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Topological Signatures of Diffusive Release in Porous Media

T0 review · 2 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper argues that persistent homology summaries of the solid phase in porous media—counts and lifetimes of loops, cavities, and connected components—are strongly associated with diffusive release behavior even after stratifying by targ

desk verdict Solid, honest pilot study of persistent homology as a release-regime descriptor, but the unmeasured porosity confounder must be fixed before the central claim is safe. read the letter →

arxiv 2607.07061 v3 pith:ORXCF4N2 submitted 2026-07-08 cs.CE cs.NAmath.NA

Donghan Kim This is my paper · ORCID
classification cs.CEcs.NAmath.NA MSC 55N3165M60
keywords persistenthomologyporousmediadiffusivereleasetopologicaldataanalysisrelease-curveclassificationfiniteelementmethodporositycontroldrugscreening
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

The paper sets out to show that the shape and multiscale organization of the solid phase in a porous medium—not just how much pore space there is—drives how quickly a dissolved tracer escapes. By computing persistent homology of the solid-offset filtration and solving a finite element diffusion problem on synthetic sphere packings, the author finds that within fixed target porosity levels, samples with richer topological features (more and longer-lived loops and cavities) tend to exhibit delayed, long-tailed release. Eighteen persistent-homology summary features classify three release regimes—early-fast, late-release, long-tail—with test accuracy between 64% and 76%, far above the 33% chance level, and the features are computed tens to hundreds of times faster than the diffusion simulation. A sympathetic reader would take this as evidence that persistent homology is a viable lightweight, interpretable screening descriptor for porous structures in applications like drug delivery and contaminant transport.

What carries the argument

The load-bearing object is the solid-offset filtration: the solid phase (union of spheres) is thickened by a distance r, and persistent homology records the birth and death of connected components (H0), loops (H1), and void-like regions (H2) as r grows; the weighted alpha complex computes the same diagrams efficiently. From each diagram, six scalar summaries (count, finite count, persistence sum, mean persistence, maximum persistence, mean birth) yield 18 features that carry the whole argument. The release curve Q(t) is generated independently by a conforming P1 finite element solve of the mixed Dirichlet–Neumann diffusion equation on a voxel-derived tetrahedral mesh, and the three regimes (

What would settle it

Measure the true pore volume of each generated sample (e.g., by voxel counting), then re-run the Kruskal–Wallis and classification analyses with actual porosity as a covariate or stratifier; if the PH–release associations vanish, the central claim is refuted.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that the multiscale topology of the solid phase carries release-behavior information that total porosity does not. In porosity-controlled comparisons (40%, 60%, 80% target porosity), Kruskal–Wallis tests show that persistence-sum and count summaries for H0, H1, and H2 differ significantly across release regimes, with the largest effect sizes for H0/H1 persistence sum (η²_H ≈ 0.27–0.54). Group means show a consistent gradient: early-fast samples have the smallest persistence counts and sums, long-tail samples the largest, with late-release in between. A multinomial logistic regression using only the 18 PH summaries achieves 0.641–0.759 test a

Load-bearing premise

The porosity-controlled result compares samples that only approximately match a prescribed target porosity, and true porosity is never measured or reported; if actual porosity varies within a target stratum and correlates with topology, the claimed 'beyond porosity' effect could be a hidden porosity effect.

Editorial extensions

If this is right

  • At any fixed porosity, two microstructures can have markedly different release curves, so porosity-based screening alone can mis-rank candidates.
  • Persistent-homology summaries can flag long-tail or slow-release candidates early in a design loop, reserving expensive FEM simulations for a shortlist.
  • The direction of association gives a design heuristic: to achieve fast release, keep the solid phase topologically simple; to delay release, introduce multi-scale loop/cavity structures.
  • Combining PH features with release-curve regression (not just regime labels) may yield fast surrogate models for Q(t) over large candidate libraries.
  • The classification accuracy (0.641–0.759) provides a baseline; adding further geometry descriptors or using the full persistence diagrams (not 18 summaries) may push accuracy higher.

Reading between the lines

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

  • If the association holds on experimentally measured microstructures, then porosity alone—the standard design metric in many drug-delivery papers—is an incomplete predictor, and PH summaries could be reported alongside porosity as a standard descriptor.
  • The implied mechanism (loops and cavities acting as topological traps that slow escape) could be tested by comparing PH features against pore-network tortuosity or effective diffusivity on the same geometries.
  • A direct extension would be to predict the entire release curve Q(t) from PH features using a regression model, and to check whether adding PH features to porosity reduces prediction error compared with porosity alone.
  • The self-referential structure (regimes defined from the same simulation and labels fitted on them) means the 64–76% accuracy partly reflects internal consistency; the true test is external validation on experimental release data.
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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

2 major / 5 minor

Summary. The paper proposes persistent homology (PH) of the solid phase as a fast, geometry-based descriptor for diffusive release from porous media. Synthetic samples are generated from overlapping spherical grains in six structural classes at target porosities of 40%, 60%, and 80% and at two voxel resolutions (16^3 and 32^3). Release curves are computed with a tetrahedral finite-element approximation of a mixed Dirichlet–Neumann diffusion problem. Release-curve regimes (early-fast, late-release, long-tail) are defined by quantiles of curve summaries. Within each target porosity, the authors compare 18 PH summaries across the three regimes using Kruskal–Wallis tests with Benjamini–Hochberg correction and report large effect sizes, especially for persistence sums. They also train a multinomial logistic regression on the PH features and report test accuracies between 0.641 and 0.759 at fixed porosities. Timing experiments show PH feature extraction is orders of magnitude faster than the finite-element release solve.

Significance. If the central claim holds—that solid-phase topology is associated with release behavior beyond pore-space amount—this is a potentially useful, interpretable screening descriptor for porous materials. The statistical machinery is generally appropriate: nonparametric tests, multiple-testing correction, effect sizes, stratified train–test splits, and balanced classes. The direction of the effect is consistent across porosity levels and grid resolutions, and the paper is appropriately cautious at the end of §4.2 that the analysis shows association, not causation. The main weakness is that the 'beyond pore-space amount' claim rests on stratification by target porosity, while actual realized porosity is never measured or reported. Since both the PH features and the release curves are functions of the same random geometry, the reported associations could be confounded by differences in realized porosity within target-porosity strata. The paper also evaluates classification only on deliberately selected extreme release-regime groups, which may overstate the screening utility. These issues are fixable within the manuscript's scope.

major comments (2)
  1. [§4.1 and §4.2 (Abstract; Table 4)] The central claim—that release behavior depends on multiscale solid organization 'even within each target-porosity level'—is supported only by stratifying on target porosity. However, §4.1 states that samples are generated only to 'approximately match a prescribed target porosity,' and no realized porosity is reported or used. Since sphere placements are random, realized porosity almost surely varies within each target stratum. If actual porosity is lower (more solid) in long-tail samples than in early-fast samples, the larger H0/H1/H2 persistence sums in Table 4 could reflect differences in solid fraction rather than in multiscale organization. Please report the distribution of realized porosity by regime and porosity level, and add a sensitivity analysis, e.g., including realized porosity as a covariate in the §4.3 multinomial model, or stratifying by binned realized porosity. Without
  2. [§4.3 (Table 5)] The reported classification accuracies (0.641–0.759) are computed on labels obtained by selecting extreme quantiles of release-curve summaries from 720 samples per porosity condition: top 180 by early release, then top 180 by final unreleased fraction, then top 180 by late gain; the remaining 180 samples are discarded. This extreme-group design makes the three classes artificially well separated, so an accuracy above the 1/3 chance level overstates the model's ability to screen a natural distribution of candidate geometries. To support the stated screening use case, evaluate the classifier on the unassigned 180 samples or on all 720 samples (e.g., by adding an 'unassigned' class or by predicting a continuous curve summary), and report those results alongside the extreme-regime accuracy. At minimum, state this selection explicitly as a limitation of §4.3.
minor comments (5)
  1. [§4.1, Table 5] The grid notation '163' and '323' should be 16^3 and 32^3; use superscripts to avoid ambiguity.
  2. [§4.1] The numerical values of t_end, t_20%, t_50%, the diffusion coefficient D, and the time step Δt are not reported. These are needed to reproduce the release curves and the regime definitions.
  3. [§4.1, Table 1] The text says structural classes are 'sampled approximately uniformly' and porosity is 'approximately match[ed]'; please describe the sampling/placement procedure and report the actual class counts and porosity ranges per condition.
  4. [Table 4] Only group means are reported. Since n = 540 per porosity level, the very small corrected q-values partly reflect sample size; reporting standard deviations or confidence intervals would help readers judge the magnitude of separation.
  5. [General] A code/data availability statement would improve reproducibility. If the finite-element solver and PH pipeline are not released, please state how the reported numbers can be obtained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: persistent homology features and FEM release curves are independently constructed, and the classification is evaluated on held-out data.

full rationale

The paper's derivation chain is not circular. The persistent homology features are computed directly from the weighted alpha complex of the solid-offset filtration (Remark 1, Section 3.2), i.e., from the geometry of the spherical solid phase. The release curves Q(t) are obtained by numerically solving the mixed Dirichlet–Neumann diffusion problem with finite elements on a voxel-derived tetrahedral mesh (Section 3.1). There is no equation linking the PH features to Q(t); the association is established empirically via Kruskal–Wallis tests and multinomial logistic regression with five stratified train–test splits. No parameter is fitted to Q(t) and then renamed as a prediction: the 18 persistence summaries are fixed scalar features, and the classifier is trained only on those features to predict curve-derived regime labels, with held-out test accuracy reported. The regime labels are indeed derived from Q(t) itself, but this is ordinary supervised classification, not circularity, and the paper explicitly discloses the associational nature of the result (Section 4.2: 'The release-curve regimes are defined from Q(t) itself... it does not by itself prove that these summaries causally determine the release curve'). There are no self-citations, no imported uniqueness theorems, and no ansatz smuggled in via prior work by the authors. The only notable weakness is that samples are generated to only 'approximately match a prescribed target porosity' and realized porosity is not reported, so the porosity-stratified comparison could be confounded if actual porosity varies systematically across regimes; however, this is a validity concern, not a circularity of the kind this analysis flags. Therefore the paper's central claim does not reduce to its inputs by construction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities, forces, or conserved quantities. Its central claim rests on synthetic sphere-pack geometries, a PDE model of diffusion, standard persistent-homology computations, and hand-chosen regime definitions. Several experimental degrees of freedom (actual porosity, structural class proportions, simulation parameters) are unstated or only approximately controlled.

free parameters (4)
  • target porosity levels = 40%, 60%, 80%
    Stratification levels chosen by the author; samples only approximately match these values, and actual porosity is never reported.
  • release-regime quantile thresholds = top 25% for early-fast; top 180 of remaining for long-tail; top 180 of remainder for late-release
    These hand-chosen thresholds define the response classes, and classification accuracy depends on them.
  • release summary time points = 20%, 50%, and 100% of an unreported tend
    Qearly, Qmid, and Qfinal are evaluated at fractions of the unstated simulation horizon; changing tend changes the regime labels.
  • simulation parameters = not reported (D, Delta t, tend, grain radii distributions, grain counts)
    Diffusion coefficient, time step, and sphere-generation hyperparameters are not specified, so the FEM release curves cannot be exactly reproduced.
assumptions (5)
  • domain assumption The solid phase is a finite union of (possibly overlapping) spheres inside the unit cube (Section 2).
    All conclusions are restricted to this synthetic geometry family; real microstructures may differ.
  • domain assumption Tracer release obeys the mixed Dirichlet–Neumann diffusion equation with uniform initial concentration (Section 2).
    The release curves are generated by this model, with no experimental validation.
  • standard math Voxelized pore geometry with a P1 tetrahedral FEM converges to the continuum solution under mesh refinement (Section 3.1, citing standard theory).
    No convergence study or analytic validation is reported for the actual release curves.
  • standard math The nerve theorem applies to the weighted alpha filtration, so that persistent homology of the solid-offset filtration is computed correctly (Remark 1).
    Standard topological result, assumed rather than verified numerically in this paper.
  • ad hoc to paper Release regimes defined by quantiles of Q summaries capture behaviorally meaningful classes (Section 4.1).
    The regime definitions are constructed for this study; no external or experimental definition of early-fast, late-release, or long-tail is used.

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Pith. "Pith review of Topological Signatures of Diffusive Release in Porous Media." pith.science (2026). https://pith.science/paper/ORXCF4N2

@misc{pith2026260707061,
  author       = {Pith},
  title        = {Pith review of: Topological Signatures of Diffusive Release in Porous Media},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ORXCF4N2}},
  note         = {Machine review of arXiv:2607.07061}
}
read the original abstract

We used persistent homology to quantify the multiscale topological and geometric organization of porous media, including solid connectivity and the formation of loop-like and cavity-like structures across spatial scales. Through statistical analysis, we show that these topological and geometric features are closely associated with diffusion-driven release behavior in porous media. In particular, even within each target-porosity level, samples with richer topological features tend to exhibit long-tailed release, indicating that release behavior depends not only on the amount of pore space but also on the multiscale organization of the solid phase. We further show that persistent homology-based features can classify release-curve regimes using a simple classification model. Notably, feature extraction is substantially faster than finite element diffusion simulations. Together, these results suggest that persistent homology provides a lightweight, interpretable, and geometry-based descriptor for screening diffusive release behavior in porous media.

Figures

Figures reproduced from arXiv: 2607.07061 by the authors.

Figure 1
Figure 1. A schematic porous medium in a square cross-section of the sample domain [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. New rings appear at ri (i = 3, 4), and the filled rings indicate that these features disappear. Roughly speaking, the lifetimes di − bi in a persistence diagram provide scale information about the corresponding topological features. For example, long lifetimes in the H1 or H2 persistence diagrams indicate geometric or topological structures that persist across a wide range of offset scales, whereas short lifetimes c… view at source ↗
Figure 3
Figure 3. Schematic illustration of the three release regimes used in this study. Early-fast samples are [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Representative release curves and weighted-alpha persistence diagrams for the three release regimes. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Porosity-controlled association between release-curve regimes and persistence-diagram summaries. [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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Reference graph

Works this paper leans on

47 extracted references

  1. [1]

    Clarendon Press, Oxford, 2 edition, 1975

    John Crank.The Mathematics of Diffusion. Clarendon Press, Oxford, 2 edition, 1975

    1975

  2. [2]

    Kluwer Academic Publishers, 1998

    Peter Grathwohl.Diffusion in Natural Porous Media: Contaminant Transport, Sorption/Desorption and Dissolution Kinetics. Kluwer Academic Publishers, 1998. doi: 10.1007/978-1-4615-5683-1. 12

    doi:10.1007/978-1-4615-5683-1 1998

  3. [3]

    Tartakovsky and Marco Dentz

    Daniel M. Tartakovsky and Marco Dentz. Diffusion in porous media: Phenomena and mechanisms. Transport in Porous Media, 130(1):105–127, 2019. doi: 10.1007/s11242-019-01262-6

    doi:10.1007/s11242-019-01262-6 2019

  4. [4]

    Wiley-VCH, Weinheim, 2 edition, 2011

    Muhammad Sahimi.Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches. Wiley-VCH, Weinheim, 2 edition, 2011. doi: 10.1002/9783527636693

    doi:10.1002/9783527636693 2011

  5. [5]

    Blunt, Branko Bijeljic, Hu Dong, Oussama Gharbi, Stefan Iglauer, Peyman Mostaghimi, Adriana Paluszny, and Christopher Pentland

    Martin J. Blunt, Branko Bijeljic, Hu Dong, Oussama Gharbi, Stefan Iglauer, Peyman Mostaghimi, Adriana Paluszny, and Christopher Pentland. Pore-scale imaging and modelling.Advances in Water Resources, 51:197–216, 2013. doi: 10.1016/j.advwatres.2012.03.003

    doi:10.1016/j.advwatres.2012.03.003 2013

  6. [6]

    Habibian, and Charles R

    Kuan-Mu Yao, Mohammad T. Habibian, and Charles R. O’Melia. Water and waste water filtration: Concepts and applications.Environmental Science & Technology, 5(11):1105–1112, 1971. doi: 10.1021/es60058a005

    doi:10.1021/es60058a005 1971

  7. [7]

    Porosity of 3d biomaterial scaffolds and osteogenesis

    Vassilis Karageorgiou and David Kaplan. Porosity of 3d biomaterial scaffolds and osteogenesis. Biomaterials, 26(27):5474–5491, 2005

    2005

  8. [8]

    Modeling of diffusion controlled drug delivery.Journal of Controlled Release, 161(2):351–362, 2012

    Juergen Siepmann and Florence Siepmann. Modeling of diffusion controlled drug delivery.Journal of Controlled Release, 161(2):351–362, 2012. doi: 10.1016/j.jconrel.2011.10.006

    doi:10.1016/j.jconrel.2011.10.006 2012

Show all 47 references
  1. [9]

    Porous carriers for controlled/modulated drug delivery.Indian Journal of Pharmaceutical Sciences, 71(6):599–607, 2009

    Gaurav Ahuja and Kamla Pathak. Porous carriers for controlled/modulated drug delivery.Indian Journal of Pharmaceutical Sciences, 71(6):599–607, 2009. doi: 10.4103/0250-474X.59540

    2009 doi

  2. [10]

    Mathematical modeling and simulation of drug release from microspheres: Implications to drug delivery systems.Advanced Drug Delivery Reviews, 58(12–13):1274–1325, 2006

    Davis Yohanes Arifin, Lai Yeng Lee, and Chi-Hwa Wang. Mathematical modeling and simulation of drug release from microspheres: Implications to drug delivery systems.Advanced Drug Delivery Reviews, 58(12–13):1274–1325, 2006. doi: 10.1016/j.addr.2006.09.007

    2006 doi

  3. [11]

    Ford Versypt, Daniel W

    Ashlee N. Ford Versypt, Daniel W. Pack, and Richard D. Braatz. Mathematical modeling of drug delivery from autocatalytically degradable PLGA microspheres–a review.Journal of Controlled Release, 165(1):29–37, 2013. doi: 10.1016/j.jconrel.2012.10.015

    2013 doi

  4. [12]

    Mathematical modelling of variable porosity coatings for controlled drug release.Medical Engineering & Physics, 45:51–60, 2017

    Sean McGinty, David King, and Giuseppe Pontrelli. Mathematical modelling of variable porosity coatings for controlled drug release.Medical Engineering & Physics, 45:51–60, 2017. doi: 10.1016/j. medengphy.2017.04.006

    2017 doi

  5. [13]

    Ml-lbm: predicting and accelerating steady state flow simulation in porous media with convolutional neural networks.Transport in Porous Media, 138(1):49–75, 2021

    Ying Da Wang, Traiwit Chung, Ryan T Armstrong, and Peyman Mostaghimi. Ml-lbm: predicting and accelerating steady state flow simulation in porous media with convolutional neural networks.Transport in Porous Media, 138(1):49–75, 2021

    2021

  6. [14]

    Machine learning models to accelerate the design of polymeric long-acting injectables

    Pauric Bannigan, Zeqing Bao, Riley J Hickman, Matteo Aldeghi, Florian H¨ase, Al´an Aspuru-Guzik, and Christine Allen. Machine learning models to accelerate the design of polymeric long-acting injectables. Nature communications, 14(1):35, 2023

    2023

  7. [15]

    Machine learning predicts drug release profiles and kinetic parameters based on tablets’ formulations.The AAPS Journal,

    Chrystalla Protopapa, Angeliki Siamidi, Amelia Adibe Eneli, Moe Elbadawi, et al. Machine learning predicts drug release profiles and kinetic parameters based on tablets’ formulations.The AAPS Journal,

  8. [16]

    Robles and Manar D

    Karla N. Robles and Manar D. Samad. Predicting early and complete drug release from long-acting injectables using explainable machine learning.International Journal of Pharmaceutics, 698:126922,

  9. [17]

    Topology and data.Bulletin of the American Mathematical Society, 46(2):255–308,

    Gunnar Carlsson. Topology and data.Bulletin of the American Mathematical Society, 46(2):255–308,

  10. [18]

    Barcodes: The persistent topology of data.Bulletin of the American Mathematical Society, 45(1):61–75, 2008

    Robert Ghrist. Barcodes: The persistent topology of data.Bulletin of the American Mathematical Society, 45(1):61–75, 2008. doi: 10.1090/S0273-0979-07-01191-3

    2008 doi

  11. [19]

    Persistence barcodes for shapes

    Gunnar Carlsson, Afra Zomorodian, Anne Collins, and Leonidas Guibas. Persistence barcodes for shapes. InProceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing, pages 124–135, 2004

    2004

  12. [20]

    Topological persistence and simplification

    Herbert Edelsbrunner, David Letscher, and Afra Zomorodian. Topological persistence and simplification. Discrete & Computational Geometry, 28(4):511–533, 2002. doi: 10.1007/s00454-002-2885-2

    2002 doi

  13. [21]

    Computing persistent homology

    Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. InProceedings of the twentieth annual symposium on Computational geometry, pages 347–356, 2004

    2004

  14. [22]

    Sheppard

    Vanessa Robins, Mohammad Saadatfar, Olaf Delgado-Friedrichs, and Adrian P. Sheppard. Percolating length scales from topological persistence analysis of micro-CT images of porous materials.Water Resources Research, 52(1):315–329, 2016. doi: 10.1002/2015WR017937

    2016 doi

  15. [23]

    Ushizima, Dmitriy Morozov, Gunther H

    Daniela M. Ushizima, Dmitriy Morozov, Gunther H. Weber, Andrea G. C. Bianchi, James A. Sethian, and E. Wes Bethel. Augmented topological descriptors of pore networks for material science.IEEE Transactions on Visualization and Computer Graphics, 18(12):2041–2050, 2012. doi: 10....

    doi

  16. [24]

    Pore geometry characterization by persistent homology theory.Water Resources Research, 54(6):4150–4163, 2018

    Fei Jiang, Takeshi Tsuji, and Tomoyuki Shirai. Pore geometry characterization by persistent homology theory.Water Resources Research, 54(6):4150–4163, 2018. doi: 10.1029/2017WR021864

    2018 doi

  17. [25]

    Herring, Vanessa Robins, and Adrian P

    Anna L. Herring, Vanessa Robins, and Adrian P. Sheppard. Topological persistence for relating microstructure and capillary fluid trapping in sandstones.Water Resources Research, 55(1):555–573,

  18. [26]

    Mitchell, Jason E

    Chul Moon, Scott A. Mitchell, Jason E. Heath, and Matthew Andrew. Statistical inference over persistent homology predicts fluid flow in porous media.Water Resources Research, 55(11):9592–9603, 2019. doi: 10.1029/2019WR025171

    2019 doi

  19. [27]

    Ciarlet.The Finite Element Method for Elliptic Problems, volume 40 ofClassics in Applied Mathematics

    Philippe G. Ciarlet.The Finite Element Method for Elliptic Problems, volume 40 ofClassics in Applied Mathematics. SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898719208

    2002 doi

  20. [28]

    Brenner and L

    Susanne C. Brenner and L. Ridgway Scott.The Mathematical Theory of Finite Element Methods, volume 15 ofTexts in Applied Mathematics. Springer, New York, 3 edition, 2008. doi: 10.1007/ 978-0-387-75934-0

    2008

  21. [29]

    Springer, Berlin, Heidelberg, 2008

    Alfio Quarteroni and Alberto Valli.Numerical Approximation of Partial Differential Equations, volume 23 ofSpringer Series in Computational Mathematics. Springer, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-85268-1

    2008 doi

  22. [30]

    Springer, Berlin, Heidelberg, 2 edition, 2006

    Vidar Thom´ee.Galerkin Finite Element Methods for Parabolic Problems, volume 25 ofSpringer Series in Computational Mathematics. Springer, Berlin, Heidelberg, 2 edition, 2006. doi: 10.1007/3-540-33122-0

    2006 doi

  23. [31]

    American Mathematical Soc., 2010

    Herbert Edelsbrunner and John Harer.Computational topology: an introduction. American Mathematical Soc., 2010

    2010

  24. [32]

    Weighted persistent homology.Involve, a Journal of Mathematics, 12(5):823–837, 2019

    Gregory Bell, Austin Lawson, Joshua Martin, James Rudzinski, and Clifford Smyth. Weighted persistent homology.Involve, a Journal of Mathematics, 12(5):823–837, 2019. 14

    2019

  25. [33]

    On the imbedding of systems of compacta in simplicial complexes.Fundamenta Mathematicae, 35(1):217–234, 1948

    Karol Borsuk. On the imbedding of systems of compacta in simplicial complexes.Fundamenta Mathematicae, 35(1):217–234, 1948. doi: 10.4064/fm-35-1-217-234

    1948 doi

  26. [34]

    Kruskal and W

    William H. Kruskal and W. Allen Wallis. Use of ranks in one-criterion variance analysis.Journal of the American Statistical Association, 47(260):583–621, 1952. doi: 10.1080/01621459.1952.10483441

    1952

  27. [35]

    Yoav Benjamini and Yosef Hochberg. Controlling the false discovery rate: A practical and powerful approach to multiple testing.Journal of the Royal Statistical Society: Series B (Methodological), 57(1): 289–300, 1995. doi: 10.1111/j.2517-6161.1995.tb02031.x

    1995

  28. [36]

    The need to report effect size estimates revisited: An overview of some recommended measures of effect size.Trends in Sport Sciences, 21(1):19–25, 2014

    Maciej Tomczak and Ewa Tomczak. The need to report effect size estimates revisited: An overview of some recommended measures of effect size.Trends in Sport Sciences, 21(1):19–25, 2014

    2014

  29. [37]

    URL https://CRAN.R-project.org/package=rstatix

    Alboukadel Kassambara.rstatix: Pipe-Friendly Framework for Basic Statistical Tests, 2023. URL https://CRAN.R-project.org/package=rstatix. R package version 0.7.2

    2023

  30. [38]

    Conditional logit analysis of qualitative choice behavior

    Daniel McFadden. Conditional logit analysis of qualitative choice behavior. In Paul Zarembka, editor, Frontiers in Econometrics, pages 105–142. Academic Press, New York, 1974

    1974

  31. [39]

    Wiley, Hoboken, NJ, 3 edition, 2013

    Alan Agresti.Categorical Data Analysis. Wiley, Hoboken, NJ, 3 edition, 2013. ISBN 9780470463635

    2013

  32. [40]

    Harris, K

    Charles R. Harris, K. Jarrod Millman, St´efan J. van der Walt, et al. Array programming with numpy. Nature, 585:357–362, 2020. doi: 10.1038/s41586-020-2649-2

    2020 doi

  33. [41]

    Oliphant, et al

    Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, et al. Scipy 1.0: Fundamental algorithms for scientific computing in python.Nature Methods, 17:261–272, 2020. doi: 10.1038/s41592-019-0686-2

    2020 doi

  34. [42]

    GUDHI Editorial Board, 3.12.0 edition,

    The GUDHI Project.GUDHI User and Reference Manual. GUDHI Editorial Board, 3.12.0 edition,

  35. [47]

    URLhttps://gudhi.inria.fr/doc/3.12.0/. 15

  36. [2009]

    doi: 10.1090/S0273-0979-09-01249-X. 13

    doi

  37. [2019]

    doi: 10.1029/2018WR022780

    doi

  38. [2025]

    doi: 10.1208/s12248-025-01101-1

    doi

  39. [2026]

    doi: 10.1016/j.ijpharm.2026.126922

    2026

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