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REVIEW 3 major objections 4 minor 103 references

Ultra-Efficient Reconstruction of Anisotropic Hyperuniform Continuous Random Fields in 2D and 3D via Generalized Spectral Filtering

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read One analytic mask generates 2D and 3D hyperuniform random fields in two FFT passes.

desk verdict Useful parameterized mask family for spectral filtering, but the Hermitian-symmetry slip, unbenchmarked speedup, and missing code make it a conditional accept at best. read the letter →

arxiv 2509.08675 v1 pith:6ZENOCNX submitted 2025-09-10 cond-mat.mtrl-sci

Liyu Zhong , Sheng Mao This is my paper · ORCID
classification cond-mat.mtrl-sci
keywords hyperuniformityspectraldensityrandomfieldsfilteringsuperellipsemicrostructurereconstructionanisotropyfastFouriertransform
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

Hyperuniform random fields suppress long-wavelength fluctuations while keeping local disorder, a combination prized for photonic, thermal, and mechanical materials. This paper claims that such fields, isotropic or anisotropic, can be generated at scale by one analytic Fourier-domain mask rather than by iterative reconstruction. The mask is a generalized superellipse inserted into a log-Gaussian spectral density, letting four parameters independently set hyperuniformity class, radial bandwidth, angular shape, and anisotropy. The authors report 1024x1024 realizations in about 0.03 seconds and 256^3 volumes in about 1.3 seconds, and show that thresholding the continuous field yields binary microstructures whose morphology tracks the mask shape. If the claim holds, large-scale simulation and inverse design of disordered hyperuniform materials becomes a single-shot FFT operation.

What carries the argument

The key object is the generalized superellipse norm K_{p,a,b}(k) = [(|k_x|/a)^p + (|k_y|/b)^p]^(1/p), which interpolates between diamond, circular, and square contours in Fourier space and carries the anisotropy ratio a/b. It is inserted into the analytic spectral density exp[alpha ln K - K^2/(2 sigma^2)], where alpha sets the power-law suppression near k = 0, sigma sets the band width, and p and a/b shape the angular envelope. This single expression replaces iterative matching of target spectral densities with a direct, parameterized mask.

What would settle it

Measure wall-clock times for the paper's 1024^2 and 256^3 examples against a Yeong-Torquato decoder and against the non-iterative spectrally shaped disorder methods cited as [42,43]; the orders-of-magnitude runtime claim stands only if the measured gap is large. Also compute the spectral density at k=0 for thresholded maps: the paper predicts a finite value, not a vanishing one.

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Extended reading notes

Core claim

The central claim is that the spectral density defined by Eq. (13), built from the generalized superellipse norm of Eq. (12), completely prescribes a continuous hyperuniform random field's second-order statistics, so the field can be reconstructed by filtering Gaussian white noise in Fourier space and taking one inverse FFT. Because the mask is analytic, alpha controls the low-wavenumber power-law and hence the hyperuniformity class, sigma controls the radial width, the exponent p controls angular shape, and a/b controls anisotropy. The authors argue this single-shot pipeline reproduces spectral features that iterative schemes target, extends directly to 3D, and leaves the field Gaussian so

Load-bearing premise

The claimed orders-of-magnitude speedup assumes the proper comparison is to iterative decoders such as Yeong-Torquato, but the paper measures no runtime for any competing method, so if the real baseline is already a fast spectral-shaping method the speedup could be small.

Editorial extensions

If this is right

  • Four independent controls let users tune class, bandwidth, angular shape, and anisotropy without disturbing the other features.
  • Thresholding a generated continuous field to a +1/-1 binary map preserves the visual shell morphology but refills the zero-wavenumber density, so strict hyperuniformity is lost under binarization.
  • Because the whole pipeline is linear and FFT-based, additional functional maps can be applied to the raw field with no additional spectral filtering cost.
  • The same parameter trends observed in 2D carry over to 3D, giving a dimension-agnostic way to tailor pore architectures.
  • The analytic mask makes the reconstruction fast enough for high-resolution parametric sweeps over microstructure families.

Reading between the lines

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

  • A natural extension is inverse design: since the mask is differentiable in alpha, sigma, p, and a/b, one could gradient-optimize these parameters against target effective properties or scattering responses.
  • The paper's speedup claim would be sharpened by a direct wall-clock comparison against the non-iterative spectral-shaping methods it cites, since those methods may be the relevant baseline rather than simulated annealing.
  • The thresholding step is identified as the point where exact hyperuniformity is lost; a future variant could replace simple sign thresholding with a hole-preserving binarization to keep the zero-wavenumber density near zero.
  • Because the fields are Gaussian by construction, higher-order correlation functions are constrained by the two-point spectrum, which may limit how far the generated microstructures can deviate from Gaussian statistics in real-space textures.
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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

3 major / 4 minor

Summary. The paper proposes a single-shot FFT-based method for generating Gaussian random fields with prescribed superellipse-shaped power spectral densities, targeting continuous and thresholded two-phase hyperuniform microstructures in 2D and 3D. The spectral density is defined analytically in Eq. (13) with parameters α (low-wavenumber exponent), σ (bandwidth), p (superellipse shape), and a/b (anisotropy). After filtering complex white noise by the square root of this mask and inverse-Fourier-transforming, the fields are demeaned and rescaled; thresholding is then explored as a route to binary morphologies. The authors report O(N log N) complexity, absolute generation times of about 0.03 s for 1024² and 1.3 s for 256³ grids, and claim orders-of-magnitude speedups over iterative reconstruction methods.

Significance. If validated, the proposed spectral mask family is practically useful: it provides explicit analytic control over hyperuniform exponent, angular shell shape, and anisotropy in a non-iterative generation pipeline, with straightforward extension to 3D. The parameterization in Eqs. (12)-(13) is transparent and easily reimplemented, which is a genuine strength. However, the core spectral-representation technique is standard, and the paper's added value rests on the mask design and on the validation and performance claims. At present, the hyperuniform-class verification is not independent of the construction, and the speedup comparison is unsupported by any measured baseline. These issues substantially affect the strength of the central claims.

major comments (3)
  1. [Sec. II B, Eqs. (8)-(9)] The assertion that 'Because χ_K is real and symmetric, f is strictly real valued' is incorrect. For the inverse discrete Fourier transform in Eq. (9) to produce a real field, the Fourier coefficients must obey F_filt(-k) = conj(F_filt(k)). Under the stated i.i.d. complex Gaussian definition of F_wn(k), F_filt(k) and F_filt(-k) are independent, so f is generically complex. The later statement that the pipeline uses 'two FFTs (one forward on white noise and one inverse on the filtered spectrum)' implies real-space white noise and hence Hermitian symmetry, contradicting the earlier definition. This is load-bearing because variance rescaling, thresholding, and all spectral analyses in Sec. III presuppose real scalar fields. Please specify Hermitian-symmetric weights explicitly (e.g., start from real-space white noise and FFT it, or enforce conjugate symmetry).
  2. [Sec. III A/III B/IV, Figs. 1-5] The hyperuniform-class claims are not independently verified. Equation (13) prescribes the target spectrum and Eq. (8) multiplies white noise by sqrt(χ), so the empirical power spectra displayed in Figs. 1-5 necessarily reproduce the input mask; the 'confirmation' of spectral shell shapes is tautological. The paper never computes the real-space variance σ_F^2(R) or its scaling per Eqs. (3) and (6), nor any independent estimate of the low-k exponent. This matters because Eq. (14) is a continuous-space asymptotic statement; on a finite periodic grid, especially for large α (e.g., α=50,100), the low-k region contains very few modes and the realized class is not guaranteed by the mask formula alone. Please add real-space variance-scaling tests for representative (α,p,a/b) combinations, or otherwise provide a non-tautological validation of the hyperuniform class.
  3. [Sec. III A and IV] The claimed 'orders-of-magnitude speedup compared to existing approaches' is not supported by any measured baseline. The text reports absolute timings (≈0.03 s for a 1024² grid, ≈1.3 s for a 256³ grid) but gives no runtime for the methods it claims to outperform, nor does it state the hardware/implementation details. Since refs. [42,43] already describe non-iterative spectral-shaping methods, the relevant baseline is unclear. Please report benchmark timings for at least one representative iterative method (e.g., simulated annealing) and, ideally, for fast spectral methods; otherwise revise the speedup claim to describe the absolute cost of the single-shot FFT pipeline.
minor comments (4)
  1. [Eq. (14)] For anisotropic masks (a≠b), the small-wavenumber expansion is χ(k) ~ C |k|^α multiplied by a direction-dependent factor (e.g., (1/a)^α along kx and (1/b)^α along ky). The exponent α is unchanged, so the hyperuniform class is unaffected, but Eq. (14) as written should be qualified.
  2. [Introduction and Related Work] Refs. [42,43] are cited among representation schemes but are not discussed as prior fast, non-iterative spectral-shaping methods. A brief comparison would strengthen the positioning of the contribution and clarify what is genuinely new.
  3. [Data Availability] The statement 'The codes and data are available upon request' is weak for a methods paper. Consider providing a public repository with the generation code and parameters used for the figures.
  4. [Terminology] The paper uses 'reconstruction' for what is essentially forward generation from a prescribed spectrum. If the method is intended for microstructure reconstruction from correlation data, the connection should be made explicit; otherwise 'generation' avoids overclaiming.

Circularity Check

1 steps flagged · score 6.0 of 10

Output spectrum is the input mask by construction; spectral-shape 'confirmations' are tautological, though thresholding and real-space morphology retain independent content.

  1. self definitional [Sec. II B, Eqs. (8)-(13); Sec. III A, Figs. 1-5]
    "Ffilt(k) = A(k)F_wn(k), A(k) = sqrt(χK(k)) ... Because χK is real and symmetric, f is strictly real valued; its autocovariance ... and spectral density satisfy the Wiener–Khinchin pair ... The target spectral density is specified analytically as χK(k) = C exp[α ln Kp,a,b(k) − Kp,a,b(k)^2/(2σ^2)]"

    The filtered coefficients are defined by multiplying white noise by the square root of the input spectral mask. By construction, the ensemble-averaged power spectrum of the generated field equals the input mask: E|F_filt|^2 = χK. Therefore, the figures that display the log-spectral density of the continuous field and describe the shell morphology (diamond, star, square, ellipse, etc.) are not independent verifications of a prediction; they are restatements of the analytic mask chosen in Eq. (13). The paper presents these as demonstrations of 'precise manipulation of spectral band shape,' but no parameter is fitted and no spectral property is derived beyond what was inserted. The independent content lies in the thresholding analysis and real-space textural observations, which are not fixed

full rationale

The paper's central generative step is a forward spectral filter: Eq. (8) sets the Fourier coefficients of the field to sqrt(mask) times complex Gaussian white noise, so the output field's second-order statistics coincide with the input mask by definition. Consequently, the extensive results showing that the generated field's power spectrum has the intended diamond, star, or square shells confirm the construction itself rather than a nontrivial prediction. This is a partial circularity because the 'spectral control' claim is built into the algorithm. However, the paper also contains genuinely independent elements: the real-space morphological transitions, the effects of ±1 thresholding on the power spectrum and hyperuniformity, and the demonstration of 2D/3D scalability are not determined solely by the linear spectral mask. The claimed orders-of-magnitude speedup is not circular, though it is under-supported by absence of measured baselines. The Hermitian-symmetry issue (i.i.d. complex weights do not guarantee a real field) is a mathematical correctness concern, not a circularity. No load-bearing self-citation was found; the one self-citation [103] merely supports the stealthy-hyperuniform context. Overall, the derivation of the spectral density from the mask is tautological, but the paper's broader empirical content gives it partial rather than total circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central construction rests on classical spectral representation and a small set of hand-picked mask parameters (alpha, sigma, p, a/b). The chosen spectral-density family is ad hoc but explicit; no hidden fitted constants are used to make the generator work. The main unverified assumption is the discrete-to-thermodynamic-limit correspondence and the representativeness of the baseline for the claimed speedup.

free parameters (4)
  • alpha (hyperuniform exponent) = varied: 0.5, 1, 2, 20, 50, 100 (2D); 20 (3D)
    Chosen by hand to set the low-wavenumber power law and hence the hyperuniform class; it is a tuning parameter, not fitted to external data.
  • sigma (Gaussian bandwidth) = 2.0 in all runs
    Chosen by hand; controls annular band width. Fixed across all examples.
  • superellipse exponent p = varied: 0.5, 1, 1.5, 2, 10, 50, 100
    Chosen by hand to shape the angular spectrum; not fitted.
  • aspect ratio a/b = 1 (isotropic) or 0.5 (anisotropic)
    Chosen by hand to control anisotropy; not fitted.
assumptions (6)
  • standard math Spectral representation theorem: Gaussian white noise filtered by sqrt(spectral density) yields a Gaussian random field with that spectral density.
    Invoked in Eqs. (8)-(11), Sec. II B; unproved in this paper but classical.
  • standard math Wiener-Khinchin theorem links autocovariance and spectral density.
    Used in Eqs. (10)-(11).
  • domain assumption The field is a zero-mean stationary Gaussian process fully characterized by its second-order statistics; non-Gaussian higher-order correlations are not controlled.
    The method only prescribes the power spectrum; the paper acknowledges morphology follows from Gaussianity and linearity.
  • domain assumption The analytic low-wavenumber expansion chi ~ C|k|^alpha in Eq. (14) controls the thermodynamic-limit hyperuniform class; discrete finite-grid realizations are assumed to inherit it.
    The paper never computes real-space variance scaling; this correspondence is a background assumption.
  • ad hoc to paper The specific spectral-density family Eq. (13) is a reasonable design space for hyperuniform microstructures.
    This functional form is introduced by the authors rather than derived; all demonstrations are within this family.
  • domain assumption FFT on a periodic grid with wavevectors k in K accurately samples the continuous spectral density; aliasing effects are ignored.
    The reconstruction uses discrete inverse FFT; no convergence or aliasing analysis is given.

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Cite this review

Pith. "Pith review of Ultra-Efficient Reconstruction of Anisotropic Hyperuniform Continuous Random Fields in 2D and 3D via Generalized Spectral Filtering." pith.science (2026). https://pith.science/paper/6ZENOCNX

@misc{pith2026250908675,
  author       = {Pith},
  title        = {Pith review of: Ultra-Efficient Reconstruction of Anisotropic Hyperuniform Continuous Random Fields in 2D and 3D via Generalized Spectral Filtering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6ZENOCNX}},
  note         = {Machine review of arXiv:2509.08675}
}
read the original abstract

Hyperuniform continuous random fields suppress large-scale fluctuations while preserving rich local disorder, making them highly attractive for next-generation photonic, thermal and mechanical materials. However, traditional reconstruction techniques often suffer from limited spectral control or excessive computational cost, especially in high-resolution 2D and 3D settings. In this work, we present an ultra-efficient generative algorithm based on generalized superellipse spectral filtering, which allows independent tuning of isotropic and anisotropic spectral envelopes without resorting to costly iterative schemes. We demonstrate our method on a comprehensive set of 2D and 3D examples, showing precise manipulation of spectral band shape and orders-of-magnitude speedup compared to existing approaches. Furthermore, we explore the effect of simple thresholding on the generated fields, analyzing the morphological features and power-spectrum characteristics of the resulting two-phase maps. Our results confirm that the proposed framework not only accelerates hyperuniform field synthesis but also provides a versatile platform for systematic study of binary microstructures derived from continuous designs. This work opens new avenues for large-scale simulation and optimized design of advanced hyperuniform materials.

Figures

Figures reproduced from arXiv: 2509.08675 by the authors.

Figure 1
Figure 1. FIG. 1. Isotropic case ( [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Anisotropic case ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Isotropic masks ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Anisotropic masks ( [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Three–dimensional reconstructions for [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

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Reference graph

Works this paper leans on

103 extracted references · 1 linked inside Pith

  1. [1]

    Modeling cell migration regu- lated by cell extracellular-matrix micromechanical cou- pling,

    Yu Zheng, Hanqing Nan, Yanping Liu, Qihui Fan, Xiaochen Wang, Ruchuan Liu, Liyu Liu, Fangfu Ye, Bo Sun, and Yang Jiao, “Modeling cell migration regu- lated by cell extracellular-matrix micromechanical cou- pling,” Physical Review E100, 043303 (2019)

    2019

  2. [2]

    Enhanced comprehensive perfor- mance of bonding interface between cfrp and steel by a novel film adhesive,

    Lu Ke, Chuanxi Li, Nanhai Luo, Jun He, Yang Jiao, and Yongming Liu, “Enhanced comprehensive perfor- mance of bonding interface between cfrp and steel by a novel film adhesive,” Composite structures229, 111393 (2019)

    2019

  3. [3]

    Liq- uid metal-filled phase change composites with tunable stiffness: Computational modeling and experiment,

    Quang-Kha Nguyen, Jiexian Ma, and Pu Zhang, “Liq- uid metal-filled phase change composites with tunable stiffness: Computational modeling and experiment,” Mechanics of Materials183, 104702 (2023)

    2023

  4. [4]

    A review article: investigations on soft materials for soft robot manipulations,

    N Elango and Ahmad Athif Mohd Faudzi, “A review article: investigations on soft materials for soft robot manipulations,” The International Journal of Advanced Manufacturing Technology80, 1027–1037 (2015)

    2015

  5. [5]

    Robust three- component elastomer–particle–fiber composites with tunable properties for soft robotics,

    Amir Mohammadi Nasab, Siavash Sharifi, Shuai Chen, Yang Jiao, and Wanliang Shan, “Robust three- component elastomer–particle–fiber composites with tunable properties for soft robotics,” Advanced Intel- ligent Systems3, 2000166 (2021)

    2021

  6. [6]

    Strongly anisotropic waveguide as a nonmagnetic left- handed system,

    Viktor A Podolskiy and Evgenii E Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left- handed system,” Physical Review B71, 201101 (2005)

    2005

  7. [7]

    Dis- persion relation for general anisotropic media,

    NJ Damaskos, AL Maffett, and PLE Uslenghi, “Dis- persion relation for general anisotropic media,” IEEE transactions on antennas and propagation30, 991–993 (1982)

    1982

  8. [8]

    Dispersion relation for electromagnetic waves in anisotropic media,

    Yakov Itin, “Dispersion relation for electromagnetic waves in anisotropic media,” Physics Letters A374, 1113–1116 (2010)

    2010

Show all 103 references
  1. [9]

    Extraordinary op- tical and transport properties of disordered stealthy hy- peruniform two-phase media,

    Jaeuk Kim and Salvatore Torquato, “Extraordinary op- tical and transport properties of disordered stealthy hy- peruniform two-phase media,” Journal of Physics: Con- densed Matter (2023)

    2023

  2. [10]

    Random heterogeneous materi- als: microstructure and macroscopic properties,

    Salvatore Torquato, “Random heterogeneous materi- als: microstructure and macroscopic properties,” Appl. Mech. Rev.55, B62–B63 (2002)

    2002

  3. [11]

    Sahimi,Heterogeneous Materials I: Linear trans- port and optical properties, Vol

    M. Sahimi,Heterogeneous Materials I: Linear trans- port and optical properties, Vol. 1 (Springer, New York, 2003)

    2003

  4. [12]

    Design of smart composite materials using topology optimiza- tion,

    Ole Sigmund and Salvatore Torquato, “Design of smart composite materials using topology optimiza- tion,” Smart Materials and Structures8, 365 (1999)

    1999

  5. [13]

    Design of ma- terials with extreme thermal expansion using a three- phase topology optimization method,

    Ole Sigmund and Salvatore Torquato, “Design of ma- terials with extreme thermal expansion using a three- phase topology optimization method,” Journal of the Mechanics and Physics of Solids45, 1037–1067 (1997)

    1997

  6. [14]

    High-throughput experiments and machine learning strategies for efficient exploration of additively manufactured inconel 625,

    Zachary S Courtright, Aditya Venkatraman, Berkay Yu- cel, Venkata Surya Karthik Adapa, Abel Diaz, and Surya R Kalidindi, “High-throughput experiments and machine learning strategies for efficient exploration of additively manufactured inconel 625,” Acta Materialia , 120875 (2025)

    2025

  7. [15]

    Inverse stochastic microstruc- ture design,

    Adam P Generale, Andreas E Robertson, Conlain Kelly, and Surya R Kalidindi, “Inverse stochastic microstruc- ture design,” Acta Materialia271, 119877 (2024)

    2024

  8. [16]

    Active learning for regression of structure–property mapping: the importance of sampling and representa- tion,

    Hao Liu, Berkay Yucel, Baskar Ganapathysubramanian, Surya R Kalidindi, Daniel Wheeler, and Olga Wodo, “Active learning for regression of structure–property mapping: the importance of sampling and representa- tion,” Digital Discovery3, 1997–2009 (2024)

    1997

  9. [17]

    A framework for data-driven analysis of materials under uncertainty: Countering the curse of dimensionality,

    Miguel A Bessa, Ramin Bostanabad, Zeliang Liu, Anqi Hu, Daniel W Apley, Catherine Brinson, Wei Chen, and Wing Kam Liu, “A framework for data-driven analysis of materials under uncertainty: Countering the curse of dimensionality,” Computer Methods in Applied Me- chanics and Eng...

    2017

  10. [18]

    Machine learning and materials in- formatics approaches for predicting transverse mechan- ical properties of unidirectional cfrp composites with mi- crovoids,

    Mengze Li, Haowei Zhang, Shuran Li, Weidong Zhu, and Yinglin Ke, “Machine learning and materials in- formatics approaches for predicting transverse mechan- ical properties of unidirectional cfrp composites with mi- crovoids,” Materials & Design224, 111340 (2022)

    2022

  11. [19]

    Inverse design of microstructures using conditional continuous normal- izing flows,

    Hossein Mirzaee and Serveh Kamrava, “Inverse design of microstructures using conditional continuous normal- izing flows,” Acta Materialia285, 120704 (2025)

    2025

  12. [20]

    Improving direct physical properties predic- tion of heterogeneous materials from imaging data via convolutional neural network and a morphology-aware generative model,

    Ruijin Cang, Hechao Li, Hope Yao, Yang Jiao, and Yi Ren, “Improving direct physical properties predic- tion of heterogeneous materials from imaging data via convolutional neural network and a morphology-aware generative model,” Computational Materials Science 150, 212–221 (2018). 9

    2018

  13. [21]

    Data-driven learning of 3-point correlation functions as microstruc- ture representations,

    Sheng Cheng, Yang Jiao, and Yi Ren, “Data-driven learning of 3-point correlation functions as microstruc- ture representations,” Acta Materialia229, 117800 (2022)

    2022

  14. [22]

    Correlation-function-based microstructure design of alloy-polymer composites for dynamic dry adhesion tun- ing in soft gripping,

    Yaopengxiao Xu, Pei-En Chen, Hechao Li, Wenxi- ang Xu, Yi Ren, Wanliang Shan, and Yang Jiao, “Correlation-function-based microstructure design of alloy-polymer composites for dynamic dry adhesion tun- ing in soft gripping,” Journal of Applied Physics131, 115104 (2022)

    2022

  15. [23]

    Computational mi- crostructure characterization and reconstruction: Re- view of the state-of-the-art techniques,

    Ramin Bostanabad, Yichi Zhang, Xiaolin Li, Tucker Kearney, L Catherine Brinson, Daniel W Apley, Wing Kam Liu, and Wei Chen, “Computational mi- crostructure characterization and reconstruction: Re- view of the state-of-the-art techniques,” Progress in Ma- terials Science95, 1–41 (2018)

    2018

  16. [24]

    Recon- struction, optimization, and design of heterogeneous materials and media: Basic principles, computational algorithms, and applications,

    Muhammad Sahimi and Pejman Tahmasebi, “Recon- struction, optimization, and design of heterogeneous materials and media: Basic principles, computational algorithms, and applications,” Physics Reports939, 1– 82 (2021)

    2021

  17. [25]

    Reconstructing ran- dom media,

    C. L. Y. Yeong and S. Torquato, “Reconstructing ran- dom media,” Phys. Rev. E57, 495 (1998)

    1998

  18. [26]

    Reconstructing random media. ii. three-dimensional media from two- dimensional cuts,

    C. L. Y. Yeong and S. Torquato, “Reconstructing random media. ii. three-dimensional media from two- dimensional cuts,” Phys. Rev. E58, 224 (1998)

    1998

  19. [27]

    Statistical reconstruction of three- dimensional porous media from two-dimensional im- ages,

    Anthony P Roberts, “Statistical reconstruction of three- dimensional porous media from two-dimensional im- ages,” Physical Review E56, 3203 (1997)

    1997

  20. [28]

    De- lineation of the space of 2-point correlations in a com- posite material system,

    SR Niezgoda, DT Fullwood, and SR Kalidindi, “De- lineation of the space of 2-point correlations in a com- posite material system,” Acta Materialia56, 5285–5292 (2008)

    2008

  21. [29]

    Pore space re- construction using multiple-point statistics,

    Hiroshi Okabe and Martin J Blunt, “Pore space re- construction using multiple-point statistics,” Journal of Petroleum Science and Engineering46, 121–137 (2005)

    2005

  22. [30]

    Modeling heterogeneous materials via two-point correlation func- tions: Basic principles,

    Yang Jiao, FH Stillinger, and S Torquato, “Modeling heterogeneous materials via two-point correlation func- tions: Basic principles,” Physical Review E76, 031110 (2007)

    2007

  23. [31]

    Modeling heterogeneous materials via two-point correlation func- tions. ii. algorithmic details and applications,

    Y Jiao, FH Stillinger, and S Torquato, “Modeling heterogeneous materials via two-point correlation func- tions. ii. algorithmic details and applications,” Physical Review E77, 031135 (2008)

    2008

  24. [32]

    A multiple-point statistics algorithm for 3d pore space reconstruction from 2d images,

    Alireza Hajizadeh, Aliakbar Safekordi, and Farhad A Farhadpour, “A multiple-point statistics algorithm for 3d pore space reconstruction from 2d images,” Advances in water Resources34, 1256–1267 (2011)

    2011

  25. [33]

    Cross- correlation function for accurate reconstruction of het- erogeneous media,

    Pejman Tahmasebi and Muhammad Sahimi, “Cross- correlation function for accurate reconstruction of het- erogeneous media,” Physical review letters110, 078002 (2013)

    2013

  26. [34]

    Multiple-point geostatistical modeling based on the cross-correlation functions,

    Pejman Tahmasebi, Ardeshir Hezarkhani, and Muham- mad Sahimi, “Multiple-point geostatistical modeling based on the cross-correlation functions,” Computa- tional Geosciences16, 779–797 (2012)

    2012

  27. [35]

    Stochastic reassembly strategy for man- aging information complexity in heterogeneous materi- als analysis and design,

    Hongyi Xu, M Steven Greene, Hua Deng, Dmitriy Dikin, Catherine Brinson, Wing Kam Liu, Craig Burkhart, George Papakonstantopoulos, Mike Poldneff, and Wei Chen, “Stochastic reassembly strategy for man- aging information complexity in heterogeneous materi- als analysis and design...

    2013

  28. [36]

    A descriptor-based design methodology for developing heterogeneous microstructural materials system,

    Hongyi Xu, Yang Li, Catherine Brinson, and Wei Chen, “A descriptor-based design methodology for developing heterogeneous microstructural materials system,” Jour- nal of Mechanical Design136(2014)

    2014

  29. [37]

    Microstruc- ture representation and reconstruction of heterogeneous materials via deep belief network for computational ma- terial design,

    Ruijin Cang, Yaopengxiao Xu, Shaohua Chen, Yong- ming Liu, Yang Jiao, and Max Yi Ren, “Microstruc- ture representation and reconstruction of heterogeneous materials via deep belief network for computational ma- terial design,” Journal of Mechanical Design139(2017)

    2017

  30. [38]

    Mi- crostructural materials design via deep adversarial learning methodology,

    Zijiang Yang, Xiaolin Li, L Catherine Brinson, Alok N Choudhary, Wei Chen, and Ankit Agrawal, “Mi- crostructural materials design via deep adversarial learning methodology,” Journal of Mechanical Design 140(2018)

    2018

  31. [39]

    A transfer learning approach for microstructure reconstruction and structure-property predictions,

    Xiaolin Li, Yichi Zhang, He Zhao, Craig Burkhart, L Catherine Brinson, and Wei Chen, “A transfer learning approach for microstructure reconstruction and structure-property predictions,” Scientific reports8, 1– 13 (2018)

    2018

  32. [40]

    A spectral den- sity function approach for active layer design of organic photovoltaic cells,

    Umar Farooq Ghumman, Akshay Iyer, Rabindra Du- lal, Joydeep Munshi, Aaron Wang, TeYu Chien, Ganesh Balasubramanian, and Wei Chen, “A spectral den- sity function approach for active layer design of organic photovoltaic cells,” Journal of Mechanical Design140 (2018)

    2018

  33. [41]

    Quantifying phase mixing and separation behaviors across length and time scales,

    Murray Skolnick and Salvatore Torquato, “Quantifying phase mixing and separation behaviors across length and time scales,” Acta Materialia , 119774 (2024)

    2024

  34. [42]

    Fast generation of spectrally shaped disorder,

    Aaron Shih, Mathias Casiulis, and Stefano Martiniani, “Fast generation of spectrally shaped disorder,” Physi- cal Review E110, 034122 (2024)

    2024

  35. [43]

    Gyromorphs: a new class of functional disordered ma- terials,

    Mathias Casiulis, Aaron Shih, and Stefano Martiniani, “Gyromorphs: a new class of functional disordered ma- terials,” arXiv preprint arXiv:2410.09023 (2024)

    2024

  36. [44]

    Torquato,Random Heterogeneous Materials: Mi- crostructure and Macroscopic Properties(Springer- Verlag, New York, 2002)

    S. Torquato,Random Heterogeneous Materials: Mi- crostructure and Macroscopic Properties(Springer- Verlag, New York, 2002)

    2002

  37. [45]

    Dominique Jeulin,Morphological models of random structures(Springer, 2021)

    2021

  38. [46]

    Designing disordered hype- runiform two-phase materials with novel physical prop- erties,

    D. Chen and S. Torquato, “Designing disordered hype- runiform two-phase materials with novel physical prop- erties,” Acta Mater.142, 152–161 (2018)

    2018

  39. [47]

    Computational design of anisotropic stealthy hyperuniform composites with en- gineered directional scattering properties,

    Wenlong Shi, David Keeney, Duyu Chen, Yang Jiao, and Salvatore Torquato, “Computational design of anisotropic stealthy hyperuniform composites with en- gineered directional scattering properties,” Physical Re- view E108, 045306 (2023)

    2023

  40. [48]

    Wenlong Shi, Yang Jiao, and Salvatore Torquato, “Three-dimensional construction of hyperuniform, non- hyperuniform, and antihyperuniform disordered hetero- geneous materials and their transport properties via spectral density functions,” Physical Review E111, 035310 (2025)

    2025

  41. [49]

    Effective stiffness tensor of composite me- dia—i. exact series expansions,

    S Torquato, “Effective stiffness tensor of composite me- dia—i. exact series expansions,” Journal of the Mechan- ics and Physics of Solids45, 1421–1448 (1997)

    1997

  42. [50]

    Effective electrical conductivity of two- phase disordered composite media,

    S Torquato, “Effective electrical conductivity of two- phase disordered composite media,” Journal of Applied Physics58, 3790–3797 (1985)

    1985

  43. [51]

    Effective elastic wave characteristics of composite media,

    Jaeuk Kim and Salvatore Torquato, “Effective elastic wave characteristics of composite media,” New Journal of Physics22, 123050 (2020)

    2020

  44. [52]

    Nonlocal effective electromagnetic wave characteristics of composite me- dia: beyond the quasistatic regime,

    Salvatore Torquato and Jaeuk Kim, “Nonlocal effective electromagnetic wave characteristics of composite me- dia: beyond the quasistatic regime,” Physical Review X 10 11, 021002 (2021)

    2021

  45. [53]

    Effective electro- magnetic wave properties of disordered stealthy hyper- uniform layered media beyond the quasistatic regime,

    Jaeuk Kim and Salvatore Torquato, “Effective electro- magnetic wave properties of disordered stealthy hyper- uniform layered media beyond the quasistatic regime,” Optica10, 965–972 (2023)

    2023

  46. [54]

    Diffusion spreadability as a probe of the microstructure of complex media across length scales,

    Salvatore Torquato, “Diffusion spreadability as a probe of the microstructure of complex media across length scales,” Physical Review E104, 054102 (2021)

    2021

  47. [55]

    Simulated diffusion spreadability for characterizing the structure and transport properties of two-phase materials,

    Murray Skolnick and Salvatore Torquato, “Simulated diffusion spreadability for characterizing the structure and transport properties of two-phase materials,” Acta Materialia250, 118857 (2023)

    2023

  48. [56]

    Rigorous bounds on the fluid permeability: Effect of polydispersivity in grain size,

    S Torquato and B Lu, “Rigorous bounds on the fluid permeability: Effect of polydispersivity in grain size,” Physics of Fluids A: Fluid Dynamics2, 487–490 (1990)

    1990

  49. [57]

    Predicting transport character- istics of hyperuniform porous media via rigorous microstructure-property relations,

    Salvatore Torquato, “Predicting transport character- istics of hyperuniform porous media via rigorous microstructure-property relations,” Advances in water resources140, 103565 (2020)

    2020

  50. [58]

    Optimization by simulated annealing,

    Scott Kirkpatrick, C Daniel Gelatt Jr, and Mario P Vecchi, “Optimization by simulated annealing,” science 220, 671–680 (1983)

    1983

  51. [59]

    Local density fluctu- ations, hyperuniformity, and order metrics,

    S. Torquato and F. H. Stillinger, “Local density fluctu- ations, hyperuniformity, and order metrics,” Phys. Rev. E68, 041113 (2003)

    2003

  52. [60]

    Hyperuniformity in point patterns and two-phase random heterogeneous media,

    C. E. Zachary and S. Torquato, “Hyperuniformity in point patterns and two-phase random heterogeneous media,” J. Stat. Mech. Theor. Exp.2009, P12015 (2009)

    2009

  53. [61]

    Hyperuniformity and its generalizations,

    S. Torquato, “Hyperuniformity and its generalizations,” Phys. Rev. E94, 022122 (2016)

    2016

  54. [62]

    Hyperuniform states of matter,

    S. Torquato, “Hyperuniform states of matter,” Phys. Rep.745, 1–95 (2018)

    2018

  55. [63]

    Designer disordered materials with large, com- plete photonic band gaps,

    Marian Florescu, Salvatore Torquato, and Paul J Stein- hardt, “Designer disordered materials with large, com- plete photonic band gaps,” Proceedings of the National Academy of Sciences106, 20658–20663 (2009)

    2009

  56. [64]

    Photonic band gap in isotropic hyper- uniform disordered solids with low dielectric contrast,

    Weining Man, Marian Florescu, Kazue Matsuyama, Polin Yadak, Geev Nahal, Seyed Hashemizad, Eric Williamson, Paul Steinhardt, Salvatore Torquato, and Paul Chaikin, “Photonic band gap in isotropic hyper- uniform disordered solids with low dielectric contrast,” Optics express21, 1...

    2013

  57. [65]

    Evolving scattering networks for engineer- ing disorder,

    Sunkyu Yu, “Evolving scattering networks for engineer- ing disorder,” Nat. Comput. Sci.1, 1 (2023)

    2023

  58. [66]

    Near-field investigation of lumi- nescent hyperuniform disordered materials,

    Nicoletta Granchi, Richard Spalding, Matteo Lodde, Maurangelo Petruzzella, Frank W Otten, Andrea Fiore, Francesca Intonti, Riccardo Sapienza, Marian Florescu, and Massimo Gurioli, “Near-field investigation of lumi- nescent hyperuniform disordered materials,” Advanced optical m...

    2022

  59. [67]

    Hearing the shape of a drum for light: isospectrality in photonics,

    Seungkyun Park, Ikbeom Lee, Jungmin Kim, Namkyoo Park, and Sunkyu Yu, “Hearing the shape of a drum for light: isospectrality in photonics,” Nanophotonics 11, 2763–2778 (2021)

    2021

  60. [68]

    Wave propagation and band tails of two- dimensional disordered systems in the thermodynamic limit,

    Michael A Klatt, Paul J Steinhardt, and Salvatore Torquato, “Wave propagation and band tails of two- dimensional disordered systems in the thermodynamic limit,” Proceedings of the National Academy of Sciences 119, e2213633119 (2022)

    2022

  61. [69]

    Over 65% sunlight absorption in a 1µm si slab with hyper- uniform texture,

    Nasim Tavakoli, Richard Spalding, Alexander Lam- bertz, Pepijn Koppejan, Georgios Gkantzounis, Chen- glong Wan, Ruslan Rohrich, Evgenia Kontoleta, A Femius Koenderink, Riccardo Sapienza,et al., “Over 65% sunlight absorption in a 1µm si slab with hyper- uniform texture,” ACS ph...

    2022

  62. [70]

    Wave transport in stealth hyperuniform materials: The diffu- sive regime and beyond,

    ´Elie Ch´ eron, Simon F´ elix, Jean-Philippe Groby, Vin- cent Pagneux, and Vicente Romero-Garc ´ ıa, “Wave transport in stealth hyperuniform materials: The diffu- sive regime and beyond,” Applied Physics Letters121, 061702 (2022)

    2022

  63. [71]

    Engineered disorder in photonics,

    Sunkyu Yu, Cheng-Wei Qiu, Yidong Chong, Salvatore Torquato, and Namkyoo Park, “Engineered disorder in photonics,” Nature Reviews Materials6, 226–243 (2021)

    2021

  64. [72]

    Biological tissue- inspired tunable photonic fluid,

    Xinzhi Li, Amit Das, and Dapeng Bi, “Biological tissue- inspired tunable photonic fluid,” Proceedings of the Na- tional Academy of Sciences115, 6650–6655 (2018)

    2018

  65. [73]

    Transport, geometrical, and topological properties of stealthy dis- ordered hyperuniform two-phase systems,

    G Zhang, FH Stillinger, and S Torquato, “Transport, geometrical, and topological properties of stealthy dis- ordered hyperuniform two-phase systems,” The Journal of chemical physics145, 244109 (2016)

    2016

  66. [74]

    Characterization of void space, large- scale structure, and transport properties of maximally random jammed packings of superballs,

    Charles Emmett Maher, Frank H Stillinger, and Salva- tore Torquato, “Characterization of void space, large- scale structure, and transport properties of maximally random jammed packings of superballs,” Physical Re- view Materials6, 025603 (2022)

    2022

  67. [75]

    Microstructure and mechan- ical properties of hyperuniform heterogeneous materi- als,

    Yaopengxiao Xu, Shaohua Chen, Pei-En Chen, Wenxi- ang Xu, and Yang Jiao, “Microstructure and mechan- ical properties of hyperuniform heterogeneous materi- als,” Physical Review E96, 043301 (2017)

    2017

  68. [76]

    Anisotropic suppression of hyperuniformity of elastic systems in media with planar disorder,

    Joaqu ´ ın Puig, Federico El ´ ıas, Jazm ´ ın Arag´ on S´ anchez, Ra´ ul Cort´ es Maldonado, Gonzalo Rumi, Gladys Nieva, Pablo Pedrazzini, Alejandro B Kolton, and Yanina Fasano, “Anisotropic suppression of hyperuniformity of elastic systems in media with planar disorder,” Comm...

    2022

  69. [77]

    Multifunctional composites for elastic and electromagnetic wave prop- agation,

    Jaeuk Kim and Salvatore Torquato, “Multifunctional composites for elastic and electromagnetic wave prop- agation,” Proceedings of the National Academy of Sci- ences117, 8764–8774 (2020)

    2020

  70. [78]

    Unexpected density fluctuations in jammed disordered sphere packings,

    Aleksandar Donev, Frank H Stillinger, and Salvatore Torquato, “Unexpected density fluctuations in jammed disordered sphere packings,” Physical review letters95, 090604 (2005)

    2005

  71. [79]

    Noise, diffusion, and hyperuniformity,

    Daniel Hexner and Dov Levine, “Noise, diffusion, and hyperuniformity,” Physical review letters118, 020601 (2017)

    2017

  72. [80]

    Hyperuniform monocrystalline structures by spinodal solid-state dewetting,

    Marco Salvalaglio, Mohammed Bouabdellaoui, Monica Bollani, Abdennacer Benali, Luc Favre, Jean-Benoit Claude, Jerome Wenger, Pietro de Anna, Francesca In- tonti, Axel Voigt,et al., “Hyperuniform monocrystalline structures by spinodal solid-state dewetting,” Physical Review Lett...

    2020

  73. [81]

    Quantum jamming transition to a correlated electron glass in 1t-tas 2,

    Y. A. Gerasimenko, I. Vaskivskyi, M. Litskevich, J. Ravnik, J. Vodeb, M. Diego, V. Kabanov, and D. Mi- hailovic, “Quantum jamming transition to a correlated electron glass in 1t-tas 2,” Nat. Mater.18, 1078–1083 (2019)

    2019

  74. [82]

    Approach to hyperuni- formity in a metallic glass-forming material exhibit- ing a fragile to strong glass transition,

    Hao Zhang, Xinyi Wang, Jiarui Zhang, Hai-Bin Yu, and Jack F Douglas, “Approach to hyperuni- formity in a metallic glass-forming material exhibit- ing a fragile to strong glass transition,” arXiv preprint arXiv:2302.01429 (2023)

    2023 arXiv

  75. [83]

    Multihyperuniform long-range order in medium-entropy alloys,

    Duyu Chen, Xinyu Jiang, Duo Wang, Houlong Zhuang, and Yang Jiao, “Multihyperuniform long-range order in medium-entropy alloys,” Acta Materialia246, 118678 11 (2023)

    2023

  76. [84]

    Avian photoreceptor patterns represent a disordered hyperuniform solution to a multiscale pack- ing problem,

    Yang Jiao, Timothy Lau, Haralampos Hatzikirou, Michael Meyer-Hermann, Joseph C Corbo, and Salva- tore Torquato, “Avian photoreceptor patterns represent a disordered hyperuniform solution to a multiscale pack- ing problem,” Physical Review E89, 022721 (2014)

    2014

  77. [85]

    The hidden order of turing patterns in arid and semi-arid vegetation ecosystems,

    Zhenpeng Ge, “The hidden order of turing patterns in arid and semi-arid vegetation ecosystems,” Proceedings of the National Academy of Sciences120, e2306514120 (2023)

    2023

  78. [86]

    Universal hyperuniform organization in looped leaf vein networks,

    Yuan Liu, Duyu Chen, Jianxiang Tian, Wenxiang Xu, and Yang Jiao, “Universal hyperuniform organization in looped leaf vein networks,” Physical Review Letters 133, 028401 (2024)

    2024

  79. [87]

    Random scalar fields and hy- peruniformity,

    Z. Ma and S. Torquato, “Random scalar fields and hy- peruniformity,” J. Appl. Phys.121, 244904 (2017)

    2017

  80. [88]

    Disordered hyperuniform heterogeneous materials,

    S. Torquato, “Disordered hyperuniform heterogeneous materials,” J. Phys. Condens. Matter28, 414012 (2016)

    2016

  81. [89]

    Hyperuniformity of quasicrystals,

    E. C. O˘ guz, J. E. S. Socolar, P. J. Steinhardt, and S. Torquato, “Hyperuniformity of quasicrystals,” Phys. Rev. B95, 054119 (2017)

    2017

  82. [90]

    Classical many-particle systems with unique disor- dered ground states,

    Ge Zhang, Frank H Stillinger, and Salvatore Torquato, “Classical many-particle systems with unique disor- dered ground states,” Physical Review E96, 042146 (2017)

    2017

  83. [91]

    Hyperuniform long-range correlations are a signature of disordered jammed hard-particle packings,

    Chase E Zachary, Yang Jiao, and Salvatore Torquato, “Hyperuniform long-range correlations are a signature of disordered jammed hard-particle packings,” Physical review letters106, 178001 (2011)

    2011

  84. [92]

    Maximally random jammed packings of platonic solids: Hyperuniform long- range correlations and isostaticity,

    Yang Jiao and Salvatore Torquato, “Maximally random jammed packings of platonic solids: Hyperuniform long- range correlations and isostaticity,” Physical Review E 84, 041309 (2011)

    2011

  85. [93]

    Hyperuni- formity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle pack- ings. i. polydisperse spheres,

    C. E. Zachary, Y. Jiao, and S. Torquato, “Hyperuni- formity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle pack- ings. i. polydisperse spheres,” Phys. Rev. E83, 051308 (2011)

    2011

  86. [94]

    Hyperuni- formity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle pack- ings. ii. anisotropy in particle shape,

    C. E. Zachary, Y. Jiao, and S. Torquato, “Hyperuni- formity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle pack- ings. ii. anisotropy in particle shape,” Phys. Rev. E83, 051309 (2011)

    2011

  87. [95]

    Anomalous local co- ordination, density fluctuations, and void statistics in disordered hyperuniform many-particle ground states,

    C. E. Zachary and S. Torquato, “Anomalous local co- ordination, density fluctuations, and void statistics in disordered hyperuniform many-particle ground states,” Phys. Rev. E83, 051133 (2011)

    2011

  88. [96]

    Hyperuniformity of critical absorbing states,

    Daniel Hexner and Dov Levine, “Hyperuniformity of critical absorbing states,” Physical review letters114, 110602 (2015)

    2015

  89. [97]

    Effect of imperfections on the hyperuniformity of many-body systems,

    J. Kim and S. Torquato, “Effect of imperfections on the hyperuniformity of many-body systems,” Phys. Rev. B 97, 054105 (2018)

    2018

  90. [98]

    Local number fluctuations in hyperuniform and non- hyperuniform systems: Higher-order moments and dis- tribution functions,

    Salvatore Torquato, Jaeuk Kim, and Michael A Klatt, “Local number fluctuations in hyperuniform and non- hyperuniform systems: Higher-order moments and dis- tribution functions,” Physical Review X11, 021028 (2021)

    2021

  91. [99]

    Simula- tion of stochastic processes by the spectral representa- tion method,

    Masanobu Shinozuka and George Deodatis, “Simula- tion of stochastic processes by the spectral representa- tion method,” Applied Mechanics Reviews44, 191–204 (1991)

    1991

  92. [100]

    Simulation of sta- tionary gaussian processes by circulant embedding of the covariance matrix,

    Andrew T. Wood and George Chan, “Simulation of sta- tionary gaussian processes by circulant embedding of the covariance matrix,” Journal of Computational and Graphical Statistics3, 409–432 (1994)

    1994

  93. [101]

    Random scalar fields and hyperuniformity,

    Zheng Ma and Salvatore Torquato, “Random scalar fields and hyperuniformity,” Journal of Applied Physics 121, 244904 (2017)

    2017

  94. [102]

    Designing dis- ordered hyperuniform two-phase materials with novel physical properties,

    Duyu Chen and Salvatore Torquato, “Designing dis- ordered hyperuniform two-phase materials with novel physical properties,” Acta Materialia142, 152–161 (2018)

    2018

  95. [103]

    Modeling disordered hyperuniform heterogeneous materials: Mi- crostructure representation, field fluctuations and effec- tive properties,

    Liyu Zhong, Sheng Mao, and Yang Jiao, “Modeling disordered hyperuniform heterogeneous materials: Mi- crostructure representation, field fluctuations and effec- tive properties,” Acta Materialia296, 121218 (2025)

    2025

Pith tools

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