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Journal articles on the topic 'Rational lattice on the torus'

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Published: 4 June 2021

Last updated: 28 January 2023

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1

Kouptsov, K. L., J. H. Lowenstein, and F. Vivaldi. "Quadratic rational rotations of the torus and dual lattice maps." Nonlinearity 15, no. 6 (September 16, 2002): 1795–842. http://dx.doi.org/10.1088/0951-7715/15/6/306.

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2

NAZIR, SHAHEEN. "ON THE INTERSECTION OF RATIONAL TRANSVERSAL SUBTORI." Journal of the Australian Mathematical Society 86, no. 2 (April 2009): 221–31. http://dx.doi.org/10.1017/s1446788708000372.

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AbstractWe show that under a suitable transversality condition, the intersection of two rational subtori in an algebraic torus (ℂ*)n is a finite group which can be determined using the torsion part of some associated lattice. We also give applications to the study of characteristic varieties of smooth complex algebraic varieties. As an example we discuss A. Suciu’s line arrangement, the so-called deleted B3-arrangement.

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3

Scavia, Federico. "Retract Rationality and Algebraic Tori." Canadian Mathematical Bulletin 63, no. 1 (July 18, 2019): 173–86. http://dx.doi.org/10.4153/s0008439519000079.

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AbstractFor any prime number $p$ and field $k$, we characterize the $p$-retract rationality of an algebraic $k$-torus in terms of its character lattice. We show that a $k$-torus is retract rational if and only if it is $p$-retract rational for every prime $p$, and that the Noether problem for retract rationality for a group of multiplicative type $G$ has an affirmative answer for $G$ if and only if the Noether problem for $p$-retract rationality for $G$ has a positive answer for all $p$. For every finite set of primes $S$ we give examples of tori that are $p$-retract rational if and only if $p\notin S$.

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4

CHAIR, NOUREDDINE. "AN EXPLICIT COMPUTATION FOR THE BOSE-FERMI EQUIVALENCE ON RIEMANN SURFACES OF GENUS g." International Journal of Modern Physics A 04, no. 17 (October 20, 1989): 4437–47. http://dx.doi.org/10.1142/s0217751x89001850.

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The instanton sum in the partition function for D bosons on a Riemann surface of genus g, with values in a general D-dimensional torus, TD = RD/ΛD is given explicitly. When the rational metric Q of the lattice, ΛD, is the identity we get the bosonization formula of Alvarez-Gaumé et al. for SO( 2D ). If Q is orthogonal, in the bosonization formula, we get the theta function associated with the quadratic form Q, if Q is generic we get rational Conformal Field Theory. Also we look for conditions on a twisted spin bundle LE, which may ensure that our partition functions arise from some generalized bosonization formulas.

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5

Opdam, Eric M. "ON THE SPECTRAL DECOMPOSITION OF AFFINE HECKE ALGEBRAS." Journal of the Institute of Mathematics of Jussieu 3, no. 4 (September 8, 2004): 531–648. http://dx.doi.org/10.1017/s1474748004000155.

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An affine Hecke algebra $\mathcal{H}$ contains a large abelian subalgebra $\mathcal{A}$ spanned by the Bernstein–Zelevinski–Lusztig basis elements $\theta_x$, where $x$ runs over (an extension of) the root lattice. The centre $\mathcal{Z}$ of $\mathcal{H}$ is the subalgebra of Weyl group invariant elements in $\mathcal{A}$. The natural trace (‘evaluation at the identity’) of the affine Hecke algebra can be written as integral of a certain rational $n$-form (with values in the linear dual of $\mathcal{H}$) over a cycle in the algebraic torus $T=\textrm{Spec}(\mathcal{A})$. This cycle is homologous to a union of ‘local cycles’. We show that this gives rise to a decomposition of the trace as an integral of positive local traces against an explicit probability measure on the spectrum $W_0\setminus T$ of $\mathcal{Z}$. From this result we derive the Plancherel formula of the affine Hecke algebra.AMS 2000 Mathematics subject classification: Primary 20C08; 22D25; 22E35; 43A32

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6

DIVAKARAN, P. P., and A. K. RAJAGOPAL. "QUANTUM THEORY OF LANDAU AND PEIERLS ELECTRONS FROM THE CENTRAL EXTENSIONS OF THEIR SYMMETRY GROUPS." International Journal of Modern Physics B 09, no. 03 (January 30, 1995): 261–94. http://dx.doi.org/10.1142/s0217979295000136.

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By Wigner’s theorem on symmetries, the total state space of a quantum system whose symmetries form the group G is the collection of all projective unitary representations of G; these are, in turn, realised as certain unitary representations of the set of all central extensions of G by U(1). Exploiting this relationship, we present in this paper a new approach to the quantum mechanics of an electron in a uniform magnetic field B, in the plane (the Landau electron) and on the 2-torus in the presence of a periodic potential V whose periodicity is that of an N×N lattice (the Peierls electron). For the Landau electron, G is the Euclidean group E(2) whose central extensions arise from the Heisenberg Lie group central extensions, determined by B, of the translation subgroup. The state space is a unitary representation of the direct product of two such groups corresponding to B and -B and the Hamiltonian is a unique element of the universal enveloping algebra of the centrally-extended E(2). The complete quantum theory of the Landau electron follows directly. For the Peierls electron, lattice translation-invariance is possible only if the flux per unit cell Φ takes rational values with denominator N. The state space is a unitary representation of the direct product of a finite Heisenberg group, which is a central extension of the translation group, and a Heisenberg Lie group, both characterised by Φ. The following new results are rigorous consequences. In the empty lattice limit V=0, the energy spectrum is the Landau spectrum with degeneracy equal to the total flux through the sample. As V moves away from zero, every Landau level splits into NΦ discrete sublevels, each of degeneracy N. More generally, for V≠0 of any strength and (periodic) form, and B such that Φ is nonintegral, every point in the spectrum has multiplicity N. The degeneracy is thus proportional to the linear size rather than the area of the sample. Throughout the paper, vector potentials and gauges are dispensed with and many misconceptions thereby removed.

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7

Muñoz, Vicente. "Torus rational fibrations." Journal of Pure and Applied Algebra 140, no. 3 (August 1999): 251–59. http://dx.doi.org/10.1016/s0022-4049(98)00004-8.

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8

Galaz-García, Fernando, Martin Kerin, Marco Radeschi, and Michael Wiemeler. "Torus Orbifolds, Slice-Maximal Torus Actions, and Rational Ellipticity." International Mathematics Research Notices 2018, no. 18 (March 24, 2017): 5786–822. http://dx.doi.org/10.1093/imrn/rnx064.

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9

LIENDO, ALVARO, and CHARLIE PETITJEAN. "UNIFORMLY RATIONAL VARIETIES WITH TORUS ACTION." Transformation Groups 24, no. 1 (November 4, 2017): 149–53. http://dx.doi.org/10.1007/s00031-017-9451-8.

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10

Gorsky, Eugene, Alexei Oblomkov, Jacob Rasmussen, and Vivek Shende. "Torus knots and the rational DAHA." Duke Mathematical Journal 163, no. 14 (November 2014): 2709–94. http://dx.doi.org/10.1215/00127094-2827126.

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11

Connelly, Robert, and William Dickinson. "Periodic planar disc packings." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372, no. 2008 (February 13, 2014): 20120039. http://dx.doi.org/10.1098/rsta.2012.0039.

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Several conditions are given when a packing of equal discs in a torus is locally maximally dense, where the torus is defined as the quotient of the plane by a two-dimensional lattice. Conjectures are presented that claim that the density of any collectively jammed packing, whose graph does not consist of all triangles, and the torus lattice is the standard triangular lattice, is at most , where n is the number of packing discs in the torus. Several classes of collectively jammed packings are presented where the conjecture holds.

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12

Carotenuto, Alessandro, and Ludwik Dąbrowski. "Spin geometry of the rational noncommutative torus." Journal of Geometry and Physics 144 (October 2019): 28–42. http://dx.doi.org/10.1016/j.geomphys.2019.05.008.

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13

Neeb, Karl-Hermann. "On the Classification of Rational Quantum Tori and the Structure of Their Automorphism Groups." Canadian Mathematical Bulletin 51, no. 2 (June 1, 2008): 261–82. http://dx.doi.org/10.4153/cmb-2008-027-7.

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AbstractAn n-dimensional quantum torus is a twisted group algebra of the group ℤn. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational n-dimensional quantum tori over any field. Moreover, we show that for n = 2 the natural exact sequence describing the automorphism group of the quantum torus splits over any field.

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14

Chamizo, Fernando, and Dulcinea Raboso. "Lattice points in the 3-dimensional torus." Journal of Mathematical Analysis and Applications 429, no. 2 (September 2015): 733–43. http://dx.doi.org/10.1016/j.jmaa.2015.04.053.

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15

Ilten, Nathan, and Milena Wrobel. "Khovanskii-finite valuations, rational curves, and torus actions." Journal of Combinatorial Algebra 4, no. 2 (June 25, 2020): 141–66. http://dx.doi.org/10.4171/jca/41.

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16

Greenlees, J. P. C., and B. Shipley. "An algebraic model for rational torus-equivariant spectra." Journal of Topology 11, no. 3 (June 22, 2018): 666–719. http://dx.doi.org/10.1112/topo.12060.

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17

Brion, M. "Rational smoothness and fixed points of torus actions." Transformation Groups 4, no. 2-3 (June 1999): 127–56. http://dx.doi.org/10.1007/bf01237356.

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18

KELLER, GERHARD, and CHRISTOPH RICHARD. "Dynamics on the graph of the torus parametrization." Ergodic Theory and Dynamical Systems 38, no. 3 (September 19, 2016): 1048–85. http://dx.doi.org/10.1017/etds.2016.53.

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Model sets are projections of certain lattice subsets. It was realized by Moody [Uniform distribution in model sets. Canad. Math. Bull. 45(1) (2002), 123–130] that dynamical properties of such a set are induced from the torus associated with the lattice. We follow and extend this approach by studying dynamics on the graph of the map that associates lattice subsets to points of the torus and then we transfer the results to their projections. This not only leads to transparent proofs of known results on model sets, but we also obtain new results on so-called weak model sets. In particular, we prove pure point dynamical spectrum for the hull of a weak model set of maximal density together with the push forward of the torus Haar measure under the torus parametrization map, and we derive a formula for its pattern frequencies.

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19

HO, CHOON-LIN. "W∞ AND SLq(2) ALGEBRAS IN THE LANDAU PROBLEM AND CHERN-SIMONS THEORY ON A TORUS." Modern Physics Letters A 10, no. 35 (November 20, 1995): 2665–73. http://dx.doi.org/10.1142/s0217732395002799.

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We discuss ω∞ and sl q(2) symmetries in Chern-Simons theory and Landau problem on a torus. It is shown that when the coefficient of the Chern-Simons term, or when the total flux passing through the torus is a rational number, there exist in general two w∞ and sl q(2) algebras, instead of one set each discussed in the literature. The general wave functions for the Landau problem with rational total flux is also presented.

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20

Hegna, C. C., and A. Bhattacharjee. "Islands in three-dimensional steady flows." Journal of Fluid Mechanics 227 (June 1991): 527–42. http://dx.doi.org/10.1017/s002211209100023x.

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We consider the problem of steady Euler flows in a torus. We show that in the absence of a direction of symmetry the solution for the vorticity contains δ-function singularities at the rational surfaces of the torus. We study the effect of a small but finite viscosity on these singularities. The solutions near a rational surface contain cat's eyes or islands, well known in the classical theory of critical layers. When the islands are small, their widths can be computed by a boundary-layer analysis. We show that the islands at neighbouring rational surfaces generally overlap. Thus, steady toroidal flows exhibit a tendency towards Beltramization.

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21

Li, Miao. "Abelian Chern-Simons theory and CFT of rational torus." Il Nuovo Cimento B 105, no. 10 (October 1990): 1113–17. http://dx.doi.org/10.1007/bf02827320.

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22

Bertram, Aaron. "Another way to enumerate rational curves with torus actions." Inventiones mathematicae 142, no. 3 (December 2000): 487–512. http://dx.doi.org/10.1007/s002220000094.

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23

Greenlees, J. P. C. "Rational torus-equivariant stable homotopy III: Comparison of models." Journal of Pure and Applied Algebra 220, no. 11 (November 2016): 3573–609. http://dx.doi.org/10.1016/j.jpaa.2016.05.001.

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24

Addas-Zanata, Salvador, and Patrice Le Calvez. "Rational mode locking for homeomorphisms of the $2$-torus." Proceedings of the American Mathematical Society 146, no. 4 (December 26, 2017): 1551–70. http://dx.doi.org/10.1090/proc/13793.

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25

JÄGER, T., and F. TAL. "Irrational rotation factors for conservative torus homeomorphisms." Ergodic Theory and Dynamical Systems 37, no. 5 (March 8, 2016): 1537–46. http://dx.doi.org/10.1017/etds.2015.112.

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We provide an equivalent characterization for the existence of one-dimensional irrational rotation factors of conservative torus homeomorphisms that are not eventually annular. It states that an area-preserving non-annular torus homeomorphism $f$ is semiconjugate to an irrational rotation $R_{\unicode[STIX]{x1D6FC}}$ of the circle if and only if there exists a well-defined speed of rotation in some rational direction on the torus, and the deviations from the constant rotation in this direction are uniformly bounded. By means of a counterexample, we also demonstrate that a similar characterization does not hold for eventually annular torus homeomorphisms.

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26

Chamizo, F., E. Cristóbal, and A. Ubis. "Lattice points in rational ellipsoids." Journal of Mathematical Analysis and Applications 350, no. 1 (February 2009): 283–89. http://dx.doi.org/10.1016/j.jmaa.2008.09.051.

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27

Fink, Alex. "Lattice games without rational strategies." Journal of Combinatorial Theory, Series A 119, no. 2 (February 2012): 450–59. http://dx.doi.org/10.1016/j.jcta.2011.10.005.

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28

VASSILEVICH, D. V. "INDUCED CHERN–SIMONS ACTION ON NONCOMMUTATIVE TORUS." Modern Physics Letters A 22, no. 17 (June 7, 2007): 1255–63. http://dx.doi.org/10.1142/s0217732307023596.

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We compute a Chern–Simons term induced by the fermions on noncommutative torus interacting with two U(1) gauge fields. For rational noncommutativity θ∝P/Q we find a new mixed term in the action which involves only those fields which are (2π)/Q periodic, like the fields in a crystal with Q2nodes.

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29

Feng, Wei, Songlin Zhao, and Ying Shi. "Rational Solutions for Lattice Potential KdV Equation and Two Semi-discrete Lattice Potential KdV Equations." Zeitschrift für Naturforschung A 71, no. 2 (February 1, 2016): 121–28. http://dx.doi.org/10.1515/zna-2015-0473.

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AbstractBy imposing reduction conditions on rational solutions for a system involving the Hirota–Miwa equation, rational solutions for lattice potential KdV equation are constructed. Besides, the rational solutions for two semi-discrete lattice potential KdV equations are also considered. All these rational solutions are in the form of Schur function type.

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30

Zverev, N. V., and A. A. Slavnov. "Nonlocal lattice fermionic models on a two-dimensional torus." Theoretical and Mathematical Physics 115, no. 1 (April 1998): 448–57. http://dx.doi.org/10.1007/bf02575503.

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31

Nowak, W. G. "The lattice point discrepancy of a torus in ℝ3." Acta Mathematica Hungarica 120, no. 1-2 (March 14, 2008): 179–92. http://dx.doi.org/10.1007/s10474-007-7129-8.

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32

de Carvalho, Edson Donizete, Waldir Silva Soares, and Eduardo Brandani da Silva. "Topological Quantum Codes from Lattices Partition on the n-Dimensional Flat Tori." Entropy 23, no. 8 (July 27, 2021): 959. http://dx.doi.org/10.3390/e23080959.

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In this work, we show that an n-dimensional sublattice Λ′=mΛ of an n-dimensional lattice Λ induces a G=Zmn tessellation in the flat torus Tβ′=Rn/Λ′, where the group G is isomorphic to the lattice partition Λ/Λ′. As a consequence, we obtain, via this technique, toric codes of parameters [[2m2,2,m]], [[3m3,3,m]] and [[6m4,6,m2]] from the lattices Z2, Z3 and Z4, respectively. In particular, for n=2, if Λ1 is either the lattice Z2 or a hexagonal lattice, through lattice partition, we obtain two equivalent ways to cover the fundamental cell P0′ of each hexagonal sublattice Λ′ of hexagonal lattices Λ, using either the fundamental cell P0 or the Voronoi cell V0. These partitions allow us to present new classes of toric codes with parameters [[3m2,2,m]] and color codes with parameters [[18m2,4,4m]] in the flat torus from families of hexagonal lattices in R2.

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33

Combot, Thierry. "Rational integrability of trigonometric polynomial potentials on the flat torus." Regular and Chaotic Dynamics 22, no. 4 (July 2017): 386–407. http://dx.doi.org/10.1134/s1560354717040049.

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34

Brion, M. "Erratum to "Rational Smoothness and Fixed Points of Torus Actions"." Transformation Groups 7, no. 1 (March 1, 2002): 107. http://dx.doi.org/10.1007/s00031-002-0007-0.

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35

Bregman, Corey. "Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles." International Mathematics Research Notices 2019, no. 13 (October 12, 2017): 4004–46. http://dx.doi.org/10.1093/imrn/rnx243.

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AbstractGiven a matrix $A\in SL(N,\mathbb{Z})$, form the semidirect product $G=\mathbb{Z}^N\rtimes_A \mathbb{Z}$ where the $\mathbb{Z}$-factor acts on $\mathbb{Z}^N$ by $A$. Such a $G$ arises naturally as the fundamental group of an $N$-dimensional torus bundle which fibers over the circle. In this article, we prove that if $A$ has distinct eigenvalues not lying on the unit circle, then there exists a finite index subgroup $H\leq G$ possessing rational growth series for some generating set. In contrast, we show that if $A$ has at least one eigenvalue not lying on the unit circle, then $G$ is not almost convex for any generating set.

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36

Brion, M. "Erratum to ?rational smoothness and fixed points of torus actions?" Transformation Groups 7, no. 1 (March 2002): 107. http://dx.doi.org/10.1007/bf01253468.

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37

Nassar, Ali, and Mark A. Walton. "Rational conformal field theory with matrix level and strings on a torus." Canadian Journal of Physics 92, no. 1 (January 2014): 65–70. http://dx.doi.org/10.1139/cjp-2013-0326.

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Study of the matrix-level affine algebra Um,K is motivated by conformal field theory and the fractional quantum Hall effect. Gannon completed the classification of Um,K modular-invariant partition functions. Here we connect the algebra U2,K to strings on 2-tori describable by rational conformal field theories. As Gukov and Vafa proved, rationality selects the complex-multiplication tori. We point out that the rational conformal field theories describing strings on complex-multiplication tori have characters and partition functions identical to those of the matrix-level algebra Um,K. This connection makes it obvious that the rational theories are dense in the moduli space of strings on Tm, and may prove useful in other ways.

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38

Page, Warren. "A Rational Approach to Lattice Polygons." College Mathematics Journal 18, no. 4 (September 1987): 316. http://dx.doi.org/10.2307/2686802.

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39

Page, Warren. "A Rational Approach to Lattice Polygons." College Mathematics Journal 18, no. 4 (September 1987): 316–18. http://dx.doi.org/10.1080/07468342.1987.11973050.

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40

Bodnár, József, and András Némethi. "Lattice cohomology and rational cuspidal curves." Mathematical Research Letters 23, no. 2 (2016): 339–75. http://dx.doi.org/10.4310/mrl.2016.v23.n2.a3.

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41

Chen, Beifang, and Vladimir Turaev. "Counting Lattice Points of Rational Polyhedra." Advances in Mathematics 155, no. 1 (October 2000): 84–97. http://dx.doi.org/10.1006/aima.2000.1931.

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42

LIU, GENQIANG, and KAIMING ZHAO. "IRREDUCIBLE HARISH CHANDRA MODULES OVER THE DERIVATION ALGEBRAS OF RATIONAL QUANTUM TORI." Glasgow Mathematical Journal 55, no. 3 (February 25, 2013): 677–93. http://dx.doi.org/10.1017/s0017089512000845.

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AbstractLet d be a positive integer, q=(qij)d×d be a d×d matrix, ℂq be the quantum torus algebra associated with q. We have the semidirect product Lie algebra $\mathfrak{g}$=Der(ℂq)⋉Z(ℂq), where Z(ℂq) is the centre of the rational quantum torus algebra ℂq. In this paper, we construct a class of irreducible weight $\mathfrak{g}$-modules $\mathcal{V}$α (V,W) with three parameters: a vector α∈ℂd, an irreducible $\mathfrak{gl}$d-module V and a graded-irreducible $\mathfrak{gl}$N-module W. Then, we show that an irreducible Harish Chandra (uniformaly bounded) $\mathfrak{g}$-module M is isomorphic to $\mathcal{V}$α(V,W) for suitable α, V, W, if the action of Z(ℂq) on M is associative (respectively nonzero).

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43

SHEN, YUNZHU, and YONGXIANG ZHANG. "STRANGE NONCHAOTIC ATTRACTORS IN A QUASIPERIODICALLY-FORCED PIECEWISE SMOOTH SYSTEM WITH FAREY TREE." Fractals 27, no. 07 (November 2019): 1950118. http://dx.doi.org/10.1142/s0218348x19501184.

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The existence of strange nonchaotic attractors (SNAs) is verified in a simple quasiperiodically-forced piecewise smooth system with Farey tree. It can be seen that more and more jumping discontinuities appear on the smooth torus and the torus becomes extremely fragmented with the change of control parameter. Finally, the torus becomes an SNA with fractal property. In order to confirm the existence of SNAs in this system, we preliminarily use the estimation of the phase sensitivity exponent, estimation of the largest Lyapunov exponent and rational approximation. SNAs are further characterized by power spectra, recurrence plots, the largest Lyapunov exponents and their variance, the distribution of the finite-time Lyapunov exponents, the spectral distribution function and scaling laws.

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44

Zaslavski, Alexander J. "Generic uniqueness of a minimal solution for variational problems on a torus." Abstract and Applied Analysis 7, no. 3 (2002): 143–54. http://dx.doi.org/10.1155/s1085337502000842.

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We study minimal solutions for one-dimensional variational problems on a torus. We show that, for a generic integrand and any rational numberα, there exists a unique (up to translations) periodic minimal solution with rotation numberα.

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45

YIN, QIAN. "Lattès maps and combinatorial expansion." Ergodic Theory and Dynamical Systems 36, no. 4 (February 11, 2015): 1307–42. http://dx.doi.org/10.1017/etds.2014.125.

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A Lattès map $f:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ is a rational map that is obtained from a finite quotient of a conformal torus endomorphism. We characterize Lattès maps by their combinatorial expansion behavior.

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46

Herppich, Elaine. "On Fano Varieties with Torus Action of Complexity 1." Proceedings of the Edinburgh Mathematical Society 57, no. 3 (April 16, 2014): 737–53. http://dx.doi.org/10.1017/s0013091513000710.

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AbstractIn this work we provide effective bounds and classification results for rational ℚ-factorial Fano varieties with a complexity-one torus action and Picard number 1 depending on the two invariants dimension and Picard index. This complements earlier work by Hausenet al., where the case of a free divisor class group of rank 1 was treated.

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47

Kaneyama, Tamafumi. "Torus-equivariant vector bundles on projective spaces." Nagoya Mathematical Journal 111 (September 1988): 25–40. http://dx.doi.org/10.1017/s0027763000000982.

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For a free Z-module N of rank n, let T = TN be an n-dimensional algebraic torus over an algebraically closed field k defined by N. Let X = TN emb (Δ) be a smooth complete toric variety defined by a fan Δ (cf. [6]). Then T acts algebraically on X. A vector bundle E on X is said to be an equivariant vector bundle, if there exists an isomorphism ft: t*E → E for each k-rational point t in T, where t: X → X is the action of t. Equivariant vector bundles have T-linearizations (see Definition 1.2 and [2], [4]), hence we consider T-linearized vector bundles.

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48

Pasquier, V. "Lattice derivation of modular invariant partition functions on the torus." Journal of Physics A: Mathematical and General 20, no. 18 (December 21, 1987): L1229—L1237. http://dx.doi.org/10.1088/0305-4470/20/18/003.

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49

Deshpande, Abhinav, and Anne E. B. Nielsen. "Lattice Laughlin states on the torus from conformal field theory." Journal of Statistical Mechanics: Theory and Experiment 2016, no. 1 (January 28, 2016): 013102. http://dx.doi.org/10.1088/1742-5468/2016/01/013102.

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50

Ioffe, Dmitry, and Bálint Tóth. "Split-and-Merge in Stationary Random Stirring on Lattice Torus." Journal of Statistical Physics 180, no. 1-6 (February 1, 2020): 630–53. http://dx.doi.org/10.1007/s10955-020-02487-2.

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