Defect Subspaces and Localized Instabilities in Cut-Cell Finite-Volume Operators

Justo E. Karell

REVIEW 3 major objections 4 minor 14 references

This paper proves that small cut-cell blow-up in explicit finite-volume schemes is governed by a single local defect number, and turns that number into a per-cell stabilization formula.

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2026-08-03 00:22 UTC pith:Y3DZRCED

load-bearing objection Genuinely new spectral analysis of cut-cell instability with a clean 1D proof and a useful defect-matrix criterion; the fixed-λ transfer in Section 9 is a real gap, but the stress-test counterexample is wrong—the 2×2 SRD block is conservative, not under-stabilizing. the 3 major comments →

arxiv 2607.28808 v1 pith:Y3DZRCED submitted 2026-07-30 math.NA cs.NA

Defect Subspaces and Localized Instabilities in Cut-Cell Finite-Volume Operators

Justo E. Karell This is my paper

classification math.NA cs.NA MSC 65M1265M0876M12

keywords cut-cell methodfinite volumestabilitystate redistributiondefect matrixeigenvalue localizationembedded boundaryCFL condition

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper aims to prove that the classic cut-cell instability in explicit finite-volume methods is a sharply localized spectral event, not a diffuse numerical failure. When an embedded boundary clips a cell to a tiny volume fraction α, the update operator develops exactly one large unstable eigenvalue per cut cell, and the corresponding eigenvector is concentrated on that cell with entries decaying like α away from it. The growth rate is set by a local defect matrix Γ, essentially the net linearized diagonal flux coefficient at the cut cell scaled by Δt/h, so the paper derives a closed-form minimum state-redistribution (SRD) blending parameter s_stab = (λ/α − 2)/(λ/α − 1) + O(α), computable from α and the background CFL at mesh-generation time. A sympathetic reader would care because this replaces trial-and-error stabilization and rerunning crashed production simulations with a cheap geometric check at each cut cell, and it gives the first mechanistic explanation of why local redistribution cures the blow-up.

Core claim

The paper's central claim is that the cut-cell instability is created by a low-dimensional defect subspace: for m small cut cells, the linearized update operator has exactly m unstable eigenvalues, each asymptotic to 1 − λ/α_k plus an O(1) correction, with eigenvectors equal to the coordinate vector e_{j_k} plus O(α). On a scalar periodic 1D mesh, the growth rate is governed by the defect matrix Γ = (Δt/h)(a_{c+1/2,c} − a_{c−1/2,c}); the unstable spectral projector converges to the m-dimensional coordinate subspace of the cut cells at rate O(α). For state redistribution, the paper proves the correction vector is aligned with the unstable eigendirection up to O(α), and derives the minimum ble

What carries the argument

The engine of the argument is the singular-block decomposition of the update operator after reordering unknowns into cut-cell ('defect') and regular coordinates: A_α = [[−α^{−p}Γ + R_α, B_α],[P_α, Q_α]]. The defect matrix Γ encodes the net linearized diagonal flux coefficient at each cut cell, scaled by Δt/h; its eigenvalues γ_k control the large eigenvalues −γ_k/α^p + O(1). A contraction-mapping/root-location argument on the characteristic polynomial pins the dominant eigenvalue, and the smallness of the eigenvector entries away from the cut cell—each step away scales like O(α)—localizes the mode. The same block structure yields the O(α^p) convergence of the unstable spectral projector to t

Load-bearing premise

The formula s_stab = (λ_α−2)/(λ_α−1)+O(α) is proved for α→0 with λ_α fixed, but the paper only verifies numerically that it transfers to fixed background CFL λ; if that transfer fails, the criterion under-stabilizes cut cells at practical volume fractions.

What would settle it

Assemble the full N×N blended operator A(s) for fixed λ (say 0.4), α = 0.05, and s = s_stab(λ/α) from the formula; compute ρ(A(s)) by power iteration or eigensolve. If ρ(A(s)) exceeds 1 by more than O(α) for any α in, say, [0.02, 0.2], the criterion under-stabilizes. A sharper check: compare s_stab to the exact threshold from the 2×2 characteristic polynomial of M_s(α) across the same α range; the difference must be O(α) for the paper's claim to hold.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

At mesh generation time, a solver can compute Γ and λ_α = λ/α at each cut cell and flag any cell with λ_α > 2 as unstable, with no time stepping and no full-matrix eigenvalue solve.
For each flagged cell, the minimum SRD blending parameter is s_stab = (λ_α − 2)/(λ_α − 1) + O(α); using at least this fraction damps the dominant mode to the unit disk.
Any stabilization method whose correction vector D satisfies |cos(D, v_u)| = 1 − O(α) acts in the unstable eigendirection and will stabilize the cut cell for the same reason as SRD, so the alignment condition serves as a certification criterion.
For m non-adjacent cut cells the unstable spectral subspace converges to the m-dimensional coordinate subspace of those cells at rate O(α), so the per-cell analysis is not disturbed by distant cut cells.
The characterization theorem delimits the scope: the structure holds for scalar periodic 1D schemes with bounded flux coefficients (p=1) and more generally for power-law scaling of flux coefficients; outside that scope the explicit formula for Γ must be replaced by the scaled limit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The paper leaves open whether the same criterion extends to non-periodic boundaries, adjacent cut cells, and systems; if the defect-matrix mechanism is generic, the per-cell formula would give immediate guidance for those settings.
Because the eigenvector decays as α^{|i−c|}, the instability is physically confined to the cut cell; a natural testable consequence is that local mesh refinement away from the boundary cannot cure it, only treatment at the cut cell can.
The alignment condition |cos(D, v_u)| = 1 − O(α) could be used as a design test for new stabilizations beyond SRD: any local redistribution whose correction is nearly parallel to the unstable eigenvector should stabilize with one scalar parameter per cut cell.
A quantitative extension would be to compute finite-α corrections to s_stab by solving the exact 2×2 characteristic polynomial of the blended block M_s(α), yielding a safety margin for practical mesh generation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Genuinely new spectral analysis of cut-cell instability with a clean 1D proof and a useful defect-matrix criterion; the fixed-λ transfer in Section 9 is a real gap, but the stress-test counterexample is wrong—the 2×2 SRD block is conservative, not under-stabilizing.

read the letter

Read the paper carefully. The core spectral analysis is real. Section 2 gives a clean contraction-map proof that the upwind cut-cell operator has a dominant eigenvalue μ_u = 1 − λ/α + O(α^{N−2}) with eigenvector localized at the cut cell, and it reproduces Berger's λ_α = 2 threshold. The defect matrix Γ in Theorem 8.1 and the singular-block framework in Section 6 are a genuinely useful abstraction: the instability is governed by a local two-number flux difference, computable without assembling the mesh-wide operator. The SRD alignment theorem and the multi-cut-cell projector convergence are also solid. This is a real analytical contribution, not a repackaging.

Where I part ways with the stress-test note: the claimed counterexample to Remark 9.4 does not survive contact with the actual blended operator. The stress-test reduces the cut-cell update to a scalar diagonal a = (1−s)(1−λ/α) + s(α−λ)/(1+α) and solves a = −1. That misses the off-diagonal entries in the SRD rows. For the 2×2 block (rows c−1 and c, columns c−1 and c), which is the correct local reduction for this localized eigenvector, the off-diagonal entries m12 = s(α−λ)/(1+α) and m21 = (1−s)λ/α + s/(1+α) are both O(1) in the fixed-λ regime. With α=0.1, λ=0.4, s=2/3, the scalar diagonal gives −1.18, but the eigenvalues of the 2×2 block are ≈0.61 and −0.985, both inside the unit disk. Solving for the threshold μ=−1 in the 2×2 block gives s≈0.662, just below the paper's 2/3. So the paper's formula is slightly conservative, not under-stabilizing, for that case.

This does not remove the legitimate gap. Theorem 9.2 is proven in the λ→0 fixed-λ_α limit, and Remark 9.4's transfer to fixed λ is argued by plausibility and numerics rather than proved. The paper's own wording admits this. Also, the eigenvector localization theorem claims O(α^{|i−c|}) for all i, but on a periodic domain the components left of the cut cell are actually O(α^{N−|i−c|}) for the present mode; the Euclidean bound still holds, so it is cosmetic. The production claim that the criterion 'prevents the class of crash' is inflated: the linear-stability statement does not cover nonlinearities, wider stencils, or non-periodic boundaries, which the paper itself lists as open.

Who should read it: anyone working on cut-cell stabilization, especially SRD or similar redistribution methods. The spectral mechanism is worth knowing even if the production claims are oversold. The paper deserves a serious referee; the 1D analysis is clean and the singular-block theorem may be broadly applicable. The referee should push for a proper proof of the fixed-λ transfer in Remark 9.4 or an explicit statement that it is a numerical heuristic.

Referee Report

3 major / 4 minor

Summary. The paper develops a spectral theory for the small-cell instability in explicit finite-volume cut-cell schemes. For the one-dimensional periodic upwind operator with one cut cell of volume fraction α, it proves a dominant eigenvalue μ_u = 1 − λ/α + O(α^{N−2}), a localized eigenvector v_u = e_c + O(α), and an instability threshold λ_α = λ/α > 2. It extends this to multiple non-adjacent cut cells (Theorem 4.2), to d dimensions (Theorem 5.1), and to an abstract singular-block framework (Theorems 6.2–6.3). Section 7 characterizes which finite-volume operators admit this structure, and Section 8 derives the defect-matrix formula Γ = (Δt/h)(a_{c+1/2,c} − a_{c−1/2,c}). The practical output is a per-cell SRD blending criterion s_stab = (λ_α − 2)/(λ_α − 1), claimed to be computable from local geometry alone and to be the minimum stabilization parameter.

Significance. If the main claims hold, the paper would give a useful algebraic explanation of a well-known numerical instability and a cheap, assembly-free way to set SRD blending parameters. The spectral results in Sections 2–8 are generally coherent: the contraction-mapping/Rouché argument for the one-cut-cell case, the Gershgorin/Stewart argument for multiple cut cells, and the Schur-complement argument in the singular-block framework are internally consistent, and they correctly reproduce Berger's known λ_α = 2 threshold. The defect-matrix formula is a genuine simplification, and the eigenvalue localization theorem directly matches previously observed numerical behavior. However, the headline application—the ‘minimum’ SRD blending criterion in Section 9—has a load-bearing gap: the transfer from the asymptotic λ→0 regime to the practical fixed-λ regime is not proved, and for the operator actually used in Section 9 the formula under-stabilizes by O(α), contradicting the reported experiments.

major comments (3)

[Section 9, Theorem 9.2 / Remark 9.4] The fixed-λ transfer of the stability criterion is not proved and, for the operator analyzed in Section 9, it is false as a sufficient condition. The proof of Theorem 9.2 takes α→0 with λ_α=λ/α fixed, hence λ→0. Using the merged-average A_SRD of Proposition 9.1, the (c,c) entry of A(s)=(1−s)A+sA_SRD is a(s)=(1−s)(1−λ/α)+s(α−λ)/(1+α). For large N the localized eigenvalue satisfies μ=a(s)+O(q^N), q=λ/(μ−1+λ), so stability requires a(s)≥−1. Solving a(s)=−1 gives the exact threshold s_exact=(λ−2α)(1+α)/(λ−α), whereas the paper's formula is s_stab=(λ−2α)/(λ−α). The difference s_exact−s_stab=α(λ−2α)/(λ−α) is O(α) and positive. For λ=0.4, α=0.1, s_stab=2/3 yields a≈−1.18, so ρ(A(s))>1; the claimed ‘minimum’ is not sufficient. The O(α) in (21) is a two-sided asymptotic error, not a stability margin, so the leading-order formula cannot be used as a sufficient bound. This issue is load-bearing for
[Section 3, Eq. (8) versus Section 9, Proposition 9.1] The SRD operator is defined inconsistently. Eq. (8) defines a correction D=Δ(αe_{c−1}−e_c) with Δ=[λp+(1−λ_α)r]/(1+α), so the stabilized update at the cut cell is U^{n+1}_c=U^{base}_c−Δ. The coefficient of U_c in A_SRD is then (1−λ_α)−(1−λ_α)^2/(1+α), not α(1−λ_α)/(1+α) as claimed in Proposition 9.1. The latter corresponds to U^{n+1}_c=(U^{base}_{c−1}+αU^{base}_c)/(1+α), the usual merged-average SRD. These are different operators: the Eq. (8) operator does not reduce Γ_SRD to O(α) in the same way. Theorem 3.1 (alignment) is proved for the Eq. (8) operator, while Theorem 9.2 (stability criterion) is derived for the merged-average operator. The paper must either reconcile these definitions or explicitly state which variant each theorem concerns; as written, the stability criterion is not tied to the SRD correction analyzed in Section 3.
[Section 10, Experiment 7] The time-domain experiment reports that for α=0.1, λ_α=4, the choice s=s_stab=2/3 keeps ∥U^n∥ bounded. This is inconsistent with the operator defined in Section 9: at that parameter value the dominant eigenvalue is ≈−1.18, so the solution should grow like 1.18^n. Either the experiment uses a different SRD implementation (which would need to be specified) or the reported boundedness is incorrect. The same section's Experiment 5(b) varies α by setting λ=λ_α α, which forces λ→0 as α→0 and therefore does not probe the fixed-λ regime claimed in Remark 9.4. The numerical verification is thus incomplete where it is needed most.

minor comments (4)

[Section 9, proof of Theorem 9.2] The limiting matrix M_s has a typo: the (2,1) entry should be (1−s)L+s, not (1−s)L+s(1−s)(1−L). The subsequent eigenvalue statement indicates the intended form.
[Section 4, Step 3 of Theorem 4.2] The text says ‘For uniform fractions α_k=α this is O(mλ α^{d_min−2})’; the preceding Proposition 4.1 gives O(λ_max α^{d_min−1}). The d_min−2 exponent appears to be a typo; the argument only needs the d_min−1 bound.
[Remark 9.3] The claim that ‘the O(α) error is below 0.01 for α≤0.1’ is contradicted by the calculation in the first major comment: for λ=0.4, α=0.1 the error in the threshold is approximately 0.067, not 0.01. This remark should be revised together with Theorem 9.2.
[Section 8, Theorem 8.1 / Remark 8.2] The notation λ_α = Γ/α is used before Γ is defined in the proof; a sentence of orientation would help. In Remark 8.2, the statement that ‘Γ=λ’ for first-order upwind follows from a_{c+1/2,c}=a and a_{c−1/2,c}=0; this is correct, but it would be clearer to note that Δt/h is absorbed into λ.

Circularity Check

0 steps flagged

No circular derivation: spectral and SRD results are obtained by explicit algebra; the only same-author input is the SRD correction formula (8), a definitional construction, and the fixed-λ transfer in Remark 9.4 is a soundness gap, not a circle.

full rationale

The derivation chain in Section 2 is self-contained: the characteristic polynomial P_α(q)=q^N-(1-α)q-α follows from the operator definition and is solved by contraction mapping and Rouché's theorem, yielding μ_u=1-λ/α+O(α^{N-2}); no fitted data, parameter, or external result is used for the eigenvalue. Corollary 2.1 reproduces Berger's known threshold but from the derived characteristic polynomial, so it is an independent re-derivation, not a renaming. The defect matrix Γ in Theorem 8.1 is obtained by substituting the linearized fluxes (17) into the conservative update (16), not by fitting; Remark 8.2 verifies consistency with the upwind special case. Theorem 9.2's s_stab is solved algebraically from the 2×2 blended limit matrix; it is not an ansatz imported from prior work. The only same-author dependency is Eq. (8), where D is taken from the submitted [14] as an explicit SRD construction; Theorem 3.1 then proves a property of that D rather than assuming the property. I therefore find no circular reduction. I do flag Remark 9.4: the transfer of the λ_α-fixed asymptotic criterion to the fixed-λ regime is asserted rather than proved from the full N×N operator, and the skeptic's example indicates a possible O(α) under-stabilization; that is a correctness/regime gap, not a circularity, and does not raise the circularity score. Experiment 8 is also honestly labeled as a consistency check rather than independent verification. The low nonzero score reflects only the same-author provenance of Eq. (8), which is not load-bearing in a circular sense.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper's results are analytic and parameter-free. The main burden is the modeling assumption that the fixed-λ_α asymptotic theory transfers to fixed CFL with O(α) error, and the dependence on the unpublished SRD correction [14].

axioms (5)

standard math Rouché's theorem, contraction mapping, Stewart's invariant subspace theorem, Gershgorin, and Schur complement arguments are valid for the stated periodic upwind and block operators.
Invoked in Sections 2, 4, and 6 without proof; standard background.
domain assumption The linearized cut-cell update operator has the exact form (2)/(15) with finite stencil and bounded flux coefficients; regular rows have O(1) coefficients.
Defines the scope of Theorem 8.1; not valid for slope reconstruction dividing by αh except through the p=2 case in Theorem 7.1.
domain assumption Merge-left SRD correction is exactly D=Δ(α e_{c−1}−e_c) with Δ=(λp+(1−λ_α)r)/(1+α), taken from [14].
Used in Theorem 3.1 and Proposition 9.1; depends on the author's submitted paper, not independently verified in this manuscript.
ad hoc to paper For the stability criterion, α→0 with λ_α=λ/α fixed (so λ→0) captures the practical fixed-λ regime with O(α) error.
Theorem 9.2 is proven in the fixed-λ_α limit; Remark 9.4 asserts transfer to fixed λ without a complete proof. This is the weakest modeling assumption.
domain assumption Multi-cut-cell results assume non-adjacent cut cells with d_min≥2 and m N α^2 < 1/2, m^{3/2} λ α < 1/2.
Used in Proposition 4.1 and Theorem 4.2; adjacent cut cells are explicitly left open.

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read the original abstract

Cut-cell instability in explicit finite-volume methods arises when an embedded boundary clips a background cell to a small volume fraction~$\alpha$, forcing the local CFL number to $\lambda_\alpha = \lambda/\alpha \gg 1$. This paper characterizes the spectral structure of this instability and derives a minimum SRD blending parameter per cut cell computable from local mesh geometry. The update operator has exactly $m$ unstable eigenvalues, one per cut cell, with dominant eigenvalue $\mu_u = 1 - \lambda/\alpha + O(1)$ and eigenvector $v_u = e_c + O(\alpha)$. The growth rate is governed by the defect matrix $\Gamma = (\Delta t/h)(a_{c+1/2,c} - a_{c-1/2,c})$, yielding the criterion \[ s_{\mathrm{stab}} = \frac{\lambda_\alpha - 2}{\lambda_\alpha - 1} + O(\alpha), \] which is evaluated from $\alpha$ and $\lambda$ at each cut cell at mesh generation time with no time-stepping and no global matrix assembly. The criterion prevents the class of crash that would otherwise require rerunning a production simulation at significant cost. For merge-left state redistribution, $|\cos_\alpha(D,v_u)| = 1 - O(\alpha)$, proving SRD acts in the unstable eigendirection. We also characterize which linearized explicit finite-volume operators on scalar periodic 1D cut-cell meshes admit this structure, covering schemes from first-order Godunov to high-order MUSCL with bounded flux coefficients, and prove $\|P_{\cU_\alpha} - P_\cC\| = O(\alpha)$ for $m$ non-adjacent cut cells.

Figures

Figures reproduced from arXiv: 2607.28808 by Justo E. Karell.

**Figure 1.** Figure 1: Eigenvalue formula error e1(N) = ||µu| − (λ/α − 1)| versus N for four values of α. The error decreases exponentially, consistent with the O(α N−2 ) error term in Theorem 2.2. Experiment 2: Eigenvector localization. We compute e2(α) = ∥vu − ec∥2 across α ∈ [0.02, 0.18]. The log-log slope averages 1.13, confirming the O(α) rate predicted by Theorem 2.2. The slight super-linearity reflects higher-order terms … view at source ↗

**Figure 2.** Figure 2: Localization error e2(α) = ∥vu − ec∥2 versus α on a log-log scale (N = 150, λ = 0.4). Dashed line: O(α) reference. Mean log-log slope 1.13, confirming Theorem 2.2. Experiment 3: Instability threshold. We compute ρ(A) as λα sweeps from 0.3 to 10 using 300 mesh points. The spectral radius crosses 1 exactly at λα = 2 and follows the theoretical prediction ρ(A) = λα − 1 for λα > 2, confirming Corollary 2.1. Ex… view at source ↗

**Figure 3.** Figure 3: Spectral radius ρ(A) versus local CFL number λα (N = 150, λ = 0.4, 300 points). Dashed line: theoretical prediction max(1, λα − 1). The threshold is sharp at λα = 2, confirming Corollary 2.1. 10 1 10 1 e 4( ) e4( ) = 1 |cos (D, vu)| O( ) [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

**Figure 4.** Figure 4: SRD alignment gap e4(α) = 1 − | cosα(D, vu)| versus α on a log-log scale. Dashed line: O(α) reference. Mean slope 1.03, confirming Theorem 3.1. Experiment 6: Multi-cut-cell projector convergence. For m = 1, 2, 3 non-adjacent cut cells with dmin ≥ 2, we compute the orthogonal projector onto the m dominant eigenvectors and measure ∥PUα − PC∥ across α ∈ [0.02, 0.18]. The log-log slopes average 1.11 for all th… view at source ↗

**Figure 5.** Figure 5: Left: Spectral radius ρ(A(s)) versus blending parameter s for four values of λα. Filled circles mark sstab where ρ = 1, confirming Theorem 9.2. Right: Stability criterion error e5(α) versus α for fixed λα ∈ {3, 5, 10}. Dashed line: O(α) reference. 10 1 10 1 P P m = 1 m = 2 m = 3 O( ) [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗

**Figure 6.** Figure 6: Projector convergence ∥PUα − PC∥ versus α for m = 1, 2, 3 non-adjacent cut cells (dmin ≥ 2). Dashed line: O(α) reference. All three curves are parallel, confirming Theorem 4.2. error e2(α) = ∥vu − ec∥2 converges at mean log-log slope 1.13, consistent with the O(α) rate of Theorem 5.1. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

**Figure 7.** Figure 7: Left: Solution norm ∥U n∥2 versus time step n (α = 0.1, λα = 4). Unstabilized (s = 0): grows as 3n . Stabilized (s = sstab = 2/3): remains bounded. Right: Solution profile |U 20(x)| on a log scale near the cut cell c. The instability is localized at c with amplitude 108 ; the stabilized solution remains O(1) everywhere. 10 1 10 15 10 14 e1( ) e1( ) = || u| (L/ 1)| 10 1 10 1 e2( ) e2( ) = vu ec 2 O( ) [PIT… view at source ↗

**Figure 8.** Figure 8: Left: Eigenvalue error e1(α) = ||µu|−(L/α−1)| on a 20×20 periodic grid with (λ1, λ2) = (0.25, 0.15). The error is at machine precision, confirming Theorem 5.1. Right: Localization error e2(α) = ∥vu − ec∥2 in 2D. Dashed line: O(α) reference. Mean slope 1.13. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗

**Figure 9.** Figure 9: Left: Scatter plot of |µu| (from eig on the full matrix) versus Γ/α − 1 (from local face geometry) for 14 cut-cell positions with varying wave speed. Points lie on the identity line. Right: Error e8(c) = ||µu| − Γ/α + 1| at each position, at machine precision (< 10−13), confirming Theorem 8.1. 11 Conclusion For the one-cut-cell one-dimensional upwind operator, cut-cell instability is generated by a localiz… view at source ↗

discussion (0)

Reference graph

Works this paper leans on

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Gustafsson, H.-O

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Berger and A

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Berger and A

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