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REVIEW 4 major objections 4 minor 33 references

Linear and uniform in time bound for the binary branching model with Moran type interactions

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The normalized empirical measure of the $N_{\min}$–$N_{\max}$ BBMMI particle system converges to the normalized Feynman–Kac semigroup with an error that is at most linear in time, and uniformly in time under exponential contraction.

desk verdict The Section 3 bound is a genuine improvement over [8] and the counter-examples are instructive, but Theorem 4 rests on an unproved coupling and the one-step estimate is imported with a missing modification. read the letter →

arxiv 2501.18208 v1 pith:QZ7KTBK2 submitted 2025-01-30 math.PR

A M G Cox , E Horton , D Villemonais (IRMA , IUF) This is my paper · ORCID
classification math.PR MSC 82C2282C8065C0560J2592D2560J80
keywords binarybranchingmodelwithMoraninteractionsNmin-NmaxFeynman-Kacsemigroupmany-to-oneformulaL2approximationerrorquasi-stationarydistributionBrownianmotionkilledatboundary
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This note strengthens the error bound for the binary branching model with Moran-type interactions (BBMMI), an interacting particle system whose population size is kept between $N_{\min}$ and $N_{\max}$ by resampling and selection events. Its main theorem replaces the exponential-in-$T$ error bound of the earlier construction with a bound that is linear in $T$, expressed as a sum of contraction coefficients $\alpha_t(f)$ weighted by a term involving the particle-size process. The practical stake is that the particle system can approximate Feynman–Kac semigroups—used for conditioned and killed Markov processes—over long time horizons without the error exploding. When the semigroup contracts exponentially to a quasi-stationary law and $h$ is bounded away from zero, the bound becomes uniform in time with the optimal $C_f/\sqrt{N_{\min}}$ rate, and the paper proves this rate for Brownian motion with drift killed on a bounded $C^2$ domain.

What carries the argument

The proof's engine is the pair $(h,\alpha_t)$: $h(x)$ records how much mass a single trajectory starting at $x$ can lose relative to the best possible starting point over one unit of time, and $\alpha_t(f)$ records how far the normalized semigroup action on $f$ is from a chosen probability measure $\nu_t$. Starting from the one-step estimate inherited from [8], the argument applies it to $Q_{T-s-1}f$ at each integer time $s$ and adds the $T$ pieces with Minkowski's inequality and the Markov property. For the Brownian domain result, the additional mechanism is a stochastic domination: the sum of the selected particles' distances to the boundary is dominated by the sum of $N_{\min}$ independent jumping reflected Brownian motions on $[0,a]$, whose inverse-sum expectation is $O(1/N_{\min})$; the paper notes the construction follows [31] and leaves its details to the reader.

What would settle it

Simulate the $N_{\min}{-}N_{\max}$ model for Brownian motion with drift in a bounded $C^2$ domain with small $N_{\min}$, and test whether the summed distance-to-boundary of the $N_{\min}$ tracked particles is stochastically dominated by the sum of independent jumping reflected Brownian motions with the stated jump-to-zero rate; a violation would break the coupling and with it the proof of Theorem 4.

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Extended reading notes

Core claim

Theorem 1 states that under Assumptions 1, 2 and 3 there is $C>0$ such that for all $T\ge 1$ and bounded measurable $f$, $$\left\|\frac{\hat m_0 Q_T f}{\hat m_0 Q_T 1_E} - \hat m_T(f)\right\|_2 \le C \sum_{s=0}^{T-1} \alpha_{T-s-1}(f)\, \mathbb E\!\left[\frac{1}{\sqrt{N_s}\,\hat m_s(h)}\right],$$ where $h(x)=\inf_{t\ge1}\delta_x Q_t 1_E / \|Q_{t-1}1_E\|_\infty$ and $\alpha_t(f)=\sup_x|\delta_x Q_t f/(\delta_x Q_t 1_E)-\nu_t(f)|$. This replaces the exponential-in-$T$ factor from the earlier BBMMI paper. Under the extra conditions that $h$ is bounded away from zero and $\sum_t \alpha_t(f)<\infty$, the right-hand side is bounded uniformly in time by $C_f/\sqrt{N_{\min}}$; Example 4 shows some such condition is needed, since the bound can fail when the semigroup does not converge uniformly.

Load-bearing premise

For the Brownian-motion application, the proof relies on a coupling between the $N_{\min}{-}N_{\max}$ process and independent jumping reflected Brownian motions such that the sum of boundary distances dominates their sum; the paper says the construction follows [31] and leaves it to the reader, so if that coupling fails, the uniform $1/\sqrt{N_{\min}}$ bound for this application is not supported.

Editorial extensions

If this is right

  • Under Assumptions 1–3, the $L^2$ error grows at most linearly in $T$, removing the exponential factor of the earlier BBMMI bound.
  • If $h$ is bounded away from zero and the coefficients $\alpha_t(f)$ are summable, the error is at most $C_f/\sqrt{N_{\min}}$ uniformly in $T$.
  • For Brownian motion with drift killed on a bounded $C^2$ domain, the bound is $C\sqrt{N_{\max}}/N_{\min}\|f\|_\infty + \mathbb E[C/(\sqrt{N_0}\,\hat m_0(\rho_D))]\|f\|_\infty$, giving the $1/\sqrt{N_{\min}}$ rate.
  • In the Moran case $N_{\min}=N_{\max}$, uniform-in-time convergence follows without the generator and carré-du-champs regularity conditions used in earlier Moran-model results.
  • Example 4 shows the uniform bound genuinely needs the semigroup's normalized action to converge; without it, even $h>0$ does not save the estimate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the omitted boundary-distance coupling can be constructed rigorously, the same strategy should transfer to other killed diffusions whose distance to the boundary is controlled by a one-dimensional reflected process, potentially relaxing the $C^2$ boundary assumption.
  • The reference measures $\nu_t$ are free parameters; choosing them time-dependent appears to offer a route to linear-in-$T$ bounds in time-inhomogeneous settings even when exponential contraction is unavailable.
  • A direct numerical check is to fix $N_{\min}$ and increase $T$: the error should plateau exactly when $\sum_t \alpha_t(f)$ converges, giving a practical diagnostic for how long the particle system can be trusted.
  • The inverse-sum lemma suggests the $O(1/N_{\min})$ rate is tied to the reflected Brownian motions' bounded drift; replacing the drift with an unbounded one might degrade the rate, so the bounded-drift assumption is likely essential.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies the binary branching model with Moran type interactions (BBMMI), specifically the Nmin-Nmax process introduced in the authors' earlier paper [8]. Its main claim is an L2 error bound for the difference between the normalized empirical measure mhat_T and the normalized Feynman-Kac semigroup m0-hat(Q_T)/m0-hat(Q_T 1_E), stated as Theorem 1: under Assumptions 1-3 the error is at most C times the sum of alpha_{T-s-1}(f) E[1/(sqrt{N_s} mhat_s(h))]. In Remark 3 this is converted into a linear-in-T bound and, when alpha is summable, a uniform-in-time O(1/sqrt{Nmin}) bound. Section 4 specializes to Brownian motion with drift killed at the boundary of a C^2 domain and claims the improved estimate C sqrt{Nmax}/Nmin plus an initial-condition term. The paper also contains two counterexamples showing that linear or uniform bounds can fail without the stated assumptions.

Significance. If the proofs were complete, the result would be a genuine improvement over the exponential-in-T bound of [8] and would extend uniform-in-time particle approximation bounds for Moran-type and Fleming-Viot-type particle systems, including an optimal O(1/sqrt{Nmin}) rate in a Brownian setting. The paper is clearly written and gives useful structural decomposition of the error into a semigroup stability term alpha and a particle-sampling term. It also provides explicit counterexamples that delineate the necessity of the hypotheses. However, the central proof currently rests on several substantial unproved ingredients: the one-step inequality (6) is said to follow by a modification of a proof in [8], and the coupling in Section 4 is described but its key domination, independence, and regularity properties are left to the reader. These omissions are load-bearing, so the significance is conditional until those details are supplied.

major comments (4)
  1. [Section 3, inequality (6)] The proof of Theorem 1 begins with inequality (6), which is introduced with the sentence 'This is obtained via a modification of the end of the proof of Theorem 2.6 in [8].' No modification is shown. Since every later bound in Theorem 1 and Remark 3 is built on (6), this is a load-bearing step and must be proved, or at least quoted with a precise derivation that the reader can verify. A reference to an unpublished 'modification' is not sufficient for a journal proof.
  2. [Section 4, construction of the coupling] The proof of Theorem 4 hinges on a coupling between the Nmin-Nmax particle system and independent jumping reflected Brownian motions R^i on [0,a] that satisfies sum_i rho_D(X^i_t) >= sum_{i=1}^{Nmin} R^i_t for all t. The text states 'The construction of such a process follows similar ideas to those presented in [31] and so we leave the details of the construction to the reader' and later asserts that the R^i are independent 'the proof is very similar to the one developed in [31]'. This is not an acceptable proof for the main new estimate: the pathwise domination is used to pass from (13) to the C sqrt{Nmax}/Nmin bound, and the independence is used in Lemma 5. The transition rules are also incompletely specified; in particular, in the hard-killing/no-resampling case the paper says a reflected Brownian motion 'is associated to a new particle, not already associated to a Brownian motion', although in that case no new particle is created. The coupling must be constructed in detail and the independence property proved.
  3. [Section 4, Lemma 5] Lemma 5 assumes without proof that the random variables R^i_1 have a bounded density f_R with respect to Lebesgue measure on [0,a] and that they are independent. The product-form identity E[1/sum_i R^i_1] = integral_0^infinity L(t)^{Nmin} dt requires exactly this independence, and the subsequent bound requires the boundedness and regularity of the density. Neither property follows from the informal description of the R^i process, especially after repeated jumps to 0 and reassociation at selection and resampling events. A proof or a precise citation for these properties must be provided.
  4. [Section 4, verification of Assumption 3] The verification of Assumption 3 in Section 4 also depends on the unproved coupling: the paper argues that if the event time tau_n had a finite limit, then the distance to the boundary of the particle system would accumulate at 0, and hence the set of R^i would accumulate at 0, 'which is not possible (by independence of the processes R^i)'. Since the independence of the R^i is not established, this argument is currently unsupported. The same issue affects the claimed validity of Assumption 3 and hence the applicability of Theorem 1 in the Brownian setting.
minor comments (4)
  1. [Section 3, proof of Theorem 1, equation (8)] In equation (8), the denominator is written with N0 and m0, but by the Markov property at time s the displayed expression should involve N_s and m_s; otherwise the final expectation E[1/(sqrt{N_s} mhat_s(h))] does not follow. This appears to be a typographical slip, but it should be corrected because the displayed inequality is otherwise not the one being iterated.
  2. [Section 3, equation (4)] In the definition of alpha_t(f), the expression 'for all >= 0' is missing the variable t; it should read 'for all t >= 0'.
  3. [Section 4, equation (14)] The notation dR^i_t = dB^i_t - ||r||_infinity dt + dL^{i,0}_t - dL^{i,a}_t is used before the local time processes L^{i,0} and L^{i,a} are formally introduced; adding a sentence defining these local times immediately after the display would improve readability.
  4. [Throughout] There are several minor typographical and formatting issues, such as 'Itoˆ's formula' and the reference to 'Annales de l’Institut Henri Poincare, Probabilites et Statistiques' in the bibliography; these should be cleaned up in revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: Theorem 1's bound is an inequality derived from prior published theorems, and the Section 4 coupling gap is an omitted proof, not a self-referential derivation.

full rationale

The central result, Theorem 1, rests on the first-step L2 estimate (6), which the paper says is 'obtained via a modification of the end of the proof of Theorem 2.6 in [8]'. This is a self-citation, but [8] is an independent published theorem with its own proof, and the target bound (5) is not identical to that input: the paper adds a Minkowski/telescoping argument and substitutes f by f−νt(f), using h and αt only through inequalities. The definitions of h and αt make the right-hand side contract, but this is bookkeeping, not a circular identity. Remark 3 and Examples 1–4 combine Theorem 1 with quasi-stationary convergence results from [3,5]; they are applications, not restatements. The only notable weakness is in Section 4, where the coupling to independent reflected Brownian motions is asserted rather than proved: the paper says 'we leave the details of the construction to the reader' and 'the Ri are independent (the proof is very similar to the one developed in [31] and we leave the details to the reader)'. This is a load-bearing proof gap for Theorem 4 and a correctness risk, but it is not a circular reduction: the asserted coupling, if proved, would supply content that is not already contained in the assumptions. No parameter is fitted and no prediction is defined in terms of the quantity it purports to bound, so there is no significant circularity.

Assumptions & free parameters 0 free parameters · 10 assumptions · 1 invented entities

The central bounds rest on domain assumptions (bounded rates, no atoms, well-defined event times), on a one-step estimate imported from the authors' prior paper [8], and on an unproved coupling of the distance-to-boundary with jumping reflected Brownian motions. The latter is the most significant unpublished input, since its details and the independence of the R_i are left to the reader.

assumptions (10)
  • domain assumption Assumption 1: branching rate b is uniformly bounded.
    Invoked in the proof of Theorem 1 to pass from sup_{t∈[0,1]} ||Q_{t+T-1}1E||∞ to C||Q_{T-1}1E||∞.
  • domain assumption Assumption 2: for every t, P_x(τ∂ = t) = 0 and inf_x P_x(τ∂ > t) > 0.
    Stated in Section 2 and used to ensure well-defined event times and uniform survival control.
  • domain assumption Assumption 3: event times τ_n → +∞ a.s.
    Formally needs proof; in Section 4 it is derived from the unproved coupling, so it is not independently established in the manuscript.
  • ad hoc to paper One-step inequality (6): obtained 'via a modification of the end of the proof of Theorem 2.6 in [8]'.
    The main ingredient of Theorem 1; the modification is not shown in this paper and relies on the authors' prior paper [8].
  • ad hoc to paper Domination coupling of Section 4: existence of independent jumping reflected Brownian motions R_i on [0,a] with Σ ρD(X_i) ≥ Σ R_i for all t.
    The construction is explicitly left to the reader; it is load-bearing for Theorem 4, Lemma 5, and verification of Assumption 3.
  • ad hoc to paper The R_i^1 have a bounded density and are independent.
    Asserted in the proof of Lemma 5 without proof; independence is non-trivial since the R_i are coupled to a common particle system.
  • domain assumption Exponential convergence criteria for QSD: conditions (A1)-(A2) of [5] imply h ≥ c>0 and α_t(f) ≤ C e^{-αt} ||f||∞.
    External result from [5], invoked in Example 1 to obtain the uniform bound.
  • domain assumption Wasserstein convergence result of [4] gives (11) and a lower bound on h.
    External result from [4], invoked in Example 2.
  • domain assumption Boundary-distance lower bound δxQt1D / sup_y δyQt−11D ≥ c0 ρD(x) from [3].
    External result from [3], invoked in Theorem 4 to control E[1/(√Ns mhat_s(h))].
  • domain assumption Coupling estimates from [31] for Fleming-Viot-type systems.
    The paper's Section 4 coupling is an adaptation of [31]; the adaptation is not detailed.
invented entities (1)
  • Jumping reflected Brownian motions (R_i)_{i≤Nmin} on [0,a]
    purpose: Auxiliary stochastic processes used to dominate the sum of distances to the boundary in the Nmin-Nmax model, yielding the L2 bound in Theorem 4.
    Introduced in Section 4 specifically for the proof. The construction is not fully specified and its existence is not independently verified; details are left to the reader.

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Pith. "Pith review of Linear and uniform in time bound for the binary branching model with Moran type interactions." pith.science (2026). https://pith.science/paper/QZ7KTBK2

@misc{pith2026250118208,
  author       = {Pith},
  title        = {Pith review of: Linear and uniform in time bound for the binary branching model with Moran type interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QZ7KTBK2}},
  note         = {Machine review of arXiv:2501.18208}
}
read the original abstract

In this note, we recall the definition of the binary branching model with Moran type interactions (BBMMI) introduced in [8]. In this interacting particle system, particles evolve, reproduce and die independently and, with a probability that may depend on the configuration of the whole system, the death of a particle may trigger the reproduction of another particle, while a branching event may trigger the death of another particle. We recall its relation to the Feynman-Kac semigroup of the underlying Markov evolution and improve on the L 2 distance between their normalisations proved in [8], when additional regularity is assumed on the process.

Figures

Figures reproduced from arXiv: 2501.18208 by the authors.

Figure 1
Figure 1. A schematic representation of the Nmin−Nmax dynamic with Nmin = 3 and Nmax = 4. The process starts with N = 4 particles at time 0. The first event is a killing, so that the number of particles goes down to N = 3 = Nmin. The next event is a killing, so that the number of particles goes down to 2 < Nmin, which triggers a resampling event: one of the 2 remaining particles (chosen uniformly at random) is duplicated, and… view at source ↗

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Works this paper leans on

33 extracted references · 32 canonical work pages

  1. [8]

    A. M. Cox, E. Horton, and D. Villemonais. Binary branching processes with Moran type interactions. Annales de l’Institut Henri Poincare, Probabilites et Statistiques, 2024

    work page 2024

  2. [31]

    Villemonais

    D. Villemonais. Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift. Electronic Journal of Probability, 16:1663–1692, 2011

    work page 2011

  3. [1]

    J. Bertoin. On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors. Journal of Functional Analysis, 277(11):108270, 2019

    work page 2019

  4. [2]

    Bertoin et al

    J. Bertoin et al. Markovian growth-fragmentation processes. Bernoulli, 23(2):1082–1101, 2017

    work page 2017

  5. [3]

    Champagnat, K

    N. Champagnat, K. A. Coulibaly-Pasquier, and D. Villemonais. Criteria for Exponential Convergence to Quasi-Stationary Distributions and Applications to Multi-Dimensional Diffusions, pages 165–182. Springer International Publishing, 2018

    work page 2018

  6. [4]

    Champagnat, E

    N. Champagnat, E. Strickler, and D. Villemonais. Uniform wasserstein convergence of penalized markov processes. arXiv preprint arXiv:2306.16051, 2023

    work page arXiv 2023

  7. [5]

    Champagnat and D

    N. Champagnat and D. Villemonais. Exponential convergence to quasi-stationary distri- bution and Q-process. Probab. Theory Related Fields, 164(1):243–283, 2016

    work page 2016

  8. [6]

    Uniform in time propagation of chaos for a Moran model

    B. Cloez and J. Corujo. Uniform in time propagation of chaos for a Moran model. arXiv e-prints, page arXiv:2107.10794, July 2021. 13

    work page Pith review arXiv 2021

Show all 33 references
  1. [7]

    A. M. Cox, S. C. Harris, E. L. Horton, and A. E. Kyprianou. Multi-species neutron transport equation. Journal of Statistical Physics, 176(2):425–455, 2019

    2019

  2. [9]

    A. M. G. Cox, E. Horton, and D. Villemonais. Binary branching processes with Moran type interactions: a formal construction of the particle system. Preprint, 2022

    2022

  3. [10]

    D. A. Dawson. Measure-valued Markov processes. In ´Ecole d’´Et´ ede Probabilit´ esde Saint-Flour XXI—1991, volume 1541 of Lecture Notes in Math., pages 1–260. Springer, Berlin, 1993

    1991

  4. [11]

    Del Moral

    P. Del Moral. Feynman-Kac formulae. Probability and its Applications (New York). Springer-Verlag, New York, 2004. Genealogical and interacting particle systems with applications

    2004

  5. [12]

    Del Moral

    P. Del Moral. Mean field simulation for Monte Carlo integration, volume 126 of Monographs on Statistics and Applied Probability. CRC Press, Boca Raton, FL, 2013

    2013

  6. [13]

    Del Moral and L

    P. Del Moral and L. Miclo. Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering. Springer, 2000

    2000

  7. [14]

    Del Moral and L

    P. Del Moral and L. Miclo. A Moran particle system approximation of Feynman–Kac formulae. Stochastic processes and their applications, 86(2):193–216, 2000

    2000

  8. [15]

    Del Moral and L

    P. Del Moral and L. Miclo. On the stability of nonlinear Feynman-Kac semigroups. Ann. Fac. Sci. Toulouse Math. (6), 11(2):135–175, 2002

    2002

  9. [16]

    Del Moral and D

    P. Del Moral and D. Villemonais. Exponential mixing properties for time inhomogeneous diffusion processes with killing. Bernoulli, 2018

    2018

  10. [17]

    Etheridge

    A. Etheridge. Some Mathematical Models from Population Genetics: ´Ecole D’´Et´ ede Probabilit´ esde Saint-Flour XXXIX-2009, volume 2012. Springer Science & Business Media, 2011

    2009

  11. [18]

    S. C. Harris and M. I. Roberts. The many-to-few lemma and multiple spines. InAnnales de l’Institut Henri Poincar´ e,Probabilit´ eset Statistiques, volume 53, pages 226–242. Institut Henri Poincar´ e, 2017

    2017

  12. [19]

    Ikeda, M

    N. Ikeda, M. Nagasawa, S. Watanabe, et al. Branching Markov processes I. Journal of Mathematics of Kyoto University, 8(2):233–278, 1968

    1968

  13. [20]

    Ikeda, M

    N. Ikeda, M. Nagasawa, S. Watanabe, et al. Branching Markov processes II. Journal of Mathematics of Kyoto University, 8(3):365–410, 1968

    1968

  14. [21]

    Ikeda, M

    N. Ikeda, M. Nagasawa, S. Watanabe, et al. Branching Markov processes III. Journal of Mathematics of Kyoto University, 9(1):95–160, 1969

    1969

  15. [22]

    P. Jagers. General branching processes as Markov fields. Stochastic Processes and their Applications, 32(2):183–212, 1989

    1989

  16. [23]

    P. Jagers. Branching processes as population dynamics. Bernoulli, 1(1-2):191–200, 1995. 14

    1995

  17. [24]

    Journel and P

    L. Journel and P. Monmarch´ e. Uniform convergence of the Fleming-Viot process in a hard killing metastable case. arXiv preprint arXiv:2207.02030, 2022

    2022 arXiv

  18. [25]

    Journel and M

    L. Journel and M. Rousset. The particle approximation of quasi-stationary distributions. part˜ i: concentration bounds in the uniform case. arXiv preprint arXiv:2412.15820, 2024

    2024 arXiv

  19. [26]

    A. Lambert. The branching process with logistic growth. Ann. Appl. Probab., 15(2):1506– 1535, 2005

    2005

  20. [27]

    J. M. Marshall. A branching process for the early spread of a transposable element in a diploid population. Journal of Mathematical Biology, 57(6):811, 2008

    2008

  21. [28]

    P. A. P. Moran. Random processes in genetics. In Mathematical proceedings of the cambridge philosophical society, volume 54, pages 60–71. Cambridge University Press, 1958

    1958

  22. [29]

    M. Rousset. On the control of an interacting particle estimation of Schr¨ odinger ground states. SIAM J. Math. Anal., 38(3):824–844 (electronic), 2006

    2006

  23. [30]

    B. A. Sevastyanov and A. M. Zubkov. Controlled branching processes. Theory of Probability & Its Applications, 19(1):14–24, 1974

    1974

  24. [32]

    Villemonais

    D. Villemonais. General approximation method for the distribution of Markov processes conditioned not to be killed. ESAIM Probab. Stat., 18:441–467, 2014

    2014

  25. [33]

    N. M. Yanev. Branching processes in cell proliferation kinetics. In Workshop on Branching Processes and Their Applications, pages 159–178. Springer, 2010. 15

    2010

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