--- title: Contraction & Expansion Values url: https://www.emergentmind.com/topics/contraction-and-expansion-values type: topic --- # Contraction & Expansion Values 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 [1001.0467][2202.00825][2109.00974][2404.05327][2208.03471][2303.03034]. ## 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 | \( \FRsubs \), \( \FRsups \) | 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 \(\Xi\), while “contraction” and “expansion” refer to alternating updates of the parameter and frequency variables in the Hybrid Expansion-Contraction algorithm [2109.00974]. In tropical diagrams, the principal contraction value is the residual conditional entropy \([Z'-0.57ex X']\), whereas expansion is parameterized by \(\lambda=[Z-0.57ex X]\) [1905.05597]. In open clusters, contraction is simply a negative measured expansion rate, i.e. a negative slope of the \(R\)–\(\mu_R\) relation [2404.05327]. In oversquashing theory for GNNs, contraction is formalized through strong data-processing coefficients, while expansion is formalized through Cheeger or spectral-expansion quantities [2208.03471]. 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 [1001.0467][1605.05323][2305.08338][1805.06644][1905.05597][2303.03034]. ## 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 \([100]\), \([110]\), and \([111]\) directions exhibit thermal contraction at low temperatures because the lowest-energy bending mode contributes negatively to CTE. The directional ordering for equal size is \[ \alpha_{[110]} > \alpha_{[111]} > \alpha_{[100]}, \] with bending-mode frequencies \(53.1\), \(41.2\), and \(39.3\ \mathrm{cm}^{-1}\), respectively. The structure ratio \[ \gamma=\frac{\text{length}}{\text{diameter}} \] is decisive: CTE decreases rapidly with increasing \(\gamma\), and for \(\gamma=1.3\) it remains negative over the whole temperature range studied. For \([111]\) SiNWs, the sign change occurs near \(120\ \mathrm{K}\) at \(\gamma=0.25\), near \(270\ \mathrm{K}\) at \(\gamma=1.0\), and does not occur at \(\gamma=1.3\) [1001.0467]. In electroviscous flow through slit-type contraction-expansion microchannels, the governing geometric value is the contraction ratio \[ d_c=\frac{A_c}{A}=\frac{W_c}{W}, \qquad 00\) and outward migration; accretion or mergers give \(v_r<0\) and inward migration. For disk galaxies with flat rotation curves and a Tully–Fisher relation, luminosity-driven mass loss yields \[ v_r \simeq 27~{\rm cm\,s^{-1}}\left(\frac{L}{L_{\rm MW}}\right)^{1/2}, \] and over \(10\ \mathrm{Gyr}\), \[ \Delta r \approx 2.7~{\rm pc}\left(\frac{t}{10~{\rm Gyr}}\right)\left(\frac{L}{L_{\rm MW}}\right)^{1/2}. \] A Milky Way analogue therefore gives \(v_r\approx 27~\mathrm{cm\,s^{-1}}\) and \(\Delta r\approx 2.7~\mathrm{pc}\) in \(10\ \mathrm{Gyr}\), whereas luminous local disks can reach \(\sim 1~\mathrm{m\,s^{-1}}\), quasar episodes can produce outward migration up to \(\sim 0.1~\mathrm{km\,s^{-1}}\), and gas outflows can raise this by another order of magnitude. The corresponding contraction estimate for accretion is \[ v_r \approx -0.8~{\rm km\,s^{-1}} \left(\frac{\dot M}{10\,M_\odot\,{\rm yr^{-1}}}\right) \left(\frac{L}{L_{\rm MW}}\right)^{1/2}. \] At \(\dot M=10\,M_\odot\,{\rm yr^{-1}}\) and \(L=L_{\rm MW}\), this gives \(v_r\approx -0.8~\mathrm{km\,s^{-1}}\) [2202.00825]. In Gaia DR3 analyses of open clusters, the expansion value is the slope of the projected \(R\)–\(\mu_R\) 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 \((-112 \pm 18)\), HSC\_598 \((-30 \pm 8)\), and HSC\_1807 \((-39 \pm 21)\)\(\ \mathrm{m\,s^{-1}\,pc^{-1}}\), while the strongest positive outlier is Sigma Orionis at \(611 \pm 69\)\(\ \mathrm{m\,s^{-1}\,pc^{-1}}\). The detected radial motions are concentrated in young clusters, with expansion rates decreasing with age and becoming negative near \(100\)–\(300\ \mathrm{Myr}\) [2404.05327]. In the NIHAO IX halo-response analysis, contraction and expansion are measured by the dark-matter mass-profile ratio \[ \frac{M_{\rm dark}^{\rm hydro}(r)}{M_{\rm dark}^{\rm DMO}(r)} = \left[\frac{V_{\rm dark}^{\rm hydro}(r)}{V_{\rm dark}^{\rm DMO}(r)}\right]^2 \] and, in shell language, by \(\rf/\ri\). At \(0.01R_{200}\), the inner dark matter can change by up to a factor \(\sim 10\). The response is organized by \[ \epsilon_{\rm SF}\equiv (\Mstar/M_{200})/(\Omegab/\Omegam), \qquad \epsilon_R\equiv \rhalf/R_{200}. \] The reported regimes are: essentially no change for \(\epsilon_{\rm SF}\lesssim 10^{-3}\); expansion at all radii for \(10^{-3}\lesssim \epsilon_{\rm SF}\lesssim 0.1\), with strongest expansion near \(\epsilon_{\rm SF}\sim 0.03\); contraction at large radii plus expansion at small radii for \(0.1\lesssim \epsilon_{\rm SF}\lesssim 0.2\); and contraction at all radii for \(\epsilon_{\rm SF}\gtrsim 0.2\). At fixed mass or \(\epsilon_{\rm SF}\), larger galaxies experience more halo expansion and smaller galaxies more halo contraction [1605.05323]. In thin-shell tectonics, planetary contraction and expansion are prescribed kinematically by the degree-zero radial displacement \[ \bar w=\bar w_0 P_0, \] with negative \(\bar w_0\) denoting contraction and positive \(\bar w_0\) denoting expansion. For constant thickness, \[ \sigma^C_{\theta\theta}=\sigma^C_{\varphi\varphi} = \frac{E}{1-\nu}\frac{w_0}{R}, \] 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 \(40\ \mathrm{km}\) global contraction for Iapetus from porosity collapse and \(1\)–\(2\ \mathrm{km}\) contraction for Mercury inferred from lobate scarps [1006.5818]. ## 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 \(14{:}40\ \mathrm{UT}\), while the earliest lateral-loop contraction begins at \(\sim 14{:}59\ \mathrm{UT}\), about \(19\) minutes later. Coronal-loop expansion and contraction speeds are in the range \(2\) to \(34\ \mathrm{km\,s^{-1}}\). The erupting flux rope itself reaches \(110\)–\(240\ \mathrm{km\,s^{-1}}\), with border expansion relative to the flux-rope center of \(\approx 65\ \mathrm{km\,s^{-1}}\) and a center speed of \(\approx 175\ \mathrm{km\,s^{-1}}\). 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 [2101.07682]. In the 2022 August 19 M1.6 eruptive flare, adjacent coronal loops first contract for \(\sim 15\) minutes and then expand gradually until \(\sim 05{:}15\ \mathrm{UT}\). The contraction speeds are \(13\)–\(40\ \mathrm{km\,s^{-1}}\) in 171 Å and \(8\)–\(54\ \mathrm{km\,s^{-1}}\) 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-\(\tanh\) function, with the first term representing contraction and the second expansion [2305.08338]. 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 \(-12\), \(-51\), \(-57\), \(-195\), and \(-320\ \mathrm{km\,s^{-1}}\); the maximum expansion speeds are \(83\), \(90\), \(94\), \(401\), and \(538\ \mathrm{km\,s^{-1}}\); and the delay of contraction relative to expansion onset ranges from \(1.8\) to \(34.6\) minutes. The contracting loops never regain their original positions, and the paper interprets the new equilibrium as a lower-energy post-eruption state [1208.0640]. 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 [2101.07682][2305.08338][1208.0640]. ## 5. Control, dynamical-systems, and information-propagation formulations In passivity optimization for linear time-invariant systems, the main value is the extremal parameter \[ \Xi, \] the largest \(\xi\) 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 \(\hat\xi_k,\xi_{k+1}\) obtained from \[ \gamma_{\omega_k}(\xi)=0, \] and expansion values are the updated frequencies \(\omega_{k+1}\). 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 \(\Xi \approx -14.4073741346323\) and a discrete-time case with \(\Xi \approx -9.3732036470\times 10^{-5}\), both computed with dramatically fewer large pencil solves than the midpoint-based baseline [2109.00974]. 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 \[ S_n(t)=\sum_{i=1}^n \Lambda_i(t), \qquad D_n(t)=\min_{1\le i\le n}\mathcal{L}_i(t)-\max_{n+1\le i\le N}\mathcal{L}_i(t), \] together with their finite-time analogues. Here \(S_n(t)\) measures instant logarithmic \(n\)-volume growth, while \(D_n(t)\) 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, \(\lambda_1=0.906\), \(\lambda_2\approx0\), \(\lambda_3=-14.573\), and the two-dimensional volume-growth condition becomes strictly positive only on sufficiently large finite windows; for the generalized Lorenz system, \(\lambda_1=2.193\), \(\lambda_2=0\), \(\lambda_3=-1.959\), \(\lambda_4=-16.567\), and all three local violations occur [1805.06644]. In oversquashing theory for message-passing GNNs, contraction is formalized by information-theoretic attenuation. The key value is the KL contraction coefficient \[ \eta_{\mathrm{KL}}(K) = \sup_{U\to X\to Y}\frac{I(U;Y)}{I(U;X)}, \] and for a binary symmetric channel, \[ \eta_{\mathrm{KL}}(\mathrm{BSC}(\delta))=(1-2\delta)^2. \] Using the noisy-circuit analogy, the paper cites the bound \[ I(X_i;Y)\le (\eta k)^{d_i}\log 2, \] where \(k\) is gate fan-in and \(d_i\) the input-output graph distance. Expansion is quantified structurally by the Cheeger constant \[ h(G)=\min_{|\mathcal S|\le n/2}\frac{|\partial \mathcal S|}{|\mathcal S|}, \] spectral gap \(d-\mu_2(A)\), and related rewiring diagnostics. The proposed RLEF and G-RLEF schemes increase spectral expansion while preserving node degrees exactly and never disconnecting the graph [2208.03471]. ## 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 \(\epsilon>0\) one can modify an admissible fan so that the distinguished arrow becomes asymptotically an isomorphism, quantified by \[ [Z'-0.57ex X']\le \epsilon, \] while preserving both the subdiagram \(X\) and the conditioned subdiagram \(X-0.57ex U\) up to asymptotically small entropy distance. At finite scale, the proof yields the bounds \[ \ikd(\Xcal,\Xcal')\le 20\,|\Gbf|, \qquad [Z'_0-0.57ex X'_0]\le 4\ln\ln|X_0|. \] The expansion theorem is a right inverse: starting from a reduced admissible fan and any \(\lambda>0\), one constructs an expanded arrow with \[ [Z-0.57ex X]=\lambda \] while preserving \([\Xcal-0.57ex U]\) exactly [1905.05597]. In matroid theory, the expansion functor is parameterized by a multiplicity vector \[ \alpha=(k_1,\dots,k_n)\in\mathbb N^n, \] which replaces each element \(x_i\) by \(k_i\) parallel copies. The expanded ground set has cardinality \(\sum_i k_i\), but rank is preserved: \[ r(M^\alpha)=r(M). \] 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 [1705.09539]. In finite-based belief change, contraction-like and expansion-like values are finitely representable model sets. If \(\baseb\) is the current finite base and \(\mSet\) the input model set, the contraction-like eviction value is selected from \[ \FRsubs(\modelsof{\baseb}\setminus \mSet,\logsys), \] the set of maximal finitely representable subsets of the ideal remainder, while the expansion-like reception value is selected from \[ \FRsups(\modelsof{\baseb}\cup \mSet,\logsys), \] 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 [2303.03034]. In one-dimensional additive combinatorics, contraction and expansion are linked by a quantitative theorem on finite sets \(A\subset\mathbb R\). If for each \(a\in A\) there is an injective map \(f_a:A\to\mathbb R\) with \(f_a(a)=a\), \(|a-f_a(x)|\le |a-x|/K\), and \(f_a(x)\) between \(a\) and \(x\), then \[ \left|\bigcup_{a\in A} f_a(A)\right| \ge \frac18 K n\bigl(1-n^{-1/K^2}\bigr), \] hence \[ \left|\bigcup_{a\in A} f_a(A)\right| \ge \frac{K|A|}{10}-O_K(1). \] An immediate corollary is \[ |A+K\cdot A|\ge \frac{K|A|}{10}-O(e^{C K^2}). \] The paper also proves that without the betweenness condition one can have \[ \left|\bigcup_{a\in A} f_a(A)\right|=2|A|, \] so the order condition is essential for linear-in-\(K\) expansion [1112.3468]. ## 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 \(\gamma\), including the all-negative \(\gamma=1.3\) regime [1001.0467]. In pseudohyperbolic dynamics, average volume expansion and subspace contraction can be violated instantaneously even when the invariant splitting is genuinely pseudohyperbolic [1805.06644]. In Oldroyd-B constrictions, low-\(De\) pressure-drop reduction continues with \(De\), but the high-\(De\) symmetric contraction-expansion geometry reaches a plateau because the normal-stress contribution cancels [2510.07166]. 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 [2404.05327]. 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 [2303.03034]. In additive-combinatorial contractions, the betweenness condition is not cosmetic but structurally necessary for the expansion lower bound [1112.3468]. 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 [2202.00825][2208.03471][2109.00974]. 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. Source: https://www.emergentmind.com/topics/contraction-and-expansion-values