Author: R Hilfer

Categories
dielectric relaxation Fractional Calculus Fractional Time Theory of Time

Applications and Implications of Fractional Dynamics for Dielectric Relaxation

R. Hilfer

in: Recent Advances in Broadband Dielectric Spectroscopy
edited by: Y. Kalmykov
Springer, Berlin, 123 (2012)
10.1007/978-94-007-5012-8
978-94-007-5011-1

submitted on
Friday, September 23, 2011

This article summarizes briefly the presentation given by the author at the NATO Advanced Research Workshop on “Broadband Dielectric Spectroscopy and its Advanced Technological Applications”, held in Perpignan, France, in September 2011. The purpose of the invited presentation at the workshop was to review and summarize the basic theory of fractional dynamics (Hilfer, Phys Rev E 48:2466, 1993; Hilfer and Anton, Phys Rev E Rapid Commun 51:R848, 1995; Hilfer, Fractals 3(1):211, 1995; Hilfer, Chaos Solitons Fractals 5:1475, 1995; Hilfer, Fractals 3:549, 1995; Hilfer, Physica A 221:89, 1995; Hilfer, On fractional diffusion and its relation with continuous time random walks. In: Pekalski et al. (eds) Anomalous diffusion: from basis to applications. Springer, Berlin, p 77, 1999; Hilfer, Fractional evolution equations and irreversibility. In: Helbing et al. (eds) Traffic and granular flow’99. Springer, Berlin, p 215, 2000; Hilfer, Fractional time evolution. In: Hilfer (ed) Applications of fractional calculus in physics. World Scientific, Singapore, p 87, 2000; Hilfer, Remarks on fractional time. In: Castell and Ischebeck (eds) Time, quantum and information. Springer, Berlin, p 235, 2003; Hilfer, Physica A 329:35, 2003; Hilfer, Threefold introduction to fractional derivatives. In: Klages et al. (eds) Anomalous transport: foundations and applications. Wiley-VCH, Weinheim, pp 17– 74, 2008; Hilfer, Foundations of fractional dynamics: a short account. In: Klafter et al. (eds) Fractional dynamics: recent advances. World Scientific, Singapore,207, 2011) and demonstrate its relevance and application to broadband dielectric spectroscopy (Hilfer, J Phys Condens Matter 14:2297, 2002; Hilfer, Chem Phys 284:399, 2002; Hilfer, Fractals 11:251, 2003; Hilfer et al., Fractional Calc Appl Anal 12:299, 2009). It was argued, that broadband dielectric spectroscopy might be useful to test effective field theories based on fractional dynamics.



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image analysis Porous Media

High precision synthetic computed tomography of reconstructed porous media

R. Hilfer, T. Zauner

Physical Review E 84, 062301 (2011)
https://doi.org/10.1103/PhysRevE.84.062301

submitted on
Tuesday, July 26, 2011

Multiscale simulation of transport in disordered and porous media requires microstructures covering several decades in length scale. X-ray and synchrotron computed tomography are presently unable to resolve more than one decade of geometric detail. Recent advances in pore scale modeling [Biswal, Held, Khanna, Wang, and Hilfer, Phys. Rev. E 80, 041301 (2009)] provide strongly correlated microstructures with several decades in microstructural detail. A carefully calibrated microstructure model for Fontainebleau sandstone has been discretized into a suite of three-dimensional microstructures with resolutions from roughly 128 μm down to roughly 500 nm. At the highest resolution the three-dimensional image consists of 35 184 372 088 832 discrete cubic volume elements with gray values between 0 and 216. To the best of our knowledge, this synthetic image is the largest computed tomogram of a porous medium available at present.



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Fractional Calculus Fractional Time

Foundations of Fractional Dynamics: A Short Account

R. Hilfer

in: Fractional Dynamics: Recent Advances
edited by: J. Klafter and S. Lim and R. Metzler
World Scientific, Singapore, 207 (2011)
https://doi.org/10.1142/8087
ISBN: 978-981-4340-58-8

submitted on
Tuesday, March 22, 2011

Applications of fractional dynamics have received a steadily increasing amount of attention during the past decade. Its foundations have found less interest. This chapter briefly reviews the physical foundations of fractional dynamics.



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image analysis Porous Media

Particle-based Rendering for Porous Media

S. Grottel, G. Reina, T. Zauner, R. Hilfer, T. Ertl

in: Proceedings of SIGRAD 2010: Content aggregation and visualization
edited by: K. Jää-Aro and T. Larsson
Link{\”o}ping Electronic Conference Proceedings, vol. 52,Link{\”o}ping University Electronic Press, Linköping, 45 (2010)

submitted on
Thursday, October 28, 2010

Particle-based modeling and simulation of granular or porous media is a widely-used tool in physics and material science to study behavior like fracture and failure under external force. Classical models use spherical particles. However, up to 108 polyhedral-shaped particles are required to achieve realistic results comparable to laboratory experiments. As contact points and exposed surfaces play important roles for the analysis, a meaningful visualization aiding the numeric analysis has to represent the exact particle shapes. For particle-based data sets with spherical particles, ray tracing has been established as the state-of-the-art approach yielding high rendering performance, optimal visual quality and good scalability. However, when rendering polyhedral-shaped particles, there is no issue with visual quality comparing polygon-based rendering approaches and ray casting, whereas the polygon-based approaches cause significantly lower fragment load. The paper at hand investigates the advantages and drawbacks of both approaches by analyzing the performance of state-of-the-art rendering methods employing vertex-buffer objects, hardware-supported instancing, geometry shader, and GPU-based ray casting.



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Mathematics

Erratum on ”Some Bounds for Alternating Mathieu Type Series”

R. Hilfer

J.Math.Inequalities 4, 615 (2010)
dx.doi.org/10.7153/jmi-04-56

submitted on
Monday, September 20, 2010



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fluid flow Porous Media Simulations

Quantitative analysis of numerical estimates for the permeability of porous media from lattice-Boltzmann simulations

A. Narvaez, Th. Zauner, F. Raischel, R. Hilfer, J. Harting

Journal of Statistical Mechanics 2010, P11026 (2010)
https://doi.org/10.1088/1742-5468/2010/11/P11026

submitted on
Sunday, May 30, 2010

During the last decade, lattice-Boltzmann simulations have been improved to become an efficient tool for determining the permeability of porous media samples. However, well-known improvements of the original algorithm are often not implemented. These include, for example, multirelaxation time schemes or improved boundary conditions, as well as different possibilities to impose a pressure gradient. This paper shows that a significant difference of the calculated permeabilities can be found unless one uses a carefully selected set-up. We present a detailed discussion of possible simulation set-ups and quantitative studies of the influence of simulation parameters. We illustrate our results by applying the algorithm to a Fontainebleau sandstone and by comparing our benchmark studies to other numerical permeability measurements in the literature.



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Porous Media Precision Simulations Simulations

Continuum-based rock model of a reservoir dolostone with four orders of magnitude in pore sizes

S. Roth, B. Biswal, G. Afshar, R. Held, P. Øren, L. Berge, R. Hilfer

AAPG Bulletin 95, 925 (2011)
DOI:10.1306/12031010092

submitted on
Friday, May 28, 2010

A continuum-based pore-scale representation of a dolomite reservoir rock is presented, containing several orders of magnitude in pore sizes within a single rock model. The macroscale rock fabric from a low-resolution x-ray microtomogram was combined with microscale information gathered from high- resolution two-dimensional electron microscope images. The low-resolution x-ray microtomogram was segmented into six separate rock phases in terms of mineralogy, matrix appearances, and open- versus crystal-filled molds. These large-scale rock phases were decorated (modeled) with geometric objects, such as different dolomite crystal types and anhydrite, according to the high-resolution information gathered from the electron microscope images. This procedure resulted in an approximate three-dimensional representation of the diage- netically transformed rock sample with respect to dolomite crystal sizes, porosity, appearance, and volume of different matrix phases and pore/matrix/cement ratio. The resulting rock model contains a pore-size distribution ranging from moldic macropores (several hundred micrometers in diameter) down to mudstone micropores ( less than 1 mm in diameter). This allows us to study the effect and contribution of different pore classes to the petrophysical properties of the rock. Higher resolution x-ray tomographs of the same rock were used as control volumes for the pore-size distribution of the model. The pore-size analysis and percolation tests performed in three dimensions at various discretization resolutions indicate pore-throat radii of 1.5 to 6 mm for the largest interconnected pore network. This also highlights the challenge to determine appropriate resolutions for x-ray imaging when the exact rock microstructure is not known.



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Fractional Calculus Mathematics Special Functions

Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions

Z. Tomovski, R. Hilfer, H.M. Srivastava

Integral Transforms and Special Functions 21, 797 (2010)
https://doi.org/10.1080/10652461003675737

submitted on
Monday, November 9, 2009

In this paper, we study a certain family of generalized Riemann–Liouville fractional derivative operators α,β Da± of order α and type β, which were introduced and investigated in several earlier works [R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000; R. Hilfer, Fractional time evolution, in Applications of Fractional Calculus in Physics, R. Hilfer, ed., World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000, pp. 87–130; R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, J. Chem. Phys. 284 (2002), pp. 399–408; R. Hilfer, Threefold introduction to fractional derivatives, in Anomalous Transport: Foundations and Applications, R. Klages, G. Radons, and I.M. Sokolov, eds., Wiley-VCH Verlag, Weinheim, 2008, pp. 17–73; R. Hilfer and L. Anton, Fractional master equations and fractal time random walks, Phys. Rev. E 51 (1995), pp. R848–R851; R. Hilfer,Y. Luchko, and Ž. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal. 12 (2009), pp. 299–318; F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: A tutorial survey, Fract. Calc. Appl. Anal. 10 (2007), pp. 269–308; T. Sandev and Ž. Tomovski, General time fractional wave equation for a vibrating string, J. Phys. A Math. Theor. 43 (2010), 055204; H.M. Srivastava and Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211 (2009), pp. 198–210]. In particular, we derive various compositional properties, which are associated with Mittag–Leffler functions and Hardy-type inequalities for the generalized fractional α,β derivative operator Da± . Furthermore, by using the Laplace transformation methods, we provide solutions of many different classes of fractional differential equations with constant and variable coefficients and some general Volterra-type differintegral equations in the space of Lebesgue integrable functions. Particular cases of these general solutions and a brief discussion about some recently investigated fractional kinetic equations are also given.



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image analysis Porous Media Precision Simulations Simulations

Towards precise prediction of transport properties from synthetic computer tomography of reconstructed porous media

B. Biswal, R.J. Held, V. Khanna, J. Wang, R. Hilfer

Physical Review E 80, 041301 (2009)
https://doi.org/10.1103/PhysRevE.80.041301

submitted on
Saturday, April 4, 2009

Transport properties of a multiscale carbonate rock are predicted from pore scale models, reconstructed using a continuum geometrical modeling technique. The method combines crystallite information from two-dimensional high-resolution images with sedimentary correlations from a three-dimensional low-resolution microcomputed tomography micro-CT-image to produce a rock sample with calibrated porosity, structural correlation, and transport properties at arbitrary resolutions. Synthetic micro-CT images of the reconstructed model match well with experimental micro-CT images at different resolutions, making it possible to predict physical transport parameters at higher resolutions.



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Porous Media Simulations Two-Phase Flow

Numerical solutions of a generalized theory for macroscopic capillarity

F. Doster, P. Zegeling, R. Hilfer

Physical Review E 81, 036307 (2010)
https://doi.org/10.1103/PhysRevE.81.036307

submitted on
Thursday, March 12, 2009

A recent macroscopic theory of biphasic flow in porous media [R. Hilfer, Phys. Rev. E 73, 016307 (2006)] has proposed to treat microscopically percolating fluid regions differently from microscopically nonpercolating regions. Even in one dimension the theory reduces to an analytically intractable set of ten coupled nonlinear partial differential equations. This paper reports numerical solutions for three different initial and boundary value problems that simulate realistic laboratory experiments. All three simulations concern a closed column containing a homogeneous porous medium filled with two immiscible fluids of different densities. In the first simulation the column is raised from a horizontal to a vertical orientation inducing a buoyancy-driven fluid flow that separates the two fluids. In the second simulation the column is first raised from a horizontal to a vertical orientation and subsequently rotated twice by 180° to compare the resulting stationary saturation profiles. In the third simulation the column is first raised from horizontal to vertical orientation and then returned to its original horizontal orientation. In all three simulations imbibition and drainage processes occur simultaneously inside the column. This distinguishes the results reported here from conventional simulations based on existing theories of biphasic flows. Existing theories are unable to predict flow processes where imbibition and drainage occur simultaneously. The approximate numerical results presented here show the same process dependence and hysteresis as one would expect from an experiment.



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image analysis Porous Media Simulations

Continuum reconstruction of the pore scale microstructure for Fontainebleau sandstone

F. Latief, B. Biswal, U. Fauzi, R. Hilfer

Physica A 389, 1607 (2010)
https://doi.org/10.1016/j.physa.2009.12.006

submitted on
Wednesday, March 11, 2009

A stochastic geometrical modeling technique is used to reconstruct a laboratory scale Fontainebleau sandstone with a sidelength of 1.5 cm. The model reconstruction is based on crystallite properties and diagenetic parameters determined from two-dimensional images. The three-dimensional pore scale microstructure of the sandstone is represented by a list of quartz crystallites defined geometrically and placed in the continuum. This allows generation of synthetic μ-CT images of the rock model at arbitrary resolutions. Quantitative microstructure comparison based on Minkowski functionals, two-point correlation function and local porosity theory indicates that this modeling technique can provide more realistic and accurate models of sandstones than many existing techniques used currently. Synthetic μ-CT images at different resolutions from a laboratory scale model of Fon- tainebleau sandstone are made available to the scientific community for resolution dependent petrophysical analysis.



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Porous Media Precision Simulations

Modeling of Multiscale Porous Media

B. Biswal, P.E. Øren, R.J. Held, S. Bakke, R. Hilfer

Image Analysis and Stereology 28, 23-34 (2009)
DOI: 10.5566/ias.v28.p23-34

submitted on
Friday, January 30, 2009

A stochastic geometrical modeling method for reconstructing three dimensional pore scale microstructures of multiscale porous media is presented. In this method the porous medium is represented by a random but spatially correlated structure of objects placed in the continuum. The model exhibits correlations with the sedimentary textures, scale dependent intergranular porosity over many decades, vuggy or dissolution porosity, a percolating pore space, a fully connected matrix space, strong resolution dependence and wide variability in the permeabilities and other properties. The continuum representation allows discretization at arbitrary resolutions providing synthetic micro-computertomographic images for resolution dependent fluid flow simulation. Model implementations for two different carbonate rocks are presented. The method can be used to generate pore scale models of a wide class of multiscale porous media.



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Fractional Calculus Mathematics

Operational Method for the Solution of Fractional Differential Equations with Generalized Riemann-Liouville Fractional Derivatives

R. Hilfer, Y. Luchko, Z. Tomovski

Fractional Calculus and Applied Analysis 12, 299 (2009)

submitted on
Wednesday, December 17, 2008

The operational calculus is an algorithmic approach for the solution of initial-value problems for differential, integral, and integro-differential equations. In this paper, an operational calculus of the Mikusiński type for a generalized Riemann-Liouville fractional differential operator with types introduced by one of the authors is developed. The traditional Riemann-Liouville and Liouville-Caputo fractional derivatives correspond to particu lar types of the general one-parameter family of fractional derivatives with the same order. The operational calculus constructed in this paper is used to solve the corresponding initial value problem for the general n-term linear equation with these generalized fractional derivatives of arbitrary orders and types with constant coefficients. Special cases of the obtained solutions are presented.



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Percolation Porous Media Two-Phase Flow

Percolation as a basic concept for macroscopic capillarity

R. Hilfer, F. Doster

Transport in Porous Media 82, 507 (2010)
https://doi.org/10.1007/s11242-009-9395-0

submitted on
Wednesday, November 12, 2008

The concepts of relative permeability and capillary pressure are crucial for the accepted traditional theory of two phase flow in porous media. Recently, a theoretical approach was introduced that does not require these concepts as input (Hilfer, Physica A, 359:119–128, 2006a; Phys. Rev. E, 73:016307, 2006b). Instead it was based on the concept of hydraulic percolation of fluid phases. This paper presents the first numerical solutions of the coupled nonlinear partial differential equations introduced in Hilfer (Phys. Rev. E, 73:016307, 2006b). Approximate numerical results for saturation profiles in one spatial dimension have been calculated. Long time limits of dynamic time-dependent profiles are compared to stationary solutions of the traditional theory. The initial and boundary conditions are chosen to model the displacement processes that occur when a closed porous column containing two immiscible fluids of different density is raised from a horizontal to a vertical position in a gravitational field. The nature of the displacement process may change locally in space and time between drainage and imbibition. The theory gives local saturations for nonpercolating trapped fluids near the endpoint of the displacement.



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Porous Media Simulations Two-Phase Flow

Modeling and simulation of macrocapillarity

R. Hilfer

in: CP1091, Modeling and Simulation of Materials
edited by: P. Garrido and P. Hurtado and J. Marro
American Institute of Physics, New York, 141 (2009)
https://doi.org/10.1063/1.3082273

submitted on
Monday, November 3, 2008

Macroscopic capillarity. or macrocapillarity for short, refers to capillary phenomena occurring during twophase and multiphase flow in porous media. Wetting phenomena and hysteresis in porous media are at present poorly understood in the sense that neither in physics nor in engineering a fully predictive theory seems to exist, that can describe or predict all observations. This paper extends the consitutive assumptions of a recent approach based on the concept of hydraulic percolation of fluid phases. The theory proposed here allows prediction of residual saturations. It can describe displacement processes in which imbibition and drainage occur simultaneously. This contrasts with the established traditional theory where capillary forces are lumped into capillary pressure function and relative permeabilities, and these functions need to be specified for each displacement process as input. Contrary to the traditional theory the approach advanced here allows to predict capillary pressure saturation relations as output.



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Mathematics Special Functions

Numerical Algorithm for Calculating the Generalized Mittag-Leffler Function

H.J. Seybold, R. Hilfer

SIAM Journal on Numerical Analysis 47, 69 (2008)
https://doi.org/10.1137/070700280

submitted on
Saturday, August 16, 2008

A numerical algorithm for calculating the generalized Mittag-Leffler function for arbitrary complex argument and real parameters is presented. The algorithm uses the Taylor series, the exponentially improved asymptotic series, and integral representations to obtain optimal stability and accuracy of the algorithm. Special care is applied to the limits of validity of the different schemes to avoid instabilities in the algorithm.



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fluid flow Porous Media

Numerical Modeling of Fluid Flow in Porous Media and in Driven Colloidal Suspensions

J. Harting, T. Zauner, R. Weber, R. Hilfer

in: High Performance Computing in Science and Engineering ’08
edited by: W. Nagel and D. Kröner and M. Resch
Springer, Berlin, 349 (2009)
10.1007/978-3-540-88303-6
ISBN: 978-3-540-88301-2

submitted on
Tuesday, July 1, 2008

This article summarizes some of our main efforts performed on the computing facilities provided by the high performance computing centers in Stuttgart and Karlsruhe. At first, large scale lattice Boltzmann simulations are utilized to support resolution dependent analysis of geometrical and transport properties of a porous sandstone model. The second part of this report focuses on Brownian dynamics simulations of optical tweezer experiments where a large colloidal particle is dragged through a polymer solution and a colloidal crystal. The aim of these simulations is to improve our understanding of structuring effects, jamming behavior and defect formation in such colloidal systems.



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Fractional Calculus Mathematics review article

Threefold Introduction to Fractional Derivatives

R. Hilfer

in: Anomalous Transport: Foundations and Applications
edited by: R. Klages and G. Radons and I. Sokolov
Wiley-VCH, Weinheim, 17-74 (2008)
ISBN: 978-3-527-40722-4

submitted on
Wednesday, January 2, 2008

Historical, mathematical and physical introduction to fractrional derivatives.



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Mathematics

Some Bounds for Alternating Mathieu Type Series

Z. Tomovski, R. Hilfer

Journal of Mathematical Inequalities 2, 17 (2008)
dx.doi.org/10.7153/jmi-02-03

submitted on
Monday, December 24, 2007

Using recently investigated integral representations for the generalized alternating Mathieu series and Mellin-Laplace type integral transforms for the generalized hypergeometric functions and the Bessel function of first kind, some bounding inequalities for the mathieu series are presented.



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Porous Media Two-Phase Flow

Dimensional Analysis of Two-phase Flow Including a Rate-dependent Capillary Pressure-Saturation Relationship

S. Manthey, M. Hassanizadeh, R. Helmig, R. Hilfer

Advances in Water Resources 31, 1137 (2008)
https://doi.org/10.1016/j.advwatres.2008.01.021

submitted on
Thursday, March 15, 2007

The macroscopic modelling of two-phase flow processes in subsurface hydrosystems or industrial applications on the Darcy scale usu ally requires a constitutive relationship between capillary pressure and saturation, the Pc(Sw) relationship. Traditionally, it is assumed that a unique relation between Pc and Sw exists independently of the flow conditions as long as hysteretic effects can be neglected. Recently, this assumption has been questioned and alternative formulations have been suggested. For example, the extended Pc(Sw) relationship by Hassanizadeh and Gray [Hassanizadeh SM, Gray WG. Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv Water Resources 1990;13(4):169–86] proposes that the difference between the phase pressures to the equilibrium capillary pressure is a linear function of the rate of change of saturation, thereby introducing a constant of proportionality, the coefficient s. It is desirable to identify cases where the extended relationship needs to be considered. Consequently, a dimensional analysis is performed on the basis of the two-phase balance equations. In addition to the well-known capillary and gravitational number, the dimensional analysis yields a new dimensionless number. The dynamic number Dy quantifies the ratio of dynamic capillary to viscous forces. Relating the dynamic to the capillary as well as the gravitational number gives the new numbers DyC and DyG, respectively. For given sets of fluid and porous medium parameters, the dimensionless numbers Dy and DyC are interpreted as functions of the characteristic length and flow velocity. The simulation of an imbibition process provides insight into the interpretation of the characteristic length scale. The most promising choice for this length scale seems to be the front width. We conclude that consideration of the extended Pc(Sw) relationship may be important for porous media with high permeability, small entry pressure and high coefficient s when systems with a small characteristic length (e.g. steep front) and small characteristic time scale are under investigation.



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