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M&C 2023 - The International Conference on Mathematics and Computational Methods Applied

to Nuclear Science and Engineering ·Niagara Falls, Ontario, Canada ·August 13 – 17, 2023

Beyond Renewal Approximations: A 1D Point Process Approach to

Linear Transport in Stochastic Media

Eugene d’Eon1

1NVIDIA

2788 San Tomas Expressway, Santa Clara, CA 95050

edeon@nvidia.com

ABSTRACT

We present a novel approximate model for monoenergetic linear transport in stochastic media

that permits correlations between successive free-path lengths of particle collisions. Our model

utilizes collision times determined by a 1D point process, chosen such that perfectly forward

scattering along a transect precisely matches ensemble-averaged statistics. In contrast, previous

renewal-based non-classical transport formulations only guarantee the accuracy of the ﬁrst col-

lision time and assume subsequent collision times are independently and identically distributed.

By accommodating non-renewal collision times, our model can account for step correlations that

emerge in most variability models, including cross-section ﬂuctuations driven by Gaussian pro-

cesses, transformed Gaussian processes, and the majority of discrete mixture models. We compare

multiple scattering predictions from our model to new two-dimensional benchmark simulations

featuring transformed Gaussian ﬂuctuations, demonstrating enhanced accuracy compared to re-

newal approximations.

KEYWORDS: Non-Classical, Stochastic, Point Process, Transect, Step Correlations

1. INTRODUCTION

Particle transport in stochastic media presents signiﬁcant challenges in various domains, including neutron

transport, heat transfer, remote sensing, and tissue optics. Even for homogeneous, monoenergetic, time-

independent problems, once the total macroscopic cross section Σt(x)is subject to randomness, the only

exact method for estimating ensemble-averaged observables, given the presence of scattering, is an expen-

sive double-Monte-Carlo simulation that requires averaging classical simulations across numerous system

realizations [1]. Developing non-classical equations to directly approximate transport in stochastic media

without generating ensemble realizations is, therefore, highly sought after.

One such approach is a renewal transport process, whose integral equation is that of random ﬂights [2,3],

shown to be equivalent to a generalized linear Boltzmann equation (GLBE) [4]. This approach was inspired

by the observation that ensemble-averaged extinction in stochastic systems is often non-exponential [5]. By

replacing the Poisson process, which produces exponentially-distributed intercollision lengths in classical

theory, with a broader renewal process [6], this non-exponential behaviour can be directly exhibited in

a transport formalism. However, in a system with scattering, the renewal assumption demands that all

subsequent free-path lengths are independent, thus disallowing correlations in successive lengths between

collisions (“step correlations”). The accuracy of this renewal assumption for modeling general stochastic

media is currently not well understood.

In this paper, we introduce a novel model for monoenergetic linear transport in stochastic media that al-

lows arbitrary step correlations, offering insights into when the renewal approximation is appropriate and

providing a more general model for when it is not. Our approach utilizes exact collision-time statistics de-

rived from the case of perfectly-forward (singular) scattering, where transport is restricted to unidirectional

ﬂow along a medium transect, yielding collision times determined precisely by a 1D point process. By

leveraging known results from time series analysis, we can efﬁciently simulate collisions along transects

Eugene d’Eon

Figure 1: In a realization with ﬂuctuating Σ(x)(greyscale), scattering events (white) along a

transect (red) are typically correlated (clumpy) and given by a 1D point process. We apply these

transect statistics along histories with general scattering (blue).

and rigorously identify which classes of stochastic media are truly renewal. To approximately treat non-

forward scattering, we propose applying these transect collision times along a general history, thus creating

an efﬁcient autoregressive transport model that directly simulates ensemble-averaged behavior by correlat-

ing future collisions with prior times but not locations along a history. We demonstrate that our model can

outperform a renewal transport process over a wide range benchmarks.

Previous discussions on step correlations in stochastic media have often attributed them to complex correla-

tions emerging when particles scatter backward into prior scenery [7]. While reversal can indeed cause step

correlations, we clearly demonstrate that nearly all forms of stochastic media, including those with Gaussian

or transformed-Gaussian density ﬂuctuations, exhibit step correlations even in the case of perfectly forward

scattering (Figure 1), and therefore step correlations can be expected to arise in nearly all systems, including

those with highly forward scattering. To assess the impact of these correlations, we present new benchmark

simulations for monoenergetic absorption and scattering in isotropic, stationary two-dimensional (Flatland)

stochastic media with transformed-Gaussian density ﬂuctuations. Comparisons between these benchmarks

and our new model reveal a reduction in error (relative to a renewal transport process) of up to an order of

magnitude.

2. MODEL DEFINITION AND MOTIVATION

In this section, we introduce our novel model, which directly extends the random-ﬂight interpretation of

classical linear transport [2,8]. We review essential results from point process literature and explore the role

of point processes in determining collision times along a particle history. We then discuss the time-series

analysis of collisions along a transect and the relationship to Cox processes, which forms the foundation of

our model.

Scope We will limit the scope of the present work to consider only time-independent, monoenergetic

linear transport in an isotropic, piecewise-homogeneous medium with deterministic scattering kernel and

deterministic single-scattering albedo c. We assume that the total macroscopic cross section Σt(x)at po-

sition xis given by a stationary random ﬁeld. Extension of our model to time-dependent problems is

straightforward. Multi-group, inhomogeneous systems, anisotropic media and stochastic albedo care not

presently supported by our model. Despite these restrictions, we feel that our work unveils important insight

regarding the role of step correlations in stochastic systems and the limitations of the GLBE.

2.1. Random Flights and Point Processes

We will determine our model by deﬁning the stochastic process governing a single particle, born at time t=

0. Speciﬁcally, we will form a generalized random ﬂight by specifying a sequence of random collision times

tialong each history. Our goal is to choose a ﬂight whose expectation directly approximates ensemble-

averaged observables in the stochastic system. The collision times ticonstitute a 1D point process N(t):

a non-negative integer random variable at each time tthat gives the number of collisions up to that time

[6]. Once N(t)is speciﬁed, the transport is completely determined, since the scattering kernel and survival

probability care assumed to be deterministic and independent of ti. Given N(t), a Monte Carlo estimator

2

Beyond Renewal Approximations: A 1D Point Process Approach to Linear Transport in Stochastic Media

for our model is readily derived by sampling collision times tifrom N(t), constructing a history by sampling

the scattering kernel at each collision, utilizing implicit capture for absorption, and terminating with roulette

or upon escape/boundary interactions.

The Poisson Process of Classical Transport To gain some familiarity with the role of point processes

in classical transport, let us recall that in a deterministic homogeneous medium, the collision times along a

history are given by a Poisson point process (PPP) with a constant rate λ(t)[9]. The PPP is a memoryless

point process, where the number of points within a time interval [ta, tb]follows a Poisson distribution with

a mean of Rtb

taλ(t)dt. Assuming motion occurs at unit speed, the rate of the Poisson process is equal to the

total macroscopic cross section λ(t)=Σt(x(t)) at the particle’s current position x(t). In a homogeneous

system, we observe a constant rate λ(t) = Σtand exponential times between collisions. However, in a

system with inhomogeneous cross section, such as a realization of a stochastic system, the rate Σt(x(t)) of

the point process at time tdepends on the current position x(t)(which, in turn, is a function of both prior

times and directions of the history). Averaging this PPP over the Σt(x)ensemble becomes intractable.

The complexity of this 3D averaging casts heavy doubt on ever ﬁnding an exact random ﬂight approach

to general stochastic media. If, however, we constrain the problem to one dimension by either considering

only absorption, or forcing scattering to be perfectly forward, exact results are possible. We consider each

of these scenarios now, in turn.

Attenuation A number of tractable results are available if we consider only absorption. In this case,

transport is restricted to a straight path up to the ﬁrst collision and the problem is one-dimensional. The

attenuation law T(t), which is simply the probability of ﬁnding no collision in [0, t), is

T(t)≡Pr {N(t) = 0}.(1)

In a deterministic medium, this probability follows from the rate of collisions λ(t) = Σt(x(t)) governing

the PPP, giving the well-known equation for attenuation

T(x) = exp −Zt

0

λ(t′)dt′= exp −Zx

0

Σt(x′)dx′.(2)

This equation illustrates a close relationship between the point process and particle transport literatures.

In fact, the point process community independently developed the concept of delta (Woodcock) tracking,

under the name “thinning algorithm”, for sampling events in an inhomogeneous PPP [10,11].

Similarities between these two studies are also found in the case of stochastic rate λ(t). For stochastic

systems, we denote the mean attenuation law (from an equilibrium/deterministic origin) as

Tu(x)≡ ⟨T(x)⟩(3)

which is the ensemble average of the deterministic result. Finding Tu(x)requires averaging Equation 2

where Σtis stochastic, which makes N(t)a doubly-stochastic Poisson process (or Cox Process) [12], which

was noted by Kostinski [5,13]. The mean attenuation law in Equation 3 is tractable for a number of ﬂuc-

tuation models including Gaussian ﬂuctuations, squared-Gaussian ﬂuctuations and discrete n-ary Markov

mixtures.

The ’u’ label on Tu(x)refers to an unconditional ensemble average over all realizations, and distinguishes

the result from an average that is conditioned on starting from t= 0 at a scattering center [14]. For brevity,

we will consider only deterministic sources, using equilibrium (asynchronous) initialization of the point

process N(t)(see [15] for more details). However, synchronous initialization of N(t)is easily adopted for

emission from spatially correlated scattering centers, which may be appropriate in neutronics [4].

3

Eugene d’Eon

2.2. Transect Statistics

A Cox process provides a precise and comprehensive representation of transport in a purely absorbing

stochastic system, as the absence of scattering reduces the problem to a one-dimensional scenario, allowing

for an equivalent 1D point process. Another instance where we can take advantage of this 1D equivalence

occurs when the scattering kernel is a singular Dirac delta peak in the forward direction, thus restricting

transport to a 1D transect (see, for example, Figure 1). In this case, N(t)is also reduced to a 1D Cox

process, accounting for all collisions along the transect instead of just the ﬁrst one. By eliminating all

angular dependence from N(t), we circumvent the intricacies of correlations arising from prior scenery.

Although perfectly forward scattering essentially nulliﬁes the scattering collisions, reverting the problem

back to one of pure absorption, showcasing these transect statistics is still a crucial requirement for any non-

classical transport model to be considered accurate. For this reason, we will employ these transect statistics

to not only establish a new transport model that embodies them but also to develop new benchmarks for

assessing non-classical formalisms.

We now introduce some necessary results from point process literature. Point processes are completely

determined by their generating functions. Given a stationary point process, we will denote its equilibrium

probability generating function as

ϕN(z;t)≡

∞

X

n=0

znPr {N(t) = n},(4)

which, for any time t, yields the required probabilities for N(t). The attenuation law along a transect can

then be expressed as

Tu(t) = ϕN(0; t),(5)

with higher-order probabilities recovered via

Pr {N(t) = n}=1

n!

∂nϕN(z;t)

∂zn

z=0

.(6)

By giving an exact account of the full point process N(t), the generating function ϕN(z;t)is a highly

useful tool for analyzing the clustering of collisions along a transect in a medium with stochastic cross

section (Figure 1). Remarkably, it turns out that determining ϕN(z;t)is no harder than determining the

attenuation law ϕN(0; t).

For a Cox process, the optical depth along a transect is

τ(t)≡Zt

0

Σt(x)dx, (7)

which is a random variable. The Cox process is then uniquely determined, given that

Pr {N(t) = k}=⟨e−τ(t)(τ(t))k/k!⟩, k ≥0.(8)

The probability generating function can then be written [15, p.219]

ϕN(z;t) = ⟨exp[(z−1)τ(t)]⟩,(9)

which corresponds to the moment generating function of the optical depth at time t, evaluated at z−1.

Since z= 0 in Equation 9 corresponds to the attenuation law of the Cox process, and because (z−1)

simply serves as a constant scaling factor for all optical depths τ(t)regardless of t, it becomes clear that

determining all collision times along the transect is, in fact, no more challenging than determining the

attenuation law itself. Said another way: ϕN(z;t)is just the attenuation law in a system where the random

ﬁeld Σt(x)is scaled by a constant (1 −z).

4

Beyond Renewal Approximations: A 1D Point Process Approach to Linear Transport in Stochastic Media

The Transect Absorption Law We can now deﬁne two new analytic benchmarks for measuring approxi-

mate non-classical models of transport. Firstly, for an absorbing and scattering system, where each collision

scatters perfectly forward with probability cand otherwise absorbs and terminates the random ﬂight, we can

solve for the ensemble-averaged attenuation of unit ﬂux ⟨ψ→(t)⟩along a transect using the generating func-

tion. Given that the attenuation for kcollisions occurring along the segment [0, t]is ck, we have

⟨ψ→(t)⟩=

∞

X

k=0

Pr {N(t) = k}ck=ϕN(c;t),(10)

which is just the generating function for the point process (Equation 4) evaluated at z=c. This result is also

equivalent to the attenuation law of a related purely absorbing system where Σt(x)is scaled by a constant

(1 −c)in order to account for the relative probability of real (absorbing) to total (real + null/scattering)

collisions. When c= 0, the system is purely absorbing and the original attenuation law is recovered, as

expected. Likewise, for c= 1, the system is lossless and ⟨ψ→(t)⟩= 1.

Time of the mth collision Another way that we can benchmark non classical models is using the time

of the mth collision along the transect. Given the equilibrium generating function ϕN(z;t)for a stationary

point process, the probability density fm(t)of the time of the mth collision/arrival along a transect (from

equilibrium t= 0 initialization) is [16, Eq.(25)]

fm(t) = −∂

∂t

m−1

X

j=0

(−1)j

j!

∂j

∂zjϕN(1 −z;t)|z=1.(11)

2.3. Non-Renewal Transport

Because transect statistics are a necessary and testable condition for any accurate theory of non-classical

transport, we can test the renewal assumption of the GLBE under this lens. This leads immediately to

the query: for what class of Σtﬂuctuations are the transect statistics renewal, which is to ask: what Cox

processes are also renewal processes? This was answered rigorously by Kingman [17]: either Σtis a two-

phase medium with one void phase and exponential chord lengths in the non-void phase, or Σtis singular

with respect to Lebesque measure. This immediately excludes renewal transport from being a generally

accurate model in all but an extremely narrow form of Markov binary mixtures or, alternatively, within

media where Σtis described by some unknown set of fractal variability models. In particular, Gaussian

and transformed-Gaussian ﬂuctuations (e.g. Figure 1) are non-renewal and always exhibit clumping of the

collision times. It is known that some Gamma and Weibull renewal processes also have a Cox process

equivalence, but the exact fractal nature of Σtthat produces them is not known [18].

Approximation by a renewal process To illustrate the non-renewal character of Cox processes with

Gaussian ﬂuctuations, we consider the Gauss-Poisson Cox process where Σt(x)is Gaussian with exponen-

tial autocovariance R(|s−t|) = r2e−y|s−t|, which has a known generating function [6, p.183]. We consider

unit mean cross section Σt(x)and keep rsmall to make negative cross sections unlikely. In Figure 2,

we use the generating function to compare the point counts from equilibrium over various time intervals

to those predicted by an equilibrium renewal process. The renewal and Cox processes agree for n= 0

(the attenuation law), but all other collision counts differ due to the lack of step correlation in the renewal

process. We observed similar inaccuracies of renewal approximations for transformed Gaussian processes

(where the Gaussian cross sections are squared or exponentiated to avoid non-negative values), and also

when comparing the collision times of the nth collision along a transect, using Equation 11.

2.4. Scattering

Transect statistics have led to several new analytical benchmarks for testing the accuracy of a given non

classical theory of transport, but only in the contrived case of purely forward scattering. To form a practical

5

Eugene d’Eon

0 2 4 6 8 10 12 14

0.001

0.010

0.100

1

t

Pr[N(t)]=0

0 2 4 6 8 10 12 14

0.001

0.005

0.010

0.050

0.100

t

Pr[N(t)]=1

0 2 4 6 8 10 12 14

0.001

0.005

0.010

0.050

0.100

t

Pr[N(t)]=2

0 2 4 6 8 10 12 14

0.005

0.010

0.050

0.100

t

Pr[N(t)]=3

0 2 4 6 8 10 12 14

10-4

0.001

0.010

0.100

t

Pr[N(t)]=5

0 2 4 6 8 10 12 14

10-4

0.001

0.010

0.100

t

Pr[N(t)]=10

Figure 2: We compare the probabilities for ﬁnding ncollisions in [0, t]for Gaussian Σt(x)with

exponential correlation R(|s−t|)=0.32e−0.1|s−t|using double-MC ground truth (blue dots), a

renewal process approximation (red dots) and analytic ground truth (continuous). While the

attenuation law n= 0 matches for the renewal approximation, all other probabilities differ due to

the lack of step correlations.

and general model of non-classical transport that exhibits transect statistics, we propose to simply apply

them along any history, regardless of the scattering kernel. Intuitively, the clustering of collision times that

arises for systems with long-range spatial correlations will be approximately achieved for a system with

highly forward scattering (Figure 1). We leave any further justiﬁcation for the proposed model to numerical

benchmark comparisons that we provide in section 3.

We do not presently consider an integral transport equation for our model. This would include a cross

section Σ(t1,· · · , tk−1;t)that is a Janossy density [19] for the point process N(t)up to the current time t

subject to the occurrence of k−1prior collisions at times ti. Such an integral equation could be written

down in principle, but it is not immediately clear to us how useful this would be. However, as described

above, a Monte Carlo estimator for our model follows directly from the model’s deﬁnition.

Sampling Transect Collision Times Cox processes can be sampled using a variety of methods [19,20].

For simplicity, in the next section, we sample a single large square tileable auxiliary realization of the

stationary random ﬁeld Σt(x)using Fourier transforms. This happens once for each piecewise homogeneous

element of the system with unique statistics. After this precomputation, traditional Monte Carlo sampling

then follows where collision times for each history are determined using delta tracking along a transect in

the auxiliary domain. So while a particle in the physical system follows a general history, a virtual particle

in the auxiliary domain begins at a random position and direction and follows a straight path in order to

determine collision times in the physical system.

Relationship to Prior Work One interesting property of our model is that it includes a number of previ-

ous transport formalisms as special cases under a common framework:

• When N(t)is chosen to be a PPP, our model describes classical transport in a deterministic medium.

• When N(t)is a mixed-Poisson process (where Σis random, but constant in each realization), our

model corresponds to an approximation known as the independent-column approximation in remote

6

Beyond Renewal Approximations: A 1D Point Process Approach to Linear Transport in Stochastic Media

sensing [21], and is an important benchmark for parametric stochastic media in the limit of inﬁnite

correlation lengths.

• When N(t)is a renewal process, N(t)depends only on the previous ti−1collision time, and the

resulting integro-differential/integral transport equations are the GLBE/random-ﬂight equations, re-

spectively [2,3,4,14,22,23].

• When N(t)is a Markov-renewal process, N(t)depends only on the previous ti−1collision time

and an additional integer state, and corresponds to the chord-length-sampling/Levermore-Pomraning

approximations for n-ary Markov mixtures (when we additionally extend the albedos cjto depend on

state j) [24].

3. TESTING THE MODEL

By construction, the accuracy of our model is only ensured in the limited case of purely forward scatter-

ing. In this limit, scattering collisions are effectively null events and the transport is equivalent to a purely

absorbing one. To test the accuracy of our model for non-forward scattering, we rely on numerical simu-

lations. We constructed a new two-dimensional benchmark for stochastic media with homogeneous mean

density, homogeneous deterministic c, and generalized Henyey-Greenstein [25] scattering parametrized by

the mean cosine −1< g < 1. The benchmark conﬁguration is illustrated in Figure 3 (left): a deterministic

unit monodirectional beam was applied along the normal to the boundary of a source-free disk domain (in

two-dimensional Flatland) and the leakage from the vacuum boundary at azimuth ϕ(regardless of outgoing

direction) was tallied and averaged over the sampled ensemble of disk realizations. We ran a suite of sim-

ulations for Gaussian and transformed-Gaussian Σtﬂuctuations with exponential correlations. In each, we

varied the radius Rof the disk, the correlation width, as well as cand g.

For each benchmark conﬁguration, we compared the double-MC ground truth result to our new model, and

also to a renewal approximation, where the ﬁrst collision is given exactly and the intercollision lengths were

determined in order to form an equilibrium renewal process [14]. For narrow correlation widths, we found

close agreement between all three models, and observed deviations as the correlation widths increased

relative to the mean free path (see Figure 5). The middle and right plots in Figure 3 illustrate selected

examples of the improved accuracy of our model over a renewal approximation for realizations of radius

R= 0.02 where Σtwas based on a transformed Ornstein-Uhlenbeck process with radial correlation e−r10,

where the Gaussian process was squared to create a non-negative Σtﬁeld. These examples show substantial

improvements over the renewal approximation, both in the highly-peaked g= 0.9case, but, remarkably,

also in the more isotropic g= 0.5conﬁguration, where it was less clear that transect statistics should apply.

We noted similar behaviour over a wide matrix of conﬁgurations, with the renewal approximation being

the worst performer overall, and in fact observed our new model consistently outperforming a renewal

approximation even in the case of backscattering with g < 0(Figure 4).

We did not perform benchmarks for Markov binary mixtures because our model reduces to the CLS algo-

rithm in such cases and comparisons to a renewal approximation have already been made in 3D [24], where

it was also noted that the renewal approximation can signiﬁcantly underperform relative to a model that

includes step correlations (i.e. CLS).

4. CONCLUSION

We have presented a novel non-classical transport model based on transect statistics, which provides a uni-

ﬁed framework for various existing transport formalisms. The model has been tested against a range of

benchmark conﬁgurations, demonstrating its accuracy and robustness even in cases of non-forward scatter-

ing. The improvements over the renewal approximation in both highly-peaked and more isotropic conﬁg-

urations highlight the model’s potential for practical applications. Our analysis of transect statistics also

provides a compelling argument against the use of renewal transport in stochastic systems, casting doubt

on the general applicability of the GLBE. Furthermore, the newly established analytical benchmarks for

nth collision time and attenuation pave the way for enhanced evaluation and development of future non-

classical transport methodologies in multiple ﬁelds.

7

Eugene d’Eon

Deterministic Source

ϕ

ϕ

g=0.9

Renewal

Ground Truth

Transect Stats

-3 -2 -1 1 2 3

0.010

0.050

0.100

0.500

1

ϕ

g=0.5

Renewal

Ground Truth

Transect Stats

-3 -2 -1 1 2 3

0.05

0.10

0.20

0.50

Figure 3: Example benchmark results for the emergent scalar ﬂux when a homogeneous disk

domain with stationary ﬂuctuations of Σt(x)is subject to a monodirectional beam at the boundary.

Note how even with a low mean cosine of scattering (g= 0.5), our model still signiﬁcantly

outperforms a renewal approximation.

-3-2-1 0 1 2 3

0.001

0.010

0.100

1

ϕ

g= -0.9

-3-2-1 0 1 2 3

0.005

0.010

0.050

0.100

0.500

1

ϕ

g= -0.7

-3-2-1 0 1 2 3

0.005

0.010

0.050

0.100

0.500

1

ϕ

g= -0.5

-3-2-1 0 1 2 3

0.005

0.010

0.050

0.100

0.500

1

ϕ

g=0

-3-2-1 0 1 2 3

0.01

0.02

0.05

0.10

0.20

0.50

ϕ

g=0.5

-3-2-1 0 1 2 3

0.01

0.05

0.10

0.50

1

ϕ

g=0.7

-3-2-1 0 1 2 3

0.005

0.010

0.050

0.100

0.500

1

ϕ

g=0.9

-3-2-1 0 1 2 3

0.001

0.010

0.100

1

ϕ

g=0.99

Figure 4: Disk benchmark values comparing ground truth (black) to our model (dots) and to a

renewal approximation (dashed) as the mean cosine of scattering gis varied. Note how our model

improves upon the renewal approximation even for g < 0, which is predominantly back scattering.

-3-2-1 0 1 2 3

0.005

0.010

0.050

0.100

0.500

ϕ

y=20

-3-2-1 0 1 2 3

0.005

0.010

0.050

0.100

ϕ

y=50

-3-2-1 0 1 2 3

0.005

0.010

0.050

0.100

ϕ

y=100

-3-2-1 0 1 2 3

5.×10-4

0.001

0.005

0.010

0.050

0.100

ϕ

y=200

Figure 5: Disk benchmark values comparing ground truth (black) to our model (dots) and to a

renewal approximation (dashed) as the correlation parameter yis varied. Note how as yis increased

(decreasing the the correlation width of the exponential autocovariance), the variabililty of the

ﬂuctuations averages away to a classical medium and all three predictions align.

8

Beyond Renewal Approximations: A 1D Point Process Approach to Linear Transport in Stochastic Media

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