Effective Complexity Reduction of the Landau-de Gennes Elastic Energy: A Quantitative Framework and Numerical Validation

Arghir-Dani Zarnescu; Razvan-Dumitru Ceuca; Simone Rusconi

REVIEW 4 major objections 6 minor 36 references

Effective Complexity Reduction of the Landau-de Gennes Elastic Energy: A Quantitative Framework and Numerical Validation

T0 review · 4 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper derives explicit thresholds on the Landau–de Gennes elastic constants below which dropping L3 or L2 leaves minimizers essentially unchanged, with error proportional to the dropped ratio.

desk verdict Useful new thresholds for reducing the LdG elastic energy, but the numerics never test the L3 boundary: the GOS value sits above the Ericksen cap, so the validation shows only that smaller is smaller. read the letter →

arxiv 2607.19920 v1 pith:YRUOIDUO submitted 2026-07-22 cond-mat.soft

Razvan-Dumitru Ceuca , Simone Rusconi , Arghir-Dani Zarnescu This is my paper

classification cond-mat.soft PACS 61.30.-v

keywords Landau-deGenneselasticconstantsone-constantapproximationgeneralizedoptimalscalingliquidcrystalsQ-tensorscaleseparationnumericalvalidation

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

This paper tries to turn a commonly used but ad-hoc simplification in liquid-crystal modeling — replacing the three-elastic-constant Landau–de Gennes energy (L1,L2,L3) with (L1,L2,0) or even the one-constant (L1,0,0) — into a quantitatively justified decision. Using the Generalized Optimal Scaling (GOS) procedure, the authors derive explicit thresholds: L3 must be much smaller than a 41st-root power-law combination of the other elastic constants, bulk coefficients, anchoring strength, and geometric lengths, and L2 must be much smaller than a 26th-root combination. Their numerical simulations of a spherical colloid in a nematic confirm that below these thresholds the relative deviation between full and reduced minimizers stays at the order of the ratio of the neglected constant to its threshold, with fitted power laws μ0 ≈ 0.174 λ3^0.820 and μ0 ≈ 0.925 λ2^0.928. If correct, this gives computational physicists a ready-to-use diagnostic for when simplified models are trustworthy, and it anchors the ubiquitous one-constant approximation in a parameter-dependent criterion rather than convention.

What carries the argument

The central object is the Generalized Optimal Scaling (GOS) procedure, an equation-free method that builds a complete set of dimensionless governing parameters from the physical parameters' units, then minimizes a cost function in log-space so that all dimensionless coefficients except the targeted one are driven to order unity. Its output is a power-law monomial threshold for the targeted parameter — for L3 a 41st root, for L2 a 26th root — and the paper's claim is that the ratio λ = P/P_crit controls the error: when λ ≪ 1, the minimizers of the full and reduced energies deviate only at order λ. The paper also proves (Appendix D) that these thresholds are invariant under the spatial and ene

What would settle it

Repeat the same comparison with the FIRE force cutoff set to 10^-10; if the measured relative deviation μ0 no longer tracks λ^0.82 or λ^0.93 at small λ but instead saturates at a noise floor, the numerical confirmation of the GOS scaling fails. Also, run the (L1,L2,0)→(L1,0,0) comparison at a different domain size (e.g., L=1 μm instead of 300 nm) or a different anchoring strength; the GOS threshold changes with these parameters, so the predicted μ0 should change accordingly.

Watch

Extended reading notes

Core claim

The claim, stated on the paper's own terms, is that both model reductions admit explicit boundaries: L3 << (L1^15 L2^15 a b c W^8 L^7 R^7)^(1/41) for (L1,L2,L3) → (L1,L2,0), and L2 << (L1^15 a b c L^7 R^7 W^8)^(1/26) for (L1,L2,0) → (L1,0,0). These thresholds are computed by GOS, which rescales space and energy so that every dimensionless coefficient except the targeted one is pushed toward unity, forcing the target constant to scale away. The numerical experiments bear out the scaling: the relative L2 deviation between minimizers is of the same order as λ = (neglected constant)/(critical threshold), decreasing to ~10^-7 when λ=10^-7, and the fitted exponents (0.82 and 0.93) are both near 1.

Load-bearing premise

The numerical verification assumes that the minimizer at force tolerance 10^-12 resolves Q-tensor differences as small as those produced by λ=10^-7, and that one spherical-colloid geometry with one initial random configuration represents the general parameter dependence of the GOS threshold — an assumption the paper itself flags in Remark 9, noting that looser tolerances inflate the measured error by one to two orders of magnitude.

Editorial extensions

If this is right

Computational physicists can pre-check whether a planned reduction is safe by evaluating one power-law monomial, and quantify the expected error before running expensive simulations.
The two reductions can be chained (L1,L2,L3)→(L1,L2,0)→(L1,0,0), with the total error bounded by roughly the sum of the two ratios; in the paper's example this justifies the one-constant approximation within 14%.
The same GOS machinery gives similar thresholds for other parameters (W, L, R, a, b, c) in Tables II and III, so the framework extends beyond elastic constants.
The fitted power laws (exponents near 1) mean that the deviation shrinks essentially linearly with the ratio λ, so a factor-of-ten smaller constant gives a factor-of-ten smaller error.

Reading between the lines

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

The two fitted exponents (0.82 and 0.93) sit close to, but not exactly at, 1; whether the deviation from unity is a finite-size effect, a consequence of the single geometry, or an intrinsic feature of GOS is not resolved by the paper, and testing a second geometry would distinguish these.
Because the thresholds depend on W, L, and R as powers, the criterion can be rephrased as a dimensionless comparison — for instance, against the anchoring extrapolation length K/W — which could connect GOS to familiar regimes in colloid science without additional computation.
The strong sensitivity of μ0 to the FIRE force tolerance (Remark 9) suggests that any practical use of this diagnostic should include a tolerance check; otherwise, numerical noise at looser tolerances will be mistaken for the physical scaling.
A natural next test is a multi-colloid or patterned-wall system, where the elastic energy landscape is qualitatively different; if the same thresholds hold there, the criterion would become a general tool for complex nematic geometries.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the paper's product — the explicit GOS thresholds in Eqs. (5) and (7) telling you when the three-constant LdG elastic energy can be reduced to two- or one-constant form — is new, dimensionally consistent, and worth having. Appendix B works through the GOS machinery carefully; Appendix D's invariance check is a nice catch; and they post the raw openQmin outputs on Zenodo, so the numerics are checkable. That part deserves credit.

The validation is where it goes soft, and the stress-test note holds up when I put the numbers in. For their parameters, L1 ≈ 3.91 pN, so the Ericksen condition caps L3 below 2L1 ≈ 7.83 pN. Their GOS threshold is L3* ≈ 9.28 pN — above the cap. So every stable material automatically satisfies Eq. (5), and the sweep never crosses the boundary it is meant to test. The λ3 = 1 row in Table IV is rescaled L3 ≈ 5.99 versus 2L1 ≈ 5.06, an inadmissible configuration by the paper's own K24 < 2K2 constraint. The data therefore show the trivial direction — smaller L3 gives smaller deviations — and the honest phrasing is that the numerics are consistent with the threshold, not that they confirm it. The L2 case is not subject to the same objection: Eq. (4) puts no upper bound on L2, so the criterion could bite, though the λ2 = 1 end needs K1 ≈ 21.5 pN, formally admissible but a strange material.

The reader's other complaint is fair: Eqs. (6) and (8) are power-law fits to the same data they are then said to confirm. GOS predicts the threshold; it does not predict the fitted exponents. The exponents are close to but not at 1, and Remark 9 concedes that at the smallest λ the measured µ0 is inflated by one to two orders of magnitude under a more realistic force tolerance — which is exactly the end of the curve that anchors the fits. Add in a single geometry, a single material, one random initial configuration (their Remark 8), and a GOS benchmark that lives in the authors' own 2025 preprint, and the phrase 'direct connection' in the abstract is more than the evidence supports.

None of this kills the paper. It is an openly written derivation with reproducible data and a practically useful rule: compute P/P*, and if it is well below 1, drop the constant. The right audience is anyone running LdG simulations who currently picks the one-constant model by habit.

My recommendation: send it to peer review. The thresholds deserve referee time. A competent referee will press on the Ericksen issue, on labeling (6) and (8) as fits rather than predictions, and on removing or flagging the non-physical λ3 = 1 point. Reframed or re-validated, this is a solid methods paper.

Referee Report

4 major / 6 minor

Summary. The paper applies the Generalized Optimal Scaling (GOS) procedure, introduced in a previous paper by two of the authors, to derive explicit power-law thresholds for reducing the Landau–de Gennes elastic energy from (L1,L2,L3) to (L1,L2,0) and from (L1,L2,0) to (L1,0,0). The thresholds are Eqs. (5) and (7). The authors then use openQmin simulations of a spherical colloid in a periodic nematic cell to measure the relative L2 deviation between full and reduced minimizers, and report log-log scaling laws (6) and (8) that they interpret as confirming the GOS predictions. The paper concludes that the GOS thresholds provide a quantitative validity criterion for the one-constant approximation.

Significance. If the quantitative claim were fully established, the paper would give practitioners a much-needed, non-ad-hoc rule for when the one-constant approximation is safe in Landau–de Gennes simulations. The GOS algebra in Appendix B is explicit and reproducible, the input tables (Appendix C) are detailed, and the raw simulation data are deposited on Zenodo. The paper also connects a formal dimensional-analysis framework to concrete parameter regimes, which is potentially useful. However, as discussed below, the numerical validation does not currently support the central claim that the GOS threshold marks the boundary of validity, and there are internal inconsistencies in the stated physical parameters.

major comments (4)

[§IV, Eq. (5), Appendix C, Table IV] The GOS threshold for the (L1,L2,L3)->(L1,L2,0) reduction lies outside the Ericksen-admissible domain. With L1≈3.913 pN, Eq. (4) requires L3<2L1≈7.826 pN, but Eq. (5) gives L3≈9.276 pN. Thus no Ericksen-valid L3 can equal or exceed the threshold; the numerical sweep includes λ3=1 (Table IV, last row), whose rescaled L3≈5.993 exceeds 2L1≈5.056 and violates the paper's own stability constraint (K24<2K2). The simulations therefore probe only the regime where L3 is small relative to the threshold and never test whether the reduction fails as L3 approaches the predicted boundary. The data show that small L3 gives small deviations, but they cannot distinguish the GOS boundary from the Ericksen stability boundary. To validate Eq. (5), the authors need a parameter set for which the threshold lies inside the admissible interval, or a direct test of the error across the boundary.
[§III, Eqs. (6) and (8), Remark 1] The error scalings (6) and (8) are fitted to the same openQmin data that they are then used to confirm. The exponents 0.820 and 0.928 and the prefactors 0.174 and 0.925 are free parameters of a log-log least-squares regression, not outputs of the GOS derivation. GOS provides thresholds, not error-scaling exponents; Remark 1's claim that the error has 'the same order of magnitude as λ_p' is not a quantitative prediction of a specific exponent. Consequently, the numerical agreement is a consistency check with a power-law ansatz, not an independent validation of a derived scaling law. Moreover, the fits include λ3=1, which is nonphysical as noted above, so the reported regression may be biased.
[§IV, Eqs. (13)-(14), Remark 6, Appendix C.2] The second reduction case contains an internal inconsistency in the physical parameters. The text states that for the (L1,L2,0)->(L1,0,0) case, K1=K3=8 pN, K2=5 pN, and K24=5 pN. Using Eq. (11), this gives L2=α(K1-K24)=1.565×(8-5)=4.695 pN, not the 1.565 pN reported in Eq. (14). A consistent set yielding L2=1.565 pN and L3=0 would require K1=K3=6 pN. This propagates to Remark 6, where λ2≈0.060 is used, whereas the stated K values would give λ2≈4.695/25.897≈0.181. The input data in Table V are consistent with L2≈4.695 pN at the initial state (dimensionless L2≈3.033), not with Eq. (14). The authors should specify the exact OF constants used for the openQmin runs and reconcile them with Eq. (14), the threshold value L2≈25.897 pN, and the numerical examples.
[§V, Remark 9] The validation is sensitive to the numerical minimization tolerance. The authors state that using a FIRE force cutoff of 10^-10 instead of 10^-12 inflates the measured µ0 by one to two orders of magnitude and changes the scaling coefficients. The smallest λ values correspond to µ0 ~10^-7, close to typical convergence noise, so the reported power laws at the low-λ end may reflect numerical artifacts rather than physical model error. A convergence study over force tolerance, lattice size, and several independent random initial conditions is needed to establish that the observed scaling is robust. As written, the evidence rests on a single random configuration and a single geometry.

minor comments (6)

[§IV, Eq. (10)] The definition of α is ambiguous: 'α=4/9S^2' should read α=4/(9S^2) to yield the stated numerical value ≈1.565.
[Appendix B, Table II] In the row for target L3, the right-hand side contains L15_3 instead of L15_1; this is inconsistent with Eq. (5) and with the dimensional balance of the inequality.
[Throughout] The symbol L3 is used both for the elastic constant and for the threshold in Eqs. (5), (7), and Appendix C. This overloaded notation is confusing; consider using L_3^c or an overbar.
[Figures 1 and 2] The figures show no error bars or confidence bands for the fits. Given the tolerance sensitivity reported in Remark 9, the authors should indicate whether the data points are deterministic for the fixed seed and provide residual or confidence information for the regression.
[Appendix C, headings] The heading 'The (L1,L2,L3)->(L1,L2,0) case' appears twice; the second section should be titled 'The (L1,L2,0)->(L1,0,0) case'.
[Appendix B, Eqs. (B4)-(B5)] 'Semipositive definite' should be 'positive semidefinite'.

Circularity Check

2 steps flagged · score 6.0 of 10

GOS thresholds rest on a self-citation chain; the numerical 'confirmation' is a regression on the same data used as validation.

self citation load bearing [Section III (GOS paragraph); Appendix B.1 (Eqs. B2-B4, B3; citations to [1, Theorems 1,2,3,5,7])]
"Generalized Optimal Scaling (GOS), developed in [1], is an advanced, equation-free methodology... We assume, as in [1], that the characteristic constants θ1 and θ2 can be written as a product of powers of the physical parameters... The existence of such optimal choice for the exponents ⃗α is given by [1, Theorem 7]."

The central thresholds (5) and (7) are not derived from first principles in this paper: they are outputs of GOS as defined in the same-authors' prior work [1]. The power-law rescaling ansatz and the existence/optimality theorem that set the 41st- and 26th-root exponents are imported by citation, not independently derived or benchmarked. The only later verification is simulations parameterized by these same thresholds (λ3 = L3/L3bar, λ2 = L2/L2bar), so the threshold formula itself is never tested independently of its self-cited source.
fitted input called prediction [Section III.A, Eq. (6); Section III.B, Eq. (8); Remark 5; Abstract]
"For the numerical values of µ0 and λ3, we pass them into the log−log scale and the Pearson coefficient for the achieved data is 0.9893... we find the following relation: log10(µ0)≈0.820 log10(λ3)−0.760, which is equivalent to µ0 ≈0.174·λ3^0.820. (6)"

The relation presented as confirming the GOS prediction is obtained by a log-log regression on the very same simulated µ0 values that are then said to confirm it. The exponent 0.820 and prefactor 0.174 are outputs of the data, not independent predictions from the scaling theory; the same holds for Eq. (8) with 0.928 and 0.925. Remark 5 then evaluates this fitted curve at the initial λ3≈0.338 to 'predict' a 7% deviation, so the numerical validation is partly an interpolation of the data used to fit the power law.

full rationale

The paper has algebraic content: the GOS threshold expressions are explicit, dimensionally consistent formulas, and real openQmin simulations were performed. However, the load-bearing threshold framework is taken from [1], whose authors overlap with the present paper, and no independent benchmark of GOS is provided. More importantly, the headline numerical 'confirmation' reduces to a fit: Eqs. (6) and (8) are regressions on the same µ0 data that are then advertised as validating the scaling theory, and Remarks 5-6 use those fitted curves to 'predict' errors. I do not count the Ericksen-admissibility problem (λ3=1 and L3≈5.99 exceeding 2L1≈5.06 in the rescaled setting, violating K24<2K2) as circularity; it is a separate correctness risk showing the boundary is never approached from the admissible side. Likewise, Remark 9's disclosure that looser FIRE tolerances inflate µ0 by one to two orders of magnitude is a data-quality limitation, not circularity. Overall, the central validation claim is partially constructed from its own outputs, so a score of 6 reflects partial circularity rather than a fully empty derivation.

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

The paper contributes two fitted power laws (Eqs. 6 and 8) and a set of GOS-derived thresholds. The thresholds rest entirely on the authors' own GOS framework, which is cited rather than re-derived or independently verified. The numerical validation is a single-point-in-parameter-space check of the error scaling in the small parameter, with the error exponents fitted from the same data used as confirmation.

free parameters (4)

error-scaling exponent in Eq. (6) = 0.820
Obtained by log-log linear regression of µ0 vs λ3 for the authors' openQmin data; not predicted by GOS.
error-scaling prefactor in Eq. (6) = 0.174
Intercept of the same log-log regression; fitted to data.
error-scaling exponent in Eq. (8) = 0.928
Obtained by log-log linear regression of µ0 vs λ2 for the authors' openQmin data; not predicted by GOS.
error-scaling prefactor in Eq. (8) = 0.925
Intercept of the same log-log regression; fitted to data.

assumptions (4)

domain assumption GOS methodology from [1] correctly identifies scale-separation regimes for the LdG functional; specifically the existence and optimality of the exponents used in Eqs. (5) and (7) (Theorems 1,2,3,5,7 in [1]).
The thresholds are derived by applying GOS, a method introduced by three of the present authors in a 2025 preprint; the present paper does not re-prove the GOS theorems and offers no independent benchmarking of the method.
domain assumption FIRE minimization in openQmin reliably finds the global minimizer (or at least the same minimizing branch) for the considered parameters, and the discrete L2 norm of the difference between full and reduced minimizers is dominated by physical model error rather than numerical noise at force tolerance
The entire numerical validation depends on this; Remark 9 admits that looser tolerances change the measured error by 1–2 orders of magnitude.
domain assumption The single chosen geometry and material parameters (L=300 nm, R=75 nm, W=5×10^-4 J/m^2, 5CB bulk coefficients, K1=K3=8 pN, K2=5 pN) are representative for validating the general parameter dependence of the GOS thresholds.
Only one point in parameter space is simulated; the thresholds' dependence on L, R, W, and bulk coefficients is never tested.
domain assumption The discrete L2 norm on the 100^3 lattice approximates the continuum L2 norm (stated in Section V).
The paper asserts this approximation, and the error metric µ0 is computed on the discrete lattice; no convergence study is provided.

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Pith. "Pith review of Effective Complexity Reduction of the Landau-de Gennes Elastic Energy: A Quantitative Framework and Numerical Validation." pith.science (2026). https://pith.science/paper/YRUOIDUO

@misc{pith2026260719920,
  author       = {Pith},
  title        = {Pith review of: Effective Complexity Reduction of the Landau-de Gennes Elastic Energy: A Quantitative Framework and Numerical Validation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YRUOIDUO}},
  note         = {Machine review of arXiv:2607.19920}
}

abstract

We revisit the elastic energy formulation of the Landau-de Gennes model for nematic liquid crystals, focusing on quantitative reductions of the multi-constant elastic energy. Building on the generalized optimal scaling procedure (GOS) introduced by Rusconi et al. in 2025, we identify explicit parameter regimes in which the three-constant model $(L_1,L_2,L_3)$ can be reduced to $(L_1,L_2,0)$ and how the two-constant model $(L_1,L_2,0)$ can be reduced to the commonly used one-constant configuration $(L_1,0,0)$. The analytical scaling predictions are tested numerically using the openQmin simulation framework, confirming that below a critical threshold for $L_3$ or $L_2$, given by GOS, the deviation from the reduced model remains of the same order of magnitude as predicted by the scaling theory. These results provide a quantitative criterion for the validity of reduced elastic models and establish a direct connection between optimal scaling arguments and numerical observations within the Landau-de Gennes framework.

Figures

Figures reproduced from arXiv: 2607.19920 by the authors.

**Figure 1.** FIG. 1: Errors for the ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 3.** FIG. 3: Initial random configuration in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6: Visual representation of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7: Visual representations of four cases from the ( [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: FIG. 8: The matrix [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: FIG. 9: The matrix [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

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Works this paper leans on

36 extracted references · 1 canonical work pages

[1]

Complexity reduction of physical models: an equation-free approach by means of scaling

Simone Rusconi and Christina Schenk and Razvan Ceuca and Arghir Zarnescu and Elena Akhmatskaya. Complexity reduction of physical models: an equation-free approach by means of scaling. 2025. arXiv:2506.05029

arXiv 2025
[2]

and Beller, Daniel A

Sussman, Daniel M. and Beller, Daniel A. , TITLE=. Frontiers in Physics , VOLUME=. 2019 , URL=. doi:10.3389/fphy.2019.00204 , ISSN=

arXiv 2019
[3]

Liquid Crystals , volume =

Miha Ravnik and Slobodan Žumer , title =. Liquid Crystals , volume =. 2009 , publisher =. doi:10.1080/02678290903056095 , URL =

doi:10.1080/02678290903056095 2009
[4]

and Podoliak, N

Bankova, D. and Podoliak, N. and Kaczmarek, M. et. al , title =. Sci Rep , volume =. 2024 , doi =

2024
[5]

Simulation data for: Effective Complexity Reduction of the Landau-de Gennes Elastic Energy , month = jul, year =

Ceuca, R. Simulation data for: Effective Complexity Reduction of the Landau-de Gennes Elastic Energy , month = jul, year =. doi:10.5281/zenodo.21476701 , url =

work page doi:10.5281/zenodo.21476701
[6]

Einstein and Yu Podolsky and N

A. Einstein and Yu Podolsky and N. Rosen. Phys.\ Rev. 1935

1935
[7]

G. P. Berman, Jr. and F. M. Izrailev, Jr. Stability of nonlinear modes. Physica D. 1983

1983
[8]

E. B. Davies and L. Parns. Trapped modes in acoustic waveguides. Q. J. Mech. Appl. Math. 1988

1988

Show all 36 references

[9]

Edward Witten. 2001. hep-th/0106109

2001 arXiv
[10]

E. Beutler. Williams Hematology. 1994

1994
[11]

Donald E. Knuth. Fundamental Algorithms. 1973b 1973

1973
[12]

J. S. Smith and G. W. Johnson. Philos. Trans. R. Soc. London, Ser. B. 2005

2005
[13]

W. J. Smith and T. J. Johnson and B. G. Miller. Surface chemistry and preferential crystal orientation on a silicon surface. 2010

2010
[14]

V. K. Smith and K. Johnson and M. O. Klein. Surface chemistry and preferential crystal orientation on a silicon surface. 2010

2010
[15]

Lower Bounds for Wishful Research Results

Ulrich \" U nderwood and Ned \ N et and Paul \= P ot. Lower Bounds for Wishful Research Results
[16]

M. P. Johnson and K. L. Miller and K. Smith. 2007

2007
[17]

AIP Conf. Proc. 2007

2007
[18]

Fifteenth Annual

Proc. Fifteenth Annual
[19]

Y. Burstyn. Proceedings of the 5th International Molecular Beam Epitaxy Conference, Santa Fe, NM. 2004

2004
[20]

Proceedings of the 2003 Particle Accelerator Conference, Portland, OR, 12-16 May 2005. 2001

2003
[21]

A. G. Agarwal. Proceedings of the Fifth Low Temperature Conference, Madison, WI, 1999. Semiconductors. 2001

1999
[22]

R. Smith. Hummingbirds are our friends. 2001

2001
[23]

J. Smith. Proc. SPIE. 2007

2007
[24]

An O(n n / \! n) Sorting Algorithm

Tom T \' e rrific. An O(n n / \! n) Sorting Algorithm
[25]

Mastering Thesis Writing

\' E douard Masterly. Mastering Thesis Writing
[26]

S. R. Kawa and S.-J. Lin. J. Geophys. Res. 2003

2003
[27]

Phidias Phony-Baloney

F. Phidias Phony-Baloney. Fighting Fire with Fire: Festooning F rench Phrases
[28]

Donald E. Knuth. Seminumerical Algorithms. 1973c 1981

1981
[29]

Jill C. Knvth. The Programming of Computer Art
[31]

Ballagh and C.M

R. Ballagh and C.M. Savage. Bose-Einstein condensation: from atomic physics to quantum fluids. Proceedings of the 13th Physics Summer School. 2000. cond-mat/0008070

2000 arXiv
[32]

Opechowski and R

W. Opechowski and R. Guccione. Introduction to the Theory of Normal Metals. Magnetism. 1965

1965
[33]

J. M. Smith. Molecular Dynamics. 1980

1980
[34]

V. E. Zakharov and A. B. Shabat. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Zh. Eksp. Teor. Fiz. 1971

1971
[35]

Daniel D. Lincoll. Semigroups of Recurrences. High Speed Computer and Algorithm Organization
[36]

Oaho and Jeffrey D

Alfred V. Oaho and Jeffrey D. Ullman and Mihalis Yannakakis. On Notions of Information Transfer in VLSI Circuits. Proc. Fifteenth Annual ACM
[37]

The Definitive Computer Manual

Larry Manmaker. The Definitive Computer Manual

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Complexity reduction of physical models: an equation-free approach by means of scaling

Simone Rusconi and Christina Schenk and Razvan Ceuca and Arghir Zarnescu and Elena Akhmatskaya. Complexity reduction of physical models: an equation-free approach by means of scaling. 2025. arXiv:2506.05029

arXiv 2025

[2] [2]

and Beller, Daniel A

Sussman, Daniel M. and Beller, Daniel A. , TITLE=. Frontiers in Physics , VOLUME=. 2019 , URL=. doi:10.3389/fphy.2019.00204 , ISSN=

arXiv 2019

[3] [3]

Liquid Crystals , volume =

Miha Ravnik and Slobodan Žumer , title =. Liquid Crystals , volume =. 2009 , publisher =. doi:10.1080/02678290903056095 , URL =

doi:10.1080/02678290903056095 2009

[4] [4]

and Podoliak, N

Bankova, D. and Podoliak, N. and Kaczmarek, M. et. al , title =. Sci Rep , volume =. 2024 , doi =

2024

[5] [5]

Simulation data for: Effective Complexity Reduction of the Landau-de Gennes Elastic Energy , month = jul, year =

Ceuca, R. Simulation data for: Effective Complexity Reduction of the Landau-de Gennes Elastic Energy , month = jul, year =. doi:10.5281/zenodo.21476701 , url =

work page doi:10.5281/zenodo.21476701

[6] [6]

Einstein and Yu Podolsky and N

A. Einstein and Yu Podolsky and N. Rosen. Phys.\ Rev. 1935

1935

[7] [7]

G. P. Berman, Jr. and F. M. Izrailev, Jr. Stability of nonlinear modes. Physica D. 1983

1983

[8] [8]

E. B. Davies and L. Parns. Trapped modes in acoustic waveguides. Q. J. Mech. Appl. Math. 1988

1988