scholarly journals New Aspects of Regions whose Tilings are Enumerated by Perfect Powers

10.37236/3186 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Tri Lai
Keyword(s):  
Triangular Lattice ◽  
Product Formula ◽  
Square Lattice ◽  
Reduction Theorem ◽  
Simple Product ◽  
By Products

In 2003, Ciucu presented a unified way to enumerate tilings of lattice regions by using a certain Reduction Theorem (J. Algebraic Combin., 2003). In this paper we continue this line of work by investigating new families of lattice regions whose tilings are enumerated by perfect powers or products of several perfect powers. We prove a multi-parameter generalization of Bo-Yin Yang's theorem on fortresses (Ph.D. thesis, MIT, 1991).  On the square lattice with zigzag paths, we consider two particular families of regions whose numbers of tilings are always a power of 3 or twice a power of 3. The latter result provides a new proof for a conjecture of Matt Blum first proved by Ciucu. We also consider several new lattices obtained by periodically applying two simple subgraph replacement rules to the square lattice. On some of those lattices, we get new families of regions whose numbers of tilings  are given by products of several perfect powers. In addition, we prove a simple product formula for the number of tilings of a certain family of regions on a variant of the triangular lattice.

Normal-Strain Induced Change in Lattice-Type for Confined Cyclohexane Films

1996 ◽  
Vol 464 ◽  
Author(s):  
J.E. Curry ◽  
J.H. Cushman
Keyword(s):  
Triangular Lattice ◽  
Chemical Potential ◽  
Potential Temperature ◽  
Computer Experiments ◽  
Square Lattice ◽  
Melting Transition ◽  
Normal Strain ◽  
Surface Forces Apparatus ◽  
Bulk Fluid ◽  
Lattice Type

ABSTRACTOne to three layer cyclohexane films confined between mica-like surfaces are studied to elucidate changes in the films' lattice-type. The laterally confined film is in equilibrium with the bulk fluid that is well into the liquid regime of its phase diagram. Monte Carlo simulations are conducted at constant chemical potential, temperature, and V=Ah, where A is the lateral area and h is the separation between the walls. One and two layers of fluid freeze as h increases. The one layer fluid has a triangular lattice, while the two layer fluid exhibits first a square lattice and then a triangular lattice with increasing surface separation. In contrast to previous studies, solidlike order is induced primarily by the strong fluid-solid interaction and is largely a function of pore width. A shift in the relative alignment of the surfaces perturbs the solidlike fluid structure but does not cause the sudden shear melting transition associated with epitaxial alignment of the fluid atoms with the surface. There is a correlation between the shear stress calculated in the computer experiments and that measured in Surface Forces Apparatus experiments.

scholarly journals Second-Order Nonlinearity in Triangular Lattice Perforated Gold Film due to Surface Plasmas Resonance

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Renlong Zhou ◽  
Xiaoshuang Chen ◽  
Yingyi Xiao ◽  
Bingju Zhou ◽  
Lingxi Wu ◽  
...  
Keyword(s):  
Fundamental Frequency ◽  
Triangular Lattice ◽  
Gold Film ◽  
Second Harmonic ◽  
Resonance Effect ◽  
Second Order ◽  
Square Lattice ◽  
Frequency Wave ◽  
Surface Plasmas ◽  
Second Order Nonlinearity

We have studied the excitation second-order nonlinearity through a triangular lattice perforated gold film instead of square lattice in many papers. Under the excitation of surface plasmas resonance effect, the second order nonlinearity exists in the noncentrosymmetric split-ring resonators arrays. Reflection of fundamental frequency wave through a triangular lattice perforated gold film is obtained. We also described the second harmonic conversion efficiencies in the second order nonlinear optical process with the spectra. Moreover, the electric field distributions of fundamental frequency above the gold film region are calculated. The light propagation through the holes results in the enhancement of the second order nonlinearity including second harmonic generation as well as the sum (difference) frequency generation.

scholarly journals Enumeration of Hybrid Domino-Lozenge Tilings II: Quasi-Octagonal Regions

10.37236/4669 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Tri Lai
Keyword(s):  
Product Formula ◽  
Cambridge University ◽  
Replacement Method ◽  
Lozenge Tilings ◽  
Simple Product ◽  
Geometric Combinatorics

We use the subgraph replacement method to prove a simple product formula for the tilings of an  8-vertex counterpart of Propp's quasi-hexagons (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999), called quasi-octagon.

scholarly journals Lozenge Tiling Function Ratios for Hexagons with Dents on Two Sides

10.37236/9363 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Daniel Condon
Keyword(s):  
Triangular Lattice ◽  
Lozenge Tilings ◽  
Simple Product ◽  
Two Sides ◽  
Product Formulas

We give a formula for the number of lozenge tilings of a hexagon on the triangular lattice with unit triangles removed from arbitrary positions along two non-adjacent, non-opposite sides. Our formula implies that for certain families of such regions, the ratios of their numbers of tilings are given by simple product formulas.

scholarly journals Spin wave theory of spin-1/2 XY model with ring exchange on a triangular lattice

2013 ◽  
Vol 91 (7) ◽  
pp. 542-547 ◽  
Author(s):  
Solomon A. Owerre
Keyword(s):  
Phase Transition ◽  
Spin Wave ◽  
Triangular Lattice ◽  
Finite Temperature ◽  
Wave Theory ◽  
Square Lattice ◽  
Xy Model ◽  
Superfluid Phase ◽  
Zero Temperature ◽  
Ring Exchange

We present the linear spin wave theory calculation of the superfluid phase of a hard-core boson J-K model with nearest neighbour exchange J and four-particle ring-exchange K at half filling on the triangular lattice, as well as the phase diagrams of the system at zero and finite temperatures. A similar analysis has been done on a square lattice (Schaffer et al. Phys. Rev. B, 80, 014503 (2009)). We find similar behaviour to that of a square lattice but with different spin wave values of the thermodynamic quantities. We also find that the pure J model (XY model), which has a well-known uniform superfluid phase with an ordered parameter [Formula: see text] at zero temperature is quickly destroyed by the inclusion of negative-K ring-exchange interactions, favouring a state with a (4π/3, 0) ordering wavevector. We further study the behaviour of the finite-temperature Kosterlitz–Thouless phase transition (TKT) in the uniform superfluid phase, by forcing the universal quantum jump condition on the finite-temperature spin wave superfluid density. We find that for K < 0, the phase boundary monotonically decreases to T = 0 at K/J = −4/3, where a phase transition is expected and TKT decreases rapidly, while for positive K, TKT reaches a maximum at some K ≠ 0. It has been shown on a square lattice using quantum Monte Carlo (QMC) simulations that for small K > 0 away from the XY point, the zero-temperature spin stiffness value of the XY model is decreased (Melko and Sandvik. Ann. Phys. 321, 1651 (2006)). Our result seems to agree with this trend found in QMC simulations for two-dimensional systems.

ANISOTROPIC STEP, SURFACE CONTACT, AND AREA WEIGHTED DIRECTED WALKS ON THE TRIANGULAR LATTICE

2002 ◽  
Vol 16 (09) ◽  
pp. 1269-1299 ◽  
Author(s):  
A. C. OPPENHEIM ◽  
R. BRAK ◽  
A. L. OWCZAREK
Keyword(s):  
Triangular Lattice ◽  
Generating Functions ◽  
Continued Fractions ◽  
Functional Equations ◽  
Square Lattice ◽  
Combinatorial Objects ◽  
Special Cases ◽  
Explicit Formulae ◽  
Directed Walks ◽  
Marked Area

We present results for the generating functions of single fully-directed walks on the triangular lattice, enumerated according to each type of step and weighted proportional to the area between the walk and the surface of a half-plane (wall), and the number of contacts made with the wall. We also give explicit formulae for total area generating functions, that is when the area is summed over all configurations with a given perimeter, and the generating function of the moments of heights above the wall (the first of which is the total area). These results generalise and summarise nearly all known results on the square lattice: all the square lattice results can be obtaining by setting one of the step weights to zero. Our results also contain as special cases those that already exist for the triangular lattice. In deriving some of the new results we utilise the Enumerating Combinatorial Objects (ECO) and marked area methods of combinatorics for obtaining functional equations in the most general cases. In several cases we give our results both in terms of ratios of infinite q-series and as continued fractions.

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.

ORDERING PROBLEM FOR THE ANTIFERROMAGNETIC POTTS MODEL: T=0 MONTE-CARLO STUDIES

1991 ◽  
Vol 05 (19) ◽  
pp. 3061-3071 ◽  
Author(s):  
A.V. BAKAEV ◽  
V.I. KABANOVICH ◽  
A.M. KURBATOV
Keyword(s):  
Monte Carlo ◽  
Long Range ◽  
Potts Model ◽  
Triangular Lattice ◽  
Lattice Models ◽  
Range Order ◽  
Square Lattice ◽  
Long Range Order ◽  
Monte Carlo Studies ◽  
Antiferromagnetic Potts Model

AF Potts model MC dynamics at T=0 is considered. It is shown that q=3 square lattice and q=4 triangular lattice models are frozen for local MC algorithm. The nature of the previously discussed long-range order phases is examined and entropically favored states are considered.

scholarly journals A Simple Product Formula for Certain Kazhdan-LusztigR-Polynomials

2006 ◽  
Vol 34 (2) ◽  
pp. 407-414 ◽  
Author(s):  
Michela Pagliacci
Keyword(s):  
Product Formula ◽  
Simple Product

Upper Bounds for the Connective Constant of Self-Avoiding Walks

1993 ◽  
Vol 2 (2) ◽  
pp. 115-136 ◽  
Author(s):  
Sven Erick Alm
Keyword(s):  
Large Class ◽  
Triangular Lattice ◽  
Upper Bounds ◽  
Square Lattice ◽  
Numerical Application ◽  
Cubic Lattice ◽  
Simple Cubic ◽  
Simple Cubic Lattice ◽  
Connective Constant ◽  
Self Avoiding Walks

We present a method for obtaining upper bounds for the connective constant of self-avoiding walks. The method works for a large class of lattices, including all that have been studied in connection with self-avoiding walks. The bound is obtained as the largest eigenvalue of a certain matrix. Numerical application of the method has given improved bounds for all lattices studied, e.g. μ < 2.696 for the square lattice, μ < 4.278 for the triangular lattice and μ < 4.756 for the simple cubic lattice.

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