time (i.e., after the instruc-
+tions for 𝑐 are given). The output of T is the measured time
+delay denoted 𝑑.
+Altogether, we validate𝑟𝑐 if none of the four algorithms (T, R, I, C)
+exceed their respective thresholds (𝜙1,𝜙2,𝜙3,𝜙4) where each thresh-
+old has been tuned accordingly. We invalidate 𝑟𝑐 if any model ex-
+ceeds its respective threshold. The false reject rate can be tuned
+by weighing the contribution of each constraint, however doing so
+will compromise the security of the system.
+In summary, validation is performed as follows:
+𝑉 (𝑟𝑐) =
+
+
+𝑝𝑎𝑠𝑠,
+T (𝑑) < 𝜙1, R(𝑟𝑐) < 𝜙2,
+I(𝑟𝑐,𝑎𝑡) < 𝜙3, C(𝑟𝑐,𝑐) < 𝜙4
+𝑓 𝑎𝑖𝑙,
+else
+(1)
+We note that a combination of validation methods for each con-
+straint can be used to increase performance, security and usability.
+For example, some verifications can be done with humans, some
+with algorithms and some with both.
+6
+DETECTION FRAMEWORK
+In this section we present the D-CAPTCHA framework which can
+be used to protect users (victims) from fake callers. A summary of
+the D-CAPTCHA framework can be found in Fig. 2.
+6.1
+1: Call Forwarding
+The very first step is to decide which calls should be forwarded to
+the system. In high risk settings, a D-CAPTCHA may be used to
+verify every caller. However, this is not practical in most settings.
+Instead, calls can be forwarded to the system using blacklists (e.g.,
+known offenders) or policies. An example policy is to forward all
+callers who are not in the victim’s address book, or to screen all
+calls during working hours.
+Alternatively, call screening can be activated by the user. For
+example, if a call arrives from an unknown number, the user can
+choose to forward it to the D-CAPTCHA system if the call is unex-
+pected. Another option is to let users forward ongoing calls if (1)
+the caller’s audio sounds strange, (2) the conversation is suspicious,
+or (3) a sensitive discussion needs to be made. For example, consider
+the scenario where a user receives a call from a friend under an
+odd pretext such as “I’m stuck in Brazil and need money to get
+out.” Here, the user can increase his/her confidence in the caller’s
+authenticity after forwarding the call through the D-CAPTCHA
+system.
+6.2
+2: Challenge Creation
+A random challenge 𝑐 is generated using the approach described
+in section 5.2.2. In addition to 𝑐, instructions for the caller are
+generated. Instructions include a list of actions to perform and a
+start indicator. For example, an instruction might be “at the tone,
+
+1Conference’17, July 2017, Washington, DC, USA
+Yasur et al.
+knock three times while introducing yourself.” The instruction is
+then converted into an audio message using TTS.
+At the start of the challenge, the caller is asked to state his/her
+name. This recording is saved as 𝑎𝑡 and shared with the victim
+for acknowledgment and with I for identity verification.d Next,
+the audio instructions are played to the caller. After playing the
+instructions, a tone is sounded. The time between the tone and the
+first audible sounds from the caller is measured and included as part
+of 𝑟𝑐 for T. After a set number of seconds, the caller’s recording is
+saved as 𝑟𝑐 and passed along for verification.
+6.3
+3: Response Verification
+The recorded response 𝑟𝑐 and its timing data are sent to T, R, C,
+and I for constraint verification. If all the algorithms yield scores
+below their respective thresholds, then 𝑎𝑡 is played to the user. If
+the user accepts the call with 𝑡 then the D-CAPTCHA is 𝑣𝑎𝑙𝑖𝑑 and
+the call is connected / resumed.
+If any of the algorithms produce a score above their threshold,
+then the call is dropped, and evidence is provided to the user. Evi-
+dence consists of an explanation of why the call was not trusted
+(e.g., information on which constraint(s) failed and to what degree)
+and playback recordings of 𝑎𝑡, 𝑐, and 𝑟𝑐 accordingly. Although the
+order which the models are executed does not matter, we can avoid
+executing redundant models if one model detects the deepfake.
+Therefore, we suggest the order T → R → C → I to potentially
+save execution time when detecting a deepfake. We also note that
+if higher security is required, then multiple D-CAPTCHAs can be
+sent out and subsequently verified to reduce the false negative rate.
+6.3.1
+Deployment. In general, the framework can be deployed
+as an app on the victim’s phone or as a service in the cloud. For
+example, onsite technicians, bankers, and the elderly can have the
+system screen calls directly on their phones. Call centers and online
+meeting rooms can use cloud resources to screen callers in waiting
+rooms (e.g., before connecting to a confidential Zoom meeting
+[48, 59]).
+6.4
+Limitations
+The main limitations of this system are its applicability and usabil-
+ity. In terms of deployment, the system must be able to interact
+with the deepfake so it can only protect against RT-DFs. More-
+over, since it is an active defense, the CAPTCHA protocol runs
+the risk of becoming a hindrance to users if not tuned correctly.
+Regardless, it’s a great solution for screening callers entering high
+security conversations and meetings in an age where calls cannot
+be trusted. Finally, the system uses deep learning models in R, I,
+and C. Just like other deep learning-based defenses, an attacker
+can potentially evade these models using adversarial examples [14].
+However, when trying to evade our system, the attacker must over-
+come a number of challenges: (1) most calls are made over noisy
+and compressed channels reducing the impact of the perturbations,
+(2) performing this attack would require real-time generation of
+adversarial examples, and (3) R, I, and C would most likely be a
+dRecall, this is done to prevent attackers from simply turning off the RT-DF during
+the challenge and using their actual voice.
+black box to the attacker, although not impervious, it cannot be
+easily queried.
+7
+THREAT ANALYSIS
+In this section, we assess the threat posed by RT-DFs by evaluating
+the quality of five state-of-the-art RT-DF models in the perspective
+of 41 volunteers.
+7.1
+Experiment Setup
+7.1.1
+RT-DF Models. We surveyed 25 voice cloning papers pub-
+lished over the last three years which can process audio in real-time
+as a sequence of frames. Of the 25 papers we selected the four recent
+works which published their source code: AdaIN-VC [15], MediumVC
+[19], FragmentVC [37] and StarGANv2-VC [36]. We also selected
+ASSEM-VC [30] which is a non-casual model as an additional com-
+parison. All works are from 2021 except AdaIN-VC which is from
+2019.
+any-to-many. StarGANv2-VC is many-to-many model which also
+works as an any-to-many model. The audio 𝑎𝑔 is created by
+passing the spectrogram of 𝑎𝑠 through an encoder-decoder
+network. To disentangle content from identity, the decoder also
+receives an encoding of 𝑎𝑠 taken from a pretrained network
+which extracts the fundamental frequencies. Finally, the decoder
+receives reference information on 𝑡 via a style encoder using
+sample 𝑎𝑡. ASSEM-VC works in a similar manner except 𝑎𝑠 and a
+TTS transcript of 𝑎𝑠 are used to generate a speaker independent
+representation before being passed to the decoder, and the
+decoder receives reference information on 𝑡 from an identify
+encoder.
+any-to-any. In AdaIN-VC, 𝑎𝑔 is created by disentangling identity
+from content. The model (1) passes a sample 𝑎𝑡 through an iden-
+tity encoder, (2) passes a source frame 𝑎(𝑖)
+𝑠
+through a content
+encoder with instance-normalization, and then (3) passes both
+outputs through a final decoder. In MediumVC, 𝑎𝑠 first normal-
+izes the voice by converting it to a common identity with an
+any-to-one VC model. The result is then encoded and passed to
+a decoder along with an identity encoding (similar to AdaIN-VC).
+FragmentVC, extracts the content of 𝑎𝑠 using a Wav2Vec 2.0
+model [8] and extracts fragments of 𝑎𝑡 using an encoder. A de-
+coder then uses attention layers to fuse the identity fragments
+into the content to produce 𝑎𝑔.
+All audio clips in this experiment were generated using the pre-
+trained models provided by the original authors. To simulate a
+realistic setting, the clips were passed through a phone filter (a
+band pass filter on the 0.3-3KHz voice range).
+7.1.2
+Experiments. To help quantify the threat of RT-DFs, we per-
+formed two experiments on a group of 41 volunteers:
+EXP1a - Quality. The goal of the first experiment was to see how
+easy it is to identify an RT-DF in the best-case scenario (when
+the victim is expecting a deepfake).
+EXP1b - Identity. The goal of this experiment was to understand
+how well RT-DF models are able to clone identities.
+In EXP1a, volunteers were asked to rate audio clips on a scale of
+1-5 (1: fake, 5: real). There were 90 audio clips presented in random
+
+Deepfake CAPTCHA: A Method for Preventing Fake Calls
+Conference’17, July 2017, Washington, DC, USA
+order: 30 real and 60 fake (12 from each of the five models). The
+clips were about 4-7 seconds long each.
+In EXP1b, we selected the top 2 models that performed the best
+in EXP1. For each model, we repeated the following trial 8 times: We
+first let the volunteer listen to two real samples of the target identity
+as a baseline. Then we played two real and two fake samples in
+random order and asked the volunteer to rate how similar their
+speakers sound compared to the speaker in the baseline.
+If a model has a positive mean opinion score (MOS) in both EXP1
+and EXP2 then it is a considerable threat. This is because it can (1)
+synthesize high quality speech (2) that sounds like the target (3) all
+in real-time.
+7.2
+Experiment Results
+EXP1a. To analyze the quality (realism) of the models, we com-
+pared the MOS scores of the deepfake audio to the MOS of the real
+audio (both scored blindly). In Fig. 3 we plot the distribution of
+each model’s MOS compared to real audio. Roughly 20-50% of the
+volunteers gave the RT-DF audio positive score with StarGANv2-VC
+having the highest quality.
+However, opinion scores are subjective. Therefore, we need to
+normalize the MOS to count how many times volunteers were
+fooled by an RT-DF. In principle, the range of scores a volunteer 𝑘
+has given to real audio captures that volunteer’s ‘trust’ range. Let
+𝜇𝑘
+𝑟𝑒𝑎𝑙 and 𝜎𝑘
+𝑟𝑒𝑎𝑙 be the mean and standard deviation on 𝑘’s scores for
+real clips. We estimate that a volunteer would likely be fooled by a
+clip if he or she scores a clip with a value greater than 𝜇𝑘
+𝑟𝑒𝑎𝑙 −𝜎𝑘
+𝑟𝑒𝑎𝑙.
+Using this measure, in Fig. 4 we present the attack success rate
+for each of the RT-DF models. We found that StarGANv2-VC has the
+highest success rate of 46% percent rate. This means that although
+current RT-DF models are not perfect, they can indeed fool people.
+We note that these results cannot be interpreted as the likelihood of
+a true RT-DF attack succeeding. This is because our volunteers were
+expecting to hear deepfakes and were therefore carefully listening
+for artifacts. A true victim would likely overlook some artifacts
+especially when put under pressure by the attacker.
+EXP1b. To analyze the ability of the models to copy identities,
+we normalized volunteer 𝑘’s scores on fake audio by computing
+𝑠𝑐𝑜𝑟𝑒−𝜇𝑘
+𝑟𝑒𝑎𝑙
+𝜎𝑘
+𝑟𝑒𝑎𝑙
+. Fig. 5 plots the distribution of the normalized scores on
+fake audio. We can see that the volunteers were mostly indecisive,
+rating some fake clips as more authentic and some as less. For the
+majority of cases (𝑠𝑐𝑜𝑟𝑒 > −1) volunteers felt that the identity was
+captured well by the top two models.
+In summary, there is a chronological trend given that the worst
+performing model AdaIN-VC is from 2019 and the best StarGANv2-VC
+is from 2021. This may indicate that the quality of RT-DF is rapidly
+improving. This raises concern, especially since the volunteers were
+expecting the attack yet could not accurately tell which clips were
+real or fake. Another insight we have is that the presence of artifacts
+can help victims identify RT-DFs. However, as quality improves,
+we expect that only way to induce significant artifacts will be by
+challenging the model.
+100
+90
+80
+70
+60
+50
+40
+30
+20
+10
+0
+Ada
+Medium
+Assem
+Fragment
+StarGan
+Real
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Model
+1
+2
+3
+4
+5
+Figure 3: RT-DF Quality - The distribution of ratings which
+the volunteers gave to each of the RT-DF models and real
+voice recordings (1: fake, 5: real).
+Ada
+Medium
+Assem
+Fragment
+StarGan
+0
+10
+20
+30
+40
+Success %
+Model
+Model
+Ada
+Medium
+Assem
+Fragment
+StarGan
+Per model, on participants who are aware of deepfake possibility
+Attack Success Rate
+Figure 4: RT-DF Quality - The percent of volunteers fooled
+by each RT-DF model, even though they were expecting a
+deepfake.
+Fragment
+Stargan
+−2
+0
+2
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+Normalized value
+Count
+Figure 5: RT-DF Identity - A histogram of the normalized
+MOS scores for how similar RT-DF audio sounds like the
+target identity 𝑡. Positive scores are cases where volunteers
+thought a fake audio sounded more like 𝑡 than an authentic
+recording of 𝑡.
+8
+D-CAPTCHA EVALUATION
+In this section, we evaluate the benefit of using a D-CAPTCHA as
+opposed to using passive defenses alone.
+8.1
+Experiment Setup
+8.1.1
+Datasets. To evaluate our system, we recorded 20 English
+speaking volunteers to create both speech and challenge-response
+datasets, summarized in Table 2:
+(D𝑟𝑒𝑎𝑙) 2498 samples of real speech (100-250 random sentences
+spoken by each of the 20 volunteers).
+(D𝑓 𝑎𝑘𝑒) 1821 samples of RT-DF voice conversion. To create this
+dataset we used StarGANv2-VC which was the top performing
+model from EXP1a. The model was trained to impersonate
+
+Conference’17, July 2017, Washington, DC, USA
+Yasur et al.
+6 of the 20 volunteers from D𝑟𝑒𝑎𝑙, and an additional 14 ran-
+dom voice actors from the VCTK dataset. The additional 14
+were added to help the model generalize better, and only the 6
+volunteers’ voices were used to make RT-DFs.
+(D𝑟𝑒𝑎𝑙,𝑟) 3317 samples of real responses (attempts at challenges).
+A sample of nine tasks were evaluated in total. The following
+tasks were performed 30 times per volunteer: sing (S), hum tune
+(HT), coughing (Co), vary volume (V), and talk & playback (P),
+and the following tasks were performed 5 times per volunteer:
+repeat accent (R), clap (Cl), speak with emotion (SE), and vary
+speed (VS).
+(D𝑓 𝑎𝑘𝑒,𝑟) 16,123 deepfake samples of RT-DF voice conversion ap-
+plied to the responses D𝑟𝑒𝑎𝑙,𝑟 using StarGANv2-VC. We did not
+convert samples from the same identity (i.e., where 𝑠 = 𝑡)
+It took each volunteer over an hour to record their data. The volun-
+teers were compensated for their time. For all train-test splits used
+in our evaluations, we made sure not to use the same identities in
+both the train and test sets.
+In addition, we also used public deepfake datasets to train the
+realism models R. These datasets were the ASVspoof-DF dataset
+[62] with 22,617 real and 15,000 fake samples, and the RITW dataset
+[42] with 19,963 real and 11,816 fake samples.
+8.1.2
+Models. Our system, when fully automated, consists of 3
+models: R, C and I. The algorithm T does not use a machine
+learning model to verify the time constraint.
+For the realism model R, we evaluated five different deepfake
+detection models: SpecRNet [25] which is a novel neural network
+architecture, inspired by RawNet2 [54], which get results compa-
+rable to state–of–the-art models despite a significant decrease in
+computational requirements. One-Class [63] is a method adapted
+from [41] based on a deep residual network ResNet-18 [21]. They
+improve and generalize the network performance using One-Class
+Softmax activations. GMM-ASVspoof [62] is a Gaussian mixture
+model (GMM) which operates on LFCCs features. This model was
+a baseline for the in ASVspoof 2021 competition. PC-DARTS [18]
+is a convolutional neural network (CNN) that tries to automati-
+cally learn the network’s architecture. This work also showed good
+results in generalizing to unseen attacks. Finally, we used Local
+Outlier Factor (LOF) which is a density-based anomaly detection
+model.
+We took the union of ASVspoof-DF and RITW and selected 80%
+at random for training the models and 10% for validation (early
+stopping). The models were tested on the baseline scenario (D𝑟𝑒𝑎𝑙
+and D𝑓 𝑎𝑘𝑒) and our proposed D-CAPTCHA scenario (D𝑟𝑒𝑎𝑙,𝑟 and
+D𝑓 𝑎𝑘𝑒,𝑟).
+For the task model C, we trained a GMM classifier on the MFCC
+features using the baseline model from [62]. One model was trained
+per task: to classify between real responses from that task and all
+other tasks as well as speech. A 70-30 train-test split was used.
+For the identity model I, we used a pre-trained voice recogni-
+tion model from the SpeechBrain toolkit [45]. The model uses the
+ECAPA-TDNN architecture to classify a speaker. Since we do not
+want I to have prior knowledge of 𝑡, we converted the model into
+an anomaly detector. Recall that we obtain a voice sample 𝑎𝑡 from
+the caller prior to the challenge. This sample is used as a reference
+to ensure that the RT-DF is not turned off during the challenge. To
+Table 2: The number of samples in each of our datasets
+Real: D𝑟𝑒𝑎𝑙
+Fake: D𝑓 𝑎𝑘𝑒
+Speech
+2498
+1821
+Real: D𝑟𝑒𝑎𝑙,𝑟
+Fake: D𝑓 𝑎𝑘𝑒,𝑟
+Repeat Accent (R)
+98
+570
+Clap (Cl)
+99
+551
+Cough (Co)
+537
+3,401
+Speak with Emotion (SE)
+98
+532
+Hum Tune (HT)
+593
+3,325
+Playback Audio (P)
+601
+3,420
+Sing (S)
+595
+334
+Vary Speed (VS)
+98
+570
+Vary Volume (V)
+598
+3,420
+Real
+Fake
+ASVspoof-DF
+22,617
+15,000
+RITW
+19,963
+11,816
+detect whether the identity of the caller has changed during the
+challenge, we compute
+I(𝑎𝑡,𝑟𝑐) = ||𝑓 ∗(𝑎𝑡) − 𝑓 ∗(𝑟𝑠)||2
+(2)
+where 𝑓 ∗ is the speaker encoding, taken from an inner layer of
+the speech recognition model. Smaller scores indicate similarity
+between the voice before the challenge and during the challenge.
+This technique of comparing speaker encodings has been done in
+the past (e.g., [40, 43]). To evaluate I, we create negative pairings
+as samples from the same identity (𝑎𝑖,𝑟𝑐,𝑖) and positive pairings as
+samples from different identities (𝑎𝑖,𝑟𝑐,𝑗), where
+𝑎𝑖,𝑎𝑗 ∈ D𝑟𝑒𝑎𝑙,
+𝑟𝑐,𝑖,𝑟𝑐,𝑗 ∈ D𝑟𝑒𝑎𝑙,𝑟
+and 𝑖 ≠ 𝑗.
+8.1.3
+Experiments. We performed four experiments:
+EXP2a R: A baseline comparison between existing solutions (pas-
+sive) and our solution (active) in detecting RT-DFs.
+EXP2b C: An evaluation of the task detection model which ensures
+that the caller indeed performed the challenge.
+EXP2c I: An evaluation of the identity model which ensures that
+the caller didn’t just turn off the RT-DF for the challenge.
+EXP2d R, C, I: An evaluation of the system end-to-end to evaluate
+the performance of the system as a whole.
+We do not evaluate T because it is just a restriction that the first
+frame of the response 𝑟𝑐 be received within approximately one
+second from the start time of the challenge.
+To measure the performance of the models, we use the area
+under the curve (AUC) and equal error rate (EER) metrics. AUC
+measures the general trade-off between the true positive rate (TPR)
+and the false positive rate (FPR). An AUC of 1.0 indicates a perfect
+classifier while an AUC of 0.5 indicates random guessing. The EER
+captures the trade-off between the FPR and the false negate rate
+(FNR). A lower EER is better.
+8.2
+Experiment Results
+8.2.1
+EXP2a (R). The goal of EXP2a was to see if our system can
+improve the detection of RT-DFs if the adversary is forced to per-
+form a task that is outside of the deepfake model’s capabilities. In
+Table 3, we compare the performance of the five deepfake detec-
+tors on (1) detecting regular deepfake speech (baseline) and on (2)
+
+Deepfake CAPTCHA: A Method for Preventing Fake Calls
+Conference’17, July 2017, Washington, DC, USA
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+True Positive Rate
+R auc: 0.864
+T&C auc: 0.985
+Co auc: 1.0
+SE auc: 0.938
+HT auc: 0.999
+P auc: 0.998
+S auc: 0.993
+VS auc: 0.963
+V auc: 0.974
+Figure 6: The performance of the task detection model C.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+True Positive Rate
+Co auc: 0.574
+HT auc: 0.688
+P auc: 0.831
+SE auc: 0.846
+S auc: 0.878
+R auc: 0.89
+T&C auc: 0.904
+VS auc: 0.926
+V auc: 0.942
+Figure 7: The performance of the unsupervised identity de-
+tection model I for different tasks.
+detecting deepfake challenges. The bold values indicate challenges
+which improved the performance of the corresponding model. We
+see that with the exception of SpecRNet, all of the detectors benefit
+from examining challenges. Overall, the best performing model was
+GMM-ASVspoof with the challenges. This means that the challenges
+provide a better way to detect RT-DFs.
+8.2.2
+EXP2b (C). If an attacker is evasive, he may try to do nothing
+instead of the challenge. It’s also possible that the attacker will try
+the challenge, but the model will output nothing because it can’t
+generate the data. Fig. 6 shows that either way, the task detector C
+can tell whether the task was performed or not with high certainty.
+8.2.3
+EXP2c (I). Another evasive strategy is where the attacker
+turns off the RT-DF while performing the challenge. In this scenario,
+we compare the identity of the caller before (𝑎𝑡) and during (𝑟𝑐) the
+challenge. In Fig. 7 we present the results of the identity detector I.
+Here we can see that the model does quite well, with the exception
+of the tasks ‘hum’ and ‘cough’ which do not carry much of the
+speaker’s identity.
+8.2.4
+EXP2d (R, I, C): D-CAPTCHA. Finally, when executing all
+three models, we must consider how the successes and failures of
+each model compound together. We set the threshold for each model
+(R, I, C) so that the FPR=0.01. We then passed through 3,317 real
+responses and 8,758 deepfake responses. Fig. 8 presents the results.
+91.9%
+93.9%
+99.4%
+91%
+99.6%
+93.2%
+99.3% 100%
+91.7%
+90%
+92.4%
+100%
+88.8%
+100%
+91.3%
+99.6% 100%
+89.2%
+Accuracy
+TPR
+0.80
+0.85
+0.90
+0.95
+1.00
+Value
+100% 100% 99.2% 100% 99.4% 99% 99.3% 100% 98.9%
+0%
+0%
+2%
+0%
+1.3%
+2.2%
+1.7%
+0%
+2.3%
+Precision
+FPR
+Accent
+Clap
+Cough
+Emotion
+Hum
+Playback
+Sing
+Speed
+Volume
+Accent
+Clap
+Cough
+Emotion
+Hum
+Playback
+Sing
+Speed
+Volume
+0.0
+0.3
+0.6
+0.9
+Task
+Value
+Figure 8: The performance of the ensure D-CAPTCHA sys-
+tem (end-to-end).
+We found that we were able to achieve a TPR of 0.89-1.00. FPR of
+0.0-2.3 and accuracy of 91-100% depending on the selected task. In
+contrast, the model which performed the best on deepfake speech
+detection (baseline) was SpecRNet with a TPR of 0.66 and accuracy
+of 71% when the FPR=0.01. Therefore, D-CAPTCHA significantly
+outperforms the baseline and provides a good defense against RT-
+DFs audio calls.
+9
+FUTURE WORK: VIDEO D-CAPTCHA
+As mentioned in the introduction, the same D-CAPTCHA system
+outlined in this paper can be applied to video-based RT-DFs as well.
+For example, to prevent imposters from joining online meetings
+(such as the cases in [48, 59]) we can forward suspicious calls to
+a D-CAPTCHA system. There are a wide variety of tasks which
+existing models and pipelines cannot handle for similar reasons
+to those listed in section 5.1. For example, the caller can be asked
+to drop/bounce objects, fold shirt, stroke hair, interact with back-
+ground, spill water, pick up objects, perform hand expressions, press
+on face, remove glasses, turn around, and so on. These tasks can
+easily be turned into challenges to detect video-based RT-DFs.
+To demonstrate the potential, we have performed some initial
+experiments and will now present some preliminary results. In
+our experiment we used a popular zero-shot RT-DF model called
+Avatarifye based on the work of [50] to reenact (puppet) a single
+photo. We were able to achieve a realistic RT-DF video at 35 fps with
+negligible distortions if the face stayed in a frontal position. How-
+ever, when we performed some of the mentioned challenges, the
+model failed and large distortions appeared. Fig., 9 in the appendix
+presents some screenshots of the video during the challenges.
+These preliminary results indicate that D-CAPTCHAs can be a
+good solution for both RT-DF audio and video calls.
+ehttps://github.com/alievk/avatarify-python
+
+Conference’17, July 2017, Washington, DC, USA
+Yasur et al.
+Table 3: The AUC and EER of deepfake detectors when used as regular deepfake detectors (baseline) and when used as R with
+the challenges.
+AUC
+Baseline
+R
+T&C
+SE
+P
+VS
+V
+S
+HT
+Co
+SpecRNet
+0.952
+0.914
+0.538
+0.796
+0.825
+0.922
+0.92
+0.834
+0.701
+0.789
+One-Class
+0.939
+0.952
+0.967
+0.941
+0.954
+0.958
+0.957
+0.948
+0.896
+0.832
+GMM-AsvSpoof
+0.949
+0.951
+0.978
+0.953
+0.97
+0.957
+0.949
+0.928
+0.949
+0.833
+PC-DARTS
+0.551
+0.568
+0.557
+0.611
+0.507
+0.586
+0.579
+0.655
+0.675
+0.635
+LOF
+0.678
+0.614
+0.93
+0.635
+0.756
+0.771
+0.824
+0.593
+0.681
+0.982
+EER
+Baseline
+R
+T&C
+SE
+P
+VS
+V
+S
+HT
+Co
+SpecRNet
+0.116
+0.163
+0.475
+0.285
+0.261
+0.155
+0.154
+0.245
+0.354
+0.281
+One-Class
+0.128
+0.123
+0.099
+0.133
+0.118
+0.112
+0.104
+0.128
+0.187
+0.259
+GMM-AsvSpoof
+0.122
+0.1
+0.071
+0.099
+0.09
+0.092
+0.115
+0.143
+0.131
+0.255
+PC-DARTS
+0.449
+0.418
+0.494
+0.386
+0.494
+0.43
+0.437
+0.366
+0.334
+0.415
+LOF
+0.326
+0.419
+0.122
+0.412
+0.262
+0.301
+0.26
+0.38
+0.382
+0.051
+Figure 9: Preliminary results showing how the D-CAPTCHA system can help prevent RT-DF video calls. Here a zero-shot
+reenactment model called Avatarify breaks the moment the caller performs an action other than basic expressions and talking.
+10
+CONCLUSION
+Deepfakes are rapidly improving in terms of quality and speed. This
+poses a significant threat as attackers are already using real-time
+deepfakes to impersonate people over calls. Current defenses use
+passive methods to identify deepfakes via their flaws. However, this
+approach may have limits as the quality of deepfakes continues
+to advance. Instead, in this work we proposed an active defense
+strategy: D-CAPTCHA. By challenging the attacker to create con-
+tent under four constraints based on practical and technological
+limitations, we can force the deepfake model to expose itself. By pro-
+tecting calls and meetings from deepfake imposters, we believe that
+this system can significantly improve the security of organizations
+and individuals.
+ACKNOWLEDGMENTS
+This work was supported by the U.S.-Israel Energy Center managed
+by the Israel-U.S. Binational Industrial Research and Development
+(BIRD) Foundation and the Zuckerman STEM Leadership Program.
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+A
+ETHICAL DISCLOSURES
+The experiments performed in this study have received our institu-
+tion’s ethical committee’s approval. All 20 volunteers whose voices
+were used to create deepfakes permitted the use of their data for
+this purpose. To protect our volunteers, the trained RT-DF voice
+models will not be shared.
+B
+ADDITIONAL FIGURES
+
+Deepfake CAPTCHA: A Method for Preventing Fake Calls
+Conference’17, July 2017, Washington, DC, USA
+0.668
+0.811
+0.834
+0.85
+0.854
+0.856
+0.86
+0.863
+0.864
+0.874
+0.88
+0.889
+0.891
+0.891
+0.895
+0.897
+0.904
+0.905
+0.906
+0.906
+0.91
+0.91
+0.911
+0.92
+0.921
+0.921
+0.922
+0.923
+0.924
+0.924
+0.926
+0.93
+0.935
+0.936
+0.938
+0.944
+HT + Co
+SE + Co
+P + Co
+Co + T&C
+S + Co
+HT + P
+R + Co
+SE + HT
+HT + S
+HT + R
+SE + P
+P + S
+HT + T&C
+T&C + P
+Co + VS
+R + P
+S + SE
+V + Co
+P + VS
+T&C + SE
+SE + R
+S + R
+T&C + R
+T&C + S
+R + VS
+VS + HT
+SE + V
+P + V
+SE + VS
+V + HT
+V + R
+VS + T&C
+T&C + V
+VS + S
+V + S
+VS + V
+0.00
+0.25
+0.50
+0.75
+AUC
+Pairs of challenges
+Figure 11: The performance of I when two challenges are requested, measured in AUC.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+True Positive Rate
+Receiver Operating Characteristic of gmm
+S AUC = 0.9284
+V AUC = 0.9494
+VS AUC = 0.9567
+Co AUC = 0.8328
+P AUC = 0.9704
+HT AUC = 0.9491
+T&C AUC = 0.9784
+SE AUC = 0.9531
+R AUC = 0.9510
+baseline AUC = 0.9489
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+True Positive Rate
+Receiver Operating Characteristic of raw-pc
+S AUC = 0.6551
+V AUC = 0.5786
+VS AUC = 0.5859
+Co AUC = 0.6349
+P AUC = 0.5070
+HT AUC = 0.6751
+T&C AUC = 0.5575
+SE AUC = 0.6109
+R AUC = 0.5683
+baseline AUC = 0.5512
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+True Positive Rate
+Receiver Operating Characteristic of SpecRNet
+S AUC = 0.8335
+V AUC = 0.9205
+VS AUC = 0.9221
+Co AUC = 0.7889
+P AUC = 0.8255
+HT AUC = 0.7006
+T&C AUC = 0.5378
+SE AUC = 0.7958
+R AUC = 0.9139
+baseline AUC = 0.9518
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+True Positive Rate
+Receiver Operating Characteristic of oneclass
+S AUC = 0.9479
+V AUC = 0.9574
+VS AUC = 0.9581
+Co AUC = 0.8324
+P AUC = 0.9542
+HT AUC = 0.8957
+T&C AUC = 0.9668
+SE AUC = 0.9408
+R AUC = 0.9523
+baseline AUC = 0.9393
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+True Positive Rate
+Receiver Operating Characteristic of LOF
+S AUC = 0.5934
+V AUC = 0.8241
+VS AUC = 0.7709
+Co AUC = 0.9823
+P AUC = 0.7561
+HT AUC = 0.6810
+T&C AUC = 0.9302
+SE AUC = 0.6355
+R AUC = 0.6139
+baseline AUC = 0.6784
+Figure 12: ROC plots for each deepfake detection model from experiment EXP2a. The bold line shows the baseline (regular
+deepfake detection) and the others show the performance on the given task.
+

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+arXiv:2301.02627v1  [math.RA]  6 Jan 2023
+Pre-Lie algebras, their multiplicative lattice, and
+idempotent endomorphisms
+Michela Cerqua and Alberto Facchini
+Abstract We introduce the notions of pre-morphism and pre-derivation for arbitrary
+non-associative algebras over a commutative ring 푘 with identity. These notions
+are applied to the study of pre-Lie 푘-algebras and, more generally, Lie-admissible
+푘-algebras. Associating with any algebra (퐴, ·) its sub-adjacent anticommutative
+algebra (퐴, [−, −]) is a functor from the category of 푘-algebras with pre-morphisms
+to the category of anticommutative 푘-algebras. We describe the commutator of two
+ideals of a pre-Lie algebra, showing that the condition (Huq=Smith) holds for pre-
+Lie algebras. This allows to make use of all the notions concerning multiplicative
+lattices in the study of the multiplicative lattice of ideals of a pre-Lie algebra. We
+study idempotent endomorphisms of a pre-Lie algebra 퐿, i.e., semidirect-product
+decompositions of 퐿 and bimodules over 퐿.
+Introduction
+The aim of this paper is to present pre-Lie algebras from the point of view of their
+multiplicative lattice of ideals, and to study their idempotent endomorphisms. Pre-
+Lie algebras were first introduced and studied in [15] by Vinberg. He applied them
+to the study of convex homogenous cones. He called “left-symmetric algebras” the
+algebras we call pre-Lie algebras in this paper.
+We present a notion of pre-morphism and pre-derivation for arbitrary non-
+associative algebras over a commutative ring 푘 with identity, and apply it to the
+study of pre-Lie 푘-algebras and, more generally, Lie-admissible 푘-algebras. Asso-
+ciating with any pre-Lie algebra (퐴, ·) its sub-adjacent Lie algebra (퐴, [−, −]) is a
+functor from the category PreL푘,푝 of pre-Lie 푘-algebras with pre-morphisms to the
+category of Lie 푘-algebras. We introduce the notion of module 푀 over a pre-Lie
+Michela Cerqua e-mail: michela.cerqua@studenti.unipd.it · Alberto Facchini
+Dipartimento di Matematica "Tullio Levi Civita", Università di Padova, 35121 Padova, Italy e-mail:
+facchini@math.unipd.it
+1
+
+2
+Michela Cerqua and Alberto Facchini
+algebra 퐿 and, like in the case of associative algebras, it is possible to do it in two
+equivalent ways, via a suitable scalar multiplication 퐿 × 푀 → 푀 or as a 푘-module
+푀 with a pre-morphism 휆: (퐿, ·) → (End(푘푀), ◦). The category of modules over
+a pre-Lie 푘-algebra (퐿, ·) is isomorphic to the category of modules over its sub-
+adjacent Lie 푘-algebra (퐿, [−, −]). We then consider the commutator of two ideals
+in a pre-Lie algebra. In particular we show that the condition (Huq=Smith) holds
+for pre-Lie algebras. With the notion of commutator at our disposal, the lattice of
+ideals of a pre-Lie algebra becomes a multiplicative lattice [6, 8]. As a consequence
+we immediately get the notions of abelian pre-Lie algebra, prime ideal, prime spec-
+trum of a pre-Lie algebra, solvable and nilpotent pre-Lie algebras, metabelian and
+hyperabelian pre-Lie algebras, centralizer, and center.
+We then consider idempotent endomorphisms of a pre-Lie algebra, because they
+immediately show what semi-direct products of pre-Lie algebras are, what the action
+of a pre-Lie algebra on another pre-Lie algebra is, and lead us to the notion of
+bimodule over a pre-Lie algebra. We study the “Dorroh extensions” of pre-Lie
+algebras. Like in the associative case, we get a category equivalence between the
+category PreL푘 and the category of pre-Lie algebras with identity and with an
+augmentation.
+1 Preliminary notions on non-associative 풌-algebras
+Let 푘 be a commutative ring with identity. In this article, a 푘-algebra is a 푘-module
+푘푀 with a further 푘-bilinear operation 푀 × 푀 → 푀, (푥, 푦) ↦→ 푥푦 (equivalently,
+a 푘-module morphism 푀 ⊗푘 푀 → 푀). A subalgebra (an ideal, resp.) of 푀 is a
+푘-submodule 푁 of 푀 such that 푥푦 ∈ 푁 for every 푥, 푦 ∈ 푁 (푥푛 ∈ 푁 and 푛푥 ∈ 푁
+for every 푥 ∈ 푀 and 푛 ∈ 푁, resp.) As usual, if 푁 is an ideal of 푀, the quotient
+푘-module 푀/푁 inherits a 푘-algebra structure. There is a one-to-one correspondence
+between the set of all ideals 푁 of 푀 and the set of all congruences on 푀, that is,
+all equivalence relations ∼ on 푀 for which 푥 ∼ 푦 and 푧 ∼ 푤 imply 푥 + 푧 ∼ 푦 + 푤,
+휆푥 ∼ 휆푦 and 푥푧 ∼ 푦푤 for every 푥, 푦, 푧, 푤 ∈ 푀 and every 휆 ∈ 푘. The opposite 푀op of
+an algebra 푀 is defined taking as multiplication in 푀op the mapping (푥, 푦) ↦→ 푦푥.
+If 푀 and 푀′ are two 푘-algebras, a 푘-linear mapping 휑: 푀 → 푀′ is a 푘-algebra
+homomorphism if 휑(푥푦) = 휑(푥)휑(푦) for every 푥, 푦 ∈ 푀. Clearly, 푘-algebras form a
+variety in the sense of Universal Algebra. Moreover, it is a variety of Ω-groups, that
+is, a variety which is pointed (i.e., it has exactly one constant) and has amongst its
+operations and identities those of the variety of groups. It follows that 푘-algebras form
+a semiabelian category. Other examples of Ω-groups are abelian groups, non-unital
+rings, commutative algebras, modules and Lie algebras.
+If 푀 is any 푘-algebra, its endomorphisms form a monoid, that is, a semigroup
+with a two-sided identity, with respect to composition of mappings ◦. A derivation
+of a 푘-algebra 푀 is any 푘-linear mapping 퐷 : 푀 → 푀 such that 퐷(푥푦) = (퐷(푥))푦+
+푥(퐷(푦)) for every 푥, 푦 ∈ 푀. For any 푘-algebra 푀, we can construct the 푘-algebra
+of derivations Der푘(푀) of the 푘-algebra 푀. Its elements are all derivations of 푀.
+
+Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms
+3
+If 푀 is any 푘-algebra and 퐷, 퐷′ are two derivations of 푀, then the composite
+mapping 퐷퐷′ is not a derivation of 푀 in general, but 퐷퐷′ − 퐷′퐷 is. Thus, for any
+푘-algebra 푀, we can define the Lie 푘-algebra Der푘 (푀) as the subset of End(푘푀)
+consisting of all derivations of 푀 with multiplication [퐷, 퐷′] := 퐷퐷′ − 퐷′퐷 for
+every 퐷, 퐷′ ∈ Der푘(푀).
+It is known that there is not a general notion of representation (or module)over our
+(non-associative) 푘-algebras. There is a notion of bimodule over a non-associative
+ring due to Eillenberg, and this notion works well for Lie algebras, but is not
+convenient in the study of Jordan algebras and alternative algebras. The situation, as
+far as modules are concerned, is the following.
+1.1 Modules over an associative 풌-algebra.
+Given any 푘-algebra 푀, we can consider, for every element 푥 ∈ 푀, the map-
+ping 휆푥 : 푀 → 푀, defined by 휆푥(푎) = 푥푎 for every 푎 ∈ 푀. The mapping
+휆: 푀 → End(푘푀) is defined by 휆: 푥 ↦→ 휆푥 for every 푥 ∈
+푀. This 휆 is a 푘-
+algebra morphism if and only if 푀 is associative. Thus, for any associative 푘-algebra
+푀, it is natural to define a left 푀-module as any 푘-module 푘 퐴 with a 푘-algebra
+homomorphism 휆: 푀 → End(푘 퐴). Similarly, we can define right 푀-modules as
+푘-modules 푘 퐴 with a 푘-algebra antihomomorphism 휌 : 푀 → End(푘 퐴). Here by 푘-
+algebra antihomomorphism 휓 : 푀 → 푀′ between two 푘-algebras 푀, 푀′ we mean
+any 푘-linear mapping 휓 such that 휓(푥푦) = 휓(푦)휓(푥) for every 푥, 푦 ∈ 푀. Clearly, a
+mapping 푀 → 푀′ is a 푘-algebra antihomomorphism if and only if it is a 푘-algebra
+homomorphism 푀op → 푀′. It follows that right 푀-modules coincide with left
+푀op-modules. More precisely, when we say that right 푀-modules coincide with left
+푀op-modules, we mean that there is a canonical category isomorphism between the
+category of all right 푀-modules and the category of all left 푀op-modules. Simi-
+larly, left 푀-modules coincide with right 푀op-modules. Also, if 푀 is commutative,
+then left 푀-modules and right 푀-modules coincide. Finally, left modules 퐴 over
+an associative 푘-algebra 푀 can be equivalently defined using, instead of the 푘-
+algebra homomorphism 휆: 푀 → End(푘 퐴), a 푘-bilinear mapping 휇: 푀 × 퐴 → 퐴,
+휇: (푚, 푎) ↦→ 푚푎, such that (푚푚′)푎 = 푚(푚′푎) for every 푚, 푚′ ∈ 푀 and 푎 ∈ 퐴.
+1.2 Modules over a Lie 풌-algebra.
+For any 푘-module 퐴 we will denote by 픤픩(퐴) the Lie 푘-algebra End(푘 퐴) of all
+푘-endomorphisms of 퐴 with the operation [−, −] defined by [ 푓 , 푔] = 푓 푔 − 푔 푓 .
+For any Lie 푘-algebra 푀 and any element 푥 ∈ 푀, the mapping 휆푥 is an element
+of the Lie 푘-algebra Der푘 (푀), usually called the adjoint of 푥, or the inner derivation
+defined by 푥, and usually denoted by ad푀 푥 instead of 휆푥, and the mapping ad: 푀 →
+
+4
+Michela Cerqua and Alberto Facchini
+Der푘(푀) ⊆ 픤픩(푀), defined by ad: 푥 ↦→ ad푀 푥 for every 푥 ∈ 푀, is a Lie 푘-algebra
+homomorphism.
+Left modules over a Lie 푘-algebra 푀 are defined as 푘-modules 퐴 with a Lie
+푘-algebra homomorphism휆: 푀 → 픤픩(퐴). Similarly, it is possible to define right 푀-
+modules as 푘-modules 퐴 with a 푘-algebra antihomomorphism 휌 : 푀 → 픤픩(퐴). But
+any Lie 푘-algebra 푀 is isomorphic to its opposite algebra 푀op via the isomorphism
+푀 → 푀op, 푥 ↦→ −푥. It follows that the category of right 푀-modules is canonically
+isomorphic to the category of left 푀-modules for any Lie 푘-algebra 푀. Therefore
+it is useless to introduce both right and left modules, it is sufficient to introduce left
+푀-modules and call them simply “푀-modules”.
+2 Pre-Lie 풌-algebras
+A pre-Lie 푘-algebra is a 푘-algebra (푀, ·) satisfying the identity
+(푥 · 푦) · 푧 − 푥 · (푦 · 푧) = (푦 · 푥) · 푧 − 푦 · (푥 · 푧)
+(1)
+for every 푥, 푦, 푧 ∈ 푀.
+For any 푘-algebra (푀, ·), defining the commutator [푥, 푦] = 푥 · 푦 − 푦 · 푥 for every
+푥, 푦 ∈ 푀, the algebra (푀, [−, −]) is anticommutative. If (푀, ·) is a pre-Lie algebra,
+one gets that (푀, [−, −]) is a Lie algebra, called the Lie algebra sub-adjacent to the
+pre-Lie algebra (푀, ·).
+Pre-Lie algebras are also called Vinberg algebras or left-symmetric algebras.
+This last name refers to the fact that in (1) one exchanges the first two variables on
+the left. A right-symmetric algebra is an algebra in which, for every 푥, 푦, 푧 ∈ 푀,
+(푥 · 푦) · 푧 − 푥 · (푦 · 푧) = (푥 · 푧) · 푦 − 푥 · (푧 · 푦). It is easily seen that the category of
+left-symmetricalgebras and the category of right-symmetricalgebras are isomorphic
+(the categorical isomorphism is given by 푀 ↦→ 푀op).
+Examples 1 (1) Every associative algebra is clearly a pre-Lie algebra.
+(2) Derivations on 푘[푥1, . . . , 푥푛]푛. Let 푘 be a commutative ring with identity,
+푛 ≥ 1 be an integer, and 푘[푥1, . . . , 푥푛] be the ring of polynomials in the 푛 indeter-
+minates 푥1, . . . , 푥푛 with coefficients in 푘. Let 퐴 be the free 푘[푥1, . . . , 푥푛]-module
+푘[푥1, . . . , 푥푛]푛 with free set {푒1, . . . , 푒푛} of generators. As a 푘-module, 퐴 is the
+free 푘-module with free set of generators the set { 푥푖1
+1 . . . 푥푖푛
+푛 푒 푗 | 푖1, . . . , 푖푛 ≥ 0, 푗 =
+1, . . . , 푛}. Consider the usual derivations of the ring 푘[푥1, . . . , 푥푛]:
+휕
+휕푥 푗
+(푥푖1
+1 . . . 푥푖푛
+푛 ) =
+�
+푥푖1
+1 . . . 푖 푗푥푖푗−1
+푗
+. . . 푥푖푛
+푛
+for 푖 푗 > 0,
+0
+for 푖 푗 = 0.
+Defineamultiplication on 퐴 setting,forevery 푢 = (푢1, . . . , 푢푛), 푣 = (푣1, . . . , 푣푛) ∈ 퐴,
+
+Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms
+5
+푣 · 푢 = (
+푛
+�
+푗=1
+푣 푗
+휕푢1
+휕푥 푗
+, . . . ,
+푛
+�
+푗=1
+푣 푗
+휕푢푛
+휕푥 푗
+).
+It is then possible to see that 퐴 is a pre-Lie 푘-algebra [2, Section 2.3].
+(3) An example of rank 2. Let 푘 be any commutative ring with identity and
+퐿 � 푘 ⊕ 푘 a free 푘-module of rank 2 with free set {푒1, 푒2} of generators. Define
+a multiplication on 퐿 setting 푒1푒1 = 2푒1, 푒1푒2 = 푒2, 푒2푒1 = 0, 푒2푒2 = 푒1, and
+extending by 푘-bilinearity. Then 퐿 is a pre-Lie 푘-algebra [13].
+(4) Rooted trees. Recall that a tree is an undirected graph in which any two vertices
+are connected by exactly one path, or equivalently a connected acyclic undirected
+graph. A rooted tree of degree 푛 is a pair (푇, 푟), where 푇 is a tree with 푛 vertices,
+and its root 푟 is a vertex of 푇. In the following we will label the vertices of 푇 with
+the numbers 1, . . . , 푛, and the root 푟 with 1.
+Let 푘 be a commutative ring with identity and T푛 be the free 푘-module with free
+set of generators the set of all isomorphism classes of rooted trees of degree 푛. Set
+T :=
+�
+푛≥1
+T푛.
+Define a multiplication on T setting, for every pair 푇1,푇2 of rooted trees,
+푇1 · 푇2 =
+�
+푣 ∈푉 (푇2)
+푇1 ◦푣 푇2,
+where 푉(푇2) is the set of vertices of 푇2, and 푇1 ◦푣 푇2 is the rooted tree obtained by
+adding to the disjoint union of 푇1 and 푇2 a further new edge joining the root vertex
+of 푇1 with the vertex 푣 of 푇2. The root of 푇1 ◦푣 푇2 is defined to be the same as the
+root of 푇2. To get a multiplication on T, extend this multiplication by 푘-bilinearity.
+Let us give an example. Suppose
+푇1 =
+1
+2
+3
+and
+푇2 =
+1
+2
+Then
+
+6
+Michela Cerqua and Alberto Facchini
+푇1 ◦1 푇2 =
+1
+2
+3
+4
+5
+and
+푇1 ◦2 푇2 =
+1
+2
+3
+4
+5
+,
+where we have relabelled the vertices of푇1. (If푇1 has 푛 vertices and푇2 has 푚 vertices,
+it is convenient to relabel in 푇1 ◦푣 푇2 the vertices 1, . . . , 푛 of 푇1 with the numbers
+푚 + 1, . . . , 푚 + 푛, respectively.) Therefore
+푇1 · 푇2 =
+1
+2
+3
+4
+5
++
+1
+2
+3
+4
+5
+In this way, one gets a pre-Lie 푘-algebra T [4, 2]. It is a graded 푘-algebra because
+T푛 · T푚 ⊆ T푛+푚 for every 푛 and 푚. It can be proved that this is the free pre-Lie
+푘-algebra on one generator [4]. (The free generator of T is the rooted tree with one
+vertex.)
+(5) Upper triangular matrices. This is an interesting example taken from [13],
+where all the details can be found. Let 푘 be a commutative ring with identity in which
+2 is invertible, and 푛 be a fixed positive integer. Let 푀 be the 푘-algebra of all 푛 × 푛
+matrices, and 푈 be the its subalgebra of upper triangular matrices. Let 휑: 푀 → 푈
+be the the 푘-linear mapping that associates with any matrix 퐴 = (푎푖 푗) ∈ 푀 the
+matrix 퐵 = (푏푖 푗) ∈ 푈, where 푏푖 푗 = 푎푖 푗 if 푎푖 푗 is above the main diagonal, 푏푖 푗 = 0 if
+푎푖 푗 is below the main diagonal, and 푏푖푖 = 푎푖푖/2 if 푎푖 푗 = 푎푖푖 is on the main diagonal.
+Also, for every 퐴 ∈ 푀, let 퐴tr be the transpose of the matrix 퐴. Define an operation
+· on 푈 setting, for every 푋,푌 ∈ 푈, 푋 · 푌 := 푋푌 + 휑(푋푌tr + 푌 푋tr). Then (푈, ·) is a
+pre-Lie 푘-algebra.
+
+Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms
+7
+As we have defined in Section 1, a 푘-algebra homomorphism 휑: 푀 → 푀′ is
+a 푘-module morphism such that 휑(푥푦) = 휑(푥)휑(푦) for every 푥, 푦 ∈ 푀. But we
+also need another notion. We say that a 푘-module morphism 휑: 푀 → 푀′, where
+푀, 푀′ are arbitrary (not-necessarily associative) 푘-algebras, is a pre-morphism if
+휑(푥푦) − 휑(푥)휑(푦) = 휑(푦푥) − 휑(푦)휑(푥) for every 푥, 푦 ∈ 푀.
+Lemma 1. A mapping 휑: 푀 → 푀′, where (푀, ·), (푀′, ·) are arbitrary 푘-algebras,
+is a pre-morphism (푀, ·) → (푀′, ·) if and only if it is a 푘-algebra morphism
+(푀, [−, −]) → (푀′, [−, −]).
+Proof. If (푀, ·), (푀′, ·) are 푘-algebras and 휑: 푀 → 푀′ is a mapping, then
+휑: (푀, ·) → (푀′, ·)
+is a pre-morphism if and only if 휑(푎푏) − 휑(푎)휑(푏) = 휑(푏푎) − 휑(푏)휑(푎) for every
+푎, 푏 ∈ 푀. This equality can be re-written as 휑(푎푏)−휑(푏푎) = 휑(푎)휑(푏)−휑(푏)휑(푎),
+that is, 휑([푎, 푏]) = [휑(푎), 휑(푏)].
+From this lemma and the definition of pre-morphism, we immediately get that:
+Lemma 2. (a) Every 푘-algebra morphism is a pre-morphism.
+(b) The composite mapping of two pre-morphisms is a pre-morphism.
+(c) The inverse mapping of a bijective pre-morphism is a pre-morphism.
+In Section 1, we already considered, for any (not-necessarily associative) 푘-
+algebra 푀, the mapping 휆: 푀 → End(푘푀), where 휆: 푥 ↦→ 휆푥, 휆푥 : 푀 → 푀,
+and 휆푥(푎) = 푥푎. Also, we had already remarked that this mapping 휆 is a 푘-algebra
+morphism if and only if 푀 is associative. The mapping 휆 is a pre-morphism if and
+only if 푀 is a pre-Lie algebra.
+There is a category of 푘-algebras with pre-morphisms, i.e., a category in which
+objects are 푘-algebras and the Hom-set of all morphisms 푀 → 푀′ consists of all
+pre-morphisms 푀 → 푀′. This category contains as a full subcategory the category
+PreL푘,푝 of pre-Lie 푘-algebras (with pre-morphisms). The category PreL푘,푝 contains
+as a subcategory the category PreL푘 of pre-Lie algebras with 푘-algebra morphisms,
+hence a fortiori the category of associative algebras with their morphisms.
+From lemma 1, we get
+Theorem 3. Associating with any 푘-algebra (퐴, ·) its sub-adjacent anticommuta-
+tive algebra (퐴, [−, −]) is a functor 푈 from the category of 푘-algebras with pre-
+morphisms to the category of anticommutative 푘-algebras.
+Notice that the functor 푈, viewed as a functor from the category PreL푘,푝 to the
+category of Lie 푘-algebras, is fully faithful. Two pre-Lie algebras 퐴, 퐴′ are iso-
+morphic in PreL푘,푝 if and only if their sub-adjacent Lie algebras are isomorphic Lie
+algebras. Two pre-Lie algebras isomorphic in PreL푘,푝 are not necessarily isomorphic
+as pre-Lie algebras. The simplest example is, over the field R of real numbers, the
+example of the two R-algebras R × R and C. They are non-isomorphic associative
+
+8
+Michela Cerqua and Alberto Facchini
+commutative 2-dimensional R-algebras, so that their sub-adjacent Lie algebras are
+both the 2-dimensional abelian Lie R-algebra. Hence R × R and C are isomorphic
+objects in PreLR,푝. All R-linear mappings R × R → C are pre-morphisms.
+Remark 4. More generally, a 푘-algebra 퐴 is said to be Lie-admissible if, setting
+[푥, 푦] = 푥푦−푦푥, one gets a Lie algebra (퐴, [−, −]). If the associator of a 푘-algebra 퐴
+is defined as (푥, 푦, 푧) = (푥푦)푧 −푥(푦푧) for all 푥, 푦, 푧 in 퐴, then being a pre-Lie algebra
+is equivalent to (푥, 푦, 푧) = (푦, 푥, 푧) for all 푥, 푦, 푧 ∈ 퐴. Being a Lie-admissible algebra
+is equivalent to
+(푥, 푦, 푧) + (푦, 푧, 푥) + (푧, 푥, 푦) = (푦, 푥, 푧) + (푥, 푧, 푦) + (푧, 푦, 푥)
+(2)
+for every 푥, 푦, 푧 ∈ 퐴. Pre-Lie algebras are Lie-admissible algebras. By lemma 1,
+the functor 푈 : (퐴, ·) ↦→ (퐴, [−, −]) is a fully faithful functor from the category of
+Lie-admissible 푘-algebras with pre-morphisms to the category of Lie 푘-algebras.
+Corresponding to the notion of pre-morphism, there is a notion of pre-derivation.
+We say that a 푘-module endomorphism 훿: 푀 → 푀, where 푀 is an arbitrary (not-
+necessarily associative) 푘-algebra, is a pre-derivation if
+훿(푥푦) − 훿(푥)푦 − 푥훿(푦) = 훿(푦푥) − 훿(푦)푥 − 푦훿(푥)
+for every 푥, 푦 ∈ 푀.
+Lemma 5. Let 푘 be a commutativering with identity, (퐴, ·) a 푘-algebra, and [−, −] :
+퐴 × 퐴 → 퐴 the operation on 퐴 defined by [푥, 푦] := 푥푦 − 푦푥 for every 푥, 푦 ∈ 퐴. Then
+a 푘-module endomorphism 훿 of 퐴 is a pre-derivation of (퐴, ·) if and only if it is a
+derivation of the 푘-algebra (퐴, [−, −]).
+Proof. The 푘-module endomorphism 훿 of 퐴 is a pre-derivation of (퐴, ·) if and only
+if 훿(푥푦) − 훿(푥)푦 − 푥훿(푦) = 훿(푦푥) − 훿(푦)푥 − 푦훿(푥), that is, 훿([푥, 푦]) = [훿(푥), 푦] +
+[푥, 훿(푦)].
+Proposition 6. (a) Every derivation of a 푘-algebra is a pre-derivation.
+(b) If 훿 and 훿′ are two pre-derivationsof a 푘-algebra 퐴, then [훿, 훿′] := 훿◦훿′−훿′◦훿
+is a pre-derivation.
+Proof. (a) is trivial, and (b) follows from lemma 5.
+Corollary 7. For any 푘-algebra 퐴, the set PreDer푘 (퐴) of all pre-derivations of 퐴
+is a Lie 푘-algebra with the operation [−, −] defined by [훿, 훿′] := 훿 ◦ 훿′ − 훿′ ◦ 훿 for
+every 훿, 훿′ ∈ PreDer푘 (퐴).
+Proof. The 푘-algebra (PreDer푘(퐴), [−, −]) is the Lie algebra of all derivations of
+the 푘-algebra (퐴, [−, −]) (lemma 5).
+Proposition 8. Let (퐴, ·) be any 푘-algebra. For every 푥 ∈ 퐴 define a 푘-module
+morphism 푑푥 : 퐴 → 퐴 setting 푑푥(푦) := 푥푦 − 푦푥 for every 푦 ∈ 퐴. The following
+conditions are equivalent:
+
+Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms
+9
+(a) 푑푥 is a pre-derivation for all 푥 ∈ 퐴, that is, the image 푑(퐴) of the mapping
+푑 : 퐴 → End(푘 퐴) is contained in PreDer푘(퐴).
+(b) The mapping 푑 is a pre-morphism of the 푘-algebra (퐴, ·) into the associative
+푘-algebra (End(푘 퐴), ◦).
+(c) The 푘-algebra (퐴, ·) is Lie-admissible.
+Proof. (a) ⇔ (c) The mapping 푑푥 : (퐴, ·) → (퐴, ·) is a pre-derivation if and only
+if the mapping 푑푥 : (퐴, [−, −]) → (퐴, [−, −]) is a derivation by lemma 5, i.e., if
+and only if 푑푥([푦, 푧]) = [푑푥(푦), 푧] + [푦, 푑푥(푧)]. Since the mapping 푑푥 is defined by
+푑푥(푦) = [푥, 푦], this is equivalent to [푥, [푦, 푧]] = [[푥, 푦], 푧] + [푦, [푥, 푧]], for every
+푥, 푦, 푧 ∈ 퐴. This proves that 푑푥 is a pre-derivation for every 푥 ∈ 퐴 if and only if
+(퐴, [−, −]) is a Lie algebra, that is, if and only if (퐴, ·) is Lie-admissible.
+(b) ⇔ (c) The mapping 푑 is a pre-morphism if and only if 푑푥푦−푑푥◦푑푦 = 푑푦푥−푑푦◦
+푑푥 for every 푥, 푦 ∈ 퐴, that is, if and only if 푑푥푦(푧) −푑푥(푑푦(푧)) = 푑푦푥(푧) −푑푦(푑푥(푧))
+for every 푥, 푦, 푧 ∈ 퐴. This is equivalent to (푥푦)푧 − 푧(푥푦) − 푑푥(푦푧 − 푧푦) = (푦푥)푧 −
+푧(푦푥) − 푑푦(푥푧 − 푧푥). An easy calculation shows that this is exactly Condition (2),
+i.e., it is equivalent to the fact that 퐴 is Lie-admissible.
+If 퐴 is a Lie-admissible 푘-algebra, the mapping 푑푥 is the inner pre-derivation of
+퐴 induced by 푥.
+3 Pre-Lie algebras are modules over the sub-adjacent Lie algebra
+Now we want to give another presentation of pre-Lie algebras, helpful to understand
+their structure.
+Let 푘 be a commutative ring with identity. Given a pre-Lie 푘-algebra (퐴, ·), we
+have already seen in the paragraph after Lemma 2 that the mapping 휆: (퐴, ·) →
+End(푘 퐴) is a pre-morphism. Apply to it the functor 푈, getting a Lie 푘-algebra
+morphism 퐿 := 푈(휆) : (퐴, [−, −]) → 픤픩(퐴) defined by 퐿 : 푎 ↦→ 휆푎 for every 푎 ∈ 퐴.
+This mapping 퐿 is set-theoretically equal to the mapping 휆. In other words, 퐿 defines
+a module structure on the 푘-module 푘 퐴, giving it the structure of a module over the
+sub-adjacent Lie 푘-algebra (퐴, [−, −]). Moreover, [푥, 푦] = 퐿(푥)(푦) − 퐿(푦)(푥).
+This construction can be inverted. Let (퐴, [−, −]) be a Lie 푘-algebra, and suppose
+that its sub-adjacent 푘-module 푘 퐴 has a module structure over the Lie algebra
+(퐴, [−, −]) via the Lie algebra morphism 퐿 : (퐴, [−, −]) → 픤픩(퐴) and that, for
+every 푥, 푦 ∈ 퐴, the condition 퐿(푥)(푦) − 퐿(푦)(푥) = [푥, 푦] holds. Define a new
+multiplication · on 퐴 setting 푥 · 푦 = 퐿(푥)(푦) for every 푥, 푦 ∈ 퐴. Then (퐴, ·) turns
+out to be a pre-Lie 푘-algebra. These two constructions are one the inverse of the
+other. More precisely, fix a Lie 푘-algebra 퐴. Then there is a category isomorphism
+between the following two categories S퐴 and M퐴, where:
+(1) S퐴 is the category whose objects are all pre-Lie 푘-algebras (퐴, ·) whose
+sub-adjacent Lie algebra is the fixed Lie algebra (퐴, [−, −]). The morphisms are all
+pre-Lie algebra homomorphisms between such pre-Lie algebras.
+
+10
+Michela Cerqua and Alberto Facchini
+(2) M퐴 is the category whose objects are all pre-Lie 푘-algebra morphisms
+퐿 : (퐴, [−, −]) → 픤픩(퐴) such that 퐿(푥)(푦) − 퐿(푦)(푥) = [푥, 푦] for every 푥, 푦 ∈ 퐴.
+The morphisms 휑 : 퐿 → 퐿′ between two objects 퐿, 퐿′ of M퐴 are the 푘-module
+morphisms 휑: 퐴 → 퐴 for which all diagrams
+푀
+푀
+푀
+푀
+휑
+퐿(푎)
+퐿′(휑(푎))
+휑
+commute, for every 푎 ∈ 퐴. See [2, Theorem 1.2.7].
+3.1 Modules over a pre-Lie 풌-algebra.
+Modules cannot be defined over arbitrary non-associative algebras, but the definition
+of pre-Lie algebra immediately suggests us how it is possible to define modules over
+a pre-Lie algebra.
+A module 푀 over a pre-Lie 푘-algebra 퐴 is any 푘-module 푀 with a 푘-bilinear
+mapping ·: 퐴 × 푀 → 푀 such that
+(푥 · 푦) · 푚 − 푥 · (푦 · 푚) = (푦 · 푥) · 푚 − 푦 · (푥 · 푚)
+(3)
+for every 푥, 푦 ∈ 퐴 and 푚 ∈ 푀.
+Like in the case of associative algebras, it is possible to equivalently define a
+module 푀 over a pre-Lie 푘-algebra (퐴, ·) as any 푘-module 푀 with a pre-morphism
+휆: (퐴, ·) → (End(푘푀), ◦).
+For instance, if 퐴 is any pre-Lie 푘-algebra and 퐼 is an ideal of 퐼, taking as 푘-
+bilinear mapping ·: 퐴 × 퐼 → 퐼 the restriction of the multiplication on 퐴, one sees
+immediately that 퐼 is a module over 퐴.
+Theorem 9. The category of modules over a pre-Lie 푘-algebra (퐴, ·) and the cate-
+gory of modules over its sub-adjacent Lie 푘-algebra (퐴, [−, −]) are isomorphic.
+Proof. Modules over the pre-Lie algebra (퐴, ·) are pairs (푘푀, 휆) with 푘 푀 a 푘-
+module and 휆: 퐴 → End(푘푀) a pre-morphism, and modules over the Lie algebra
+(퐴, [−, −]) are pairs (푘 푀, 휆) with 푘푀 a 푘-module and 휆: (퐴, [−, −]) → 픤픩(푀) a
+Lie 푘-algebra morphism. By Lemma 1, they are the same pairs.
+Notice that we could have obtained the results in Section 3 in a different way:
+every pre-Lie algebra is clearly a module over itself, hence, applying Theorem 9, to
+every pre-Lie algebra (퐴, ·) there corresponds a module 퐴푘 over the sub-adjacent
+Lie algebra (퐴, [−, −]), that is, a Lie algebra morphism 퐿 : (퐴, [−, −]) → 픤픩(퐴),
+and [푥, 푦] = 퐿(푥)(푦) − 퐿(푦)(푥) for every 푥, 푦 ∈ 퐴.
+
+Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms
+11
+Also notice that the modules we have defined in this section over a pre-Lie algebra
+are left modules. We don’t consider right modules because the definition of pre-Lie
+algebra is not right/left symmetric, that is, the opposite of a pre-Lie algebra is not a
+pre-Lie algebra.
+4 Commutator of two ideals. (Huq=Smith) for pre-Lie algebras
+The sum of two ideals of a pre-Lie 푘-algebra 퐴, i.e., their sum as 푘-submodules of
+퐴, is an ideal of 퐴, and any intersection of a family of ideals of 퐴 is an ideal of 퐴.
+It follows that the set I(퐴) of all ideals of a pre-Lie algebra 퐴 is a complete lattice
+with respect to ⊆, and it is a sublattice of the lattice of all 푘-submodules of 퐴푘, hence
+I(퐴) is a modular lattice. Moreover, the ideal of 퐴 generated by a subset 푋 of 퐴 is
+the intersection of all the ideals of 퐴 that contain 푋.
+We now need a notion of commutator of two ideals of a pre-Lie algebra. The
+variety V of pre-Lie 푘-algebras is a Barr-exact category, is a variety of Ω-groups,
+is protomodular and is semi-abelian [12, Example (2)]. More precisely, pre-Lie
+algebras have an underlying group structure with respect to their addition, so that
+they have the Mal’tsev term 푝(푥, 푦, 푧) = 푥 − 푦 + 푧. See [5, Proposition 5.3.1]. Notice
+that 푝(푝(푥, 푦, 0), 푥, 푦)) = 0 for every 푥, 푦 ∈ 퐴, hence the variety V of pre-Lie
+algebras is protomodular by [5, Proposition 3.1.8]. Moreover, 푝 has the property
+that 푝(푝(푥, 푦, 푡), 푡, 푧) = 푝(푥, 푦, 푧) for all 푥, 푦, 푧, 푡 ∈ 퐴 (semi-associativity), so V is
+semi-abelian by [5, Proposition 5.3.3].
+We want to show that the Huq and the Smith commutators of two ideals of a
+pre-Lie 푘-algebra coincide. Recall that in the case of the semi-abelian variety V of
+pre-Lie algebras, the Huq commutator of two ideals 퐼 and 퐽 of a pre-Lie algebra 퐴 is
+the smallest ideal [퐼, 퐽]퐻 of 퐴 for which there is a well-defined canonical morphism
+퐼 × 퐽 → 퐴/[퐼, 퐽]퐻 such that (푖, 0) ↦→ 푖 + [퐼, 퐽]퐻 and (0, 푗) ↦→ 푗 + [퐼, 퐽]퐻 for every
+푖 ∈ 퐼 and 푗 ∈ 퐽. That is, [퐼, 퐽]퐻 is the smallest ideal of 퐴 for which the mapping
+퐼 × 퐽 → 퐴/[퐼, 퐽]퐻, defined by (푖, 푗) ↦→ 푖 + 푗 + [퐼, 퐽]퐻 for every 푖 ∈ 퐼 and 푗 ∈ 퐽, is
+a pre-Lie algebra morphism.
+Proposition 10. The Huq commutator [퐼, 퐽]퐻 of two ideals 퐼 and 퐽 of a pre-Lie
+algebra 퐴 is the ideal of 퐴 generated by the subset { 푖푗, 푗푖 | 푖 ∈ 퐼, 푗 ∈ 퐽 }.
+Proof. The mapping ¯휎 : 퐼 × 퐽 → 퐴/[퐼, 퐽]퐻, defined by (푖, 푗) ↦→ 푖 + 푗 + [퐼, 퐽]퐻,
+is a pre-Lie 푘-algebra morphism if and only if it respects multiplication, that is, if
+and only if ¯휎((푖, 푗) · (푖′, 푗′)) ≡ ¯휎(푖, 푗) ¯휎(푖′, 푗′) for every (푖, 푗), (푖′, 푗′) ∈ 퐼 × 퐽, that
+is, if and only if 푖푖′ + 푗 ��′ ≡ (푖 + 푗)(푖′ + 푗′) modulo [퐼, 퐽]퐻. Hence ¯휎 is a pre-Lie
+algebra morphism if and only if 푖푗′ + 푗푖′ ≡ 0 modulo [퐼, 퐽]퐻, i.e., if and only if
+푖푗′ + 푗푖′ ∈ [퐼, 퐽]퐻. The conclusion follows immediately.
+The Smith commutator in the Mal’tsev variety V (see [11]) can be defined, for
+a pre-Lie 푘-algebra 퐴 with Mal’tsev term 푝(푥, 푦, 푧) and two ideals 퐼, 퐽 of 퐴, as the
+smallest ideal [퐼, 퐽]푆 of 퐴 for which the function
+
+12
+Michela Cerqua and Alberto Facchini
+푝 : {(푥, 푦, 푧) | 푥 ≡ 푦
+(mod 퐼), 푦 ≡ 푧
+(mod 퐽)} → 퐴/[퐼, 퐽]푆,
+defined by 푝(푥, 푦, 푧) = 푥 − 푦 + 푧 + [퐼, 퐽]푆 is a pre-Lie algebra morphism.
+Theorem 11. The Smith commutator [퐼, 퐽]푆 of two ideals 퐼 and 퐽 of a pre-Lie
+algebra 퐴 is the ideal of 퐴 generated by the subset { 푖푗, 푗푖 | 푖 ∈ 퐼, 푗 ∈ 퐽 }. Hence
+Huq=Smith for pre-Lie algebras.
+Proof. The mapping 푝 : { (푏 + 푖, 푏, 푏 + 푗) | 푏 ∈ 퐴, 푖 ∈ 퐼, 푗 ∈ 퐽 } → 퐴/[퐼, 퐽]푆 is a
+pre-Lie algebra morphism if and only if for every 푏, 푏′ ∈ 퐴, 푖, 푖′ ∈ 퐼, 푗, 푗′ ∈ 퐽, one
+has
+푝((푏+푖, 푏, 푏+푗)(푏′+푖′, 푏′, 푏′+푗′)) ≡ 푝(푏+푖, 푏, 푏+푗)푝(푏′+푖′, 푏′, 푏′+푗′)
+(mod [퐼, 퐽]푆),
+that is, 푝((푏 +푖)(푏′ +푖′), 푏푏′, (푏 + 푗)(푏′ + 푗′)) ≡ (푏 +푖 + 푗)(푏′ +푖′ + 푗′) mod[퐼, 퐽]푆.
+Equivalently,if and only if 0 ≡ 푖푗′+ 푗푖′ mod[퐼, 퐽]푆. Thereforethe Smith commutator
+[퐼, 퐽]푆 of the two ideals 퐼 and 퐽 is the ideal of 퐴 generated by the subset { 푖푗, 푗푖 |
+푖 ∈ 퐼, 푗 ∈ 퐽 }. In particolar, [퐼, 퐽]퐻 = [퐼, 퐽]푆.
+From now on we will not distinguish between the Huq commutator [퐼, 퐽]퐻 and
+the Smith commutator [퐼, 퐽]푆. We will simply call it the commutatorof the two ideals
+퐼 and 퐽. Notice that the commutator is commutative, in the sense that [퐼, 퐽] = [퐽, 퐼].
+Let us briefly discuss the structure of this ideal [퐼, 퐽]. It is clear that if 푋 is any
+subset of a pre-Lie 푘-algebra 퐴, the ideal ⟨푋⟩ of 퐴 generated by 푋, that is, the
+intersection of all the ideals of 퐴 that contain 푋, can be also described as the union
+⟨푋⟩ = �
+푛≥0 푋푛 of the following ascending chain 푋0 ⊆ 푋1 ⊆ . . . of 푘-submodules of
+퐴: 푋0 is the 푘-submodule of 퐴 generated by 푋; given 푋푛, set 푋푛+1 = 푋푛+퐴푋푛+푋푛퐴,
+where 퐴푋푛 denotes the set of all finite sums of products 푎푥 with 푎 ∈ 퐴 and 푥 ∈ 푋푛,
+and similarly for 푋푛퐴. In the case of the ideal [퐼, 퐽] this specializes as follows:
+Proposition 12. Let 퐼 and 퐽 be ideals of a pre-Lie 푘-algebra 퐴. Then
+[퐼, 퐽] = 퐼퐽 +
+�
+푛≥0
+푆푛,
+where 푆푛 = ((. . . (((퐽퐼)퐴)퐴) . . . )퐴)퐴 and in 푆푛 there are 푛 factors equal to 퐴 on
+the right of the factor J퐼.
+Proof. Step 1: 퐴(퐼퐽) ⊆ 퐼퐽.
+By Property (1), we have that 퐴(퐼퐽) ⊆ (퐴퐼)퐽 + (퐼퐴)퐽 + 퐼(퐴퐽) ⊆ 퐼퐽.
+Step 2: 퐴(퐽퐼) ⊆ 퐽퐼.
+From Step 1, by symmetry.
+Step 3: 퐴푆푛 ⊆ 푆푛 + 푆푛+1 for every 푛 ≥ 0.
+Induction on 푛. Step 2 gives the case 푛 = 0. Suppose that 퐴푆푛 ⊆ 푆푛 + 푆푛+1
+for some 푛 ≥ 0. Then 퐴푆푛+1 = 퐴(푆푛퐴) ⊆ (퐴푆푛)퐴 + (푆푛퐴)퐴 + 푆푛(퐴퐴) ⊆ (푆푛 +
+푆푛+1)퐴 + 푆푛+2 + 푆푛+1 = 푆푛+1 + 푆푛+2.
+
+Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms
+13
+Step 4: 푆푛퐴 = 푆푛+1.
+By definition.
+Step 5: (퐼퐽)퐴 ⊆ 퐼퐽 + 푆0 + 푆1.
+In fact, (퐼퐽)퐴 ⊆ 퐼(퐽퐴) + (퐽퐼)퐴 + 퐽(퐼퐴) ⊆ 퐼퐽 + 푆1 + 푆0.
+Final Step.
+Clearly, 퐼퐽 + �
+푛≥0 푆푛 is a 푘-submodule of 퐴 that contains 퐼퐽 and 퐽퐼 and is
+contained in the ideal generated by 퐼퐽 ∪퐽퐼. Hence it remains to show that it is closed
+by left and right multiplication by elements of 퐴. This is proved in Steps 1, 3, 4 and
+5.
+Now that we have a good notion of commutator of two ideals 퐼 and 퐽 of a
+pre-Lie 푘-algebra 퐴, we can introduce the multiplicative lattice of all ideals of
+퐴: it is the complete modular lattice I(퐴) of all ideals of 퐴 endowed with the
+commutator of ideals. Notice that, trivially, [퐼, 퐽] ⊆ 퐼 ∩ 퐽. As a consequence of
+looking at pre-Lie algebras from the point of view of multiplicative lattices, we
+immediately get the notions of prime ideal of a pre-Lie 푘-algebra 퐴, (Zariski) prime
+spectrum of 퐴, semiprime ideal, abelian pre-Lie algebra, idempotent (=perfect) pre-
+Lie algebra, derived series, solvable pre-Lie algebra, lower central series, nilpotent
+pre-Lie algebra, 푚-system, 푛-system, hyperabelian pre-Lie algebra, metabelian pre-
+Lie algebra, Jacobson radical, centralizer of an ideal, center of a pre-Lie 푘-algebra,
+hypercenter. See the next Section 5 and [8, 9, 6, 7].
+Notice that the monotonicity condition holds for our commutator of ideals of a
+pre-Lie algebra 퐴, in the sense that if 퐼 ≤ 퐼′ and 퐽 ≤ 퐽′ are ideals of 퐴, then
+[퐼, 퐽] ≤ [퐼′, 퐽′].
+Also notice that the description of the commutator in Proposition 12 reduces, in
+the case of 퐼 = 퐽 = 퐴, to the equality [퐴, 퐴] = 퐴2 = 퐴퐴. Here 퐴2 is the image of
+the 푘-module morphism 휇: 퐴 ⊗푘 퐴 → 퐴 induced by the 푘-bilinear multiplication
+of 퐴.
+5 The commutator is not associative
+In this section we will show that the commutator of ideals in a pre-Lie algebra 퐴 is
+not associative in general, that is, if 퐼, 퐽, 퐾 are ideals of 퐴, it is not necessarily true
+that [퐼, [퐽, 퐾]] = [[퐼, 퐽], 퐾]. In our example, the algebra 퐴 will be factor algebra
+퐴 := T/푃, where T is the pre-Lie algebra of rooted trees of Example 4 in Section 2,
+and 푃 is the ideal of T generated by all rooted trees with at least 5 vertices. Such 푃
+is the 푘-submodule of T generated by all rooted trees with at least 5 vertices. The
+rooted trees with at most 4 vertices up to isomorphism are
+
+14
+Michela Cerqua and Alberto Facchini
+푣 =
+1
+,
+푒 =
+1
+2
+,
+푎 =
+1
+2
+3
+,
+푏 =
+1
+2
+3
+,
+푐 =
+1
+2
+3
+4
+,
+푑 =
+1
+2
+3
+4
+,
+푓 =
+1
+2
+3
+4
+,
+푔 =
+1
+2
+3
+4
+.
+Hence our pre-Lie 푘-algebra 퐴 is eight dimensional, and we will denote by 푣, 푒, 푎, 푏,
+푐, 푑, 푓 , 푔 the images in 퐴 of the corresponding rooted trees. That is, we will say that
+{푣, 푒, 푎, 푏, 푐, 푑, 푓 , 푔} is a free set of generators for the free 푘-module 퐴. From the
+multiplication in T defined in Example 4 of Section 2, we get that the multiplication
+table in 퐴 is
+푣
+푒
+푎
+푏
+푐 푑 푓 푔
+푣 푒 푎 + 푏 푐 + 2 푓 푓 + 푑 + 푔 0 0 0 0
+푒 푏 푓 + 푔
+0
+0
+0 0 0 0
+푎 푑
+0
+0
+0
+0 0 0 0
+푏 푔
+0
+0
+0
+0 0 0 0
+푐 0
+0
+0
+0
+0 0 0 0
+푑 0
+0
+0
+0
+0 0 0 0
+푓 0
+0
+0
+0
+0 0 0 0
+푔 0
+0
+0
+0
+0 0 0 0
+
+Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms
+15
+From the multiplication table we see that 퐴2 = 퐴퐴 has {푒, 푏, 푑, 푔, 푎, 푓 , 푐} as a set
+of generators, and is a seven dimensional free 푘-module.
+Now [퐴2, 퐴2] = �
+푛≥0(. . . ((퐴2 · 퐴2) · 퐴) · · · · · 퐴) · 퐴, where there are 푛 factors
+equal to 퐴 on the right. But, always from the multiplication table, one sees that 퐴2·퐴2
+is generated by 푓 + 푔. Moreover ( 푓 + 푔)퐴 = 0 and 퐴( 푓 + 푔) = 0. Therefore [퐴2, 퐴2]
+is one dimension as a free 푘-module, and its free set of generators is { 푓 + 푔}.
+Similarly, [퐴2, 퐴] = 퐴 · 퐴2 + �
+푛≥1(. . . ((퐴2 · 퐴) · 퐴) · · · · · 퐴) · 퐴, where there
+are 푛 + 1 factors equal to 퐴 on the right. From the multiplication table, we see that
+퐴 · 퐴2 is generated by 푎 + 푏, 푓 + 푔, 푐 + 2 푓 , 푓 + 푑 + 푔. Also, 퐴2 · 퐴 is generated by
+{푏, 푑, 푔, 푓 + 푔}, (퐴2 · 퐴) · 퐴 is generated by 푔, and ((퐴2 · 퐴) · 퐴) · 퐴 = 0. Therefore
+[퐴2, 퐴] is the 푘-module generated by 푏, 푑, 푔, 푓 , 푎, 푐 and is six dimensional. It follows
+that [퐴2, 퐴]· 퐴 is generated by {푑, 푔}, 퐴·([퐴2, 퐴]) is generated by {푐+2 푓 , 푓 +푑+푔},
+and ([퐴2, 퐴] · 퐴) · 퐴 = 0. From these equalities we get that [[퐴2, 퐴], 퐴] is generated
+by {푑, 푔, 푐 + 2 푓 , 푓 + 푑 + 푔}. Equivalently, [[퐴2, 퐴], 퐴] is generated by {푑, 푔, 푓 , 푐}
+and is four dimensional. In particular [퐴2, 퐴2] ≠ [[퐴2, 퐴], 퐴].
+Let’s illustrate in detail some of the notions that immediately derive from the
+commutative multiplication [−, −] (the commutator) in the multiplicative lattice
+I(퐴).
+First of all, a pre-Lie 푘-algebra 퐴 is abelian if the commutator of 퐴 and itself is
+zero: [퐴, 퐴] = 0. This is equivalent to saying that 푖푗 = 0 for every 푖, 푗 ∈ 퐴. That
+is, a pre-Lie algebra (퐴, ·) is abelian if and only if 푥 · 푦 = 0 for every 푥, 푦 ∈ 퐴.
+(This is equivalent to requiring that the addition +: 퐴 × 퐴 → 퐴 is a pre-Lie algebra
+morphism.)
+By definition,an ideal 퐼 of a pre-Lie 푘-algebra 퐴 is prime if it is properly contained
+in 퐴 and, for every ideal 퐽, 퐾 of 퐴, [퐽, 퐾] ⊆ 퐼 implies 퐽 ⊆ 퐼 or 퐾 ⊆ 퐼. An ideal 퐼
+of a pre-Lie 푘-algebra 퐴 is semiprime if, for every ideal 퐽 of 퐴, [퐽, 퐽] ⊆ 퐼 implies
+that 퐽 ⊆ 퐼. An ideal of 퐴 is semiprime if and only if it is the intersection of a family
+of prime ideals (if and only if it is the intersection of all the ideals of 퐴 that contain
+it). An ideal 푃 of a pre-Lie 푘-algebra 퐴 is prime if and only if the lattice I(퐴/푃) is
+uniform and 퐴/푃 has no non-zero abelian ideal.
+Remark 13. Instead of the commutator [퐼, 퐽] of two ideals 퐼 and 퐽, we could have
+taken two other “product of ideals” in a pre-Lie 푘-algebra: we could consider the
+product 퐼퐽, i.e., the ���-submodule of 퐴 generated by all products 푖푗, which is a
+푘-submodule but not an ideal of 퐴 in general, or the ideal ⟨퐼퐽⟩ generated by the
+submodule 퐼퐽. Notice that 퐼퐽 ⊆ ⟨퐼퐽⟩ ⊆ [퐼, 퐽] = ⟨퐼퐽⟩ + ⟨퐽퐼⟩, where the last
+equality follows from Proposition 12. Correspondingly, we would have had three
+different notions of “prime ideal”. In the next proposition (essentially contained in
+[8, Example 3.7]) we prove that these three notions of “prime ideal” coincide:
+Proposition 14. The following conditions are equivalent for an ideal 푃 of a pre-Lie
+algebra 퐴:
+(a) If 퐼, 퐽 are ideals of 퐴 and 퐼퐽 ⊆ 푃, then either 퐼 ⊆ 푃 or 퐽 ⊆ 푃.
+(b) If 퐼, 퐽 are ideals of 퐴 and ⟨퐼퐽⟩ ⊆ 푃, then either 퐼 ⊆ 푃 or 퐽 ⊆ 푃.
+(c) If 퐼, 퐽 are ideals of 퐴 and [퐼, 퐽] ⊆ 푃, then either 퐼 ⊆ 푃 or 퐽 ⊆ 푃.
+
+16
+Michela Cerqua and Alberto Facchini
+Proof. The implications (a) ⇒ (b) ⇒ (c) follow immediately from the fact that
+퐼퐽 ⊆ ⟨퐼퐽⟩ ⊆ [퐼, 퐽].
+(c) ⇒ (a). Let 푃 satisfy condition (c) and fix two ideals 퐼, 퐽 of 퐴 such that 퐼퐽 ⊆ 푃.
+Since 푃 is an ideal, it follows that ⟨퐼퐽⟩ ⊆ 푃. Also, [⟨퐽퐼⟩, ⟨퐽퐼⟩] = ⟨⟨퐽퐼⟩⟨퐽퐼⟩⟩ ≤
+⟨퐼퐽⟩ ≤ 푃. From (c), we get that ⟨퐽퐼⟩ ≤ 푃, so that [퐼, 퐽] = ⟨퐼퐽⟩ + ⟨퐽퐼⟩ ≤ 푃. From
+(c) again, we get that either 퐼 ⊆ 푃 or 퐽 ⊆ 푃.
+Proposition 14 shows that if the pre-Lie algebra 퐴 is an associative algebra, then
+this notion of prime ideal coincide with the notion of prime ideal in an associative
+algebra. Proposition 12 shows that, for every pair (퐼, 퐽) of ideals of a pre-Lie algebra
+퐴, one has [퐼, 퐽] = 퐼퐽 +⟨퐽퐼⟩ = 퐽퐼 +⟨퐼퐽⟩. Also, Step 5 in the proof of that Proposition
+shows that one always has that 퐼퐽 + 퐽퐼 + (퐼퐽)퐴 = 퐼퐽 + 퐽퐼 + (퐽퐼)퐴.
+A pre-Lie 푘-algebra 퐴 is idempotent (or perfect) if [퐴, 퐴] = 퐴, that is, if 퐴2 = 퐴
+(last paragraph of Section 4).
+Given any pre-Lie algebra 퐴, let Spec(퐴) be the set of all its prime ideals. For
+every 퐼 ∈ I(퐴), set 푉(퐼) = { 푃 ∈ Spec(퐴) | 푃 ⊇ 퐼 }. Then the family of all subsets
+푉(퐼) of Spec(퐴), 퐼 ∈ I(퐴), is the family of all the closed sets for a topology on
+Spec(퐴). With this topology, the topological space Spec(퐴) is the (Zariski) prime
+spectrum of 퐴, and is a sober space [8]. It is not a spectral space in the sense of
+Hochster in general. For instance, if 퐵 is a Boolean ring without identity, then 퐵 is a
+pre-Lie algebra, but its prime spectrum is not compact.
+If the pre-Lie algebra 퐴 is an associative algebra, then this notion of prime
+spectrum coincide with the “standard notion” of prime spectrum of an associative
+algebra 퐴, where the points of the spectrum are the prime ideals of 퐴 and the closed
+sets are the subsets 푉(퐼) of the spectrum. To tell the truth, there is not a “standard
+notion” of prime spectrum of an associative algebra that extends the classical notion
+of prime spectrum for commutative associative algebras with identity. There are
+several such notions as it is shown in [1] and [14]. For instance, the points of the
+spectrum could be the completely prime ideals of 퐴, or the spectrum of 퐴 could be
+defined to be the Zariski spectrum of the commutative ring 퐴/[퐴, 퐴], where [퐴, 퐴]
+now denotes the ideal of 퐴 generated by all elements 푎푏 − 푏푎.
+A pre-Lie 푘-algebra 퐴 is hyperabelian if it has no prime ideal. For instance,
+abelian pre-Lie algebras are hyperabelian.
+Let 퐴 be a pre-Lie 푘-algebra. The lower central series (or descending central
+series) of 퐴 is the descending series
+퐴 = 퐴1 ≥ 퐴2 ≥ 퐴3 ≥ . . . ,
+where 퐴푛+1 := [퐴푛, 퐴] for every 푛 ≥ 1. If 퐴푛 = 0 for some 푛 ≥ 1, then 퐴 is
+nilpotent. (Notice that it is not necessary to distinguish between left nilpotency and
+rightnilpotency,becausethecommutatoriscommutative,thatis,[퐴푛, 퐴] = [퐴, 퐴푛].)
+The derived series of 퐴 [8, Definition 6.1] is the descending series
+퐴 := 퐴(0) ≥ 퐴(1) ≥ 퐴(2) ≥ . . . ,
+
+Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms
+17
+where 퐴(푛+1) := [퐴(푛), 퐴(푛)] for every 푛 ≥ 0. The pre-Lie algebra 퐴 is solvable if
+퐴(푛) = 0 for some integer 푛 ≥ 0. It is metabelian if 퐴(2) = 0.
+In a multiplicative lattice an element is semisimple if it is the join of a set of
+minimal idempotent elements. (An element 푚 of a lattice 퐿 is minimal if, for every
+푥 ∈ 퐿, 푥 ≤ 푚 implies 푥 = 푚 or 푥 = 0, that is, if it is minimal in the partially ordered
+set 퐿 \ {0}. An element 푒 of a multiplicative lattice 퐿 is idempotent if 푒 · 푒 = 푒).
+“Minimal idempotent element” of 퐿 means minimal element of 퐿 \ {0} that is also
+an idempotent element. Notice that for a minimal element 푥 ∈ 퐿 either 푥 · 푥 = 푥 or
+푥 · 푥 = 0, i.e., minimal elements are either idempotent or abelian.
+The Jacobson radical of 퐿 is the meet of the set of all maximal elements 푎 of
+퐿 \ {1} with 1 · 1 ̸≤ 푎. The radical is the join of the set of all solvable elements of 퐿.
+6 Idempotent endomorphisms, semidirect products of pre-Lie
+algebras, and actions
+Let 푒 be an idempotent endomorphism of a pre-Lie 푘-algebra 퐴. Then 퐴 = ker(푒) ⊕
+푒(퐴) (direct sum as 푘-modules), where the kernel ker(푒) of 푒 is an ideal of 퐴 and its
+image 푒(퐴) is a pre-Lie sub-푘-algebra of 퐴. If there is a direct-sum decomposition
+퐴 = 퐼 ⊕ 퐵 as 푘-module of a pre-Lie 푘-algebra 퐴, where 퐼 is an ideal of 퐴 and 퐵 is a
+pre-Lie sub-푘-algebra of 퐴, we will say that 퐴is the semidirect product of 퐼 and 퐵. We
+are interested in semidirect products because, for any algebraic structure, idempotent
+endomorphisms are in one-to-one correspondence with semidirect products and are
+related to the notion of action of the structure on another structure, and bimodules.
+The proof of the following proposition is elementary.
+Proposition 15. Let 퐴 be a pre-Lie 푘-algebra, 퐼 an ideal of 퐴 and 퐵 a pre-Lie
+sub-푘-algebra of 퐴. The following conditions are equivalent:
+(1) 퐴 = 퐼 ⊕ 퐵 as a 푘-module.
+(2) For every 푎 ∈ 퐴, there are a unique 푖 ∈ 퐼 and a unique 푏 ∈ 퐵 such that
+푎 = 푖 + 푏.
+(3) There exists a pre-Lie 푘-algebra morphism 퐴 → 퐵 whose restriction to 퐵 is
+the identity and whose kernel is 퐼.
+(4) There is an idempotent pre-Lie 푘-algebra endomorphism of 퐴 whose image
+is 퐵 and whose kernel is 퐼.
+It is now clear that there is a one-to-one correspondence between the set of all
+idempotent endomorphisms of a pre-Lie 푘-algebra 퐴 and the set of all pairs (퐼, 퐵),
+where 퐼 is an ideal of 퐴, 퐵 is a pre-Lie sub-푘-algebra of 퐴, and 퐴 is the direct sum
+of 퐼 and 퐵 as a 푘-module.
+Let us first consider inner semidirect product. Suppose that (퐴, ·) is a pre-Lie
+푘-algebra that is a semidirect product of its ideal 퐼 and its pre-Lie sub-푘-algebra 퐵.
+Then there is a pre-morphism 휆: (퐵, ·) → (End(퐼푘), ◦), given by multiplying on the
+
+18
+Michela Cerqua and Alberto Facchini
+left by elements of 퐵 (this follows from the fact that every ideal is a module, as we
+have already remarked in Section 3.1). Also, there is a 푘-module morphism 휌 : 퐵 →
+End(퐼푘), given by multiplying on the right by elements of 퐵, that is, 휌 : 푏 ↦→ 휌푏,
+where 휌푏(푖) = 푖 · 푏 for every 푖 ∈ 퐼. Moreover, Identity (1), applied to elements 푥, 푧
+in 퐵 and 푦 ∈ 퐼, can be re-written as 휌푎(휆푏(푖)) − 휆푏(휌푎(푖)) = (휌푎 ◦ 휌푏 − 휌푏·푎)(푖)
+for every 푎, 푏 ∈ 퐵 and 푖 ∈ 퐼. Identity (1), applied to elements 푥 in 퐵 and 푦, 푧 ∈ 퐼,
+can be re-written as 휆푎(푖) · 푗 − 휆푎(푖 · 푗) = 휌푎(푖) · 푗 − 푖 · 휆푎( 푗) for every 푎 ∈ 퐵 and
+푖, 푗 ∈ 퐼. Finally, the same identity (1), applied to elements 푧 in 퐵 and 푥, 푦 ∈ 퐼, can
+be re-written as 휌푎(푥 · 푦) − 푥 · 휌푎(푦) = 휌푎(푦 · 푥) − 푦 · 휌푎(푥) for every 푎 ∈ 퐵 and
+푖, 푗 ∈ 퐼.
+Conversely, for outer semidirect product:
+Theorem 16. Let 퐼 and 퐵 be pre-Lie 푘-algebras and (휆, 휌) a pair of 푘-linear map-
+pings 퐵 → End(퐼푘) such that:
+(a) 휆: (퐵, ·) → (End(퐼푘), ◦) is a pre-morphism.
+(b) 휌푎 ◦ 휆푏 − 휆푏 ◦ 휌푎 = 휌푎 ◦ 휌푏 − 휌푏·푎 for every 푎, 푏 ∈ 퐵.
+(c) 휆푎(푖) · 푗 − 휆푎(푖 · 푗) = 휌푎(푖) · 푗 − 푖 · 휆푎( 푗) for every 푎 ∈ 퐵 and 푖, 푗 ∈ 퐼.
+(d) 휌푎(푖 · 푗) − 푖 · 휌푎( 푗) = 휌푎( 푗 · 푖) − 푗 · 휌푎(푖) for every 푎 ∈ 퐵 and 푖, 푗 ∈ 퐼.
+On the 푘-module direct sum 퐼 ⊕ 퐵 define a multiplication ∗ setting
+(푖, 푏) ∗ ( 푗, 푐) = (푖 · 푗 + 휆푏( 푗) + 휌푐(푖), 푏 · 푐)
+for every (푖, 푏), ( 푗, 푐) ∈ 퐼 ⊕ 퐵. Then (퐼 ⊕ 퐵, ∗) is a pre-Lie 푘-algebra.
+Proof. For every 푎, 푏, 푐 ∈ 퐵 and 푥, 푦, 푧 ∈ 퐼 we have that
+((푥, 푎) ∗ (푦, 푏)) ∗ (푧, 푐) = (푥 · 푦 + 휆푎(푦) + 휌푏(푥), 푎 · 푏) ∗ (푧, 푐) =
+= ((푥 · 푦) · 푧 + 휆푎(푦) · 푧 + 휌푏(푥) · 푧 + 휆푎·푏(푧)+
++휌푐(푥 · 푦 + 휆푎(푦) + 휌푏(푥)), (푎 · 푏) · 푐)
+(4)
+and
+(푥, 푎) ∗ ((푦, 푏) ∗ (푧, 푐)) = (푥, 푎) ∗ (푦 · 푧 + 휆푏(푧) + 휌푐(푦), 푏 · 푐) =
+= (푥 · (푦 · 푧) + 푥 · 휆푏(푧) + 푥 · 휌푐(푦)+
++휆푎(푦 · 푧 + 휆푏(푧) + 휌푐(푦)) + 휌푏·푐(푥), 푎 · (푏 · 푐)).
+(5)
+The difference of (4) and (5) is
+((푥 · 푦) · 푧 − 푥 · (푦 · 푧) + 휆푎(푦) · 푧 − 휆푎(푦 · 푧)+
++휌푏(푥) · 푧 − 푥 · 휆푏(푧) + 휆푎·푏(푧) − (휆푎 ◦ 휆푏)(푧)+
++휌푐(푥 · 푦) − 푥 · 휌푐(푦) + 휌푐(휆푎(푦)) − 휆푎(휌푐(푦)) + 휌푐(휌푏(푥)) − 휌푏·푐(푥)),
+(푎 · 푏) · 푐 − 푎 · (푏 · 푐)).
+Similarly,
+((푦, 푏) ∗ (푥, 푎)) ∗ (푧, 푐) − (푦, 푏) ∗ ((푥, 푎) ∗ (푧, 푐)) =
+= ((푦 · 푥) · 푧 − 푦 · (푥 · 푧) + 휆푏(푥) · 푧 − 휆푏(푥 · 푧) + 휌푎(푦) · 푧 − 푦 · 휆푎(푧)+
++휆푏·푎(푧) − (휆푏 ◦ 휆푎)(푧) + 휌푐(푦 · 푥) − 푦 · 휌푐(푥) + 휌푐(휆푏(푥)) − 휆푏(휌푐(푥))+
++휌푐(휌푎(푦)) − 휌푎·푐(푦)), (푏 · 푎) · 푐 − 푏 · (푎 · 푐)).
+
+Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms
+19
+Hence, for the proof, it suffices to show that
+휆푎(푦) · 푧 − 휆푎(푦 · 푧) + 휌푏(푥) · 푧 − 푥 · 휆푏(푧) + 휆푎·푏(푧) − (휆푎 ◦ 휆푏)(푧)+
++휌푐(푥 · 푦) − 푥 · 휌푐(푦) + 휌푐(휆푎(푦)) − 휆푎(휌푐(푦)) + 휌푐(휌푏(푥)) − 휌푏·푐(푥)) =
+= 휆푏(푥) · 푧 − 휆푏(푥 · 푧) + 휌푎(푦) · 푧 − 푦 · 휆푎(푧)+
++휆푏·푎(푧) − (휆푏 ◦ 휆푎)(푧)+
++휌푐(푦 · 푥) − 푦 · 휌푐(푥) + 휌푐(휆푏(푥)) − 휆푏(휌푐(푥))+
++휌푐(휌푎(푦)) − 휌푎·푐(푦)).
+(6)
+Now
+휆푎(푦) · 푧 − 휆푎(푦 · 푧) = 휌푎(푦) · 푧 − 푦 · 휆푎(푧)
+by hypotheses (c);
+휌푏(푥) · 푧 − 푥 · 휆푏(푧) = 휆푏(푥) · 푧 − 휆푏(푥 · 푧)
+by hypotheses (c);
+휆푎·푏(푧) − (휆푎 ◦ 휆푏)(푧) = 휆푏·푎(푧) − (휆푏 ◦ 휆푎)(푧) by hypotheses (a);
+휌푐(푥 · 푦) − 푥 · 휌푐(푦) = 휌푐(푦 · 푥) − 푦 · 휌푐(푥)
+by hypotheses (d);
+휌푐(휆푎(푦)) − 휆푎(휌푐(푦)) = 휌푐(휌푎(푦)) − 휌푎·푐(푦)) by hypotheses (b);
+휌푐(휌푏(푥)) − 휌푏·푐(푥)) = 휌푐(휆푏(푥)) − 휆푏(휌푐(푥)) by hypotheses (b).
+Summing up these equalities one gets Equality (6).
+Hence the theorem characterises the four properties that an action (휆, 휌), that
+is, a pair of 푘-linear mappings 퐵 → End(퐼푘), must have in order to construct the
+semidirect product of a pre-Lie 푘-algebra 퐵 acting on a pre-Lie 푘-algebra 퐼.
+6.1 Bimodules over a pre-Lie algebra
+The most important case of semidirect product is probably when the pre-Lie algebra
+퐼 is abelian, i.e., the case where the action, that is, the pair (휆, 휌) of 푘-linear mappings
+퐵 → End(퐼푘), is an action of the pre-Lie 푘-algebra 퐵 on a 푘-module 푀. In other
+words, when 퐼 is a 퐵-bimodule. Let us be more precise, giving the precise definition
+of what a bimodule over a pre-Lie algebra must be:
+Definition 17. Let 퐴 be a pre-Lie 푘-algebra. A bimodule over 퐴 is a 푘-module 푀푘
+with a pair (휆, 휌) of 푘-linear mappings 퐴 → End(푀푘) such that:
+(a) 휆: (퐴, ·) → (End(푀푘), ◦) is a pre-morphism (that is, 푀 is a module over 퐴).
+(b) 휌푎 ◦ 휆푏 − 휆푏 ◦ 휌푎 = 휌푎 ◦ 휌푏 − 휌푏·푎 for every 푎, 푏 ∈ 퐵.
+Notice that Conditions (c) and (d) of Theorem 16 are always trivially satisfied
+because in this case the 푘-module 푀 is viewed as an abelian pre-Lie algebra, that is,
+with null multiplication. This definition already appears, for instance, in [13]. Notice
+the nice interpretation of condition (b) given in that paper: In condition (b) the left
+hand side 휌푎 ◦ 휆푏 − 휆푏 ◦ 휌푎 describes how far the action is from associativity (for
+bimodules over an associative algebra, it is always required to be zero); the right hand
+side 휌푎◦휌푏−휌푏·푎 describes how far 휌 is from being a 푘-algebra antihomomorphism.
+
+20
+Michela Cerqua and Alberto Facchini
+6.2 Adjoining the identity to a pre-Lie algebra
+The class of pre-Lie algebras contains the class of associative algebras. For asso-
+ciative algebras, it is very natural to consider associative algebras with an identity,
+and when there is not an identity, to adjoin one. This construction is often called
+the “Dorroh extension”. Let’s show that this is possible for pre-Lie algebras as well.
+We will see in fact that a more appropriate name for our class of algebras, instead
+of “pre-Lie algebras”, would have been “pre-associative algebras”. Adjoining an
+identity to a pre-Lie 푘-algebra 퐴 is exactly our semidirect product of the pre-Lie
+푘-algebra 푘 acting on the pre-Lie 푘-algebra 퐴. Let’s be more precise.
+An identity in a pre-Lie 푘-algebra 퐴 is an element, which we will denote by 1퐴,
+such that 푎 · 1퐴 = 1퐴 · 푎 = 푎 for every 푎 ∈ 퐴. If 퐴 has an identity, we will say that 퐴
+is unital. An element 푒 of 퐴 is idempotent if 푒2 := 푒 · 푒 = 푒. The zero of 퐴 is always
+an idempotent element of 퐴, and the identity, when it exists, is also an idempotent
+element of 퐴.
+Let 퐴 be any fixed pre-Lie 푘-algebra. Then the associative commutative ring 푘 is
+a pre-Lie 푘-algebra, and there is a one-to-one correspondence between the set of all
+the pre-Lie 푘-algebra morphisms 푘 → 퐴 and the set of all idempotent elements of
+퐴. For any idempotent element 푒 of 퐴 the corresponding morphism 휑푒 : 푘 → 퐴 is
+defined by 휑푒(휆) = 휆푒 for every 휆 ∈ 푘. Conversely, for any morphism 휑: 푘 → 퐴
+the corresponding idempotent element of 퐴 is 휑(1).
+For any fixed pre-Lie 푘-algebra 퐴 it is possible to construct the semidirect product
+of 푘 acting on 퐴 via the pair (휆, 휌) of 푘-module morphisms 푘 → End(퐴푘) for which
+휆훼 = 휌훼 is multiplication by 훼 for all 훼 ∈ 푘. Then the four conditions (a), (b), (c),
+(d) of Theorem 16 are all automatically satisfied, and the corresponding semidirect
+product is the 푘-module direct sum 퐴 ⊕ 푘 with the multiplication defined by
+(푥, 훼)(푦, 훽) = (푥 · 푦 + 훽푥 + 훼푦, 훼훽)
+for every (푥, 훼), (푦, 훽) ∈ 퐴 ⊕ 푘. Hence 퐴 ⊕ 푘 becomes a pre-Lie 푘-algebra with
+identity (0, 1). The Lie algebra sub-adjacent this pre-Lie algebra 퐴 ⊕ 푘 is the direct
+sum of the Lie algebra (퐴, [−, −]) and the abelian Lie algebra 푘. We will denote this
+semidirect product by 퐴#푘.
+Now let PreL푘,1 be the category of all unital pre-Lie 푘-algebras. Its objects are the
+pre-Lie 푘-algebras 퐴 with an identity. Its morphisms 푓 : 퐴 → 퐵 are the 푘-algebra
+morphisms 푓 such that 푓 (1퐴) = 1퐵. There is also a further category involved.It is the
+category PreL푘,1,푎 of all unital pre-Lie 푘-algebras with an augmentation. Its objects
+are all the pairs (퐴, 휀퐴), where 퐴 is a unital pre-Lie 푘-algebra and 휀퐴: 퐴 → 푘 is a
+morphism in PreL푘,1 that is a left inverse for 휑1퐴:
+푘
+휑1퐴 � 퐴
+휀퐴 �푘.
+The morphisms 푓 : (퐴, 휀퐴) → (퐵, 휀퐵) are the morphisms 푓 : 퐴 → 퐵 in PreL푘,1
+such that 휀퐵 푓 = 휀퐴. For instance, the 푘-algebra 퐴#푘 is clearly a unital 푘-algebra with
+
+Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms
+21
+augmentation: the augmentation is the canonical projection 휋2 : 퐴#푘 = 퐴 ⊕ 푘 → 푘
+onto the second summand.
+It is easy to see that:
+Theorem 18. There is a category equivalence 퐹: PreL푘 → PreL푘,1,푎 that associates
+with any object 퐴 of PreL푘 the 푘-algebra with augmentation 퐹(퐴) := (퐴#푘, 휋2).
+The quasi-inverse of 퐹 is the functor PreL푘,1,푎 → PreL푘, that associates with
+each unital pre-Lie 푘-algebra with augmentation (퐴, 휀퐴) the kernel ker(휀퐴) of the
+augmentation.
+References
+1. M. Ben-Zvi, A. Ma and M. Reyes, A Kochen-Specker theorem for integer matrices and
+noncommutative spectrum functors, J. Algebra 491 (2017), 28–313.
+2. M. Cerqua, “Pre-Lie algebras”, Master Thesis in Math., University of Padua, 2022.
+3. Chengming Bai, An Introduction to Pre-Lie Algebras, In “Algebra and Applications 1, Non-
+associative algebras and categories”, A. Makhlouf (Ed.), ISTE Ltd, 2020, pp. 245–273.
+4. F. Chapoton and N. Livernet, Pre-Lie algebras and the rooted trees operad, International
+Mathematics Research Notices 2001 (8) (2001), 395–408.
+5. F. Borceux and D. Bourn, “Mal’cev, protomodular, homological and semi-abelian categories”,
+Mathematics and Its Applications 566, Kluwer, 2004.
+6. A. Facchini, Algebraic structures from the point of view of complete multiplicative lattices,
+available at: http://arxiv.org/abs/2201.03295 , accepted for publication in “Rings, Quadratic
+Forms, and their Applications in Coding Theory”, Contemporary Math., 2022.
+7. A. Facchini and C. A. Finocchiaro, Multiplicative lattices: maximal implies prime, and related
+questions, submitted for publication (2022).
+8. A. Facchini, C. A. Finocchiaro and G. Janelidze, Abstractly constructed prime spectra, Algebra
+Universalis 83 (2022), no. 1, Paper No. 8, 38 pp.
+9. A. Facchini, F. de Giovanni and M. Trombetti, Spectra of groups, published online on 5 June
+2022 in Algebr. Represent. Theory (2022).
+10. A. Facchini and L. Heidari Zadeh, Algebras with a bilinear form, and Idempotent endomor-
+phisms, submitted for publication, 2022, available at http://arxiv.org/abs/2210.08230
+11. G. Janelidze, and G. M. Kelly, Central extensions in Mal’tsev varieties, Theory Appl. Categ.
+7 (2000), 219–226.
+12. G. Janelidze, L. Márki and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168
+(2002), no. 2–3, 367–386.
+13. A. Nijenhuis, On a classof common propertiesof some different types of algebras, Nieuw Arch.
+Wisk. (3) 17 (1969), 17–46. French translation in Enseign. Math. (2) 14 (1968), 225–277.
+14. M. Reyes, Obstructing extensions of the functor Spec to noncommutative rings, Israel
+J. Math. 192 (2012), 667–698.
+15. E. B. Vinberg, The theory of convex homogeneous cones, Trudy Mosk. Mat. Obshch. 12
+(1963), 303–358. English transl.: Trans. Mosc. Math. Soc. 12 (1963), 340–403.
+

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+arXiv:2301.11789v1  [math-ph]  27 Jan 2023
+A radiation and propagation problem for
+a Helmholtz equation with a compactly
+supported nonlinearity
+Lutz Angermann∗
+January 30, 2023
+The present work describes some extensions of an approach, originally devel-
+oped by V.V. Yatsyk and the author, for the theoretical and numerical analysis
+of scattering and radiation effects on infinite plates with cubically polarized lay-
+ers. The new aspects lie on the transition to more generally shaped, two- or
+three-dimensional objects, which no longer necessarily have to be represented in
+terms a Cartesian product of real intervals, to more general nonlinearities (in-
+cluding saturation) and the possibility of an efficient numerical approximation of
+the electromagnetic fields and derived quantities (such as energy, transmission
+coefficient, etc.). The paper advocates an approach that consists in transform-
+ing the original full-space problem for a nonlinear Helmholtz equation (as the
+simplest model) into an equivalent boundary-value problem on a bounded do-
+main by means of a nonlocal Dirichlet-to-Neumann (DtN) operator. It is shown
+that the transformed problem is equivalent to the original one and can be solved
+uniquely under suitable conditions. Morever, the impact of the truncation of the
+DtN operator on the resulting solution is investigated, so that the way to the
+numerical solution by appropriate finite element methods is available.
+Keywords: Scattering, radiation, nonlinear Helmholtz equation, nonlinearly polarizable medium,
+DtN operator, truncation
+AMS Subject Classification (2022): 35 J 05 35 Q 60 78 A 45
+1 Introduction
+The present work deals with the mathematical modeling of the response of a penetrable two-
+or three-dimensional object (obstacle), represented by a bounded domain, to the excitation
+∗Dept. of Mathematics, Clausthal University of Technology, Erzstr. 1, D-38678 Clausthal-Zellerfeld, Ger-
+many, lutz.angermann@tu-clausthal.de
+1
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+by an external electromagnetic field. A special aspect of the paper is that, in contrast to
+many other, thematically comparable works, nonlinear constitutive laws of this object are
+in the foreground.
+A standard example are the so-called Kerr nonlinearities. It is physically known, but also
+only little investigated mathematically that sufficiently strong incident fields, under certain
+conditions, cause effects such as frequency multiplication, which cannot occur in the linear
+models frequently considered in the literature. On the other hand, such effects are interest-
+ing in applications, which is why a targeted exploitation, for example from a numerical or
+optimization point of view, first requires thorough theoretical investigation.
+A relatively simple mathematical model for this is a nonlinear Helmholtz equation, which
+results from the transition from the time-space formulation of Maxwell’s equations to the
+frequency-space formulation together with further simplifications. Although some interesting
+nonlinear effects cannot be modeled by means of a single scalar equation alone, its inves-
+tigation is of own importance, for example from the aspect of variable coefficients, and on
+the other hand its understanding is also the basis for further development, for example for
+systems of nonlinear Helmholtz equations, see, e.g., [AY19]. The latter is also the reason
+why we consider a splitted nonlinearity and not concentrate the nonlinearity in one term as
+is obvious.
+The Helmholtz equation with nonlinearities has only recently become the focus of mathe-
+matical investigations. However, problems are mainly dealt with in which the nonlinearities
+are globally smooth, while here a formulation as a transmission problem is used that allows
+less smooth transitions at the object boundary. In addition, we allow more general nonlin-
+earities than the Kerr nonlinearities mentioned, in particular saturation effects can be taken
+into account.
+Starting from a physically oriented problem description as a full-space problem, we derive a
+weak formulation on a bounded domains using the well-known technique of DtN operators,
+and show its equivalence to the weakly formulated original problem. Since the influence
+of the external field only occurs indirectly in the weak formulation, we also give a second
+variant of the weak formulation that better clarifies this influence and which we call the
+input-output formulation.
+Since the DtN operators are non-local, their practical application (numerics) causes prob-
+lems, which is why a well-known truncation technique is used.
+This raises the problem
+of proving the well-posedness of the reduced problem and establishing a connection (error
+estimate) of the solution of the reduced problem to the original problem. Although these
+questions in the linear case have been discussed in the literature for a relatively long time,
+they even for the linear case seemed to have been treated only selectively and sometimes
+only very vaguely. The latter concerns in particular the question of the independence of
+the stability constant from the truncation parameter. In this work, both stability and er-
+ror estimates are given for the two- and three-dimensional case, whereby a formula-based
+relationship between the discrete and the continuous stability constant is established.
+Another difference to many existing, especially older works is that the present paper works
+with variational (weak) formulations but not with integral equations. Unfortunately, the
+complete tracking of the dependence of the occurring parameters on the wave number (so-
+called wavenumber-independent bounds) has not yet been included.
+It has already been mentioned that, for the linear situation, in connection with scattering
+problems or with problems that are formulated from the very beginning in bounded domains
+2
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+(e.g., with impedance boundary conditions), there is an extensive and multi-threaded body
+of literature that is beyond the scope of this article to list. Transmission problems of the
+type considered here are rarely found in the literature.
+Nevertheless, without claiming completeness, a few works should be mentioned here that had
+an influence on the present results and whose bibliographies may be of help. A frequently
+cited work that deals with linear scattering problems in two dimensions and also served
+as the motivation for the present work is [HNPX11], which, however, does not discuss the
+dependence of the stability constant on the truncation parameter. A number of later works
+by other authors quote this work, but sometimes assume results that cannot be found in the
+original.
+The work that comes closest to our intentions is [Koy07], where the exterior Dirichlet
+boundary-value problem for the linear Helmholtz equation is considered.
+In this paper,
+no separate, parameter-uniform stability estimate of the truncated problem is given, but
+the truncation error is included in the error estimate of a finite element approximation. A
+similar work is [Koy09], but in which another boundary condition at the boundary of the
+auxiliary domain is considered, the so-called modified DtN condition.
+Among the more recent papers, works by Mandel [Man19], Chen, Ev´equoz & Weth [CEW21],
+and Maier & Verf¨urth [MV22] should be mentioned, especially because of the cited sources.
+In his cumulative habilitation thesis, which contains further references, Mandel examines ex-
+istence and uniqueness questions for solutions of systems of nonlinear Helmholtz equations
+in the full-space case. Scattering or transmission problems are not considered. Chen et al.
+consider the scattering problem with quite high regularity assumptions to the superlinear
+nonlinearities, but without truncation approaches and not in the context of variational solu-
+tions. Maier & Verf¨urth, who focus mainly on multiscale aspects for a nonlinear Helmholtz
+equation over a bounded domain with impedance boundary conditions, give an instructive
+review of the literature on nonlinear Helmholtz equations.
+The structure of the present work is based on the program outlined above. After the problem
+formulation in Section 2, the exterior auxiliary problem required for truncation is discussed,
+after which the weak formulation and equivalence statement follow in Section 4. Section 5
+is dedicated to the existence and uniqueness of the weak solution, where in particular the
+assumptions on the nonlinear terms are discussed. The final section then deals with the
+properties of the truncated problem – uniform (with respect to the truncation parameter)
+well-posedness and estimate of the truncation error.
+2 Problem formulation
+Let Ω ⊂ Rd be a bounded domain with a Lipschitz boundary ∂Ω. It represents a medium
+with a nonlinear behaviour with respect to electromagnetic fields. Since Ω is bounded, we
+can choose an open Euclidean d-ball BR ⊂ Rd of radius R > supx∈Ω |x| with center in the
+origin such that Ω ⊂ BR. The complements of Ω and BR are denoted by Ωc := Rd \ Ω
+Bc
+R := Rd \ BR, resp., the open complement of BR is denoted by B+
+R := Rd \ BR (the overbar
+over sets denotes their closure in Rd), and the boundary of BR, the sphere, by SR := ∂BR
+(cf. Fig. 1). The open complement of Ω is denoted by Ω+ := Rd \ Ω. By ν we denote the
+outward-pointing (w.r.t. either Ω or BR) unit normal vector on ∂Ω or SR, respectively.
+Trace operators will be denoted by one and the same symbol γ; the concrete meaning (e.g.,
+3
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Ω
+SR
+uinc
+Figure 1: The nonlinear medium Ω is excited by an incident field uinc (d = 2)
+traces on the common interface of an interior and exterior domain) will be clear from the
+context.
+The classical direct problem of radiation and propagation of an electromagnetic field – ac-
+tually just one component of it – by/in the penetrable obstacle Ω is governed by a nonlinear
+Helmholtz equation with a variable complex-valued wave coefficient:
+− ∆u(x) − κ2c(x, u) u = f(x, u)
+for (almost) all x ∈ Rd,
+(1)
+where the wavenumber κ > 0 is fixed. The physical properties of the obstacle Ω are described
+by the coefficient c : Rd × C → C (physically the square of the refractive index) and the
+right-hand side f :
+Rd × C → C. In general, both functions are nonlinear and have the
+following properties:
+supp(1 − c(·, w)) = Ω
+and
+supp f(·, w) ⊂ Ω
+for all w ∈ C.
+(2)
+The function 1 − c is often called the contrast function. Basically we assume that c and
+f are Carath´eodory functions, i.e. the mapping x �→ c(x, v) is (Lebesgue-)measurable for
+all v ∈ C, and the mapping v �→ c(x, v) is continuous for almost all x ∈ Rd. These two
+conditions imply that x �→ c(x, v(x)) is measurable for any measurable v. The same applies
+to f.
+The unknown total field u : Rd → C should have the following structure:
+u =
+�
+urad + uinc
+in Ωc,
+utrans
+in Ω,
+(3)
+where urad : Ωc → C is the unknown radiated/scattered field, utrans : Ω → C denotes the
+unknown transmitted field, and the incident field uinc ∈ H1
+loc(Ω+) is given. The incident
+field is usually a (weak) solution of either the homogeneous or inhomogeneous Helmholtz
+equation (even in the whole space). Typically it is generated either by concentrated sources
+located in a bounded region of Ω+ or by sources at infinity, e.g. travalling waves.
+Example 1 (d = 2). The incident plane wave, whose transmission and scattering is inves-
+tigated, is given by
+uinc(x) := αinc exp(i(Φx1 − Γx2)), x = (x1, x2)⊤ ∈ B+
+R
+4
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+with amplitude αinc and angle of incidence ϕinc, |ϕinc| < π, where Φ := κ sin ϕinc is the
+longitudinal wave number and Γ :=
+√
+κ2 − Φ2 = κ cos ϕinc the transverse wave number. In
+polar coordinates is then
+uinc(r, ϕ) = αinc exp(i(Φr cos ϕ − Γr sin ϕ))
+= αinc exp(iκr(sin ϕinc cos ϕ − cos ϕinc sin ϕ))
+= αinc exp(iκr sin(ϕinc − ϕ)),
+(r, ϕ) ∈ B+
+R.
+The radiated/scattered field urad should satisfy an additional condition, the so-called Som-
+merfeld radiation condition:
+lim
+|x|→∞ |x|(d−1)/2 �
+ˆx · ∇urad − iκurad�
+= 0
+(4)
+uniformly for all directions ˆx := x/|x|, where ˆx·∇urad denotes the derivative of urad in radial
+direction ˆx, cf. [CK13, eq. (3.7) for d = 3, eq. (3.96) for d = 2]. Physically, the condition (4)
+allows only outgoing waves at infinity; mathematically it guaranties the uniqueness of the
+solution uscat : B+
+R → C of the following exterior Dirichlet problem
+−∆uscat − κ2uscat = 0
+in B+
+R,
+uscat = fSR
+on SR,
+lim
+|x|→∞ |x|(d−1)/2 �
+ˆx · ∇uscat − iκuscat�
+= 0,
+(5)
+where fSR : SR → C is given. We mention that, in the context of classical solutions (i.e.
+uscat ∈ C2(B+
+R)) to problem (5), Rellich [Rel43] has shown that the condition (4) can be
+weakened to the following integral version:
+lim
+|x|→∞
+�
+SR
+��ˆx · ∇uscat − iκuscat��2 ds(x) = 0.
+In the context of weak solutions (i.e. uscat ∈ H1
+loc(B+
+R)), an analogous equivalence statement
+can be found in [McL00, Thm. 9.6].
+3 The exterior problem in Bc
+R
+For a given fSR ∈ C(SR) and d = 3, the unique solvability of problem (5) in C2(B+
+R)∩C(Bc
+R)
+is proved, for example, in [CK13, Thm. 3.21]. In addition, if fSR is smoother, say fSR ∈
+C∞(SR), then the normal derivative of uscat on the boundary SR is a well-defined continuous
+function [CK13, Thm. 3.27]. These assertions remain valid in the case d = 2, see [CK13,
+Sect. 3.10].
+Therefore, by solving (5) for given fSR ∈ C∞(SR), a mapping can be introduced that takes
+the Dirichlet data on SR to the corresponding Neumann data on SR, i.e.
+fSR �→ TκfSR := ˆx · ∇uscat��
+SR ,
+(6)
+see, e.g., [CK19, Sect. 3.2].
+5
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Furthermore, it is well-known that the mapping Tκ can be extended to a bounded linear
+operator Tκ : Hs+1/2(SR) → Hs−1/2(SR) for any |s| ≤ 1/2 [CWGLS12, Thm. 2.31] (we keep
+the notation already introduced for this continued operator). This operator is called the
+Dirichlet-to-Neumann operator, in short DtN operator, or capacity operator.
+Since the problem (5) is considered in a spherical exterior domain, an explicit series represen-
+tation of the solution is available using standard separation techniques in polar or spherical
+coordinates, respectively. The term-by-term differentiation of this series thus also provides
+a series representation of the image of Tκ.
+The solution of the problem (5) in the two-dimensionsional case (here with uscat replaced by
+u) is given by [Mas87, Proposition 2.1], [KG89, eq. (30)]:
+u(x) = u(rˆx) = u(r, ϕ) =
+�
+n∈Z
+H(1)
+n (κr)
+H(1)
+n (κR)
+fn(R)Yn(ˆx) =
+�
+n∈Z
+H(1)
+n (κr)
+H(1)
+n (κR)
+fn(R)Yn(ϕ),
+x = rˆx ∈ Sr, r > R, ϕ ∈ [0, 2π]
+(7)
+(identifying u(x) with u(r, ϕ) and Yn(ˆx) with Yn(ϕ) for x = rˆx = r(cos ϕ, sin ϕ)⊤), where
+(r, ϕ) are the polar coordinates, H(1)
+n
+are the cylindrical Hankel functions of the first kind of
+order n [DLMF22, Sect. 10.2]1, Yn are the circular harmonics defined by
+Yn(ϕ) = einϕ
+√
+2π
+,
+n ∈ Z,
+fn(R) are the Fourier coefficients of fSR defined by
+fn(R) := (fSR(R·), Yn)S1 =
+�
+S1
+fSR(Rˆx)Yn(ˆx)ds(ˆx) =
+� 2π
+0
+fSR(R, ϕ)Yn(ϕ)dϕ,
+(8)
+and ds(ˆx) is the Lebesgue arc length element.
+Now we formally differentiate the representation (7) with respect to r to obtain the outward
+normal derivative of u:
+ˆx · ∇u(x) = ∂u
+∂r (rˆx) = κ
+�
+n∈Z
+H(1)′
+n
+(κr)
+H(1)
+n (κR)
+fn(R)Yn(ˆx),
+x = rˆx ∈ Sr, r > R.
+Setting fR := u|SR and letting x in this representation approach the boundary SR, we can
+formally define the (extended) DtN operator by
+Tκu(x) := 1
+R
+�
+n∈Z
+Zn(κR)un(R)Yn(ˆx),
+x = Rˆx ∈ SR,
+(9)
+where
+Zn(ξ) := ξ H(1)′
+n
+(ξ)
+H(1)
+n (ξ)
+,
+1Instead of (4) [Mas87] considered the ingoing Sommerfeld condition and thus obtained a representation
+in terms of the cylindrical Hankel functions of the second kind. Note that H(2)
+n (−ξ) = −(−1)nH(1)
+n (ξ)
+[DLMF22, (10.11.5)].
+6
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+and un(R) are the Fourier coefficients of u|SR analogously to (8). The admissibility of this
+procedure has been proven in many sources in the classical context, for example [CK19,
+Sect. 3.5]. For the present case, in the paper [Ern96, Thm. 1] it was shown that the operator
+Tκ : Hs+1/2(SR) → Hs−1/2(SR) is bounded for any s ∈ N0. Ernst’s result was extended to
+all s ≥ 0 in [HNPX11, Thm. 3.1].
+In the case d = 3, the solution of the problem (5) is given by [KG89, eq. (33)]:
+u(x) = u(rˆx) = u(r, ϕ, θ) =
+�
+n∈N0
+�
+|m|≤n
+h(1)
+n (κr)
+h(1)
+n (κR)
+f m
+n (R)Y m
+n (ˆx)
+=
+�
+n∈N0
+�
+|m|≤n
+h(1)
+n (κr)
+h(1)
+n (κR)
+f m
+n (R)Y m
+n (ϕ, θ),
+x ∈ Sr, r > R, (ϕ, θ) ∈ [0, 2π] × [0, π]
+(10)
+(identifying u(x) with u(r, ϕ, θ) and Y m
+n (ˆx) with Y m
+n (ϕ, θ) for x = rˆx = r(cos ϕ sin θ,
+sin ϕ sin θ, cos θ)⊤), where (r, ϕ, θ) are the spherical coordinates, h(1)
+n are the spherical Hankel
+functions of the first kind of order n [DLMF22, Sect. 10.47], Y m
+n are the spherical harmonics
+defined by
+Y m
+n (ϕ, θ) =
+�
+2n + 1
+4π
+(n − |m|)!
+(n + |m|)! P |m|
+n (cos θ)eimϕ,
+n ∈ N0, |m| ≤ n,
+(identifying Y m
+n (ˆx) with Y m
+n (ϕ, θ) for ˆx = (cos ϕ sin θ, sin ϕ sin θ, cos θ)⊤), where P m
+n are the
+associated Legendre functions of the first kind [DLMF22, Sect. 14.21], f m
+n (R) are the Fourier
+coefficients defined by
+f m
+n (R) = (fSR(R·), Y m
+n )S1 =
+�
+S1
+fSR(Rˆx)Y m
+n (ˆx)ds(ˆx)
+=
+� 2π
+0
+� π
+0
+fSR(R, ϕ, θ)Y m
+n (ϕ, θ) sin θdθdϕ,
+(11)
+and ds(ˆx) is the Lebesgue surface area element.
+Proceeding as in the two-dimensional case, we get
+ˆx · ∇u(x) = ∂u
+∂r (rˆx) = κ
+�
+n∈N0
+�
+|m|≤n
+h(1)
+n (κr)
+h(1)
+n (κR)
+f m
+n (R)Y m
+n (ˆx),
+x = rˆx ∈ Sr, r > R.
+Setting fR := u|SR and letting r → R, we can define the (extended) DtN operator by
+Tκu(x) = 1
+R
+�
+n∈N0
+�
+|m|≤n
+zn(κR)um
+n (R)Y m
+n (ˆx),
+x = Rˆx ∈ SR,
+(12)
+where
+zn(ξ) := ξ h(1)′
+n (ξ)
+h(1)
+n (ξ)
+,
+7
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+and um
+n (R) are the Fourier coefficients of u|SR analogously to (11). The admissibility of this
+procedure is proved in [CK19, Thm. 2.15] or [N´ed01, Thm. 2.6.2], for example. For the
+present situation there is a boundedness result for d = 3 analogous to [HNPX11, Thm. 3.1]
+in [N´ed01, Thm. 2.6.4]. In summary, the following statement applies to both dimensions.
+Theorem 2. The DtN operator Tκ : Hs+1/2(SR) → Hs−1/2(SR) is bounded for any s ≥ 0.
+Remark 3. A more refined analysis of the DtN operator in the case s = 0 results in a sharp
+estimate of the its norm w.r.t. the wavenumber [BSW16, Thm. 1.4]: Given κ0 > 0, there
+exists a constant C > 0 independent of κ such that
+∥Tκv∥−1/2,2,SR ≤ Cκ∥v∥1/2,2,SR
+for all v ∈ H1
+loc(B+
+R)
+and
+κ ≥ κ0.
+The result from [BSW16, Thm. 1.4] applies to more general domains, for the present situation
+it already follows from the proof of Lemma 23 (see the estimates (46), (47) for s = 0, where
+the bounds do not depend on N).
+At the end of this section we give a collection of some properties of the coefficient functions
+in the representations (9), (12) which will be used in some of the subsequent proofs.
+Lemma 4. For all ξ > 0, the following holds:
+−n ≤ Re Zn(ξ) ≤ −1
+2,
+0 < Im Zn(ξ) < ξ
+for all |n| ∈ N,
+−1
+2 ≤ Re Z0(ξ) < 0,
+ξ < Im Z0(ξ),
+−(n + 1) ≤ Re zn(ξ) ≤ −1,
+0 < Im zn(ξ) ≤ ξ
+for all n ∈ N,
+Re z0(ξ) = −1,
+Im z0(ξ) = ξ.
+Proof. For the case d = 2, the estimates can be found in [SW07, eq. (2.34)]. The other
+estimates can be found in [N´ed01, Thm. 2.6.1], see also [SW07, eqs. (2.22), (2.23)]. Although
+only 0 ≤ Im zn(ξ) is specified in the formulation of the cited theorem, the strict positivity
+follows from the positivity of the function qℓ in [N´ed01, eq. (2.6.34)], as has been mentioned
+in [MS10].
+Corollary 5. For all ξ > 0, the following holds:
+|Zn(ξ)|2 ≤ (1 + n2)(1 + |ξ|2)
+for all |n| ∈ N,
+|zn(ξ)|2 ≤ (1 + n2)(2 + |ξ|2)
+for all n ∈ N0.
+Proof. The estimates of the real and imaginary parts of Zn from Lemma 4 immediately
+imlpy that
+1
+1 + n2|Zn(ξ)|2 =
+1
+1 + n2
+�
+| Re Zn(ξ)|2 + | Im Zn(ξ)|2�
+≤
+1
+1 + n2
+�
+n2 + |ξ|2�
+≤ 1 +
+|ξ|2
+1 + n2 ≤ 1 + |ξ|2,
+n ∈ N.
+Since H(1)
+−n(ξ) = (−1)nH(1)
+n (ξ), n ∈ N [DLMF22, eq. (10.4.2)], the estimate is also valid for n
+such that −n ∈ N.
+8
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Analogously we obtain from Lemma 4 that
+1
+1 + n2|zn(ξ)|2 =
+1
+1 + n2
+�
+| Re zn(ξ)|2 + | Im zn(ξ)|2�
+≤
+1
+1 + n2
+�
+(1 + n)2 + |ξ|2�
+≤ 2 +
+|ξ|2
+1 + n2 ≤ 2 + |ξ|2.
+4 Weak formulations of the interior problem
+Now we turn to the consideration of the problem (1)–(4).
+In the classical setting it can be formulated as follows: Given uinc ∈ H1
+loc(Ω+), determine the
+transmitted field utrans : Ω → C and the radiated/scattered field urad : Ωc → C satisfying
+−∆utrans − κ2c(·, utrans) utrans = f(·, utrans)
+in Ω,
+−∆urad − κ2urad = 0
+in Ω+,
+utrans = urad + uinc
+on ∂Ω,
+ν · ∇utrans = ν · ∇urad + ν · ∇uinc
+on ∂Ω
+(13)
+and the radiation condition (4). Note that the incident field is usually a (weak) solution
+of either the homogeneous or inhomogeneous Helmholtz equation in Ω+, i.e. the second
+equation in (13) can be replaced by
+− ∆u − κ2u = f inc
+in Ω+,
+(14)
+where f inc : Ω+ → C is an eventual source density. For simplicity we do not include the
+case of a nontrivial source density in our investigation, but the subsequent theory can be
+easily extended by adding an appropriate linear functional, say ℓsrc, on the right-hand side
+of the obtained weak formulations (see (15) or (19) later).
+In order to give a weak formulation of (13) with the modification (14) in the case f inc = 0,
+we introduce the (complex) linear function spaces
+H1
+comp(Ω+) :=
+�
+v ∈ H1(Ω+) : supp v is compact
+�
+,
+VRd := {v ∈ L2(Rd) : v|Ω ∈ H1(Ω), v|Ω+ ∈ H1
+loc(Ω+) : γv|Ω = γv|Ω+ on ∂Ω},
+WRd := {v ∈ L2(Rd) : v|Ω ∈ H1(Ω), v|Ω+ ∈ H1
+comp(Ω+) : γv|Ω = γv|Ω+ on ∂Ω}
+(note the comment at the beginning of Section 2 on the notation for trace operators) and
+multiply the first equation of (13) by the restriction v|Ω of an arbitrary element v ∈ VRd and
+(14) by the restriction v|Ω+ of v ∈ VRd, respectively, and integrate py parts:
+(∇utrans, ∇v)Ω − (ν · ∇utrans, ∇v)∂Ω − κ2(c(·, utrans)utrans, v)Ω = (f(·, utrans), v)Ω,
+(∇u, ∇v)Ω − (ν · ∇u, ∇v)∂Ω+ − κ2(u, v)Ω+ = 0.
+9
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Here we use the notation, for any domain M ⊂ Rd with boundary ∂M and appropriately
+defined functions on M or ∂M,
+(∇w, ∇v)M :=
+�
+M
+∇w · ∇vdx,
+(w, v)M :=
+�
+M
+wvdx,
+(w, v)∂M :=
+�
+∂M
+wvds(x)
+(the overbar over functions denotes complex conjugation). Taking into consideration the last
+transmission condition in (13), the relationsship ν|Ω = −ν|Ω+, and the fact that the last but
+one transmission condition in (13) is included in the definition of the space VRd, we define a
+bivariate nonlinear form on VRd × WRd by
+aRd(w, v) := (∇w, ∇v)Ω + (∇w, ∇v)Ω+ − κ2(c(·, w)w, v)Rd,
+cf., e.g., [Wlo87, Example 21.8].
+Definition 6. Given uinc ∈ H1
+loc(Ω+), a weak solution to the problem (1)–(4) is defined as
+an element u ∈ VRd that has the structure (3), satisfies the variational equation
+aRd(u, v) = (f(·, u), v)Rd
+for all v ∈ WRd
+(15)
+and the Sommerfeld radiation condition (4).
+A second weak formulation can be obtained if we do not replace the second Helmholtz
+equation in (13) by (14). Then the first step in the derivation of the weak formulation reads
+as
+(∇utrans, ∇v)Ω − (ν · ∇utrans, ∇v)∂Ω − κ2(c(·, utrans)utrans, v)Ω = (f(·, utrans), v)Ω,
+(∇urad, ∇v)Ω − (ν · ∇urad, ∇v)∂Ω+ − κ2(urad, v)Ω+ = 0.
+The last transmission condition in (13) allows to rewrite the first equation as
+(∇utrans, ∇v)Ω − (ν · ∇urad, ∇v)∂Ω − κ2(c(·, utrans)utrans, v)Ω
+= (f(·, utrans), v)Ω + (ν · ∇uinc, ∇v)∂Ω,
+leading to the weak formulation
+(∇u0, ∇v)Ω+(∇u0, ∇v)Ω+−κ2(c(·, u0)u0, v)Rd = (f(·, u0), v)Rd+(ν·∇uinc, v)∂Ω
+for all v ∈ WRd
+(16)
+with respect to the structure
+u0 :=
+�
+urad
+in Ωc,
+utrans
+in Ω,
+(17)
+where urad ∈ H1
+loc(Ω+), utrans ∈ H1(Ω).
+The advantage of this formulation is that it clearly separates the unknown and the known
+parts of the fields, so we call this formulation the input-output formulation. The disadvantage
+10
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+is that the natural function space of the solution u0 is not a linear space due to the last but
+one transmission condition in (13).
+Instead of the problem (1)–(4) we want to solve an equivalent problem in the bounded
+domain BR, that is, we define
+V := {v ∈ L2(BR) : v|Ω ∈ H1(Ω), v|BR\Ω ∈ H1(BR \ Ω) : γv|Ω = γv|BR\Ω on ∂Ω}
+and look for an element u ∈ V such that
+−∆utrans − κ2c(·, utrans) u = f(·, utrans)
+in Ω,
+−∆u − κ2u = 0
+in BR \ Ω,
+utrans = urad + uinc
+on ∂Ω,
+ν · ∇utrans = ν · ∇urad + ν · ∇uinc
+on ∂Ω,
+ˆx · ∇urad = Tκurad
+on SR
+(18)
+formally holds. Now the weak formulation of problem (18) reads as follows:
+Find u ∈ V such that
+(∇u, ∇v)Ω + (∇u, ∇v)BR\Ω − κ2(c(·, u)u, v)BR − (Tκu, v)SR
+= (f(·, u), v)BR − (Tκuinc, v)SR + (ˆx · ∇uinc, v)SR
+(19)
+for all v ∈ V holds.
+Lemma 7. The weak formulations (15) and (19) of the problems (1)–(4) and (18), resp.,
+are equivalent.
+Proof. First let u ∈ V (Rd) be a weak solution to (1)–(4), i.e. it satisfies (15). Then its
+restriction to BR belongs to V .
+To demonstrate that this restriction satisfies the weak formulation (19), we construct the
+radiating solution uBc
+R′ of the homogeneous Helmholtz equation outside of a smaller ball BR′
+such that Ω ⊂ BR′ ⊂ BR and uBc
+R′
+���
+SR′ = (u − uinc)|SR′. This solution can be constructed
+in the form of a series expansion in terms of Hankel functions as explained in the previous
+section.
+By elliptic regularity (see, e.g., [McL00, Thm. 4.16], [Eva15, Sect. 6.3.1]), the
+solution of this problem satisfies the Helmholtz equation in Bc
+R′. Moreover, by uniqueness
+[N´ed01, Thm. 2.6.5], it coincides with u − uinc = urad in Bc
+R′.
+Now we choose a finite partition of unity covering BR, denoted by {ϕj}J [Wlo87, Sect. 1.2],
+such that its index set J can be decomposed into two disjoint subsets J1, J2 as follows:
+BR′ ⊂ int
+� �
+j∈J1
+supp ϕj
+�
+,
+�
+j∈J1
+supp ϕj ⊂ BR,
+�
+j∈J2
+supp ϕj ⊂ Bc
+R′.
+For example, we can choose {ϕj}J1 to consist of one element, say ϕ1, namely the usual
+mollifier function with support B′, where the open ball B′ (centered at the origin) lies
+between BR′ and BR, i.e. BR′ ⊂ B′ = int (supp ϕ1), supp ϕ1 ⊂ BR. Then the second part
+consists of a finite open covering of the spherical shell BR \ B′.
+11
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Then we take, for any v ∈ V , the product v1 := v �
+j∈J1 ϕj. This is an element of V , too,
+with support in BR, and it can be continued by zero to an element of W(Rd) (keeping the
+notation). Hence we can take it as a test function in the weak formulation (15) and obtain
+aRd(u, v1) = (f(·, u), v1)Rd.
+This is equal to
+(∇u, ∇v1)Ω + (∇u, ∇v1)BR\Ω − κ2(c(·, u)u, v1)BR − (Tκu, v1)SR
+= (f(·, u), v1)BR − (Tκuinc, v1)SR + (ˆx · ∇uinc, v1)SR
+due to the properties of the support of v1 (in particular, all terms “living” on SR are equal
+to zero).
+Since the homogeneous Helmholtz equation is satisfied in �
+j∈J2 supp ϕj ⊂ Bc
+R′, we can
+proceed as follows. We continue the test function v2 := v �
+j∈J2 ϕj by zero into the complete
+ball BR and have
+(f(·, u), v2)BR\Ω = 0 = (−∆u − κ2u, v2)BR\BR′
+= (∇u, ∇v2)BR\BR′ − κ2(u, v2)BR\BR′ − (ν · ∇u, v2)∂(BR\BR′)
+= (∇u, ∇v2)BR\BR′ − κ2(u, v2)BR\BR′ − (ˆx · ∇u, v2)SR.
+Now, taking into consideration the properties of the support of v2, we easily obtain the
+following relations:
+(∇u, ∇v2)BR\BR′ = (∇u, ∇v2)Ω + (∇u, ∇v2)BR\Ω,
+(u, v2)BR\BR′ = (c(·, u)u, v2)BR,
+(ˆx · ∇u, v2)SR = (ˆx · ∇urad, v2)SR + (ˆx · ∇uinc, v2)SR
+= (Tκurad, v2)SR + (ˆx · ∇uinc, v2)SR
+= (Tκu, v2)SR − (Tκuinc, v2)SR + (ˆx · ∇uinc, v2)SR,
+where the treatment of the last term makes use of the construction of the Dirichlet-to-
+Neumann map Tκ.
+Adding both relations and observing that v = v1+v2, we arrive at the variational formulation
+(19).
+Conversely, let u ∈ V be a solution to (19). To continue it into Bc
+R, similar to the first
+part of the proof we construct the radiating solution uBc
+R of the Helmholtz equation outside
+BR such that uBc
+R
+��
+SR = (u − uinc)|SR and set u := uBc
+R + uinc in B+
+R. Hence we have that
+Tκu =
+∂uBc
+R
+∂ˆx + Tκuinc.
+Now we take an element v ∈ W(Rd). Its restriction to BR is an element of V and thus can
+be taken as a test function in (19):
+(∇u, ∇v)Ω + (∇u, ∇v)BR\Ω − κ2(c(·, u)u, v)BR − (Tκu, v)SR
+= (f(·, u), v)BR − (Tκuinc, v)SR + (ˆx · ∇uinc, v)SR.
+(20)
+12
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Since v has a compact support, we can choose a ball B ⊂ Rd centered at the origin such
+that BR ∪ supp v ⊂ B. The homogeneous Helmholtz equation is obviously satisfied in the
+spherical shell B \ BR:
+−∆uBc
+R − κ2uBc
+R = 0.
+We multiply this equation by the complex conjugate of the test function v ∈ V , then integrate
+over the shell, and apply the first Green’s formula:
+(∇uBc
+R, ∇v)B\BR − κ2(uBc
+R, v)B\BR − (ν · ∇uBc
+R, v)∂(B\BR) = 0.
+Now we observe that
+(∇uBc
+R, ∇v)B\BR = (∇uBc
+R, ∇v)B+
+R,
+(uBc
+R, v)B\BR = (uBc
+R, v)B+
+R,
+(ν · ∇uBc
+R, v)∂(B\BR) = −(ˆx · ∇uBc
+R, v)SR = −(Tκu − Tκuinc, v)SR
+where the minus sign in the last line results from the change in the orientation of the outer
+normal (once w.r.t. the shell, once w.r.t. BR) and the construction of uBc
+R. So we arrive at
+(∇uBc
+R, ∇v)B+
+R − κ2(uBc
+R, v)B+
+R + (Tκu, v)SR = (Tκuinc, v)SR.
+(21)
+Finally, since the incident field satisfies the homogeneous Helmholtz equation in the spherical
+shell, too, we see by an analogous argument that the variational equation
+(∇uinc, ∇v)B+
+R − κ2(uinc, v)B+
+R = −(ˆx · ∇uinc, v)SR
+(22)
+holds.
+Adding the variational equations (20) – (22), we arrive at the variational formulation (15).
+5 Existence and uniqueness of a weak solution
+In this section we investigate the existence and uniqueness of the weak solution of the interior
+problem (18). We define the sesquilinear form
+a(w, v) := (∇w, ∇v)Ω + (∇w, ∇v)BR\Ω − κ2(w, v)BR − (Tκw, v)SR
+for all w, v ∈ V,
+(23)
+the nonlinear form
+n(w, v) := κ2(c(·, w) − 1)w, v)BR + (f(·, w), v)BR
+− (Tκuinc, v)SR + (ˆx · ∇uinc, v)SR
+(24)
+and reformulate (19) as follows: Find u ∈ V such that
+a(u, v) = n(u, v)
+for all v ∈ V.
+(25)
+On the space V , we use the standard seminorm and norm:
+|v|V :=
+�
+∥∇v∥2
+0,2,Ω + ∥∇v∥2
+0,2,BR\Ω
+�1/2
+,
+∥v∥V :=
+�
+|v|2
+V + ∥v∥2
+0,2,BR
+�1/2 .
+(26)
+13
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+For κ > 0, the following so-called wavenumber dependent norm on V is also common:
+∥v∥V,κ :=
+�
+|v|2
+V + κ2∥v∥2
+0,2,BR
+�1/2 .
+(27)
+It is not difficult to verify that the standard norm and the wavenumber dependent norm are
+equivalent on V , i.e. it holds
+C−∥v∥V ≤ ∥v∥V,κ ≤ C+∥v∥V
+for all v ∈ V,
+(28)
+where the equivalence constants depend on κ in the following way: C− := min{1; κ} and
+C+ := max{1; κ}. We now proceed to examine the linear aspects of the problem (25).
+Lemma 8. The sesquilinear form a is bounded on V .
+Proof. Applying to each addend in the definition of a the appropriate Cauchy-Bunyakovsky-
+Schwarz inequality, we obtain
+|a(w, v)| ≤ |w|V |v|V + κ2∥w∥0,2,BR∥v∥0,2,BR
++ ∥Tκw∥−1/2,2,SR∥v∥1/2,2,SR
+for all w, v ∈ V.
+According to Thm. 2 the DtN operator Tκ is bounded, i.e. there exists a constant CTκ > 0
+such that
+∥Tκw∥−1/2,2,SR ≤ CTκ∥w∥1/2,2,SR
+for all w ∈ V.
+It remains to apply a trace theorem [McL00, Thm. 3.37]:
+|a(w, v)| ≤ |w|V |v|V + κ2∥w∥0,2,BR∥v∥0,2,BR + CTκC2
+tr∥w∥1,2,BR\Ω∥v∥1,2,BR\Ω
+≤ |w|V |v|V + κ2∥w∥0,2,BR∥v∥0,2,BR + CTκC2
+tr∥w∥V ∥v∥V
+≤ min{(max{1, κ2} + CTκC2
+tr)∥w∥V ∥v∥V , (1 + CTκC2
+tr)∥w∥V,κ∥v∥V,κ}
+for all w, v ∈ V.
+Lemma 9. Given κ0 > 0 and R0 > 0, assume that κ ≥ κ0 (cf. Rem. 3) and R ≥ R0. In
+addition, κ0 ≥ 1 is required for d = 2. Then the sesquilinear form a satisfies a G˚arding’s
+inequality of the form
+Re a(v, v) ≥ ∥v∥2
+V,κ − 2κ2∥v∥2
+0,2,BR
+for all v ∈ V.
+Proof. From the definitions of a and the wavenumber dependent norm it follows immediately
+that
+Re a(v, v) = ∥v∥2
+V,κ − 2κ2∥v∥2
+0,2,BR − Re (Tκv, v)SR
+≥ ∥v∥2
+V,κ − 2κ2∥v∥2
+0,2,BR + CR−1∥v∥2
+0,2,SR
+≥ ∥v∥2
+V,κ − 2κ2∥v∥2
+0,2,BR,
+where the first estimate follows from [MS10, Lemma 3.3] with a constant C > 0 depending
+soleley on κ0 > 0 and R0 > 0.
+14
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Next we discuss the solvability and stability of the problem (25) for the case that the right-
+hand side is just an antilinear continuous functional ℓ :
+V → C. The linear problem of
+finding u ∈ V such that
+a(u, v) = ℓ(v)
+for all v ∈ V
+(29)
+holds can be formulated equivalently as an operator equation in the dual space V ∗ of V
+consisting of all continuous antilinear functionals from V to C. Namely, if we define the
+linear operator A : V → V ∗ by
+Aw(v) := a(w, v)
+for all w, v ∈ V,
+(30)
+problem (29) is equivalent to solving the operator equation
+Au = ℓ
+(31)
+for u ∈ V .
+Note that A is a bounded operator by Lemma 8.
+Theorem 10. Under the assumptions of Lemma 9, the problem (31) is uniquely solvable for
+any ℓ ∈ V ∗.
+Proof. The basic ideas of the proof are taken from the proof of [MS10, Thm 3.8]. Since the
+embedding of V into L2(BR) is compact by the compactness theorem of Rellich–Kondrachov
+[McL00, Thm. 3.27] together with Tikhonov’s product theorem [KN63, Thm. 4.1], the com-
+pact perturbation theorem [McL00, Thm. 2.34] together with Lemma 9 imply that the
+Fredholm alternative [McL00, Thm. 2.27] holds for the equation (31).
+Hence it is sufficient to demonstrate that the homogeneous adjoint problem (cf. [McL00,
+p. 43]) of finding u ∈ V such that a(v, u) = 0 holds for all v ∈ V only allows for the trivial
+solution.
+So suppose u ∈ V is a solution of the homogeneous adjoint problem. We take v := u and
+consider the imaginary part of the resulting equation:
+0 = Im a(u, u) = − Im (Tκu, u)SR = Im (Tκu, u)SR.
+Then [MS10, Lemma 3.3] implies u = 0 on SR. Then u satisfies the variational equation
+(∇u, ∇v)Ω + (∇u, ∇v)BR\Ω − κ2(u, v)BR = 0
+for all v ∈ V,
+i.e. it is a weak solution of the homogeneous interior transmission Neumann problem for the
+wave equation on BR. On the other hand, u can be extended to the whole space Rd by
+zero to an element ˜u ∈ V (Rd), and this element can be interpreted as a weak solution of a
+homogeneous full-space transmission problem, for instance in the sense of [TW93, Problem
+(P)]. Then it follows from [TW93, Lemma 7.1] that ˜u = 0 und thus u = 0.
+Since a Fredholm operator has a closed image [McL00, p. 33], it follows from the Open
+Mapping Theorem and Thm. 10 (cf. [McL00, Cor. 2.2]) that the inverse operator A−1 is
+bounded, i.e. there exists a constant C(R, κ) > 0 such that
+∥u∥V,κ = ∥A−1ℓ∥V,κ ≤ C(R, κ)∥ℓ∥V ∗
+for all ℓ ∈ V ∗.
+15
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Then it holds
+1
+C(R, κ) ≤ ∥ℓ∥V ∗
+∥u∥V,κ
+=
+sup
+v∈V \{0}
+|ℓ(v)|
+∥u∥V,κ∥v∥V,κ
+=
+sup
+v∈V \{0}
+|a(u, v)|
+∥u∥V,κ∥v∥V,κ
+.
+This estimate proves the following result.
+Lemma 11. Under the assumptions of Lemma 9, the sesquilinear form a satisfies an inf-sup
+condition:
+β(R, κ) :=
+inf
+w∈V \{0}
+sup
+v∈V \{0}
+|a(w, v)|
+∥w∥V,κ∥v∥V,κ
+> 0.
+Now we turn to the nonlinear situation and concretize the assumptions regarding the Cara-
+th´eodory functions c and f.
+Lemma 12. Let pf ∈
+�
+[2, ∞),
+d = 2,
+[2, 6],
+d = 3, and assume there exist nonnegative functions mf, gf ∈
+L∞(Ω) such that
+|f(x, ξ)| ≤ mf(x)|ξ|pf−1 + gf(x)
+for all (x, ξ) ∈ Ω × C.
+Then vf(·, w) ∈ L1(Ω) for all w, v ∈ V .
+Proof. Since f is a Carath´eodory function, the composition f(·, w) is measurable and it
+sufficies to estimate the integral of |vf(·, w)|. Moreover, it suffices to consider the term
+mfv|w|pf−1 in more detail. By H¨older’s inequality for three functions, it holds that
+∥vf(·, w)∥0,1,Ω ≤ ∥mf∥0,∞,Ω∥v∥0,pf,Ω∥wpf−1∥0,q,Ω
+with 1
+pf
++ 1
+q = 1.
+The Lpf-norm of v is bounded thanks to the embedding V |Ω ⊂ Lpf(Ω) for the allowed values
+of pf [AF03, Thm. 4.12]. Since |wpf−1|q = |w|p
+f, the Lq-norm of wpf−1 is bounded by the
+same reasoning.
+Lemma 13. Let pc ∈
+�
+[2, ∞),
+d = 2,
+[2, 6],
+d = 3, and assume there exist nonnegative functions mc, gc ∈
+L∞(Ω) such that
+|c(x, ξ) − 1| ≤ mc(x)|ξ|pc−2 + gc(x)
+for all (x, ξ) ∈ Ω × C.
+Then zv(c(·, w) − 1) ∈ L1(Ω) for all z, w, v ∈ V .
+Proof. Similar to the proof of Lemma 12 it is sufficient to consider the term mczv|w|pc−2 in
+more detail. By H¨older’s inequality for four functions, it holds that
+∥zv(c(·, w) − 1)∥0,1,Ω ≤ ∥mc∥0,∞,Ω∥z∥0,pc,Ω∥v∥0,pc,Ω∥wpc−2∥0,q,Ω
+with 2
+pc
++ 1
+q = 1.
+The Lpc-norms of z, v are bounded thanks to the embedding theorem [AF03, Thm. 4.12].
+Since |wpc−2|q = |w|p
+c, the Lq-norm of wpc−2 is bounded by the same reasoning.
+16
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Corollary 14. Under the assumptions of Lemma 12 and Lemma 13, resp., the following
+estimates hold for all z, w, v ∈ V :
+|(f(·, w), v)Ω| ≤ C
+pf
+emb∥mf∥0,∞,Ω∥w∥
+pf−1
+1,2,Ω∥v∥1,2,Ω
++
+�
+|Ω|d ∥gf∥0,∞,Ω∥v∥0,2,Ω,
+(32)
+|((c(·, w) − 1)z, v)Ω| ≤ Cpc
+emb∥mc∥0,∞,Ω∥w∥pc−2
+1,2,Ω∥z∥1,2,Ω∥v∥1,2,Ω
++ ∥gc∥0,∞,Ω∥z∥0,2,Ω∥v∥0,2,Ω,
+(33)
+where |Ω|d is the d-volume of Ω.
+Proof. Replace v by v in Lemmata 12, 13 to get the first addend of the bounds. The estimate
+of the second addend is trivial.
+Example 15. An important example for the nonlinearities is
+c(x, ξ) :=
+�
+1,
+(x, ξ) ∈ Ω+ × C,
+ε(L)(x) + α(x)|ξ|2,
+(x, ξ) ∈ Ω × C,
+with given ε(L), α ∈ L∞(Ω), and f = 0. Here pc = 4, which is within the range of validity of
+Lemma 13, and mc = |α|, gc = |ε(L) − 1|.
+The estimates from Corollary 14 show that the first two terms on the right-hand side of the
+variational equation (25) can be considered as values of nonlinear mappings from V to V ∗,
+i.e. we can define
+ℓcontr : V → V ∗
+by
+⟨ℓcontr(w), v⟩ := κ2(c(·, w) − 1)w, v)Ω,
+ℓsrc : V → V ∗
+by
+⟨ℓsrc(w), v⟩ := (f(·, w), v)Ω
+for all w, v ∈ V.
+Furthermore, if uinc ∈ H1
+loc(Ω+) is such that additionally ∆uinc belongs to L2,loc(Ω+) (where
+∆uinc is understood in the distributional sense), the last two terms on the right-hand side of
+(24) form an antilinear continuous functional on ℓinc ∈ V ∗:
+⟨ℓinc, v⟩ := (ˆx · ∇uinc − Tκuinc, v)SR
+for all v ∈ V.
+This is a consequence of Thm. 2 and the estimates before the trace theorem [KA21, Thm. 6.13].
+Hence
+∥ℓinc∥V ∗ ≤ ˜Ctr[∥∆uinc∥0,2,BR\Ω + ∥uinc∥0,2,BR\Ω] + CTκC2
+tr∥uinc∥1,2,BR\Ω,
+where ˜Ctr is the norm of the trace operator defined in [KA21, eq. (6.39)].
+However, it is more intuitive to utilize the estimate
+∥ℓinc∥V ∗ ≤ Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,2,SR.
+(34)
+The reason for this is that the bound can be interpreted as a measure of the deviation of the
+function uinc from a radiating solution of the corresponding Helmholtz equation. In other
+words: If the function uinc satisfies the boundary value problem (5) with fSR := uinc|SR, then
+the functional ℓinc is not present.
+17
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Consequently, setting
+F(w) := ℓcontr(w) + ℓsrc(w) + ℓinc
+for all w ∈ V,
+we obtain a nonlinear operator F : V → V ∗, and the problem (25) is then equivalent to the
+operator equation
+Au = F(u)
+in V ∗,
+and further, by Lemma 11, equivalent to the fixed-point problem
+u = A−1F(u)
+in V.
+(35)
+In order to prove the subsequent existence and uniqueness theorem, we specify some addi-
+tional properties of the nonlinearities c and f.
+Definition 16. The functions c and f are said to generate locally Lipschitz continuous Ne-
+mycki operators in V if the following holds: For some parameters pc, pf ∈
+�
+[2, ∞),
+d = 2,
+[2, 6],
+d = 3,,
+there exist Carath´eodory functions Lc : Ω×C×C → (0, ∞) and Lf : Ω×C×C → (0, ∞) such
+that the composition operators Ω × V × V → Lqc(Ω) : (x, w, v) �→ Lc(x, w, v), Ω × V × V →
+Lqf(Ω) : (x, w, v) �→ Lf(x, w, v) are bounded for qc, qf > 0 with
+3
+pc + 1
+qc =
+2
+pf + 1
+qf = 1, and
+|c(x, ξ) − c(x, η)| ≤ Lc(x, ξ, η)|ξ − η|,
+|f(x, ξ) − f(x, η)| ≤ Lf(x, ξ, η)|ξ − η|
+(36)
+for all (x, ξ, η) ∈ Ω × C × C.
+Remark 17. If the nonlinearities c and f generate locally Lipschitz continuous Nemycki
+operators in the sense of the above Definition 16, the assumptions of Lemmata 12, 13 can
+be replaced by the requirement that there exist functions wf, wc ∈ V such that f(·, wf) ∈
+Lpf /(pf −1)(Ω) and c(·, wf) ∈ Lpc/(pc−2)(Ω), respectively.
+Proof. Indeed, similar to the proofs of the two lemmata mentioned, we have that
+∥vf(·, w)∥0,1,Ω ≤ ∥vf(·, wf)∥0,1,Ω + ∥v(f(·, w) − f(·, wf))∥0,1,Ω
+≤ ∥vf(·, wf)∥0,1,Ω + ∥vLf(·, w, wf)|w − wf|∥0,1,Ω
+≤ ∥v∥0,pf,Ω∥f(·, wf)∥0,˜qf,Ω + ∥v∥0,pf,Ω∥Lf(·, w, wf)∥0,qf,Ω∥w − wf∥0,pf,Ω
+≤
+�
+∥f(·, wf)∥0,˜qf,Ω + ∥Lf(·, w, wf)∥0,qf,Ω(∥w∥V + ∥wf∥V )
+�
+∥v∥V ,
+∥zvc(·, w)∥0,1,Ω ≤ ∥zvc(·, wc)∥0,1,Ω + ∥zv(c(·, w) − c(·, wc))∥0,1,Ω
+≤ ∥zvc(·, wc)∥0,1,Ω + ∥zvLc(·, w, wc)|w − wc|∥0,1,Ω
+≤ ∥z∥0,pc,Ω∥v∥0,pc,Ω∥c(·, wc)∥0,˜qc,Ω
++ ∥z∥0,pc,Ω∥v∥0,pc,Ω∥Lc(·, w, wc)∥0,qc,Ω∥w − wc∥0,pc,Ω
+≤ [∥c(·, wc)∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(∥w∥V + ∥wc∥V )] ∥z∥V ∥v∥V
+with
+1
+pf + 1
+˜qf = 1 and
+2
+pc + 1
+˜qc = 1.
+18
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Theorem 18. Under the assumptions of Lemma 9, let the functions c and f generate locally
+Lipschitz continuous Nemycki operators in V and assume that there exist functions wf, wc ∈
+V such that f(·, wf) ∈ Lpf/(pf −1)(Ω) and c(·, wf) ∈ Lpc/(pc−2)(Ω), respectively.
+Furthermore let uinc ∈ H1
+loc(Ω+) be such that additionally ∆uinc ∈ L2,loc(Ω+) holds.
+If there exist numbers ̺ > 0 and LF ∈ (0, β(R, κ)) such that the following two conditions
+κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ̺
++
+�
+∥f(·, wf)∥0,˜qf,Ω + ∥Lf(·, w, wf)∥0,qf,Ω(̺ + ∥wf∥V )
+�
+(37)
++ Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,2,SR ≤ ̺β(R, κ),
+κ2 [∥Lc(·, w, v)∥0,qc,Ω̺ + ∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )]
++ ∥Lf(·, w, v)∥0,qf,Ω ≤ LF
+(38)
+are satisfied for all w, v ∈ Kcl
+̺ := {v ∈ V : ∥v∥V ≤ ̺}, then the problem (35) has a unique
+solution u ∈ Kcl
+̺ .
+Proof. First we mention that Kcl
+̺ is a closed nonempty subset of V .
+Next we show that A−1F(Kcl
+̺ ) ⊂ Kcl
+̺ . To this end we make use of the estimates given in
+the proof of Remark 17 and obtain
+∥F(w)∥V ∗ ≤ ∥ℓcontr(w)∥V ∗ + ∥ℓsrc(w)∥V ∗ + ∥ℓinc∥V ∗
+≤ κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(∥w∥V + ∥wc∥V )] ∥w∥V
++
+�
+∥f(·, wf)∥0,˜qf,Ω + ∥Lf(·, w, wf)∥0,qf,Ω(∥w∥V + ∥wf∥V )
+�
++ ∥ℓinc∥V ∗
+≤ κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ̺
++
+�
+∥f(·, wf)∥0,˜qf,Ω + ∥Lf(·, w, wf)∥0,qf,Ω(̺ + ∥wf∥V )
+�
++ Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,2,SR .
+Hence the assumption (37) implies ∥A−1F(w)∥V ≤ ̺.
+It remains to show that the mapping A−1F is a contraction.
+We start with the consideration of the contrast term. From the elementary decomposition
+(c(·, w) − 1)w − (c(·, v) − 1)v = (c(·, w) − c(·, v))w + (c(·, v) − 1)(w − v)
+we see that
+∥ℓcontr(w) − ℓcontr(v)∥V ∗
+≤ κ2∥Lc(·, w, v)∥0,qc,Ω∥w − v∥V ∥w∥V
++ κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω∥w − wc∥V ] ∥w − v∥V
+≤ κ2∥Lc(·, w, v)∥0,qc,Ω∥w − v∥V ̺
++ κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ∥w − v∥V
+≤ κ2 [∥Lc(·, w, v)∥0,qc,Ω̺ + ∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ∥w − v∥V .
+The estimate of the source term follows immediately from the properties of f:
+∥ℓsrc(w) − ℓsrc(v)∥V ∗ ≤ ∥Lf(·, w, v)∥0,qf,Ω∥w − v∥V .
+19
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+From
+∥F(w) − F(v)∥V ∗ ≤ ∥ℓcontr(w) − ℓcontr(v)∥V ∗ + ∥ℓsrc(w) − ℓsrc(v)∥V ∗
+and assumption (38) we thus obtain
+∥F(w) − F(v)∥V ∗ ≤ LF∥w − v∥V .
+In summary, Banach’s fixed point theorem can be applied (see e.g. [Eva15, Sect. 9.2.1]) and
+we conclude that the problem (35) has a unique solution u ∈ Kcl
+̺ .
+If we introduce the function space
+˜V := {v ∈ L2(BR) : v|Ω ∈ H1(Ω), v|BR\Ω ∈ H1(BR \ Ω)}
+equipped with the norm
+∥v∥ ˜V :=
+�
+∥v∥2
+1,2,Ω + ∥v∥2
+1,2,BR\Ω
+�1/2
+for all v ∈ ˜V ,
+the ball Kcl
+̺ appearing in the above theorem can be interpreted as a ball in ˜V of radius ̺
+with center in
+u0 :=
+�
+0
+in Ω,
+−uinc
+in BR \ Ω.
+Indeed, for u of the form (3), it holds that
+∥u − u0∥2
+˜V = ∥utrans∥2
+1,2,Ω + ∥urad + uinc∥2
+1,2,BR\Ω = ∥u∥2
+V .
+This means that the influence of the incident field uinc on the radius ̺ in Thm. 18 depends
+only on the deviation of uinc from a radiating field measured by ∥ℓinc∥V ∗, but not directly on
+the intensity of uinc. In other words, if the incident field uinc is radiating (i.e., it also satisfies
+the Sommerfeld radiation condition (4) and thus ℓinc = 0), the radius ̺ does not depend
+on uinc. In particular, uinc can be a strong field, which is important for the occurence of
+generation efffects of higher harmonics [AY19].
+Example 19 (Example 15 continued). The identity
+c(·, ξ) − c(·, η) = α (|ξ|2 − |η|2) = α (|ξ| + |η|)(|ξ| − |η|)
+for all ξ, η ∈ C and the inequality ||ξ| − |η|| ≤ |ξ − η| show that
+|c(·, ξ) − c(·, η)| ≤ |α|(|ξ| + |η|)|ξ − η|
+holds, hence we can set Lc(·, ξ, η) := |α|(|ξ| + |η|). With pc = qc = 4, c generates a locally
+Lipschitz continuous Nemycki operator in V . Furthermore we may choose wc = 0. Then:
+∥c(·, wc) − 1∥0,˜qc,Ω = ∥ε(L) − 1∥0,2,Ω,
+∥Lc(·, w, v)∥0,qc,Ω = ∥α(|w| + |v|)∥0,4,Ω ≤ ∥αw∥0,4,Ω + ∥αv∥0,4,Ω
+≤ ∥α∥0,∞,Ω [∥w∥0,4,Ω + ∥v∥0,4,Ω] ≤ Cemb∥α∥0,∞,Ω [∥w∥V + ∥v∥V ] ,
+∥Lc(·, w, wc)∥0,qc,Ω = ∥αw∥0,4,Ω ≤ Cemb∥α∥0,∞,Ω∥w∥V .
+20
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Hence the validity of the following conditions is sufficient for (37), (38):
+κ2 �
+∥ε(L) − 1∥0,2,Ω + Cemb∥α∥0,∞,Ω̺2�
+̺
++ Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,2,SR ≤ ̺β(R, κ),
+κ2 �
+∥ε(L) − 1∥0,2,Ω + 3Cemb∥α∥0,∞,Ω̺2�
+≤ LF.
+A consideration of these condition shows that there can be different scenarios for which
+they can be fulfilled. In particular, one of the smallness requirements concerns the product
+∥α∥0,∞,Ω̺3.
+Example 20 (saturated Kerr nonlinearity). Another important example for the nonlineari-
+ties is [Akh98]
+c(x, ξ) :=
+�
+1,
+(x, ξ) ∈ Ω+ × C,
+ε(L)(x) + α(x)|ξ|2/(1 + γ|ξ|2),
+(x, ξ) ∈ Ω × C,
+with given ε(L), α ∈ L∞(Ω), saturation parameter γ > 0, and f = 0. Based on the identity
+|ξ|2
+1 + γ|ξ|2 −
+|η|2
+1 + γ|η|2 = (1 + γ|η|2)|ξ|2 − (1 + γ|ξ|2)|η|2
+(1 + γ|ξ|2)(1 + γ|η|2)
+=
+|ξ|2 − |η|2
+(1 + γ|ξ|2)(1 + γ|η|2)
+for all ξ, η ∈ C we obtain
+����
+|ξ|2
+1 + γ|ξ|2 −
+|η|2
+1 + γ|η|2
+���� = (|ξ| + |η|) ||ξ| − |η||
+(1 + γ|ξ|2)(1 + γ|η|2) ≤ (|ξ| + |η|)|ξ − η|.
+Hence on Ω we arrive at the same Lipschitz function as in the previous Example 19, that is
+Lc(x, ξ, η) :=
+�
+0,
+(x, ξ, η) ∈ Ω+ × C × C,
+|α|(|ξ| + |η|),
+(x, ξ, η) ∈ Ω × C × C.
+Moreover, since
+c(x, wc) = c(x, 0) =
+�
+0,
+(x, ξ) ∈ Ω+ × C,
+ε(L),
+(x, ξ) ∈ Ω × C.
+we get the same sufficient conditions.
+6 The modified boundary value problem
+Since the exact DtN operator is represented as an infinite series (see (9), (12)), it is practically
+necessary to truncate this nonlocal operator and consider only finite sums
+Tκ,Nu(x) := 1
+R
+�
+|n|≤N
+Zn(κR)un(R)Yn(ˆx),
+x = Rˆx ∈ SR ⊂ R2,
+(39)
+Tκ,Nu(x) = 1
+R
+N
+�
+n=0
+�
+|m|≤n
+zn(κR)um
+n (R)Y m
+n (ˆx),
+x = Rˆx ∈ SR ⊂ R3
+(40)
+21
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+for some N ∈ N0. The map Tκ,N is called the truncated DtN operator, and N is the truncation
+order of the DtN operator.
+The replacement of the exact DtN operator Tκ in the problem (18) by the truncated DtN
+operator Tκ,N introduces a perturbation, hence we have to answer the question of existence
+and uniqueness of a solution to the following problem:
+Find uN ∈ V such that
+aN(uN, v) = nN(uN, v)
+for all v ∈ V.
+(41)
+holds, where aN and nN are the forms defined by (23), (24) with Tκ replaced by Tκ,N.
+The next result is the counterpart to Lemmata 8, 9. Here we formulate a different version
+of G˚arding’s inequality compared to the case d = 2 considered in [HNPX11, Thm. 4.4].
+Lemma 21. The sesquilinear form aN
+(i) is bounded, i.e. there exists a constant C > 0 independent of N such that
+|aN(w, v)| ≤ C∥w∥V ∥v∥V
+for all w, v ∈ V,
+and
+(ii) satisfies a G˚arding’s inequality in the form
+Re aN(v, v) ≥ ∥v∥2
+V,κ − 2κ2∥v∥2
+0,2,BR
+for all v ∈ V.
+Proof. (i) If the proof of [MS10, eq. (3.4a)] is carried out with finitely many terms of the
+expansion of Tκ only, the statement follows easily. Alternatively, Lemma 23 with s = 0 can
+also be used.
+(ii) As in the proof of Lemma 9, the definitions of aN and the wavenumber dependent norm
+yield
+Re aN(v, v) = ∥v∥2
+V,κ − 2κ2∥v∥2
+0,2,BR − Re (Tκ,Nv, v)SR.
+Hence it remains to estimate the last term. In the case d = 2, we have (see (39))
+Tκ,Nv(x) := 1
+R
+�
+|n|≤N
+Zn(κR)vn(R)Yn(ˆx),
+x = Rˆx ∈ SR.
+Then, using the L2(S1)-orthonormality of the circular harmonics [Zei95, Prop. 3.2.1], we get
+−(Tκ,Nv, v)SR = − 1
+R
+�
+|n|≤N
+Zn(κR)(vn(R)Yn, vn(R)Yn)SR
+= − 1
+R
+�
+|n|≤N
+Zn(κR)|vn(R)|2(Yn, Yn)SR
+= −
+�
+|n|≤N
+Zn(κR)|vn(R)|2(Yn, Yn)S1
+= −
+�
+|n|≤N
+Zn(κR)|vn(R)|2.
+22
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Hence, by Lemma 4,
+− Re (Tκ,Nv, v)SR =
+�
+|n|≤N
+(− Re Zn(κR))
+�
+��
+�
+≥1/2
+|vn(R)|2 + (− Re Z0(κR))
+�
+��
+�
+>0
+|v0(R)|2
+≥ 1
+2
+�
+|n|≤N
+|vn(R)|2 ≥ 0.
+The case d = 3 can be treated similarly. From
+Tκ,Nv(x) = 1
+R
+N
+�
+n=0
+�
+|m|≤n
+zn(κR)vm
+n (R)Y m
+n (ˆx)
+(see (40)), we immediately obtain, using the L2(S1)-orthonormality of the spherical harmon-
+ics [CK19, Thm. 2.8] that
+−(Tκ,Nv, v)SR = − 1
+R
+N
+�
+n=0
+�
+|m|≤n
+zn(κR)(vm
+n (R)Y m
+n , vm
+n (R)Y m
+n )SR
+= − 1
+R
+N
+�
+n=0
+�
+|m|≤n
+zn(κR)|vm
+n (R)|2(Y m
+n , Y m
+n )SR
+= −R
+N
+�
+n=0
+�
+|m|≤n
+zn(κR)|vm
+n (R)|2(Y m
+n , Y m
+n )S1
+= −R
+N
+�
+n=0
+�
+|m|≤n
+zn(κR)|vm
+n (R)|2,
+and Lemma 4 implies
+− Re (Tκ,Nv, v)SR = R
+N
+�
+n=0
+�
+|m|≤n
+(− Re zn(κR))
+�
+��
+�
+≥1
+|vm
+n (R)|2 ≥ R
+N
+�
+n=0
+�
+|m|≤n
+|vm
+n (R)|2 ≥ 0.
+In both cases we obtain the same G˚arding’s inequality as in the original (untruncated)
+problem Lemma 9.
+The next result is the variational version of the truncation error estimate. It closely follows
+the lines of the proof of [HNPX11, Thm. 3.3], where an estimate of ∥(Tκ − Tκ,N)v∥s−1/2,2,SR,
+s ∈ R, was proved in the case d = 2.
+Lemma 22. For given w, v ∈ H1/2(SR) it holds that
+���((Tκ − Tκ,N)w, v)SR
+��� ≤ c(N, w, v)∥w∥1/2,2,SR∥v∥1/2,2,SR,
+where c(N, w, v) ≥ 0 and limN→∞ c(N, w, v) = 0.
+23
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Proof. We start with the two-dimensional situation. So let
+w(x) = w(Rˆx) =
+�
+|n|∈N0
+wn(R)Yn(ˆx),
+v(x) = v(Rˆx) =
+�
+|k|∈N0
+vk(R)Yk(ˆx),
+x ∈ SR,
+(42)
+be series representations of w|SR, v|SR with the Fourier coefficients
+wn(R) = (w(R·), Yn)S1 =
+�
+S1
+w(Rˆx)Yn(ˆx)ds(ˆx),
+vk(R) = (v(R·), Yk)S1 =
+�
+S1
+v(Rˆx)Yk(ˆx)ds(ˆx).
+The norm on the Sobolev space Hs(SR), s ≥ 0, can be defined as follows [LM72, Ch. 1,
+Rem. 7.6]:
+∥v∥2
+s,2,SR := R
+�
+n∈Z
+(1 + n2)s|vn(R)|2.
+(43)
+Then, by (39), the orthonormality of the circular harmonics [Zei95, Prop. 3.2.1] and (43),
+���((Tκ − Tκ,N)w, v)SR
+��� = 1
+R
+������
+�
+|n|,|k|>N
+�
+Zn(κR)wn(R)Yn(R−1·), vk(R)Yk(R−1·)
+�
+SR
+������
+=
+������
+�
+|n|,|k|>N
+Zn(κR) (wn(R)Yn, vk(R)Yk)S1
+������
+=
+������
+�
+|n|>N
+Zn(κR)wn(R)vn(R)
+������
+=
+������
+�
+|n|>N
+Zn(κR)
+(1 + n2)1/2(1 + n2)1/4wn(R)(1 + n2)1/4vn(R)
+������
+≤ max
+|n|>N
+����
+Zn(κR)
+(1 + n2)1/2
+����
+�
+|n|>N
+��(1 + n2)1/4wn(R)(1 + n2)1/4vn(R)
+��
+≤ max
+|n|>N
+����
+Zn(κR)
+(1 + n2)1/2
+����
+
+ �
+|n|>N
+(1 + n2)1/2 |wn(R)|2
+
+
+1/2
+×
+
+ �
+|n|>N
+(1 + n2)1/2 |vn(R)|2
+
+
+1/2
+≤ 1
+R max
+|n|>N
+����
+Zn(κR)
+(1 + n2)1/2
+���� ˜c(N, w, v)∥w∥1/2,2,SR∥v∥1/2,2,SR,
+24
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+where
+˜c(N, w, v)2 :=
+�
+|n|>N(1 + n2)1/2|wn(R)|2
+�
+|n|∈N0(1 + n2)1/2|wn(R)|2
+�
+|n|>N(1 + n2)1/2|vn(R)|2
+�
+|n|∈N0(1 + n2)1/2|vn(R)|2.
+The coefficient ˜c(N, w, v) tends to zero for N → ∞ thanks to (43), (45).. Corollary 5 implies
+the estimate
+1
+1 + n2|Zn(κR)|2 ≤ max{|Z0(κR)|2, 1 + |κR|2},
+|n| ∈ N0,
+hence we can set
+c(N, w, v) := ˜c(N, w, v)
+R
+max{|Z0(κR)|, (1 + |κR|2)1/2}.
+The investigation of the case d = 3 runs similarly. So let
+w(x) = w(Rˆx) =
+�
+n∈N0
+�
+|m|≤n
+wm
+n (R)Y m
+n (ˆx),
+v(x) = v(Rˆx) =
+�
+k∈N0
+�
+|l|≤k
+vl
+k(R)Y l
+k(ˆx),
+x ∈ SR,
+(44)
+be series representations of w|SR, v|SR with the Fourier coefficients
+wm
+n (R) = (w(R·), Y m
+n )S1 =
+�
+S1
+w(Rˆx)Y m
+n (ˆx)ds(ˆx),
+vl
+k(R) = (v(R·), Y l
+k)S1 =
+�
+S1
+v(Rˆx)Y l
+k(ˆx)ds(ˆx).
+The norm on the Sobolev space Hs(SR), s ≥ 0, can be defined as follows [LM72, Ch. 1,
+Rem. 7.6]:
+∥v∥2
+s,2,SR := R2 �
+n∈N0
+�
+|m|≤n
+(1 + n2)s|vm
+n (R)|2.
+(45)
+Then, by (40), the orthonormality of the spherical harmonics [CK19, Thm. 2.8] and (45),
+���((Tκ − Tκ,N)w, v)SR
+��� = 1
+R
+������
+�
+n,k>N
+�
+|m|≤n,|l|≤k
+�
+zn(κR)wm
+n (R)Y m
+n (R−1·), vl
+k(R)Y l
+k(R−1·)
+�
+SR
+������
+= R
+������
+�
+n,k>N
+�
+|m|≤n,|l|≤k
+zn(κR)
+�
+wm
+n (R)Y m
+n , vl
+k(R)Y l
+k)
+�
+S1
+������
+= R
+������
+�
+n>N
+�
+|m|≤n
+zn(κR)wm
+n (R)vm
+n (R)
+������
+= R
+������
+�
+n>N
+�
+|m|≤n
+zn(κR)
+(1 + n2)1/2(1 + n2)1/4wm
+n (R)(1 + n2)1/4vm
+n (R)
+������
+25
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+≤ R max
+n>N
+����
+zn(κR)
+(1 + n2)1/2
+����
+�
+n>N
+�
+|m|≤n
+��(1 + n2)1/4wm
+n (R)(1 + n2)1/4vm
+n (R)
+��
+≤ R max
+n>N
+����
+zn(κR)
+(1 + n2)1/2
+����
+
+�
+n>N
+�
+|m|≤n
+(1 + n2)1/2 |wm
+n (R)|2
+
+
+1/2
+×
+
+�
+n>N
+�
+|m|≤n
+(1 + n2)1/2 |vm
+n (R)|2
+
+
+1/2
+≤ 1
+R max
+n>N
+����
+zn(κR)
+(1 + n2)1/2
+���� ˜c(N, w, v)∥w∥1/2,2,SR∥v∥1/2,2,SR,
+where
+˜c(N, w, v)2 :=
+�
+n>N
+�
+|m|≤n(1 + n2)1/2 |wm
+n (R)|2
+�
+|n|∈N0
+�
+|m|≤n(1 + n2)1/2 |wm
+n (R)|2
+�
+n>N
+�
+|m|≤n(1 + n2)1/2 |vm
+n (R)|2
+�
+|n|∈N0
+�
+|m|≤n(1 + n2)1/2 |vm
+n (R)|2.
+Thanks to Corollary 5 we can define
+c(N, w, v) := ˜c(N, w, v)
+R
+�
+2 + |κR|2�1/2 .
+Lemma 23. For s ∈ [0, 1/2) and w ∈ H1−s(BR \ Ω), v ∈ H1+s(BR \ Ω) it holds that
+|(Tκ,Nw, v)SR| ≤ Cbl∥w∥1−s,2,BR\Ω∥v∥1+s,2,BR\Ω,
+where the constant Cbl ≥ 0 does not depend on N.
+Proof. We start with the two-dimensional situation as in the proof of Lemma 22. If w, v
+have the representations (42), then, by (39), the orthonormality of the circular harmonics
+[Zei95, Prop. 3.2.1] and (43),
+|(Tκ,Nw, v)SR| = 1
+R
+������
+�
+|n|,|k|≤N
+�
+Zn(κR)wn(R)Yn(R−1·), vk(R)Yk(R−1·)
+�
+SR
+������
+=
+������
+�
+|n|,|k|≤N
+Zn(κR) (wn(R)Yn, vk(R)Yk)S1
+������
+=
+������
+�
+|n|≤N
+Zn(κR)wn(R)vn(R)
+������
+=
+������
+�
+|n|≤N
+Zn(κR)
+(1 + n2)1/2(1 + n2)(1/2−s)/2wn(R)(1 + n2)(1/2+s)/2vn(R)
+������
+≤ max
+|n|≤N
+����
+Zn(κR)
+(1 + n2)1/2
+����
+�
+|n|≤N
+��(1 + n2)(1/2−s)/2wn(R)(1 + n2)(1/2+s)/2vn(R)
+��
+26
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+≤ max
+|n|≤N
+����
+Zn(κR)
+(1 + n2)1/2
+����
+
+ �
+|n|≤N
+(1 + n2)1/2−s |wn(R)|2
+
+
+1/2
+×
+
+ �
+|n|≤N
+(1 + n2)1/2+s |vn(R)|2
+
+
+1/2
+≤ 1
+R max
+|n|≤N
+����
+Zn(κR)
+(1 + n2)1/2
+���� ∥w∥1/2−s,2,SR∥v∥1/2+s,2,SR.
+Corollary 5 implies the estimate
+1
+1 + n2|Zn(κR)|2 ≤ max{|Z0(κR)|2, 1 + |κR|2},
+|n| ∈ N0,
+hence
+|(Tκ,Nw, v)SR| ≤ 1
+R max{|Z0(κR)|, (1 + |κR|2)1/2}∥w∥1/2−s,2,SR∥v∥1/2+s,2,SR.
+(46)
+By the trace theorem [McL00, Thm. 3.38], we finally arrive at
+|(Tκ,Nw, v)SR| ≤ C2
+tr
+R max{|Z0(κR)|, (1 + |κR|2)1/2}∥w∥1−s,2,BR\Ω∥v∥1+s,2,BR\Ω.
+The investigation of the case d = 3 runs similarly. So let w, v have the representations (44),
+then, by (40), the orthonormality of the spherical harmonics [CK19, Thm. 2.8] and (45),
+|(Tκ,Nw, v)SR| = 1
+R
+������
+N
+�
+n,k=0
+�
+|m|≤n,|l|≤k
+�
+zn(κR)wm
+n (R)Y m
+n (R−1·), vl
+k(R)Y l
+k(R−1·)
+�
+SR
+������
+= R
+������
+N
+�
+n,k=0
+�
+|m|≤n,|l|≤k
+zn(κR)
+�
+wm
+n (R)Y m
+n , vl
+k(R)Y l
+k)
+�
+S1
+������
+= R
+������
+N
+�
+n=0
+�
+|m|≤n
+zn(κR)wm
+n (R)vm
+n (R)
+������
+= R
+������
+N
+�
+n=0
+�
+|m|≤n
+zn(κR)
+(1 + n2)1/2(1 + n2)(1/2−s)/2wm
+n (R)(1 + n2)(1/2+s)/2vm
+n (R)
+������
+≤ R max
+n∈N0
+����
+zn(κR)
+(1 + n2)1/2
+����
+N
+�
+n=0
+�
+|m|≤n
+��(1 + n2)(1/2−s)/2wm
+n (R)(1 + n2)(1/2+s)/2vm
+n (R)
+��
+≤ R max
+n∈N0
+����
+zn(κR)
+(1 + n2)1/2
+����
+
+
+N
+�
+n=0
+�
+|m|≤n
+(1 + n2)1/2−s |wm
+n (R)|2
+
+
+1/2
+×
+
+
+N
+�
+n=0
+�
+|m|≤n
+(1 + n2)1/2+s |vm
+n (R)|2
+
+
+1/2
+27
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+≤ 1
+R max
+n∈N0
+����
+zn(κR)
+(1 + n2)1/2
+���� ∥w∥1/2−s,2,SR∥v∥1/2+s,2,SR.
+Corollary 5 yields
+|(Tκ,Nw, v)SR| ≤ 1
+R
+�
+2 + |κR|2�1/2 ∥w∥1/2−s,2,SR∥v∥1/2+s,2,SR.
+(47)
+By the trace theorem [McL00, Thm. 3.38], we finally arrive at
+|(Tκ,Nw, v)SR| ≤ C2
+tr
+R
+�
+2 + |κR|2�1/2 ∥w∥1−s,2,BR\Ω∥v∥1+s,2,BR\Ω.
+Theorem 24. Under the assumptions of Lemma 9, given an antilinear continuous functional
+ℓ : V → C, there exists a constant N∗ > 0 such that for N ≥ N∗ the problem
+Find uN ∈ V such that
+aN(uN, v) = ℓ(v)
+for all v ∈ V
+(48)
+is uniquely solvable.
+Proof. First we show that the problem (48) has at most one solution. We start as in the
+proof of [HNPX11, Thm. 4.5] and argue by contradiction, i.e. we suppose the following:
+∀N∗ ∈ N
+∃N = N(N∗) ≥ N∗
+and
+uN = uN(N∗) ∈ V
+such that
+aN(uN, v) = 0
+for all v ∈ V
+and ∥uN∥V = 1.
+(49)
+However, the subsequent discussion differs significantly from the proof of [HNPX11, Thm. 4.5].
+We apply an argument the idea of which goes back to Schatz [Sch74].
+First we assume there exists a solution uN ∈ V of (48) and derive an a priori estimate of
+the error ∥u − uN∥V , where u ∈ V is the solution of (29), see Thm. 10. Since aN satisfies a
+G˚arding’s inequality (Lemma 21(ii)), we have, making use of (28),
+C2
+−∥u − uN∥2
+V − 2κ2∥u − uN∥2
+0,2,BR ≤ Re aN(u − uN, u − uN).
+Since
+aN(u − uN, v) = aN(u, v) − aN(uN, v)
+= a(u, v)
+� �� �
+=ℓ(v)
++aN(u, v) − a(u, v) − aN(uN, v)
+�
+��
+�
+=ℓ(v)
+= ((Tκ − Tκ,N)u, v)SR ,
+we obtain
+C2
+−∥u − uN∥2
+V − 2κ2∥u − uN∥2
+0,2,BR ≤ η1∥u − uN∥V
+(50)
+with
+η1 := sup
+v∈V
+Re ((Tκ − Tκ,N)u, v)SR
+∥v∥V
+.
+Now we consider the following auxiliary adjoint problem (cf. [McL00, p. 43]):
+28
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Find wN ∈ V such that
+a(v, wN) = (v, u − uN)BR
+for all v ∈ V.
+(51)
+Since A is a Fredholm operator (see the proof of Thm. 10), the adjoint problem possesses a
+unique solution wN ∈ V . Then
+∥u − uN∥2
+0,2,SR = a(u − uN, wN) = a(u, wN) − a(uN, wN)
+= a(u, wN) − aN(uN, wN)
+�
+��
+�
+=ℓ(wN)−ℓ(wN)=0
++aN(uN, wN) − a(uN, wN)
+= ((Tκ − Tκ,N)uN, wN)SR.
+In particular, this relation shows that ((Tκ − Tκ,N)uN, wN)SR is real. With
+η2 := sup
+v∈V
+((Tκ − Tκ,N)uN, v)SR
+∥v∥V
+we obtain
+∥u − uN∥2
+0,2,BR ≤ η2∥wN∥V ≤ η2C−1
+− C(R, κ)∥u − uN∥V ∗.
+The continuous embedding V ⊂ V ∗ yields
+∥u − uN∥2
+0,2,BR ≤ η2C−1
+− C(R, κ)Cemb∥u − uN∥V .
+Applying this estimate in (50), we get
+C2
+−∥u − uN∥2
+V − 2κ2η2C−1
+− C(R, κ)Cemb∥u − uN∥V ≤ η1∥u − uN∥V .
+Now, if ∥u − uN∥V ̸= 0, we finally arrive at
+C2
+−∥u − uN∥V ≤ η1 + 2κ2η2C−1
+− C(R, κ)Cemb.
+(52)
+Clearly this inequality is true also for ∥u − uN∥V = 0 so that we can remove this interim
+assumption.
+Thanks to Lemma 22 we have that
+���((Tκ − Tκ,N)u, v)SR
+��� ≤ c(N, u, v)∥u∥1/2,2,SR∥v∥1/2,2,SR ≤ c(N, u, c)C2
+tr∥u∥V ∥v∥V ,
+hence
+η1 ≤ c+(N, u)C2
+tr∥u∥V
+with
+c+(N, u) := sup
+v∈V
+c(N, u, v),
+(53)
+where limN→∞ c+(N, u) = 0. Note that, as can be seen from the proof of Lemma 22, the
+second fractional factor in the representation of ˜c(N, w, v) can be estimated from above by
+one without losing the limit behaviour for N → ∞. Consequently, η1 can be made arbitrarily
+small provided N is large enough.
+In order to estimate η2 we cannot apply Lemma 22 directly since the second argument in
+the factor c(N, uN, v) depends on N, too. Therefore we give a more direct estimate.
+29
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+Namely, let v ∈ V have the representation (42) or (44), respectively. Then we define
+VN|SR :=
+�
+span|n|≤N{Yn(R−1·)},
+d = 2,
+spann=0...N,|m|≤n{Y m
+n (R−1·)},
+d = 3,
+and introduce an orthogonal projector
+PN : V |SR → VN|SR : v �→ PNv :=
+��
+|n|≤N vn(R)Yn(R−1·),
+d = 2,
+�N
+n=0
+�
+|m|≤n vm
+n (R)Y m
+n (R−1·),
+d = 3.
+Then it holds that VN|SR ⊂ ker(TκPN − Tκ,N).
+Indeed, if d = 2 and v ∈ VN|SR, then
+PNv = v = �
+|n|≤N vn(R)Yn(R−1·) and
+TκPNv = Tκv = 1
+R
+�
+|n|≤N
+Zn(κR)vn(R)Yn(R−1·) = Tκ,Nv.
+An analogous argument applies in the case d = 3.
+Now we return to the estimate of η2 and write, for uN ∈ V ,
+(Tκ − Tκ,N)uN = (Tκ − TκPN)uN + (TκPN − Tκ,N)uN = Tκ(id −PN)uN,
+where we have used the above property. The advantage of this approach is that we can apply
+a wellknown estimate of the projection error. The proof of this estimate runs similarly to
+the proof of Lemma 22 but only without the coefficients Zn or zn, respectively:
+��((id −PN)w, v)SR
+�� =
+������
+�
+|n|,|k|>N
+�
+wn(R)Yn(R−1·), vk(R)Yk(R−1·)
+�
+SR
+������
+= R
+������
+�
+|n|,|k|>N
+(wn(R)Yn, vk(R)Yk)S1
+������
+= R
+������
+�
+|n|>N
+wn(R)vn(R)
+������
+= R
+������
+�
+|n|>N
+1
+(1 + n2)1/2(1 + n2)1/4wn(R)(1 + n2)1/4vn(R)
+������
+≤ max
+|n|>N
+R
+(1 + n2)1/2
+�
+|n|>N
+��(1 + n2)1/4wn(R)(1 + n2)1/4vn(R)
+��
+≤
+R
+(1 + N2)1/2
+
+ �
+|n|>N
+(1 + n2)1/2 |wn(R)|2
+
+
+1/2
+×
+
+ �
+|n|>N
+(1 + n2)1/2 |vn(R)|2
+
+
+1/2
+≤
+1
+(1 + N2)1/2∥w∥1/2,2,SR∥v∥1/2,2,SR.
+30
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+The same estimate holds true for d = 3. Then we get, by Remark 3 (or Lemma 23),
+���((Tκ − Tκ,N)uN, v)SR
+��� =
+��(Tκ(id −PN)uN, v)SR
+��
+≤
+Cκ
+(1 + N2)1/2∥uN∥1/2,2,SR∥v∥1/2,2,SR
+≤
+CC2
+trκ
+(1 + N2)1/2∥uN∥V ∥v∥V ,
+thus
+η2 ≤
+CC2
+trκ
+(1 + N2)1/2∥uN∥V .
+Using this estimate and (53) in (52), we obtain
+C2
+−∥u − uN∥V ≤ c+(N, u)C2
+tr∥u∥V + 2κ2C−1
+− C(R, κ)Cemb
+CC2
+trκ
+(1 + N2)1/2∥uN∥V .
+(54)
+Now we appply this estimate to the solutions uN of the homogeneous truncated problems in
+(49). By Thm. 10, the homogeneous linear interior problem (29) (i.e. ℓ = 0) has the solution
+u = 0, and the above estimate implies
+C2
+−∥uN∥V ≤ 2κ2C−1
+− C(R, κ)Cemb
+CC2
+trκ
+(1 + N2)1/2∥uN∥V ,
+which is a contradiction to ∥uN∥V = 1 for all N.
+Although the proof of Thm. 24 allows an analogous conclusion as in Lemma 11 that the
+truncated bilinear form aN satisfies an inf-sup condition, such a conclusion is not fully
+satisfactory since the question remains whether and how the inf-sup constant depends on N
+or not. However, at least for sufficiently large N, a positive answer can given.
+Lemma 25. Under the assumptions of Lemma 9, there exists a number N∗ ∈ N such that
+βN∗(R, κ) :=
+inf
+w∈V \{0}
+sup
+v∈V \{0}
+|aN(w, v)|
+∥w∥V,κ∥v∥V,κ
+> 0
+is independent of N ≥ N∗.
+In the proof a formula is given that expresses βN∗(R, κ) in terms of β(R, κ).
+Proof. We return to the proof of Thm. 24 and mention that the estimate (54) is valid for
+solutions u, uN of the general linear problems (29) (or, equally, (31)) and (48), respectively.
+By the triangle inequality,
+∥uN∥V ≤ ∥u∥V + ∥u − uN∥V
+≤ ∥u∥V + c+(N, u)C−2
+− C2
+tr∥u∥V + 2κ2C−3
+− C(R, κ)Cemb
+CC2
+trκ
+(1 + N2)1/2∥uN∥V .
+31
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+If N∗ is sufficiently large such that
+κ2C−3
+− C(R, κ)Cemb
+CC2
+trκ
+(1 + N2)1/2 ≤ 1
+4
+and
+c+(N, u)C−2
+− C2
+tr ≤ 1
+for all N ≥ N∗,
+then, by Lemma 11,
+∥uN∥V ≤ 4∥u∥V ≤ 4
+C−
+∥u∥V,κ ≤ ∥ℓ∥V ∗.
+That is, the sesquilinear form aN satisfies an inf-sup condition
+βN∗(R, κ) :=
+inf
+w∈V \{0}
+sup
+v∈V \{0}
+|aN(w, v)|
+∥w∥V,κ∥v∥V,κ
+> 0
+with βN∗(R, κ) := C−β(R, κ)
+4C+
+independent of N ≥ N∗.
+Analogously to (30) we introduce the truncated linear operator AN : V → V ∗ by
+ANw(v) := aN(w, v)
+for all w, v ∈ V.
+By Lemma 21, AN is a bounded operator, and Lemma 25 implies that AN has a bounded
+inverse:
+∥w∥V,κ ≤ βN∗(R, κ)−1∥ANw∥∗
+for all w ∈ V.
+Furthermore, we define a nonlinear operator FN : V → V ∗ by
+FN(w)(v) := ℓcontr(w) + ℓsrc(w) + ℓinc
+N
+for all w ∈ V,
+where
+⟨ℓinc
+N , v⟩ := (ˆx · ∇uinc − Tκ,Nuinc, v)SR.
+The problem (41) is then equivalent to the operator equation
+ANu = FN(u)
+in V ∗,
+and further to the fixed-point problem
+u = A−1
+N FN(u)
+in V.
+(55)
+Theorem 26. Under the assumptions of Lemma 9, let the functions c and f generate locally
+Lipschitz continuous Nemycki operators in V and assume that there exist functions wf, wc ∈
+V such that f(·, wf) ∈ Lpf/(pf −1)(Ω) and c(·, wf) ∈ Lpc/(pc−2)(Ω), respectively.
+Furthermore let uinc ∈ H1
+loc(Ω+) be such that additionally ∆uinc ∈ L2,loc(Ω+) holds.
+If there exist numbers ̺ > 0 and LF ∈ (0, βN∗(R, κ)) (where N∗ and βN∗(R, κ) are from
+Lemma 25) such that the following two conditions
+κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ̺
++
+�
+∥f(·, wf)∥0,˜qf,Ω + ∥Lf(·, w, wf)∥0,qf,Ω(̺ + ∥wf∥V )
+�
++ Ctr∥ˆx · ∇uinc − Tκ,Nuinc∥−1/2,2,SR ≤ ̺βN∗(R, κ),
+κ2 [∥Lc(·, w, v)∥0,qc,Ω̺ + ∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )]
++ ∥Lf(·, w, v)∥0,qf,Ω ≤ LF
+are satisfied for all w, v ∈ Kcl
+̺ , then the problem (35) has a unique solution uN ∈ Kcl
+̺ for all
+N ≥ N∗.
+Proof. Analogously to the proof of Thm. 18.
+32
+
+Resonant compactly supported nonlinearities
+January 30, 2023
+7 Conclusion
+A mathematical model together with an investigation of existence and uniqueness of its
+solution for radiation and propagation effects on compactly supported cubic nonlinearities
+is presented. The full-space problem is reduced to an equivalent truncated local problem,
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+regard to stability and errors) is studied. The results form the basis for the use of numerical
+methods, e.g., FEM, for the approximate solution of the original problem with controllable
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+

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