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arxiv: 2607.00050 · v1 · pith:C7SDDOZGnew · submitted 2026-06-29 · 🌊 nlin.CG · econ.TH

Revising price coordination in the classical and neoclassical economics based on elementary cellular automata

Igor Lugo , Martha G. Alatriste-Contreras This is my paper

Pith reviewed 2026-07-02 20:21 UTC · model grok-4.3

classification 🌊 nlin.CG econ.TH
keywords elementary cellular automataprice coordinationclassical economicsneoclassical economicsShannon entropyMonte Carlo simulationSpearman correlationmarket clearing
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The pith

Elementary cellular automata simulations indicate that classical economics supports stable price coordination through logical interactions while neoclassical economics does not.

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

This paper uses elementary cellular automata to simulate price coordination under classical and neoclassical economic theories. It shows that classical rules, based on human interactions and logical choices from objective data, produce stable patterns that clear markets. Neoclassical rules lack any mechanism, treating individuals as passive recipients of unpredictable prices. The study employs Shannon entropy to assess uncertainty in coordination patterns and Monte Carlo methods with Spearman correlation to test statistical significance. These findings suggest classical economics provides a consistent framework for price coordination.

Core claim

By modeling economic theory with elementary cellular automata, simple rules of price interactions generate stable patterns under classical economics and unstable or absent coordination under neoclassical economics, revealing that classical theory emphasizes logical human interactions while neoclassical treats price coordination as an unpredictable event without mechanism.

What carries the argument

Elementary cellular automata rules representing price interactions, where classical rules incorporate logical choices based on objective data and neoclassical rules do not.

If this is right

  • Classical economics can be modeled to produce consistent market-clearing price patterns via agent interactions.
  • Neoclassical economics lacks a proposed mechanism for price coordination in this framework.
  • Shannon entropy can quantify the uncertainty and stability of generated price coordination patterns.
  • Monte Carlo simulations with Spearman correlation can evaluate the statistical significance of price coordination in these models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be applied to test other economic theories for their coordination mechanisms.
  • If the rule mappings are accurate, neoclassical models might require additional components to explain price formation.
  • Simple local interaction rules in automata can lead to global economic coordination behaviors.

Load-bearing premise

The specific elementary cellular automata rules chosen are assumed to accurately capture the essential differences between classical and neoclassical approaches to price coordination.

What would settle it

A simulation where neoclassical rules generate stable, statistically significant price coordination patterns similar to classical ones would challenge the paper's distinction between the two approaches.

Figures

Figures reproduced from arXiv: 2607.00050 by Igor Lugo, Martha G. Alatriste-Contreras.

Figure 1
Figure 1. Figure 1: Examples of classes in elementary CAs. (a) Class 1. (b) Class 2. (c) Class 3. (d) Class 4. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Price interactions and their actualization based on a market balance. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Classical Rule 01. (a) random initial condition. (b) simple initial condition. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Classical Rule 02. (a) random initial condition. (b) simple initial condition. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Classical Rule 03. (a) random initial condition. (b) simple initial condition. [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Examples of class 1 of elementary CAs. Rule 1. Table 4 shows that (a) and (b) cases [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Examples of class 2 of elementary CAs. Rule 218. Table 5 shows that (a) and (b) cases [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Examples of class 3 of elementary CAs. Rule 30. Table 6 shows that (a) is statistically [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
read the original abstract

The coordination of prices in economics is one of the most complex phenomena. In particular, the classical and neoclassical approaches related to the economic theory provide some insights into such a complex coordination based on different formulations. However, these formulations have not been successful for explaining simple mechanisms to understand and predict a set of prices that theoretically clears all markets. Consequently, elementary cellular automata can contribute to clarify such a coordination problem by using simple computational rules to describe the theoretical bases of the classical and neoclassical economics. Therefore, we propose to use this type of cellular automata for explaining different escenarios of price coordination in which simple rules of price interactions generate stable and unstable patterns of coordination. We used an explorative data analysis based on the Shannon entropy for computing the uncertainty related to such generated patterns of coordination, and a Monte Carlo simulation approximation based on a Spearman correlation for evaluating the statistical significance of such price coordination. Findings suggested that the classical economics provides a consistent approach for understanding the coordination of prices because it emphasizes human interactions based on logical choices related to an objective data. On the other hand, the neoclassical approach does not propose any type of mechanism for describing the price coordination. The neoclassical individual is just a spectator and receiver of the unpredictable and supposed event of price coordination. As a result, by modeling the economic theory based on computational concepts, we reveal facts and believes behind the classical and neoclassical economics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims that elementary cellular automata (ECA) can model price coordination mechanisms in classical versus neoclassical economics. Using Shannon entropy to quantify uncertainty in generated patterns and Monte Carlo Spearman correlations for statistical significance, the simulations are said to show that classical economics produces consistent, low-entropy coordination patterns reflecting logical human interactions with objective data, while neoclassical economics yields unstable patterns with no intrinsic coordination mechanism, leaving the individual as a passive recipient of unpredictable prices.

Significance. If a non-arbitrary mapping from economic primitives to ECA rules could be established, the approach might supply a computational testbed for comparing coordination claims across economic schools. The use of entropy measures and Monte Carlo tests on pattern stability is a reasonable exploratory step, but the current absence of any derivation or validation for the rule-to-theory encoding prevents the results from bearing on the stated theoretical distinction.

major comments (3)
  1. [Methodology / model construction] The central claim requires that chosen ECA update rules faithfully encode the paper's stated distinction (classical as 'human interactions based on logical choices related to an objective data' versus neoclassical as having 'no type of mechanism'). No section derives the rule-to-mechanism mapping from economic primitives (e.g., supply-demand logic or agent optimization) or tests robustness against alternative plausible mappings. Without that link, entropy differences are artifacts of rule selection rather than evidence for the economic conclusion.
  2. [Simulation setup and results] The abstract and results state conclusions from simulations and statistical tests but supply no description of rule selection, initial conditions, lattice size, number of steps, or how the two economic schools are encoded in the automata. This omission makes it impossible to assess whether the reported entropy and Spearman outcomes support the comparative claim.
  3. [Discussion and conclusions] The interpretive conclusion that 'the neoclassical individual is just a spectator' appears to be an overlay placed on the simulation outputs rather than generated by the automata rules themselves; the paper does not demonstrate that the instability is a necessary consequence of the neoclassical formulation rather than a property of the particular rules chosen.
minor comments (2)
  1. [Abstract] Abstract contains spelling/usage issues: 'escenarios' should be 'scenarios'; 'believes' should be 'beliefs'.
  2. [Methods] Notation for the ECA rules and entropy calculations is not defined explicitly before use in the statistical sections.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments. The feedback identifies key areas where the manuscript requires greater transparency in model construction, simulation parameters, and interpretive framing. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Methodology / model construction] The central claim requires that chosen ECA update rules faithfully encode the paper's stated distinction (classical as 'human interactions based on logical choices related to an objective data' versus neoclassical as having 'no type of mechanism'). No section derives the rule-to-mechanism mapping from economic primitives (e.g., supply-demand logic or agent optimization) or tests robustness against alternative plausible mappings. Without that link, entropy differences are artifacts of rule selection rather than evidence for the economic conclusion.

    Authors: We agree that the manuscript does not derive the ECA rules from formal economic primitives such as supply-demand equations or optimization. The rules were selected on conceptual grounds to capture interactive logical choices versus absence of mechanism. This remains an exploratory mapping. In revision we will add an explicit subsection justifying each rule choice from the theoretical descriptions in the paper and will discuss the exploratory status together with possible alternative mappings. revision: yes

  2. Referee: [Simulation setup and results] The abstract and results state conclusions from simulations and statistical tests but supply no description of rule selection, initial conditions, lattice size, number of steps, or how the two economic schools are encoded in the automata. This omission makes it impossible to assess whether the reported entropy and Spearman outcomes support the comparative claim.

    Authors: The original text omitted these parameters. We will expand the methods section to report the specific ECA rules (Wolfram numbers), initial conditions, lattice sizes, time steps, and the precise encoding of classical versus neoclassical behavior so that the entropy and correlation results can be evaluated and replicated. revision: yes

  3. Referee: [Discussion and conclusions] The interpretive conclusion that 'the neoclassical individual is just a spectator' appears to be an overlay placed on the simulation outputs rather than generated by the automata rules themselves; the paper does not demonstrate that the instability is a necessary consequence of the neoclassical formulation rather than a property of the particular rules chosen.

    Authors: The conclusion is interpretive and links observed instability to the chosen encoding of a missing coordination mechanism. We will revise the discussion to show how the instability follows from the absence of local interaction rules in the neoclassical encoding and will explicitly note the limitation that other encodings could produce different behavior. revision: partial

Circularity Check

1 steps flagged

ECA rule assignment to economic theories is definitional; stability/entropy differences follow by construction from the chosen mapping

specific steps
  1. self definitional [Abstract (model setup and findings paragraph)]
    "we propose to use this type of cellular automata for explaining different escenarios of price coordination in which simple rules of price interactions generate stable and unstable patterns of coordination. ... Findings suggested that the classical economics provides a consistent approach for understanding the coordination of prices because it emphasizes human interactions based on logical choices related to an objective data. On the other hand, the neoclassical approach does not propose any type of mechanism for describing the price coordination."

    The rules are stipulated to embody the very distinction (logical objective-data interactions vs. no mechanism) that the entropy results are then said to 'suggest' or 'reveal.' The output patterns inherit their economic interpretation solely from the input labeling; the statistical tests therefore confirm the modeling premise rather than test an independent claim.

full rationale

The paper selects specific elementary cellular automata rules to stand for 'classical' (logical human interactions on objective data) versus 'neoclassical' (no mechanism, spectator) price coordination, then runs Shannon entropy and Monte Carlo Spearman tests on the generated patterns. The central finding—that classical yields consistent coordination while neoclassical does not—directly restates the interpretive labels attached to the rules at input. No section derives the rule-to-mechanism correspondence from economic primitives or validates it against alternatives; the conclusion is therefore the modeling choice restated in simulation language.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the central claim rests on an unstated mapping from economic concepts to specific automata rules and on standard statistical tools whose application details are not provided.

axioms (2)
  • standard math Shannon entropy quantifies uncertainty in generated coordination patterns
    Used to compute uncertainty related to price coordination patterns.
  • standard math Monte Carlo simulation with Spearman correlation can evaluate statistical significance of coordination
    Applied to assess significance of price coordination.

pith-pipeline@v0.9.1-grok · 5787 in / 1273 out tokens · 36157 ms · 2026-07-02T20:21:53.866929+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    (2012), The network takeover

    Barabási, AL. (2012), The network takeover. Nature Phys 8, 14--16. https://doi.org/10.1038/nphys2188 https://doi.org/10.1038/nphys2188

    work page doi:10.1038/nphys2188 2012

  2. [2]

    Collected Works

    von Neumann, J. (1963), The General and Logical Theory of Automata, in J.von Neumann, "Collected Works" (ed. A.H.Taub), Vol. 5, p.288--328, https://www.vordenker.de/ggphilosophy/jvn\_the-general-and-logical-theory-of-automata.pdf https://www.vordenker.de/ggphilosophy/jvn_the-general-and-logical-theory-of-automata.pdf

    work page 1963

  3. [3]

    (1966), Theory of Self-Reproducing Automata, (ed

    von Neumann, J. (1966), Theory of Self-Reproducing Automata, (ed. A.W.Burks), Univ. of Illinois press (1966). https://cba.mit.edu/events/03.11.ASE/docs/VonNeumann.pdf https://cba.mit.edu/events/03.11.ASE/docs/VonNeumann.pdf

    work page 1966

  4. [4]

    (1970), Essays on Cellular Automata, ed

    von Neumann, J. (1970), Essays on Cellular Automata, ed. A.W.Burks, Univ. of Illinois press,

    work page 1970

  5. [5]

    (2002), A New Kind of Science, Wolfram Media, Inc

    Wolfram, S. (2002), A New Kind of Science, Wolfram Media, Inc

    work page 2002

  6. [6]

    Wolfram, S. (2020). A Class of Models with the Potential to Represent Fundamental Physics, Complex Systems, 29(2), 2020 pp. 107--536. https://doi.org/10.25088/ComplexSystems.29.2.107 https://doi.org/10.25088/ComplexSystems.29.2.107

    work page doi:10.25088/complexsystems.29.2.107 2020

  7. [7]

    Turing, A. M. (1952), The chemical basis of morphogenesis, Philos. Trans. R. Soc. London, Ser. B 237, 37

    work page 1952

  8. [8]

    and Prigogine, I., (1977), Self-Organization in Nonequilibrium Systems (Wiley, New York)

    Nicolis, G. and Prigogine, I., (1977), Self-Organization in Nonequilibrium Systems (Wiley, New York)

    work page 1977

  9. [9]

    Hegselmann, R. (1996). Cellular Automata in the Social Sciences. In: Hegselmann, R., Mueller, U., Troitzsch, K.G. (eds) Modelling and Simulation in the Social Sciences from the Philosophy of Science Point of View. Theory and Decision Library, vol 23. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8686-3\_12 https://doi.org/10.1007/978-94-015-8686-3_12

    work page doi:10.1007/978-94-015-8686-3 1996

  10. [10]

    Lugo, I., Alatriste-Contreras, M.G. (2022). Intervention strategies with 2D cellular automata for testing SARS-CoV-2 and reopening the economy. Sci Rep 12, 13481. https://doi.org/10.1038/s41598-022-17665-3 https://doi.org/10.1038/s41598-022-17665-3

    work page doi:10.1038/s41598-022-17665-3 2022

  11. [11]

    Lugo, I., Alatriste-Contreras, M.G. (2022). The personal space and the collective behavior of crowd disasters, BioRxive: https://doi.org/10.1101/2024.09.05.611443 https://www.biorxiv.org/content/10.1101/2024.09.05.611443v2

    work page doi:10.1101/2024.09.05.611443 2022

  12. [12]

    (1984), The Evolution of Cooperation

    Axelrod, R. (1984), The Evolution of Cooperation. New York: Basic Books

    work page 1984

  13. [13]

    Krugman, P. (2022). The Self Organizing Economy. Wiley-Blackwell

    work page 2022

  14. [14]

    Shannon, C. (1948). A Mathematical Theory of Communication. Bell System Technical Journal 27(3)

    work page 1948

  15. [15]

    and Peng, R.D

    Leek, J.T. and Peng, R.D. (2015),What is the question?Science, 347,1314--1315(2015). doi:10.1126/science.aaa6146 https://www.science.org/doi/10.1126/science.aaa6146

    work page doi:10.1126/science.aaa6146 2015

  16. [16]

    and Graves, G.R

    Gotelli, N.J. and Graves, G.R. (1996) Null Models in Ecology. Smithsonian Institution Press, Washington, DC, USA

    work page 1996

  17. [17]

    Farine, D.R. (2017), A guide to null models for animal social network analysis, Methods in Ecology and Evolution 8, 1309--1320, https://doi.org/10.1111/2041-210X.12772 https://doi.org/10.1111/2041-210X.12772

    work page doi:10.1111/2041-210x.12772 2017

  18. [18]

    and Zamagni, S

    Becchetti, L., Bruni, L. and Zamagni, S. (2020). Chapter 1 - Economics: What it studies, with what methods, and how it evolved, The Microeconomics of Wellbeing and Sustainability, Academic Press, pages 1--49

    work page 2020

  19. [19]

    (2016), Complexity and the Art of Public Policy: Solving Society's Problems from the Bottom Up, Princeton University Press

    Colander, D, and Kupers, R. (2016), Complexity and the Art of Public Policy: Solving Society's Problems from the Bottom Up, Princeton University Press

    work page 2016

  20. [20]

    Smith, A.(1756), The Theory of Moral Sentiments, London: A. Miller

    work page

  21. [21]

    Smith, A.(1776), The Wealth of Nations, London: A. Miller

    work page

  22. [22]

    (1848), Principles of Political Economy, London: Longmans, Green

    Mill, J.S. (1848), Principles of Political Economy, London: Longmans, Green

    work page

  23. [23]

    (1890), Principles of Economics

    Marshall, A. (1890), Principles of Economics. 9th edition. London Macmillan

    work page

  24. [24]

    (1920), The Economics of Welfare London, Reprint

    Pigou, A.C. (1920), The Economics of Welfare London, Reprint. New Brunswick, N.J.: Transaction

    work page 1920

  25. [25]

    Rouge Editeur, Librairie de l'Universite http://digamo.free.fr/walras96.pdf

    Walras, L (1896), Éléments d'économie politique pure; ou, Théorie de la richesse sociale, Lausanne, F. Rouge Editeur, Librairie de l'Universite http://digamo.free.fr/walras96.pdf

    work page

  26. [26]

    (1909), Manual of Political Economy: A Critical and Variorum Edition, Oxford University Press

    Pareto, V. (1909), Manual of Political Economy: A Critical and Variorum Edition, Oxford University Press

    work page 1909

  27. [27]

    (1951) An extension of the basic theorems of classical welfare economics, Proceedings of the Second Berkeley Symposium on Mathemalical Statisties and Probability, ed J

    Arrow, K.J. (1951) An extension of the basic theorems of classical welfare economics, Proceedings of the Second Berkeley Symposium on Mathemalical Statisties and Probability, ed J. Neyman, University of California Press, 507--532

    work page 1951

  28. [28]

    Econometrica 19, 273--92

    Debreu, G (1951), The coefficient of resource utilization. Econometrica 19, 273--92

    work page 1951

  29. [29]

    Frege, G. (1879), Begriflsschriit, a formula language, modeled upon that of arithmetic, for pure thought https://dn720006.ca.archive.org/0/items/gottlob-frege-begriffsschrift-english/Gottlob\

    work page

  30. [30]

    (2014), Computational Logic, in Computational Logic edited by Jörg H

    Siekmann, J. (2014), Computational Logic, in Computational Logic edited by Jörg H. Siekmann, North Holland publications, pp: 15--30 https://www.sciencedirect.com/science/chapter/handbook/abs/pii/B9780444516244500010

    work page 2014

  31. [31]

    (2010), Complex Economics, Individual and Collective Rationality, London, Routledge

    Kirman, A. (2010), Complex Economics, Individual and Collective Rationality, London, Routledge

    work page 2010

  32. [32]

    and Kirman, A

    Wilson, D. and Kirman, A. (2016), Complexity and Evolution: Toward a New Synthesis for Economics, Cambridge, MA, The MIT Press

    work page 2016

  33. [33]

    (2007) Matplotlib: A 2D Graphics Environment, Computing in Science & Engineering, vol

    Hunter, J.D. (2007) Matplotlib: A 2D Graphics Environment, Computing in Science & Engineering, vol. 9, no. 3, pp. 90--95

    work page 2007

  34. [34]

    Harris, C.R., Millman, K.J., van der Walt, S.J. et al. (2020), Array programming with NumPy. Nature 585, 357--362 (2020). DOI: 10.1038/s41586-020-2649-2

    work page doi:10.1038/s41586-020-2649-2 2020

  35. [35]

    Promoting an open research culture

    Nosek BA, Alter G, Banks GC, Borsboom D, Bowman SD, Breckler SJ, Buck S, Chambers CD, Chin G, Christensen G, Contestabile M, Dafoe A, Eich E, Freese J, Glennerster R, Goroff D, Green DP, Hesse B, Humphreys M, Ishiyama J, Karlan D, Kraut A, Lupia A, Mabry P, Madon T, Malhotra N, Mayo-Wilson E, McNutt M, Miguel E, Paluck EL, Simonsohn U, Soderberg C, Spellm...

    work page doi:10.1126/science.aab2374 2015

  36. [36]

    (1987), Capital, Volume 1 2 3

    Marx, K. (1987), Capital, Volume 1 2 3. International Publishers Company, Incorporated

    work page 1987

  37. [37]

    (1969), Models of Segregation, The American Economic Review, Papers and Proceedings of the Eighty-first Annual Meeting of the American Economic Association,Vol

    Schelling, T.C. (1969), Models of Segregation, The American Economic Review, Papers and Proceedings of the Eighty-first Annual Meeting of the American Economic Association,Vol. 59, No. 2, pp. 488--493

    work page 1969

  38. [38]

    (1978), Micromotives and Macrobehavior (Fels Lectures on Public Policy Analysis), W

    Schelling, T.C. (1978), Micromotives and Macrobehavior (Fels Lectures on Public Policy Analysis), W. W. Norton & Company

    work page 1978

  39. [39]

    Axelrod, R. (1997). The Dissemination of Culture: A Model with Local Convergence and Global Polarization: A Model with Local Convergence and Global Polarization. Journal of Conflict Resolution, 41(2), 203--226. https://journals.sagepub.com/doi/10.1177/0022002797041002001 https://journals.sagepub.com/doi/10.1177/0022002797041002001

    work page doi:10.1177/0022002797041002001 1997

  40. [40]

    Lugo, I., and Martínez-Mekler, G. (2022). Theoretical study of the effect of ports in the formation of city systems. J. shipp. trd. 7, 16. https://doi.org/10.1186/s41072-022-00117-6 https://doi.org/10.1186/s41072-022-00117-6

    work page doi:10.1186/s41072-022-00117-6 2022

  41. [41]

    (1995), Microeconomic Theory, Oxford University Press

    Mas-Colell, A. (1995), Microeconomic Theory, Oxford University Press

    work page 1995

  42. [42]

    Wolfram, S. (1983). Statistical Mechanics of Cellular Automata. Rev. Mod. Phys. 55, 601-644

    work page 1983

  43. [43]

    (2009), Cellular Automata Ordered by Entropy, Wolfram Demonstrations Project

    de Souza Carvalho, D. (2009), Cellular Automata Ordered by Entropy, Wolfram Demonstrations Project. https://demonstrations.wolfram.com/cellularAutomataOrderedByEntropy/ Cellular Automata Ordered by Entropy

    work page 2009

  44. [44]

    (2006), Entropy and Diversity

    Jost, L. (2006), Entropy and Diversity. Oikos, 2006, Vol. 113, No. 2 (May, 2006), pp. 363--375 Published by: Wiley on behalf of Nordic Society Oikos Stable

    work page 2006

  45. [45]

    and Goethel, D.R

    Kerr, L.A. and Goethel, D.R. (2014). Simulation Modeling as a Tool for Synthesis of Stock Identification Information, Editor(s): Steven X. Cadrin, Lisa A. Kerr, Stefano Mariani, Stock Identification Methods (Second Edition), Academic Press, pages 501--533, doi:10.1016/B978-0-12-397003- 9.00021-7 doi:10.1016/B978-0-12-397003- 9.00021-7

    work page doi:10.1016/b978-0-12-397003- 2014

  46. [46]

    and Coutiño-Vázquez, B.G

    Lugo, I., Alatriste-Contreras, M.G. and Coutiño-Vázquez, B.G. (2025), Inclusive education via empathy propagation in schools of students with special education needs, arXiv:2511.16379 [cs.CY]. https://arxiv.org/abs/2511.16379 https://arxiv.org/abs/2511.16379

    work page arXiv 2025

  47. [47]

    and Rao, G

    Molugaram, K. and Rao, G. S. (2017), Random Variables, in Statistical Techniques for Transportation Engineering, Editor(s): Kumar Molugaram, G. Shanker Rao, Chapter 4, pp. 113--279, Butterworth-Heinemann

    work page 2017