Coupled dynamics of wall pressure and transpiration, with implications for the modeling of tailored surfaces and turbulent drag reduction

Anthony Leonard; Beverley McKeon; Simon Toedtli

REVIEW 3 major objections 5 minor 50 references

Coupled dynamics of wall pressure and transpiration, with implications for the modeling of tailored surfaces and turbulent drag reduction

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Wall-pressure phase decides whether transpiration cuts or boosts drag.

desk verdict First solid nonlinear-DNS evidence that wall pressure–transpiration phase tracks drag-reducing vs. drag-increasing responses, though the phase is diagnosed rather than imposed, so the causal claim stays correlational. read the letter →

arxiv 2411.14616 v1 pith:SYBKSWKI submitted 2024-11-21 physics.flu-dyn

Simon Toedtli , Anthony Leonard , Beverley McKeon This is my paper · ORCID

classification physics.flu-dyn

keywords turbulentchannelflowdragreductionwalltranspirationpressurephasedifferencevarying-phaseoppositioncontrolspanwiserollerstailoredsurfaces

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 tries to establish that the effect of wall transpiration on a turbulent channel flow is governed by one wall quantity: the phase difference between the transpiration and the wall pressure. Using direct numerical simulations with a controller that shifts the streamwise phase of the transpiration, it shows that slow, streamwise-elongated “streak” transpiration suppresses the near-wall cycle and reduces drag when it is in phase with the wall pressure, and amplifies it otherwise; short, wide “roller” transpiration generates spanwise rollers and drag increase when it is out of phase. The phase relation is itself a dynamical outcome of the coupled pressure–transpiration system, and the paper identifies conditions under which the relation is robust. If correct, the result unifies active and passive flow control: riblets, porous and permeable walls can be understood as transpiration boundary conditions whose drag effect is set by the same two scale families and phase relations.

What carries the argument

The machinery is the phase difference between wall pressure and transpiration at each Fourier mode, $\Delta\theta_\kappa(t)=\angle\hat p_\kappa(y_w,t)-\angle\hat v_\kappa(y_w,t)$, together with the decomposition of the pressure into fast (linear source), slow (nonlinear source), and Stokes (boundary-condition) components. The varying-phase opposition control law $\hat v_\kappa(y_w)=-\hat A_d\hat v_\kappa(y_d)$ with complex gain $\hat A_d$ makes the phase shift $\angle\hat A_d$ an effective streamwise shift of the transpiration relative to the sensor signal, so one control parameter sweeps the phase range. The FIK decomposition of the friction coefficient into a weighted integral of Reynolds stresses turns drag change into a sum over wavenumbers, and circular statistics (the circular mean) reduce noisy phase histograms at each scale to a single number.

What would settle it

Run a DNS or experiment with streak-scale transpiration at a higher Reynolds number, e.g. $Re_\tau\approx 1000$, where the slow pressure is relatively stronger, and measure the circular mean of $\Delta\theta_\kappa$ at the energetic streak scales: if drag is reduced while the mean phase difference is not near zero, the central mapping is falsified. Alternatively, measure the wall-pressure/transpiration phase on a drag-reducing riblet surface at its effective virtual wall: in-phase streak scales should appear if the claimed dynamical equivalence with tailored surfaces holds.

Watch

Extended reading notes

Core claim

The central claim is that the wall-pressure/transpiration phase difference $\Delta\theta_\kappa = \angle\hat p_\kappa(y_w)-\angle\hat v_\kappa(y_w)$ at a spatial scale $\kappa=(k_x,k_z)$ selects among the three canonical responses in a low-$Re_\tau$ turbulent channel flow. At streak scales, which are associated with the near-wall cycle, $\Delta\theta_\kappa\approx 0$ suppresses the cycle and reduces drag, while other phases amplify it. At roller scales, $\Delta\theta_\kappa\approx\pi$ coincides with the emergence of spanwise rollers and a large drag increase. The paper further shows that these phase relations are not imposed but emerge from the coupled dynamics: the fast pressure picks up a robust phase because its Green’s-function weight sits in the wall-normal layer where the controller constrains $\hat v_\kappa$; the slow pressure’s nonlinear source term is typically too decorrelated to set a robust phase; and the Stokes pressure sets a robust phase only when the temporal frequency content of the transpiration is approximately sparse, as happens when an amplified eigenmode dominates.

Load-bearing premise

The load-bearing premise is that the in-phase/out-of-phase dichotomy for the total wall pressure survives the slow pressure component; the paper itself notes it is unclear how the drag-reducing controller establishes a robust overall phase relation given the large directional variation of the slow pressure, so if slow pressure dominates in other regimes or at higher Reynolds number the mapping could break down.

Editorial extensions

If this is right

Drag reduction by transpiration requires streak-scale actuation that is in phase with the wall pressure; small phase shifts do this, while larger positive or negative shifts amplify the near-wall cycle and increase drag.
Roller-scale transpiration with an out-of-phase pressure–transpiration relation generates spanwise rollers and drag increase, and this is dynamically equivalent to riblets past the viscous regime and to porous or permeable walls.
Wall pressure can serve as a wall-based proxy for the background flow state in control design, but pressure data from uncontrolled canonical flows will not transfer because transpiration makes the Stokes pressure leading-order.
Passive pressure-driven tailored surfaces will struggle to reduce drag unless they impose a scale-dependent response and an in-phase relation between pressure and transpiration, for example through complex-valued permeability or resonator-type surfaces.

Reading between the lines

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

The phase-difference criterion could be tested as a wall-only sensor metric in experiments: measure the circular mean of $\Delta\theta_\kappa$ on a riblet or porous surface at its effective virtual wall, and check whether drag-reducing configurations show in-phase streak scales.
Because the slow pressure is stochastic and its direction varies, the robust in-phase relation in the drag-reducing case may be carried mainly by the Stokes and fast components; at higher $Re_\tau$ or with different controller gains the total-pressure phase could decorrelate from $\angle\hat A_d$, so the mapping may need a statistical rather than deterministic statement.
The same logic suggests a concrete design target for meta-material surfaces: the surface response to wall pressure should be scale-dependent, with zero phase at streak scales and $\pi$ phase at roller scales, which is testable beyond the paper’s simulations.

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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

You should read this one. It is the first full nonlinear DNS evidence that the phase between wall pressure and transpiration separates the three canonical drag responses: in-phase streak scales suppress the near-wall cycle, out-of-phase roller scales energize spanwise rollers and increase drag. The two clean cases (N25 and P50) make the point convincingly, and the analysis is careful throughout—validated DNS and pressure Poisson solvers, proper circular statistics, and a genuinely useful decomposition into fast, slow, and Stokes pressure. The Green's function argument for why fast pressure locks to the transpiration phase, and why slow pressure usually does not, is the most instructive part of the paper. The Stokes pressure result—that transpiration makes it leading-order even at high Reynolds number—deserves attention on its own.

The soft spots are real but mostly minor. The biggest one, which the stress-test note correctly identifies, is that the controller manipulates the phase shift ∠A_d relative to the sensor signal; the p–v phase Δθ is measured after the fact. So the data cannot strictly distinguish "Δθ controls the response" from "∠A_d controls the response and Δθ is a covarying marker." The paper partially closes this gap with the scale-restricted controllers and the mechanistic domain-of-dependence analysis, and it is honest about the remaining gap ("It would be interesting to verify..."). But the abstract and summary do claim more than the experimental design supports. That is worth flagging in review, not rejecting over.

Other soft spots: one Reynolds number, three in-depth phase shifts, and the N75 case has large circular variance and is appropriately caveated. The admitted difficulty in explaining how N25 establishes a robust total-pressure phase when the slow-pressure direction wanders (§5.2) is a secondary concern, not a fatal one. No public code or data, which makes reproduction harder but is common for DNS papers. The self-citation pattern is legitimate—the prior resolvent and DNS results are independently obtained and the current paper stands on its own.

Who is this for? Anyone working on opposition control, porous/permeable surfaces, riblets, or wall-based drag reduction. It deserves a proper peer review at JFM or a comparable venue; the causal gap and the small parameter coverage should be addressed, but the core observation is solid and new.

Referee Report

3 major / 5 minor

Summary. This paper performs DNS of a low-Reynolds-number turbulent channel flow (Re_b = 5600, Re_tau,0 = 180) with varying-phase opposition control, in which the complex controller gain A_d is chosen so that its phase angle A_d prescribes a streamwise shift of the wall transpiration. The authors analyze the wall-pressure field and its phase difference Delta_theta_kappa = angle(p_hat_kappa(y_w)) - angle(v_hat_kappa(y_w)) (Eq. 5.1) using circular statistics. They decompose the pressure into fast, slow, and Stokes components and show that transpiration makes the Stokes pressure leading-order. They identify two scale families, streak scales (exemplified by kappa_s = (0.5,11)) and roller scales (exemplified by kappa_r = (6.5,0)). For controller N25 (angle A_d = -pi/4, 21% drag reduction), streak-scale transpiration is in-phase with the wall pressure and suppresses the near-wall cycle; for P50 (angle A_d = +pi/2, drag increase), roller-scale transpiration is out-of-phase and energizes spanwise rollers; for N75 (-3pi/4), the phase data have large circular variance. The paper proposes conditions for robust pressure-transpiration phase relations based on the Green's function domain of dependence and temporal frequency sparsity, and draws analogies to porous, permeable, and riblet surfaces.

Significance. If the claimed phase dichotomy holds, the wall-pressure/transpiration phase difference would be a useful wall-based parameter unifying active opposition control and passive tailored surfaces, and it is experimentally testable. The study is careful in several respects: the DNS and pressure Poisson solvers are validated against Lee and Moser (2015); circular statistics are used appropriately for phase data; scale-restricted controllers in Section 3.3 and Appendix A support the attribution of drag changes to specific scale families; and the paper states its own limitations, such as the large circular variance in the N75 case and the stochasticity of the slow pressure. It also makes falsifiable predictions for riblets and porous walls. The principal caveat is that Delta_theta is diagnosed post hoc and never independently imposed, so the central claim that Delta_theta controls the response is correlational; moreover, the drag-reducing N25 case is not fully explained by the proposed mechanism.

major comments (3)

[Abstract and §7.1, with Eqs. (2.3)–(2.5) and (5.1)] The paper states that the phase difference Delta_theta_kappa 'controls' or 'parametrizes' the flow response, but in every simulation only the controller phase angle A_d is prescribed; Delta_theta_kappa is measured after the simulation from the resulting pressure and transpiration fields. Figure 3 maps the drag ratio against angle A_d, not against Delta_theta. The data are therefore equally consistent with angle A_d being the causal parameter and Delta_theta being a covarying marker. This distinction matters because Section 7.2's porous-wall equivalence assumes that imposing an out-of-phase relation is what generates rollers. I request either rewriting the central claims in explicitly correlational or falsifiable language, or adding a test in which Delta_theta is imposed independently of angle A_d (for example, a pressure-driven boundary condition with an adjustable complex impedance) and the response is shown to track Delta_theta.
[§5.2 and Figs. 7 and 10] The paper explicitly states: 'Given the significant variation in the direction of the slow pressure, it remains somewhat unclear how controller N25 establishes a robust overall phase relationship.' This matters because N25 is the only drag-reducing case and is the linchpin of the in-phase branch. At kappa_s the slow and Stokes pressures dominate (Fig. 7e), yet the proposed mechanisms for a robust phase relation do not apply there: the fast pressure is the smallest component at that scale, and Fig. 12a shows broadband temporal content rather than the sparsity invoked for the Stokes pressure. The robust total-pressure phase for N25 is therefore not explained by the paper's framework. Please provide a quantitative explanation, for example a time-resolved vector addition of the pressure components or a conditional analysis, or explicitly restrict the claimed mechanism to the cases it actually covers.
[§6.3, Figs. 10 and 15] The asserted dichotomy that streak scales attenuate the near-wall cycle when in-phase and amplify it 'otherwise' rests on one clean amplifying streak-scale case, N75, whose mean phase difference is approximately +pi/2 rather than pi and whose circular variance is acknowledged to be large. The phase difference spectra in Fig. 13(a) also vary substantially from scale to scale. Without additional phase shifts (for example angle A_d = -pi/2, -pi/8, +pi/4) analyzed with the same Delta_xi_t,kappa and Delta_theta_kappa metrics, the smooth transition from suppression to amplification remains speculative. I recommend either adding such cases or softening the abstract and Section 7.1 to state the observed correlation for the three cases rather than a general phase-controlled dichotomy.

minor comments (5)

[§5.1 and Fig. 10] The histograms are informative, but the paper would benefit from reporting quantitative circular variance or bootstrap confidence intervals for the circular means. This would sharpen the claim that N25 and P50 are statistically strong while N75 is ambiguous.
[§5.3, Eq. (5.6)] The explanation that the Green's function contributes an additional phase of pi because it is negative is correct, but the sign conventions are easy to confuse with the +pi appearing in the control law, Eq. (2.5). A brief explicit statement distinguishing the two sources of pi would improve readability.
[Abstract and §7.1] The phrase 'parametrized by the wall pressure' could be misread as implying the wall pressure alone is the parameter; the actual parameter is the phase difference between wall pressure and transpiration defined in Eq. (5.1). Please clarify.
[§7.1] The text contains a typo: 'conduced' should be 'conducted'.
[Throughout] There are several encoding artifacts and minor typos, for example 'G ´omez-de Segura' in the body text and 'futher' near Eq. (2.12). A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the p–v phase is a diagnosed DNS observable, not a fitted input, and the self-cited prior DNS/stability results are independent evidence.

full rationale

The paper's own equations separate the manipulated quantity from the diagnosed quantity. Equation (2.5) prescribes the transpiration phase through the controller, ∠Vκ(y_w) = ∠Â_d + ∠Vκ(y_d) + π, while the phase difference Δθκ in Eq. (5.1) is measured from the resulting DNS pressure and velocity fields in Sec. 5.2 and Fig. 10. No step fits Δθκ to the drag outcomes; the drag data in Fig. 3 and the scale-by-scale FIK decomposition in Sec. 6.1 are independent measurements and an exact identity, respectively. The Green's-function phase relations (Eq. 5.6) and the Stokes-pressure approximation (Eq. 5.8) are analytical consequences of the pressure Poisson equation, and the paper states only sufficient conditions for robust phase relations ('Robust phase relations between wall transpiration and fast or slow pressure can result if the pressure source term correlates with Vκ(y_w) over the wall-normal layer where the weight function is non-zero'), which are then checked against DNS rather than assumed. The self-citations to Toedtli et al. (2019a, 2020) supply prior DNS drag curves and a linear-stability eigenvalue; these are external, reproducible data, not an assumption equivalent to the present claim that the p–v phase controls the flow response. The admitted limitation in Sec. 5.2 ('Given the significant variation in the direction of the slow pressure, it remains somewhat unclear how controller N25 establishes a robust overall phase relationship') and the explicitly untested porous-wall equivalence in Sec. 7.2 ('It would be interesting to verify if Vκ and pκ are out-of-phase when other configurations, such as riblets, generate spanwise rollers') bear on causal completeness and robustness, not on circularity. The central claim is correlational rather than causal, but correlation is not circular reduction: the paper does not define Δθκ in terms of drag, does not fit Δθκ to force the drag correlation, and does not import an unverified uniqueness theorem from the self-cited work. Therefore no load-bearing derivation step reduces to its own inputs.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard fluid mechanics assumptions (incompressible Navier-Stokes, pressure Poisson recovery, Green's function decomposition) and the specific opposition-control law that makes the phase shift interpretable as a streamwise shift. No new physical entities are introduced. The drag-contributing scale threshold is an explicit analysis choice, not a fitted parameter. External dependencies are prior results from the same group (Toedtli et al. 2019a, 2020) for drag data and eigenvalue interpretation.

free parameters (2)

Gamma threshold = 0.05% of turbulent friction coefficient
Used to define the contour of drag-contributing scales in section 6.1; the authors state conclusions are insensitive to its value.
Sensor location y_d+ = 15
Fixed from prior opposition-control studies (Toedtli et al. 2019a); determines the constrained wall-normal region for the fast-pressure phase argument.

assumptions (5)

domain assumption Incompressible Navier-Stokes equations with no-slip tangential and permeable normal boundary conditions are an adequate model for the transpiration-controlled channel flow.
The DNS solves these equations; control enters as a wall boundary condition.
standard math Pressure fluctuations are recovered from the Poisson equation with Neumann conditions and split into fast, slow, and Stokes components.
Decomposition follows Kim (1989); it underpins the phase analysis in sections 5 and 6.
domain assumption The controller gain of Eq. (2.4) gives the phase shift a physical interpretation as a streamwise shift and does not qualitatively bias the flow response.
The gain suppresses oblique modes; the scale-family decomposition depends on this restriction.
standard math Circular statistics, specifically the circular mean of Eq. (5.4), are appropriate for phase-difference data, and phase is meaningful only where spectral amplitudes are non-zero.
Used throughout section 5; the non-zero amplitude caveat is stated in section 5.5.
domain assumption The linearized Navier-Stokes eigenvalue result of Toedtli et al. (2020) is applicable to interpreting the roller-scale response in the nonlinear DNS.
Cited in sections 5.4 and 7.1; the amplified eigenvalue's wave speed is used to explain the sparse Stokes-pressure frequency content.

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Cite this review

Pith. "Pith review of Coupled dynamics of wall pressure and transpiration, with implications for the modeling of tailored surfaces and turbulent drag reduction." pith.science (2026). https://pith.science/paper/SYBKSWKI

@misc{pith2026241114616,
  author       = {Pith},
  title        = {Pith review of: Coupled dynamics of wall pressure and transpiration, with implications for the modeling of tailored surfaces and turbulent drag reduction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SYBKSWKI}},
  note         = {Machine review of arXiv:2411.14616}
}

read the original abstract

Wall-based active and passive flow control for drag reduction in low Reynolds number (Re) turbulent flows can lead to three typical phenomena: i) attenuation or ii) amplification of the near-wall cycle, and iii) generation of spanwise rollers. The present study conducts direct numerical simulations (DNS) of a low Re turbulent channel flow and demonstrates that each flow response can be generated with a wall transpiration at two sets of spatial scales, termed "streak" and "roller" scales. The effect of the transpiration is controlled by its relative phase to the background flow, which can be parametrized by the wall pressure. Streak scales i) attenuate the near-wall cycle if transpiration and wall-pressure are approximately in-phase or ii) amplify it otherwise, and iii) roller scales energize spanwise rollers when transpiration and wall pressure are out-of-phase. The dynamics of the wall pressure and transpiration are coupled and robust relative phase relations, which are required to trigger the flow responses, can result if the source term of the linear fast or nonlinear slow pressure correlates with the wall transpiration over a scale-dependent height or if the temporal frequency content of the wall transpiration is approximately sparse. The importance of each condition depends on the relative magnitude of the pressure components, which is significantly altered by the transpiration. The analogy in flow response suggests that transpiration with the two scale families and their phase relations to the wall pressure represent fundamental building blocks for flows over tailored surfaces including riblets, porous, and permeable walls.

Figures

Figures reproduced from arXiv: 2411.14616 by the authors.

**Figure 1.** Interpretation of the phase shift as a streamwise shift in physical domain. The black curve represents an example sensor measurement, while the red curve shows the actuator response for two example phase shifts. of eq. (2.6) in two regards: first, it selects the smaller of 𝐴1 and 𝐴2 to generate the wall transpiration, and second, it enforces the same transpiration amplitude at 𝜿 and 𝜿˜. The equal amplitude suppresse… view at source ↗

**Figure 2.** Validation of the present DNS and pressure Poisson equation solver (solid lines) against the data of Lee & Moser (2015) (open circles) for a canonical channel flow at (Re𝜏)0 = 180. dependent phase shift in Fourier domain. And second, unlike the present controller, they do not suppress oblique waves in the actuation input. 2.4. Direct numerical simulation The flow response to varying-phase opposition control is studi… view at source ↗

**Figure 3.** Drag reduction of varying-phase opposition control with sensors located at 𝑦 + 𝑑 = 15. Filled symbols inside the green shaded area indicate drag reduction (𝜉 < 1), while open symbols in the red region indicate unchanged or increasing drag (𝜉 ⩾ 1). The area inside the black square in fig. 3a is magnified in fig. 3b. the friction coefficient ratio 𝜉, which is interpreted as follows 𝜉 ∈ ( [0, 1) drag reduction (smaller… view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: Left column: instantaneous spatial structure of the transpiration 𝑣 + (𝑦𝑤) at the wall. Right column: time-averaged actuation spectrum Φ+ 𝑣𝑣 (𝑦𝑤). The green contour lines denote Φ+ 𝑣𝑣/max(Φ+ 𝑣𝑣) = (0.15, 0.3, 0.45) and the blue and red square indicate the example spati…

**Figure 5.** Figure 5: Left column: Instantaneous spatial structure of 𝑝 + (𝑦𝑤), right column: time averaged wall pressure fluctuation spectrum Φ+ 𝑝 𝑝. The lines outline the levels Φ+ 𝑞𝑞/max(Φ+ 𝑞𝑞) = (0.15, 0.3, 0.45) of 𝑣 (solid green lines, replotted from fig. 4) and 𝑝 (dashed pink lines).…

**Figure 6.** Figure 6: Example Green’s function kernels for the wall pressure (𝑦 = −1) at various values of 𝜅. The Green’s function is plotted versus the integration variable 𝑎. actuator input. This hints at the importance of nonlinear interactions, either through the slow pressure or energy…

**Figure 7.** Figure 7: Total variance (top row) and example scale contribution (bottom row) to the fast (dotted orange line), slow (dashed blue line), and Stokes pressure (dash-dotted purple line). The gray lines in fig. 7a outline the reference profiles for a canonical turbulent channel flo…

**Figure 8.** Figure 8: Instantaneous spatial structure of 𝑝 + 𝑣 + at the wall. near-wall cycle for 𝜿𝑠 and generation of spanwise rollers for 𝜿𝑟 , lead to fundamentally different source terms for the pressure components and therefore to different relative magnitudes. Some aspects of this will…

**Figure 9.** Figure 9: Circular data (𝜃1, 𝜃2) and their linear (𝜃˜, red cross) and circular mean (𝜃, green square). 5.1. Circular statistics Phase differences or, more generally, phase angles in the complex plane are periodic quantities and take on values in the interval [0, 2𝜋) or any multi…

**Figure 10.** Figure 10: Histograms of the phase difference between wall pressure and transpiration. Each column represents a different phase shift and scale. The rows show from top to bottom: total, fast, slow, and Stokes pressure. The red symbol indicates the circular mean of each histogram…

**Figure 11.** Figure 11: Representative instantaneous phase profile ∠𝑣ˆ (solid black line) and product of Green’s function kernel 𝐺 and mean shear 𝑆 normalized by the maximum value (solid gray line). The two plots show the same spatial scale 𝜿𝑠 at different ∠𝐴ˆ 𝑑. clustered about the circular…

**Figure 12.** Figure 12: Power spectral density estimate E [Φ] of 𝜕𝑣ˆ𝜿/𝜕𝑡 for two different scales and phase shifts. Both spectra are normalized by their respective maximum values. The dotted vertical line in fig. 12b indicates the wavespeed of the amplified eigenmode reported in Toedtli et a…

**Figure 13.** Figure 13: Phase difference spectra Δ𝜃𝜿/𝜋 of the example control configurations. The contour lines are the same as in figs. 4 and 5 and label Φ+ 𝑞𝑞/max(Φ+ 𝑞𝑞) = (0.15, 0.3, 0.45) for 𝑣 (solid green lines) and 𝑝 (dashed pink lines). domain of dependence apply broadly and underlie…

**Figure 14.** Figure 14: Relative turbulent drag contribution for the uncontrolled reference flow (fig. 14a) and various controlled flows (figs. 14b to 14d). The spatial scales contained within the black solid contour Γ contribute at least 0.05% of the turbulent drag and sum to the total valu…

**Figure 15.** Figure 15: Left column: relative change in turbulent drag contribution Δ𝜉𝑡𝜿, right column: phase difference spectra Δ𝜃𝜿/𝜋. Both quantities are only shown for the scales within the black contour Γ defined in fig. 14. The green contour lines are the same as in fig. 4 and label Φ+ …

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

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