arxiv: 2605.12184 · v1 · submitted 2026-05-12 · 🧮 math-ph · cond-mat.str-el· math.MP· quant-ph

Recognition: 1 theorem link

· Lean Theorem

Local Topological Quantum Order and Spectral Gap Stability for the AKLT Models on the Hexagonal and Lieb Lattices

Amanda Young , Bruno Nachtergaele , Andrew Jackson

Authors on Pith no claims yet

Pith reviewed 2026-05-13 03:02 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.str-elmath.MPquant-ph

keywords AKLT modellocal topological quantum orderspectral gap stabilityhexagonal latticeLieb latticepolymer representationground state indistinguishability

0 comments

The pith

The AKLT models on the hexagonal and Lieb lattices have ground states satisfying local topological quantum order.

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

The paper establishes that the ground states of the AKLT spin models on the hexagonal and Lieb lattices meet the local topological quantum order condition. This means local observables measured in large but finite regions match the expectations of the unique infinite-volume ground state, with the mismatch shrinking exponentially as the observable sits farther from the region's edge. A sympathetic reader would care because this indistinguishability implies the energy gap above the ground state stays open even after the Hamiltonian receives small, decaying perturbations. The argument proceeds by adapting the polymer expansion of the ground states to these two lattices so that the approximation error can be bounded uniformly.

Core claim

The central claim is that, for sequences of increasing finite volumes that absorb the lattice, the expectation of any local observable in a finite-volume ground state differs from its value in the unique infinite-volume ground state by an amount that decays exponentially in the distance from the observable's support to the volume boundary, and the decay rate is independent of the volume. This strong form of ground-state indistinguishability directly yields the local topological quantum order property; the spectral gap above the ground state then remains stable under sufficiently weak perturbations of sufficient spatial decay.

What carries the argument

The modified polymer representation of the ground states, obtained by adapting the expansion originally introduced for the AKLT model to the geometry of the hexagonal and Lieb lattices, which supplies the uniform exponential bound on the finite-to-infinite volume error.

Load-bearing premise

The polymer representation of the ground states admits the necessary modifications for these lattices to yield uniform exponential decay of the approximation error between finite- and infinite-volume states.

What would settle it

A direct numerical or analytic computation, on successively larger finite hexagonal or Lieb lattices, that shows the error for a local observable placed at fixed distance from the boundary fails to decay exponentially with that distance.

Figures

Figures reproduced from arXiv: 2605.12184 by Amanda Young, Andrew Jackson, Bruno Nachtergaele.

**Figure 3.** Figure 3: An illustration of the polymer types comprising PN,K and P bulk N,K. Namely, γ1, γ2 ∈ Pbulk N,K, and γ1, γ2, γ3, γ4 ∈ PN,K . For any 0 ≤ K ≤ N − 2 let LN,K = {γ ∈ LΓ : γ ⊆ ΛN,K} WN,K = {γ ∈ WΓ : γ ⊆ ΛN,K ∧ ep(γ) ⊆ ∂ΛN,K} Wbulk N,K = {γ ∈ WΓ : γ ⊆ ΛN,K ∧ ep(γ) ⊆ ∂˚ΛK} where we use the convention that ∂˚ΛK = ∅ if K = 0, implying that Wbulk N,0 = ∅ and ∂ΛN,0 = ∂ΛN . Then, given any A ∈ A˚ΛK , the polymer sets… view at source ↗

**Figure 4.** Figure 4: Three walks γ ∈ WN,K with the boundary ∂ΛN,K colored to show the bipartition of Λ. The walk has even length if and only if the endpoints have the same coloring. This illustrates that the parity of |γ| only depends on (1) if γ crosses a corner of ΛN,K and (2) whether or not the γ connects the inner and outer boundaries, ∂ΛN and ∂˚ΛK. R(l) for even l with 4 ≤ l ≤ 20 l R(l) 4 1 6 1 8 4 10 9 12 26 14 75 16 215… view at source ↗

**Figure 5.** Figure 5: An illustration of the regions Ci(δ), Co(δ), and B(δ) and the decomposition of a path γ into pieces γ = λ1 ∨λ˜ 1 ∨. . . λ˜ 3 ∨λ4. Note that λi has an excursion into B(δ) of with length less than ℓ0. possible γ ′ that intersect γ can then be counted from considering the edges γ can share with the translations and reflections of this γ ′ . Now, fix 8 ≤ l ′ ≤ 20 and γ ∈ Pl with l > 6. For any global walk λ ∈… view at source ↗

**Figure 6.** Figure 6: The labeled corners Ci,i+1 ⊂ Ci(δ) in red and the trapezoids Ti ⊂ Ci(δ) in gray with trapezoidal boundaries in blue. The constraint N − K ≥ 53 ≥ ℓ0/2 + 2δ guarantees that each λj intersects only one of the corridors, Ci(δ) or Co(δ). Moreover, if γ ′ ∤ γ it has to intersect λj for some j. Thus, if we show (5.33) |Xl ′(λj )| ≤ |λj |/2 + ℓ0/2 for all j, then one has (5.34) |Xl ′(γ)| ≤ X p j=1 |Xl ′(λj )| ≤ 1 … view at source ↗

**Figure 7.** Figure 7: The region Tλj in shaded gray and walks αK+1 and βK for a walk λj that intersects Ci(δ). and the intersection is understood with respect to their edge sets. Let βK ∈ WΓ be the arc of γ (K) enclosed by Tλj plus the the first l ′ − 2 edges extending out from Tλj , see [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗

**Figure 8.** Figure 8: Illustration for each of the three summands in (7.5). The vertices xi , xj connected by a pair of edges in each image corresponds to a term (Ωxi · Ωxj ) in the associated summand. The vertex and edge sets of this volume satisfies VΛN,K ∩ V˚ΛK = ∂ΛK, EΛN,K = EΛN \ E˚ΛK . For any m ≥ 0, we denote by Λ(m) N ,˚Λ (m) N ,Λ (m) N,K ⊆ Γ (m) the finite graphs obtained by decorated each edge of its undecorated count… view at source ↗

**Figure 9.** Figure 9: The three collections of trails in S(E). Note that all three sets visit the degree four vertex v twice, but every edge is visited exactly once. If one were to first integrate (7.6) over dΩv, the set {γ1, γ2} is the collection of polymers associated from the term (Ωx1 · Ωx4 )(Ωx2 · Ωx3 ) in (7.5), {γ3} is the set associated to (Ωx1 · Ωx2 )(Ωx3 · Ωx4 ), and {γ4} is the set associated to (Ωx1 · Ωx3 )(Ωx2 · Ωx… view at source ↗

read the original abstract

We prove that the ground state of the AKLT models on the hexagonal lattice and the Lieb lattice satisfy the local topological quantum order (LTQO) condition. This will be a consequence of proving that the finite volume ground states are indistinguishable from a unique infinite volume ground state. Concretely, we identify a sequence of increasing and absorbing finite volumes for which any finite volume ground state expectation is well approximated by the infinite volume state with error decaying at a uniform exponential rate in the distance between the support of the observable and boundary of the finite volume. As a corollary to the LTQO property, we obtain that the spectral gap above the ground state in these models is stable under general small perturbations of sufficient decay. We prove these results by a detailed analysis of the polymer representation of the ground states state derived by Kennedy, Lieb and Tasaki (1988) with the necessary modifications required for proving the strong form of ground state indistinguishability needed for LTQO.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

They've shown LTQO for AKLT on the hexagonal and Lieb lattices by adapting the 1988 polymer expansion, with the key being whether the decay stays uniformly exponential after the lattice tweaks.

read the letter

The main point is that the authors prove local topological quantum order for the AKLT models on the hexagonal and Lieb lattices. They show the finite volume ground states are close to a unique infinite volume ground state with uniform exponential error decay in the distance to the boundary. This is new because these lattices had not been treated before with this property. The 1988 Kennedy-Lieb-Tasaki polymer representation is adapted here with lattice-specific changes to the polymer activities and boundary terms to make the indistinguishability hold. The work does well by providing the detailed analysis needed for the different site coordinations in these models. The hexagonal lattice has uniform degree three, while the Lieb has a checkerboard-like structure with mixed degrees, so the modifications are not trivial. The soft spot, if there is one, lies in confirming that the exponential decay remains uniform and free of polynomial prefactors or lattice-dependent rates after the modifications. The stress-test note flags that the altered polymer shapes could impact the convergence, but the paper claims to handle this through careful choice of the absorbing volumes and control of the expansion. This is for specialists in rigorous quantum spin systems who follow the development of topological order proofs. Someone extending these results to other 2D lattices would benefit from the explicit adaptations shown. It deserves a serious referee because the claims are grounded in prior work and add specific new cases with potential implications for gap stability. I recommend sending it for peer review.

Referee Report

1 major / 1 minor

Summary. The paper proves that the ground states of the AKLT models on the hexagonal and Lieb lattices satisfy the local topological quantum order (LTQO) condition. This follows from showing that finite-volume ground state expectations are indistinguishable from those of a unique infinite-volume ground state, with the error decaying exponentially in the distance from the observable support to the volume boundary. The proof involves lattice-specific modifications to the polymer representation of the ground states from Kennedy, Lieb, and Tasaki (1988). As a corollary, the spectral gap is stable under small, sufficiently decaying perturbations.

Significance. If the central claims hold, this result is significant for the field of quantum spin systems and topological phases. It extends the LTQO property and gap stability to the hexagonal lattice (relevant for 2D antiferromagnets) and the Lieb lattice (with its unique coordination). The manuscript's strength lies in providing the detailed modifications to the 1988 polymer expansion to achieve the required uniform exponential decay, which is a non-trivial technical achievement. This could serve as a model for analyzing other lattices where direct application of prior results fails due to geometry.

major comments (1)

[Polymer representation analysis (detailed in the main technical section)] The central claim of uniform exponential decay ||<O>_Λ - <O>_∞|| ≤ C exp(−dist(supp(O),∂Λ)/ξ) with ξ, C independent of the absorbing sequence Λ_n relies on the modified polymer expansion. The paper needs to explicitly verify that for the hexagonal lattice's 3-regular structure and the Lieb lattice's mixed 2/4-regular vertices, the polymer activities satisfy submultiplicativity with constants allowing convergence in a radius that ensures no lattice-dependent prefactors or loss of uniformity in the decay rate. This is load-bearing for the indistinguishability and thus for LTQO.

minor comments (1)

[Abstract] The abstract is dense and could benefit from separating the main result from the method and corollary for better readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for recognizing the significance of extending LTQO and gap stability to the hexagonal and Lieb lattices. We address the major comment below and will incorporate revisions to improve clarity.

read point-by-point responses

Referee: [Polymer representation analysis (detailed in the main technical section)] The central claim of uniform exponential decay ||<O>_Λ - <O>_∞|| ≤ C exp(−dist(supp(O),∂Λ)/ξ) with ξ, C independent of the absorbing sequence Λ_n relies on the modified polymer expansion. The paper needs to explicitly verify that for the hexagonal lattice's 3-regular structure and the Lieb lattice's mixed 2/4-regular vertices, the polymer activities satisfy submultiplicativity with constants allowing convergence in a radius that ensures no lattice-dependent prefactors or loss of uniformity in the decay rate. This is load-bearing for the indistinguishability and thus for LTQO.

Authors: We thank the referee for this observation, which correctly identifies a load-bearing technical requirement. In Sections 3 and 4 of the manuscript we adapt the Kennedy-Lieb-Tasaki polymer representation to the specific geometries: for the hexagonal lattice we use its uniform 3-regular structure to bound the polymer weights, while for the Lieb lattice we treat the mixed 2- and 4-coordinated vertices separately when enumerating connected clusters. We derive explicit submultiplicative bounds on the activities whose constants depend only on the maximum degree and the AKLT parameter range, yielding a convergence radius that is uniform across both lattices and independent of the absorbing sequence Λ_n. This ensures the exponential decay rate ξ and prefactor C remain free of lattice-dependent factors that would compromise LTQO. To make the verification fully explicit and address the referee's request, we will add a short lemma (or dedicated paragraph) in the main technical section that tabulates the activity bounds and confirms submultiplicativity for each lattice. revision: yes

Circularity Check

0 steps flagged

No circularity: LTQO follows from analysis of external KLT 1988 polymer representation with lattice modifications

full rationale

The paper derives LTQO for the AKLT models on the hexagonal and Lieb lattices by establishing uniform exponential indistinguishability between finite-volume and infinite-volume ground states. This is achieved through a detailed analysis of the polymer representation originally derived in the independent 1988 work of Kennedy, Lieb, and Tasaki, with explicit modifications for the target lattices to control boundary effects and ensure the required decay rate. The key citation has no author overlap with the present paper, the modifications are constructed and verified within the current work rather than presupposed, and no steps reduce a claimed prediction or uniqueness result to a fitted parameter or self-citation by construction. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the existence and uniqueness of the infinite-volume ground state together with the validity of the modified polymer expansion for the chosen lattices; no free parameters or new entities are introduced.

axioms (2)

domain assumption The infinite-volume AKLT ground state on the given lattices is unique and gapped.
Invoked to define the target state against which finite-volume approximations are measured.
domain assumption The polymer representation of Kennedy, Lieb and Tasaki (1988) extends to the hexagonal and Lieb lattices with the stated modifications.
Central technical assumption enabling the exponential decay estimates.

pith-pipeline@v0.9.0 · 5480 in / 1297 out tokens · 109345 ms · 2026-05-13T03:02:39.707997+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

[1]

Abdul-Rahman, M

H. Abdul-Rahman, M. Lemm, A. Lucia, B. Nachtergaele, and A. Young. A class of two-dimensional AKLT models with a gap. In A. Young H. Abdul-Rahman, R. Sims, editor,Analytic Trends in Mathematical Physics, volume 741 ofContemporary Mathematics, pages 1–21. American Mathematical Society, 2020. arXiv:1901.09297

work page arXiv 2020
[2]

Affleck, T

I. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki. Rigorous results on valence-bond ground states in antiferro- magnets.Phys. Rev. Lett., 59:799, 1987

work page 1987
[3]

Affleck, T

I. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki. Valence bond ground states in isotropic quantum antiferro- magnets.Comm. Math. Phys., 115(3):477–528, 1988

work page 1988
[4]

Arovas, A

D.P. Arovas, A. Auerbach, and F.D.M. Haldane. Extended heisenberg models of antiferromagnetism: Analogies to the fractional quantum hall effect.Phys. Rev. Lett., 60:531–534, 1988

work page 1988
[5]

Bachmann, E

S. Bachmann, E. Hamza, B. Nachtergaele, and A. Young. Product Vacua and Boundary State models ind dimensions.J. Stat. Phys., 160:636–658, 2015

work page 2015
[6]

M. Bishop. Spectral gaps for the two-species product vacua and boundary states models on the d-dimensional lattice.J. Stat. Phys., 75:418–455, 2019

work page 2019
[7]

Bishop, B

M. Bishop, B. Nachtergaele, and A. Young. Spectral gap and edge excitations ofd-dimensional PVBS models on half-spaces.J. Stat. Phys., 162:1485–1521, 2016

work page 2016
[8]

J. E. Bj¨ ornberg, P. M¨ uhlbacher, B. Nachtergaele, and D. Ueltschi. Dimerization in quantum spin chains with O(n) symmetry.Commun. Math. Phys., 387:1151–1189, 2021

work page 2021
[9]

Classification of the anyon sectors of Kitaev’s quantum double model

Alex Bols and Siddharth Vadnerkar. Classification of the anyon sectors of Kitaev’s quantum double model. Commun. Math. Phys., 406:188, 2025

work page 2025
[10]

Bourne and Y

C. Bourne and Y. Ogata. The classification of symmetry protected topological phases of one-dimensional fermion systems.Forum of Mathematics, Sigma, 9:E25, 2021

work page 2021
[11]

Bravyi, M

S. Bravyi, M. Hastings, and S. Michalakis. Topological quantum order: stability under local perturbations.J. Math. Phys., 51:093512, 2010

work page 2010
[12]

Bravyi and M

S. Bravyi and M. B. Hastings. A short proof of stability of topological order under local perturbations.Commun. Math. Phys., 307:609, 2011

work page 2011
[13]

M. Cha, P. Naaijkens, and B. Nachtergaele. On the stability of charges in infinite quantum spin systems.Commun. Math. Phys., 373:219–264, 2020. arXiv:1804.03203

work page arXiv 2020
[14]

Cubitt, D

T. Cubitt, D. Perez-Garcia, and M. M. Wolf. Undecidability of the spectral gap.Forum of Mathematics, Pi, 10:1–102, 2022. arXiv:1502.04573

work page arXiv 2022
[15]

S. X. Cui, D. Dawei, X. Han, G. Penington, D. Ranard, B. C. Rayhaun, and Z. Shangnan. Kitaev’s quantum double model as an error correcting code.Quantum, 4:331, 2020. arXiv:1908.02829

work page arXiv 2020
[16]

De Roeck and M

W. De Roeck and M. Salmhofer. Persistence of exponential decay and spectral gaps for interacting fermions. Commun. Math. Phys., 365:773–796, 2019. 58 T. JACKSON, B. NACHTERGAELE, AND A. YOUNG

work page 2019
[17]

Del Vecchio, J

S. Del Vecchio, J. Fr¨ ohlich, A. Pizzo, and S. Rossi. Lie-Schwinger block-diagonalization and gapped quantum chains with unbounded interactions.Commun. Math. Phys., 381:1115–1152, 2021. arXiv:1908.07450

work page arXiv 2021
[18]

Del Vecchio, J

S. Del Vecchio, J. Fr¨ ohlich, A. Pizzo, and S. Rossi. Local interative block-diagonalization of gapped Hamiltonians: a new tool in singular perturbation theory.J. Math. Phys., 63:073503, 2022. arXiv:2007.07667

work page arXiv 2022
[19]

Fannes, B

M. Fannes, B. Nachtergaele, and R. F. Werner. Ground states of VBS models on Cayley trees.J. Stat. Phys., 66:939–973, 1992

work page 1992
[20]

Friedli and Y

S. Friedli and Y. Velenik.Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction. Cambridge University Press, 2017

work page 2017
[21]

Fr¨ ohlich and A

J. Fr¨ ohlich and A. Pizzo. Lie-Schwinger block-diagonalization and gapped quantum chains.Commun. Math. Phys., 375:2039–2069, 2020. arXiv:1812.02457

work page arXiv 2039
[22]

T. Jackson. Long-range antiferromagnetic order in the AKLT model on trees and treelike graphs. arXiv:2511.21453, 2025

work page arXiv 2025
[23]

T. Jackson. SU(N) conjugate-neighbor AKLT models. In preparation, 2026

work page 2026
[24]

Local topological order and boundary algebras.Forum of Mathematics, Sigma, 13:e135, 2025

Corey Jones, Pieter Naaijkens, David Penneys, Daniel Wallick, and Masaki Izumi. Local topological order and boundary algebras.Forum of Mathematics, Sigma, 13:e135, 2025

work page 2025
[25]

Kapustin and N

A. Kapustin and N. Sopenko. Local Noether theorem for quantum lattice systems and topological invariants of gapped states.J. Math. Phys., 63:091903, 2022. arXiv:2201.01327

work page arXiv 2022
[26]

Kapustin, N

A. Kapustin, N. Sopenko, and B. Yang. A classification of phases of bosonic quantum lattice systems in one dimension.J. Math. Phys., 62:081901, 2021

work page 2021
[27]

Kennedy, E.H

T. Kennedy, E.H. Lieb, and H. Tasaki. A two-dimensional isotropic quantum antiferromagnet with unique disordered ground state.J. Stat. Phys., 53:383–415, 1988

work page 1988
[28]

A. Kitaev. Anyons in an exactly solved model and beyond.Annals of Physics, 321:2–111, 2006

work page 2006
[29]

Koteck´ y and D

R. Koteck´ y and D. Preiss. Cluster expansion for abstract polymer models.Commun. Math. Phys., 103:491–498, 1986

work page 1986
[30]

Lemm, A.W

M. Lemm, A.W. Sandvik, and L. Wang. Existence of a spectral gap in the Affleck-Kennedy-Lieb-Tasaki model on the hexagonal lattice.Phys. Rev. Lett., 124:177204, 2020. arXiv:1910.11810

work page arXiv 2020
[31]

Levin and X.-G

M.A. Levin and X.-G. Wen. String-net condensation: A physical mechanism for topological phases.Phys. Rev. B, 71:045110, 2005

work page 2005
[32]

Lucia, A

A. Lucia, A. Moon, and A. Young. Stability of the spectral gap and ground state indistinguishability for a decorated AKLT model.Ann. H. Poincar´ e, 25:3603–3648, 2024. arXiv:2209.01141

work page arXiv 2024
[33]

Lucia, A

A. Lucia, A. and A. Young. A nonvanishing spectral gap for AKLT models on generalized decorated graphs.J. Math. Phys., 64:041902, 2023

work page 2023
[34]

Michalakis and J.P

S. Michalakis and J.P. Zwolak. Stability of frustration-free Hamiltonians.Commun. Math. Phys., 322:277–302, 2013

work page 2013
[35]

Naaijkens

P. Naaijkens. Kosaki-Longo index and classification of charges in 2D quantum spin models.J. Math. Phys., 54(081901), 2013

work page 2013
[36]

Nachtergaele, R

B. Nachtergaele, R. Sims, and A. Young. Quasi-locality bounds for quantum lattice systems and perturbations of gapped ground states II. Perturbations of frustration-free spin models with gapped ground states.Ann. H. Poincar´ e, 23:393–511, 2022

work page 2022
[37]

Nachtergaele, R

B. Nachtergaele, R. Sims, and A. Young. Stability of the bulk gap for frustration-free topologically ordered quantum lattice systems.Lett. Math. Phys., 114:24, 2024. arXiv:2102.07209

work page arXiv 2024
[38]

Y. Ogata. A classification of pure states on quantum spin chains satisfying the split property with on-site finite group symmetries.Trans. Amer. Math. Soc. Series B, 8:39–65, 2021. ArXiv:1908.08621

work page arXiv 2021
[39]

Y. Ogata. AZ 2-index of symmetry protected topological phases with reflection symmetry for quantum spin chains.Commun. Math. Phys., 385:1245–1272, 2021. arXiv:1904.01669

work page arXiv 2021
[40]

Pomata and T.-C

N. Pomata and T.-C. Wei. AKLT models on decorated square lattices are gapped.Phys. Rev. B, 100:094429, 2019

work page 2019
[41]

Pomata and T.-C

N. Pomata and T.-C. Wei. Demonstrating the Affleck-Kennedy-Lieb-Tasaki spectral gap on 2d degree-3 lattices. Phys. Rev. Lett., 124:177203, 2020. arXiv:1911.01410

work page arXiv 2020
[42]

Ueltschi

D. Ueltschi. Cluster expansions and correlation functions.Moscow Math. J., 4:511–522, 2004

work page 2004
[43]

Affleck-Kennedy-Lieb-Tasaki state on a honeycomb lattice is a universal quantum computational resource.Phys

Tzu-Chieh Wei, Ian Affleck, and Robert Raussendorf. Affleck-Kennedy-Lieb-Tasaki state on a honeycomb lattice is a universal quantum computational resource.Phys. Rev. Lett., 106:070501, 2011

work page 2011
[44]

Hybrid valence-bond states for universal quantum computation.Phys

Tzu-Chieh Wei, Poya Haghnegahdar, and Robert Raussendorf. Hybrid valence-bond states for universal quantum computation.Phys. Rev. A, 90:042333, 2014

work page 2014
[45]

D. A. Yarotsky. Ground states in relatively bounded quantum perturbations of classical lattice systems.Commun. Math. Phys., 261:799–819, 2006. LTQO FOR AKLT MODELS ON HEXAGONAL AND LIEB LATTICES 59 (Thomas Jackson)Department of Mathematical Sciences, United Arab Emirates University, AL Ain, UAE (Bruno Nachtergaele)Department of Mathematics and Center for ...

work page 2006