scholarly journals Lozenge tiling constrained codes

2014 ◽  
Vol 27 (4) ◽  
pp. 521-542 ◽  
Author(s):  
Bane Vasic ◽  
Anantha Krishnan
Keyword(s):  
Triangular Lattice ◽  
Constrained Systems ◽  
Square Lattice ◽  
Theoretical Treatment ◽  
Two Dimensional ◽  
Run Length ◽  
One Dimensional ◽  
Constrained Codes ◽  
Capacity Bounds ◽  
Decoder Design

While the field of one-dimensional constrained codes is mature, with theoretical as well as practical aspects of code- and decoder-design being well-established, such a theoretical treatment of its two-dimensional (2D) counterpart is still unavailable. Research has been conducted on a few exemplar 2D constraints, e.g., the hard triangle model, run-length limited constraints on the square lattice, and 2D checkerboard constraints. Excluding these results, 2D constrained systems remain largely uncharacterized mathematically, with only loose bounds of capacities present. In this paper we present a lozenge constraint on a regular triangular lattice and derive Shannon noiseless capacity bounds. To estimate capacity of lozenge tiling we make use of the bijection between the counting of lozenge tiling and the counting of boxed plane partitions.

scholarly journals Cluster Structure of Optimal Solutions in Bipartitioning of Small Worlds

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1319
Author(s):  
Adam Lipowski ◽  
António L. Ferreira ◽  
Dorota Lipowska
Keyword(s):  
Cluster Structure ◽  
Finite Size ◽  
Lattice Models ◽  
Square Lattice ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Small Worlds ◽  
Replica Symmetry ◽  
One Dimensional ◽  
Optimal Partitions

Using simulated annealing, we examine a bipartitioning of small worlds obtained by adding a fraction of randomly chosen links to a one-dimensional chain or a square lattice. Models defined on small worlds typically exhibit a mean-field behavior, regardless of the underlying lattice. Our work demonstrates that the bipartitioning of small worlds does depend on the underlying lattice. Simulations show that for one-dimensional small worlds, optimal partitions are finite size clusters for any fraction of additional links. In the two-dimensional case, we observe two regimes: when the fraction of additional links is sufficiently small, the optimal partitions have a stripe-like shape, which is lost for a larger number of additional links as optimal partitions become disordered. Some arguments, which interpret additional links as thermal excitations and refer to the thermodynamics of Ising models, suggest a qualitative explanation of such a behavior. The histogram of overlaps suggests that a replica symmetry is broken in a one-dimensional small world. In the two-dimensional case, the replica symmetry seems to hold, but with some additional degeneracy of stripe-like partitions.

scholarly journals Band Gap of Two-dimensional Layered Cylindrical Photonic Crystal Slab and Slow Light of W1 Waveguide

2021 ◽  
Author(s):  
Zhi-Wei Wang ◽  
Ya-Ting Xiang ◽  
Hai-Feng Zhang
Keyword(s):  
Refractive Index ◽  
Triangular Lattice ◽  
Slow Light ◽  
Large Range ◽  
Square Lattice ◽  
Two Dimensional ◽  
Photonic Crystal Slab ◽  
Slowing Down ◽  
Number Of Layers ◽  
Photonic Band

Abstract In this paper, we apply the scatterers of cylindrical rings to a two-dimensional photonic crystals (PCs) slab. The effects of the number of layers, the thickness, the index, and the height of the cylindrical layers on the photonic band gaps (PBGs) of such slab with different lattice arrangements are studied. It turns out that our new structure helps to obtain a large range of the PBGs. The maximum bandwidth is obtained with the value of 0.1497 (2πc/a). The PBGs are moved to the lower frequencies with the augment of thickness, refractive index, and height. The choice of height, refractive index, and thickness is a trade-off, and adding the number of dielectric layers is not always positively correlated with the area of PBGs. In addition, in the W1 waveguide with a triangular lattice layout, we obtain a slow light of 0.026×c. Compared with the square lattice, the triangular lattice is more suitable for slowing down the speed of light.

A Two-Dimensional Coordination Polymer Formed from Cobalt(II) and an Extended Dipyridyl Ligand

2020 ◽  
Vol 73 (6) ◽  
pp. 547 ◽  
Author(s):  
Hydar A. AL-Fayaad ◽  
Rashid G. Siddique ◽  
Kasun S. Athukorala Arachchige ◽  
Jack K. Clegg
Keyword(s):  
Coordination Polymer ◽  
Cell Volume ◽  
Crystal Structures ◽  
Unit Cell Volume ◽  
Suzuki Coupling ◽  
Unit Cell ◽  
Square Lattice ◽  
Two Dimensional ◽  
Cobalt Ions ◽  
One Dimensional

The synthesis of the extended dipyridyl ligand 4,4′-(2,5-dimethyl-1,4-phenylene)dipyridine (L) in an improved yield via the palladium catalysed Suzuki coupling of 4-(4,4,5,5-tetramethyl-1,3,2-dioxaborolan-2-yl)pyridine (1) and 1,4-dibromo-2,5-dimethylbenzene (2) is reported along with its use to form a two-dimensional coordination polymer [Co2L2(OAc)4(H2O)2]n. The coordination polymer consists of one-dimensional chains of octahedral cobalt ions bridged by acetate ligands which are connected to form two dimensional sheets with square lattice (sql) topology via the dipyridyl ligands (L). The structure contains small voids totalling ~6.6% of the unit cell volume. The crystal structures 1, L, L·2H2O, and L·2HNO3 are also reported.

scholarly journals Simulation of Two Dimensional Photonic Band Gaps

2013 ◽  
Vol 24 ◽  
pp. 58-88
Author(s):  
S.E. Dissanayake ◽  
K.A.I.L. Wijewardena Gamalath
Keyword(s):  
Triangular Lattice ◽  
Expansion Method ◽  
Dielectric Medium ◽  
Band Gaps ◽  
Square Lattice ◽  
Honeycomb Lattice ◽  
Two Dimensional ◽  
Filling Fraction ◽  
Telecommunication Wavelength ◽  
Photonic Band

The plane wave expansion method was implemented in modelling and simulating the band structures of two dimensional photonic crystals with square, triangular and honeycomb lattices with circular, square and hexagonal dielectric rods and air holes. Complete band gaps were obtained for square lattice of square GaAs rods and honeycomb lattice of circular and hexagonal GaAs rods as well as triangular lattice of circular and hexagonal air holes in GaAs whereas square lattice of square or circular air holes in a dielectric medium ε = 18 gave complete band gaps. The variation of these band gaps with dielectric contrast and filling factor gave the largest gaps for all configurations for a filling fraction around 0.1.The gap maps presented indicated that TM gaps are more favoured by dielectric rods while TE gaps are favoured by air holes. The geometrical gap maps operating at telecommunication wavelength λ = 1.55 μm showed that a complete band gap can be achieved for triangular lattice with circular and hexagonal air holes in GaAs and for honeycomb lattice of circular GaAs rods.

On the Riemann property of angular lattice sums and the one-dimensional limit of two-dimensional lattice sums

2008 ◽  
Vol 464 (2100) ◽  
pp. 3327-3352 ◽  
Author(s):  
Ross C McPhedran ◽  
I.J Zucker ◽  
Lindsay C Botten ◽  
Nicolae-Alexandru P Nicorovici
Keyword(s):  
Critical Line ◽  
Quadratic Function ◽  
Quadratic Functions ◽  
Square Lattice ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Lattice Sums ◽  
The One ◽  
Complex Powers

We consider a general class of two-dimensional lattice sums consisting of complex powers s of inverse quadratic functions. We consider two cases, one where the quadratic function is negative definite and another more restricted case where it is positive definite. In the former, we use a representation due to H. Kober, and consider the limit u →∞, where the lattice becomes ever more elongated along one period direction (the one-dimensional limit). In the latter, we use an explicit evaluation of the sum due to Zucker and Robertson. In either case, we show that the one-dimensional limit of the sum is given in terms of ζ (2 s ) if Re( s )>1/2 and either ζ (2 s −1) or ζ (2−2 s ) if Re( s )<1/2. In either case, this leads to a Riemann property of these sums in the one-dimensional limit: their zeros must lie on the critical line Re( s )=1/2. We also comment on a class of sums that involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle. We show that certain of these sums can have their zeros on the critical line but not in a neighbourhood of it; others are identically zero on it, while still others have no zeros on it.

scholarly journals QUANTUM MODEL OF INTERACTING "STRINGS" ON THE SQUARE LATTICE

2000 ◽  
Vol 15 (01) ◽  
pp. 105-131
Author(s):  
H. E. BOOS
Keyword(s):  
Square Lattice ◽  
Quantum Model ◽  
Free Fermion ◽  
Two Dimensional ◽  
Fermion Model ◽  
One Dimensional ◽  
Single String ◽  
Free Fermion Model ◽  
Xy Spin Chain ◽  
The One

The model which is the generalization of the one-dimensional XY-spin chain for the case of the two-dimensional square lattice is considered. The subspace of the "string" states is studied. The solution to the eigenvalue problem is obtained for the single "string" in cases of the "string" with fixed ends and "string" of types (1, 1) and (1, 2) living on the torus. The latter case has the features of a self-interacting system and does not seem to be integrable while the previous two cases are equivalent to the free-fermion model.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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