Contraction & Expansion Values

• Contraction and expansion values are domain-specific observables that quantify inward versus outward responses, as seen in thermal contraction of nanowires and stellar migration.
• They serve as key metrics in diverse fields, enabling evaluation of phenomena in material science, astrophysical dynamics, microfluidic transport, and GNN optimization.
• Their measurement depends on context, geometry, and representability, which necessitates tailored methods and correction of observational biases.

to=arxiv_search 大发快三是国家json սխալ? The literature surveyed here suggests that “contraction and expansion values” do not identify a single invariant. Instead, the phrase denotes domain-specific observables that quantify inward versus outward response, negative versus positive growth, narrowing versus widening geometry, contraction versus expansion of information or state-space structure, or nearest finitely representable approximants to ideal semantic updates. Depending on context, the relevant quantity is a coefficient of thermal expansion, a stellar radial drift speed, an expansion-rate slope, a passivity threshold, a pressure-drop correction, a Lyapunov-type growth rate, a graph-expansion constant, or a selected model set (Jiang et al., 2010, Loeb, 2022, Mitchell et al., 2021, Jadhav et al., 2024, Banerjee et al., 2022, Guimarães et al., 2023).

1. General character of contraction and expansion values

Domain	Representative quantity	Operational meaning
Silicon nanowires	$\alpha$	negative or positive coefficient of thermal expansion
Galaxies	$v_r \equiv dr/dt$	outward stellar migration or inward migration
Open clusters	slope of $R$–$\mu_R$	projected expansion rate; negative values are contraction
Passive systems	$\Xi$, $\hat\xi_k$, $\omega_{k+1}$	passivity threshold plus HEC contraction/expansion iterates
GNNs	$\eta_{\mathrm{KL}}(K)$, $h(G)$	information contraction and graph expansion
Belief change	$\FRsubs$, $v_r \equiv dr/dt$0	maximal representable subsets and minimal representable supersets

Several papers explicitly deny that “contraction” or “expansion” names a standalone scalar. In passive-system optimization, the main scalar is the extremal passivity parameter $v_r \equiv dr/dt$1, while “contraction” and “expansion” refer to alternating updates of the parameter and frequency variables in the Hybrid Expansion-Contraction algorithm (Mitchell et al., 2021). In tropical diagrams, the principal contraction value is the residual conditional entropy $v_r \equiv dr/dt$2, whereas expansion is parameterized by $v_r \equiv dr/dt$3 (Matveev et al., 2019). In open clusters, contraction is simply a negative measured expansion rate, i.e. a negative slope of the $v_r \equiv dr/dt$4–$v_r \equiv dr/dt$5 relation (Jadhav et al., 2024). In oversquashing theory for GNNs, contraction is formalized through strong data-processing coefficients, while expansion is formalized through Cheeger or spectral-expansion quantities (Banerjee et al., 2022).

This diversity implies that any encyclopedia-level treatment must begin with the state space on which “inward” and “outward” are defined: atomic displacement, orbital radius, enclosed dark-matter mass, loop height, tangent-volume growth, graph bottlenecks, categorical arrows, or finitely representable model sets (Jiang et al., 2010, Dutton et al., 2016, Zhang et al., 2023, Kuptsov et al., 2018, Matveev et al., 2019, Guimarães et al., 2023).

2. Condensed-matter and microfluidic transport realizations

In silicon nanowires, contraction and expansion are quantified by the coefficient of thermal expansion (CTE). All SiNWs studied in the $v_r \equiv dr/dt$6, $v_r \equiv dr/dt$7, and $v_r \equiv dr/dt$8 directions exhibit thermal contraction at low temperatures because the lowest-energy bending mode contributes negatively to CTE. The directional ordering for equal size is

$v_r \equiv dr/dt$9

with bending-mode frequencies $R$0, $R$1, and $R$2, respectively. The structure ratio

$R$3

is decisive: CTE decreases rapidly with increasing $R$4, and for $R$5 it remains negative over the whole temperature range studied. For $R$6 SiNWs, the sign change occurs near $R$7 at $R$8, near $R$9 at $\mu_R$0, and does not occur at $\mu_R$1 (Jiang et al., 2010).

In electroviscous flow through slit-type contraction-expansion microchannels, the governing geometric value is the contraction ratio

$\mu_R$2

with the outer sections fixed at width $\mu_R$3 and the contracted section at $\mu_R$4. The study examines $\mu_R$5. Over that range, the total electrical potential and pressure drop increase maximally by $\mu_R$6 and $\mu_R$7, respectively. The electroviscous correction factor

$\mu_R$8

is more nuanced: its largest overall enhancement relative to non-electroviscous flow is $\mu_R$9 at $\Xi$0, $\Xi$1, $\Xi$2, not at the strongest contraction $\Xi$3 (Dhakar et al., 2023).

For steady Oldroyd-B flow through a slowly varying planar constriction, the relevant contraction-expansion value is the pressure-drop response of a symmetric contraction-expansion geometry. With

$\Xi$4

the paper focuses on $\Xi$5. The Newtonian pressure drop is

$\Xi$6

At low $\Xi$7, the pressure drop decreases monotonically with $\Xi$8. At high $\Xi$9, unlike a contraction-only geometry, the constriction reaches a plateau because the elastic normal-stress contribution vanishes and only elastic shear stresses remain: $\hat\xi_k$0 For $\hat\xi_k$1,

$\hat\xi_k$2

or about $\hat\xi_k$3 of the Newtonian value. The exit-channel pressure-gradient relaxation length is $\hat\xi_k$4 for the constriction, versus $\hat\xi_k$5 for the corresponding contraction, so downstream relaxation is much shorter after the symmetric contraction-expansion geometry (Kedem et al., 8 Oct 2025).

In sinusoidal expansion-contraction microchannels,

$\hat\xi_k$6

the expansion and contraction points have full wall spacing $\hat\xi_k$7 and $\hat\xi_k$8. The perturbation theory uses $\hat\xi_k$9, while DPD and MPCD provide mesoscale particle-based estimates. The maximum velocity at the expansion decreases monotonically with $\omega_{k+1}$0; at the contraction it initially increases and then decreases. The normalized volumetric flow rate decreases with increasing $\omega_{k+1}$1, reaching $\omega_{k+1}$2 for the most constricted channels. DPD and MPCD slightly overpredict velocities and $\omega_{k+1}$3, consistent with inferred wall-slip velocities of $\omega_{k+1}$4 and $\omega_{k+1}$5, respectively (Koulaxizis et al., 28 Aug 2025).

3. Astrophysical and planetary systems

In secular galactic dynamics, expansion and contraction are defined by the stellar radial migration speed

$\omega_{k+1}$6

For nearly circular adiabatic orbits,

$\omega_{k+1}$7

Mass loss gives $\omega_{k+1}$8 and outward migration; accretion or mergers give $\omega_{k+1}$9 and inward migration. For disk galaxies with flat rotation curves and a Tully–Fisher relation, luminosity-driven mass loss yields

$\eta_{\mathrm{KL}}(K)$0

and over $\eta_{\mathrm{KL}}(K)$1,

$\eta_{\mathrm{KL}}(K)$2

A Milky Way analogue therefore gives $\eta_{\mathrm{KL}}(K)$3 and $\eta_{\mathrm{KL}}(K)$4 in $\eta_{\mathrm{KL}}(K)$5, whereas luminous local disks can reach $\eta_{\mathrm{KL}}(K)$6, quasar episodes can produce outward migration up to $\eta_{\mathrm{KL}}(K)$7, and gas outflows can raise this by another order of magnitude. The corresponding contraction estimate for accretion is

$\eta_{\mathrm{KL}}(K)$8

At $\eta_{\mathrm{KL}}(K)$9 and $h(G)$0, this gives $h(G)$1 (Loeb, 2022).

In Gaia DR3 analyses of open clusters, the expansion value is the slope of the projected $h(G)$2–$h(G)$3 relation, after correcting proper motions for perspective effects. Positive slope implies expansion; negative slope implies contraction. The study identifies 18 expanding clusters and 3 contracting clusters. The contracting systems are HSC_151 $h(G)$4, HSC_598 $h(G)$5, and HSC_1807 %%%%8$\omega_{k+1}$8%%%%7, while the strongest positive outlier is Sigma Orionis at $h(G)$8$h(G)$9. The detected radial motions are concentrated in young clusters, with expansion rates decreasing with age and becoming negative near $\FRsubs$0–$\FRsubs$1 (Jadhav et al., 2024).

In the NIHAO IX halo-response analysis, contraction and expansion are measured by the dark-matter mass-profile ratio

$\FRsubs$2

and, in shell language, by $\FRsubs$3. At $\FRsubs$4, the inner dark matter can change by up to a factor $\FRsubs$5. The response is organized by

$\FRsubs$6

The reported regimes are: essentially no change for $\FRsubs$7; expansion at all radii for $\FRsubs$8, with strongest expansion near $\FRsubs$9; contraction at large radii plus expansion at small radii for $v_r \equiv dr/dt$00; and contraction at all radii for $v_r \equiv dr/dt$01. At fixed mass or $v_r \equiv dr/dt$02, larger galaxies experience more halo expansion and smaller galaxies more halo contraction (Dutton et al., 2016).

In thin-shell tectonics, planetary contraction and expansion are prescribed kinematically by the degree-zero radial displacement

$v_r \equiv dr/dt$03

with negative $v_r \equiv dr/dt$04 denoting contraction and positive $v_r \equiv dr/dt$05 denoting expansion. For constant thickness,

$v_r \equiv dr/dt$06

so contraction gives isotropic compression and expansion isotropic tension. Variable thickness changes the faulting pattern: equatorial thinning yields east-west thrust faults under contraction and east-west normal faults under expansion; polar thinning yields north-south thrusts or normal faults. The paper cites about $v_r \equiv dr/dt$07 global contraction for Iapetus from porosity collapse and $v_r \equiv dr/dt$08–$v_r \equiv dr/dt$09 contraction for Mercury inferred from lobate scarps (Beuthe, 2010).

4. Solar and coronal-plasma realizations

In the 02 March 2015 prominence eruption, expansion and contraction values are measured from AIA time-distance slopes. The prominence or flux rope starts erupting at about $v_r \equiv dr/dt$10, while the earliest lateral-loop contraction begins at $v_r \equiv dr/dt$11, about $v_r \equiv dr/dt$12 minutes later. Coronal-loop expansion and contraction speeds are in the range $v_r \equiv dr/dt$13 to $v_r \equiv dr/dt$14. The erupting flux rope itself reaches $v_r \equiv dr/dt$15–$v_r \equiv dr/dt$16, with border expansion relative to the flux-rope center of $v_r \equiv dr/dt$17 and a center speed of $v_r \equiv dr/dt$18. The measured delay is used to argue that the loop contraction is not the eruption driver but a consequence of the launched flux rope, consistent with side-vortex development (Devi et al., 2021).

In the 2022 August 19 M1.6 eruptive flare, adjacent coronal loops first contract for $v_r \equiv dr/dt$19 minutes and then expand gradually until $v_r \equiv dr/dt$20. The contraction speeds are $v_r \equiv dr/dt$21–$v_r \equiv dr/dt$22 in 171 Å and $v_r \equiv dr/dt$23–$v_r \equiv dr/dt$24 in 193 Å. Expansion speeds are not tabulated numerically; the paper states only that expansion is gradual and slower than contraction, and that the final heights are equal to or lower than the initial heights. The contraction-expansion trend is modeled by a two-$v_r \equiv dr/dt$25 function, with the first term representing contraction and the second expansion (Zhang et al., 2023).

Across five eruptions from four sigmoidal active regions, a distinct dual structure is reported: a cold contracting component and a warm or hot erupting component. Expansion always begins before contraction. The maximum contraction speeds are $v_r \equiv dr/dt$26, $v_r \equiv dr/dt$27, $v_r \equiv dr/dt$28, $v_r \equiv dr/dt$29, and $v_r \equiv dr/dt$30; the maximum expansion speeds are $v_r \equiv dr/dt$31, $v_r \equiv dr/dt$32, $v_r \equiv dr/dt$33, $v_r \equiv dr/dt$34, and $v_r \equiv dr/dt$35; and the delay of contraction relative to expansion onset ranges from $v_r \equiv dr/dt$36 to $v_r \equiv dr/dt$37 minutes. The contracting loops never regain their original positions, and the paper interprets the new equilibrium as a lower-energy post-eruption state (Liu et al., 2012).

Taken together, these solar studies suggest that contraction values in eruptive coronae are often response variables of neighboring or overlying loops, while the primary expanding structure is the erupting flux rope, bubble, or CME-associated front (Devi et al., 2021, Zhang et al., 2023, Liu et al., 2012).

5. Control, dynamical-systems, and information-propagation formulations

In passivity optimization for linear time-invariant systems, the main value is the extremal parameter

$v_r \equiv dr/dt$38

the largest $v_r \equiv dr/dt$39 for which the parameterized transfer function remains passive. The Hybrid Expansion-Contraction algorithm alternates a contraction phase, which solves a scalar root problem in the parameter, and an expansion phase, which updates the auxiliary frequency variable. In this setting, contraction values are the iterates $v_r \equiv dr/dt$40 obtained from

$v_r \equiv dr/dt$41

and expansion values are the updated frequencies $v_r \equiv dr/dt$42. The sequence of parameter estimates is monotone decreasing and converges locally Q-quadratically under the stated smoothness and nondegeneracy assumptions. Reported examples include a continuous-time case with $v_r \equiv dr/dt$43 and a discrete-time case with $v_r \equiv dr/dt$44, both computed with dramatically fewer large pencil solves than the midpoint-based baseline (Mitchell et al., 2021).

In Lyapunov analysis of pseudohyperbolic attractors, contraction and expansion values are instantaneous or finite-time growth rates of tangent directions and tangent volumes. The central quantities are

$v_r \equiv dr/dt$45

together with their finite-time analogues. Here $v_r \equiv dr/dt$46 measures instant logarithmic $v_r \equiv dr/dt$47-volume growth, while $v_r \equiv dr/dt$48 measures domination between the expanding and contracting subspaces. The main conclusion is that average pseudohyperbolic expansion and contraction can fail instantaneously: first-subspace volumes can contract, second-subspace directions can expand, and contraction in the first subspace can be stronger than in the second. For the Lorenz system, $v_r \equiv dr/dt$49, $v_r \equiv dr/dt$50, $v_r \equiv dr/dt$51, and the two-dimensional volume-growth condition becomes strictly positive only on sufficiently large finite windows; for the generalized Lorenz system, $v_r \equiv dr/dt$52, $v_r \equiv dr/dt$53, $v_r \equiv dr/dt$54, $v_r \equiv dr/dt$55, and all three local violations occur (Kuptsov et al., 2018).

In oversquashing theory for message-passing GNNs, contraction is formalized by information-theoretic attenuation. The key value is the KL contraction coefficient

$v_r \equiv dr/dt$56

and for a binary symmetric channel,

$v_r \equiv dr/dt$57

Using the noisy-circuit analogy, the paper cites the bound

$v_r \equiv dr/dt$58

where $v_r \equiv dr/dt$59 is gate fan-in and $v_r \equiv dr/dt$60 the input-output graph distance. Expansion is quantified structurally by the Cheeger constant

$v_r \equiv dr/dt$61

spectral gap $v_r \equiv dr/dt$62, and related rewiring diagnostics. The proposed RLEF and G-RLEF schemes increase spectral expansion while preserving node degrees exactly and never disconnecting the graph (Banerjee et al., 2022).

6. Formal, categorical, combinatorial, and epistemic uses

In tropical commutative diagrams of probability spaces, arrow contraction and arrow expansion are structural operations rather than Euclidean shrinkage and growth. The contraction theorem states that for every $v_r \equiv dr/dt$63 one can modify an admissible fan so that the distinguished arrow becomes asymptotically an isomorphism, quantified by

$v_r \equiv dr/dt$64

while preserving both the subdiagram $v_r \equiv dr/dt$65 and the conditioned subdiagram $v_r \equiv dr/dt$66 up to asymptotically small entropy distance. At finite scale, the proof yields the bounds

$v_r \equiv dr/dt$67

The expansion theorem is a right inverse: starting from a reduced admissible fan and any $v_r \equiv dr/dt$68, one constructs an expanded arrow with

$v_r \equiv dr/dt$69

while preserving $v_r \equiv dr/dt$70 exactly (Matveev et al., 2019).

In matroid theory, the expansion functor is parameterized by a multiplicity vector

$v_r \equiv dr/dt$71

which replaces each element $v_r \equiv dr/dt$72 by $v_r \equiv dr/dt$73 parallel copies. The expanded ground set has cardinality $v_r \equiv dr/dt$74, but rank is preserved: $v_r \equiv dr/dt$75 Graphic, binary, and transversal matroids are preserved under arbitrary expansion, and partition matroids arise as expansions of uniform matroids. The contraction functor collapses equivalence classes of elements with identical exchange behavior, and the paper proves that a matroid satisfies White’s conjecture if and only if its contraction does (Rahmati-asghar, 2017).

In finite-based belief change, contraction-like and expansion-like values are finitely representable model sets. If $v_r \equiv dr/dt$76 is the current finite base and $v_r \equiv dr/dt$77 the input model set, the contraction-like eviction value is selected from

$v_r \equiv dr/dt$78

the set of maximal finitely representable subsets of the ideal remainder, while the expansion-like reception value is selected from

$v_r \equiv dr/dt$79

the set of minimal finitely representable supersets of the ideal union. The representation theorems characterize maxichoice eviction by success, inclusion, vacuity, finite retainment, and uniformity, and maxichoice reception by success, persistence, vacuity, finite temperance, and uniformity. In finite propositional logic the semantic remainder or union is exactly finitely representable; in Horn logic one generally obtains nearest representable approximants instead (Guimarães et al., 2023).

In one-dimensional additive combinatorics, contraction and expansion are linked by a quantitative theorem on finite sets $v_r \equiv dr/dt$80. If for each $v_r \equiv dr/dt$81 there is an injective map $v_r \equiv dr/dt$82 with $v_r \equiv dr/dt$83, $v_r \equiv dr/dt$84, and $v_r \equiv dr/dt$85 between $v_r \equiv dr/dt$86 and $v_r \equiv dr/dt$87, then

$v_r \equiv dr/dt$88

hence

$v_r \equiv dr/dt$89

An immediate corollary is

$v_r \equiv dr/dt$90

The paper also proves that without the betweenness condition one can have

$v_r \equiv dr/dt$91

so the order condition is essential for linear-in-$v_r \equiv dr/dt$92 expansion (Breuillard et al., 2011).

7. Comparative themes and recurring caveats

A recurring theme is that local contraction or expansion need not determine global behavior. In silicon nanowires, the low-temperature bending mode enforces negative CTE, but the net sign at higher temperature depends on growth direction and $v_r \equiv dr/dt$93, including the all-negative $v_r \equiv dr/dt$94 regime (Jiang et al., 2010). In pseudohyperbolic dynamics, average volume expansion and subspace contraction can be violated instantaneously even when the invariant splitting is genuinely pseudohyperbolic (Kuptsov et al., 2018). In Oldroyd-B constrictions, low-$v_r \equiv dr/dt$95 pressure-drop reduction continues with $v_r \equiv dr/dt$96, but the high-$v_r \equiv dr/dt$97 symmetric contraction-expansion geometry reaches a plateau because the normal-stress contribution cancels (Kedem et al., 8 Oct 2025).

A second theme is that the sign or magnitude of a contraction/expansion value is often conditional on geometry, observational projection, or representability. Open-cluster expansion rates require explicit correction of perspective artefacts, since uncorrected proper motions can create false contraction signatures (Jadhav et al., 2024). In finite-based belief change, exact semantic subtraction or union may fail to be finitely representable, so contraction and expansion values are defined only through maximal or minimal representable approximants (Guimarães et al., 2023). In additive-combinatorial contractions, the betweenness condition is not cosmetic but structurally necessary for the expansion lower bound (Breuillard et al., 2011).

A third theme is that “expansion” frequently denotes outward transport or broader reach rather than literal geometric growth. In galaxies it is outward secular migration of stars; in GNNs it is bottleneck-avoiding graph structure; in passivity optimization it is the auxiliary-variable update that sharpens a threshold computation (Loeb, 2022, Banerjee et al., 2022, Mitchell et al., 2021). This suggests that contraction and expansion values are best understood as signed or ordered response variables whose semantics are inherited from the configuration space of the underlying problem rather than from a universal geometric archetype.