Forests, colorings and acyclic orientations of the square lattice

1999 ◽  
Vol 3 (2-4) ◽  
pp. 417-429 ◽  
Author(s):  
C. Merino ◽  
D. J. A. Welsh
Keyword(s):  
Square Lattice ◽  
Acyclic Orientations

scholarly journals Improved Bounds for the Number of Forests and Acyclic Orientations in the Square Lattice

10.37236/1697 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
N. Calkin ◽  
C. Merino ◽  
S. Noble ◽  
M. Noy
Keyword(s):  
Tutte Polynomial ◽  
Square Lattice ◽  
Transfer Matrices ◽  
Counting Problems ◽  
Acyclic Orientations

In a recent paper Merino and Welsh (1999) studied several counting problems on the square lattice $L_n$. There the authors gave the following bounds for the asymptotics of $f(n)$, the number of forests of $L_n$, and $\alpha(n)$, the number of acyclic orientations of $L_n$: $$3.209912 \le \lim_{n\to\infty} f(n)^{1/n^2} \le 3.84161$$ and $$22/7 \le \lim_{n\to\infty} \alpha(n)^{1/n^2} \le 3.70925.$$ In this paper we improve these bounds as follows: $$3.64497 \le \lim_{n\to\infty} f(n)^{1/n^2} \le 3.74101$$ and $$3.41358 \le \lim_{n\to\infty} \alpha(n)^{1/n^2} \le 3.55449.$$ We obtain this by developing a method for computing the Tutte polynomial of the square lattice and other related graphs based on transfer matrices.

Influence of Flexible Side-Chains on the Breathing Phase Transition of Pillared Layer MOFs: A Force Field Investigation

2020 ◽  
Author(s):  
Julian Keupp ◽  
Johannes P. Dürholt ◽  
Rochus Schmid
Keyword(s):  
First Principles ◽  
Good Accuracy ◽  
Field Investigation ◽  
Square Lattice ◽  
Side Chain ◽  
Second Step ◽  
Side Chains ◽  
Dynamics Simulations ◽  
Complex Phenomena ◽  
Closed Pore

The prototypical pillared layer MOFs, formed by a square lattice of paddle-<br>wheel units and connected by dinitrogen pillars, can undergo a breathing phase<br>transition by a “wine-rack” type motion of the square lattice. We studied this not<br>yet fully understood behavior using an accurate first principles parameterized force<br>field (MOF-FF) for larger nanocrystallites on the example of Zn 2 (bdc) 2 (dabco) [bdc:<br>benzenedicarboxylate, dabco: (1,4-diazabicyclo[2.2.2]octane)] and found clear indi-<br>cations for an interface between a closed and an open pore phase traveling through<br>the system during the phase transformation [Adv. Theory Simul. 2019, 2, 11]. In<br>conventional simulations in small supercells this mechanism is prevented by periodic<br>boundary conditions (PBC), enforcing a synchronous transformation of the entire<br>crystal. Here, we extend this investigation to pillared layer MOFs with flexible<br>side-chains, attached to the linker. Such functionalized (fu-)MOFs are experimen-<br>tally known to have different properties with the side-chains acting as fixed guest<br>molecules. First, in order to extend the parameterization for such flexible groups,<br>1a new parametrization strategy for MOF-FF had to be developed, using a multi-<br>structure force based fit method. The resulting parametrization for a library of<br>fu-MOFs is then validated with respect to a set of reference systems and shows very<br>good accuracy. In the second step, a series of fu-MOFs with increasing side-chain<br>length is studied with respect to the influence of the side-chains on the breathing<br>behavior. For small supercells in PBC a systematic trend of the closed pore volume<br>with the chain length is observed. However, for a nanocrystallite model a distinct<br>interface between a closed and an open pore phase is visible only for the short chain<br>length, whereas for longer chains the interface broadens and a nearly concerted trans-<br>formation is observed. Only by molecular dynamics simulations using accurate force<br>fields such complex phenomena can be studied on a molecular level.

A STUDY OF GEOMETRICALLY STABLE PATTERNS ON A SQUARE LATTICE ECONOMY

2018 ◽  
Vol 74 (4) ◽  
pp. 411-425
Author(s):  
Yosuke KOGURE ◽  
Mikihisa ONDA ◽  
Minoru OSAWA ◽  
Yuki TAKAYAMA ◽  
Kiyohiro IKEDA
Keyword(s):  
Square Lattice

scholarly journals Publisher Correction: Enhanced thermal Hall effect in the square-lattice Néel state

2021 ◽  
Author(s):  
Rhine Samajdar ◽  
Mathias S. Scheurer ◽  
Shubhayu Chatterjee ◽  
Haoyu Guo ◽  
Cenke Xu ◽  
...  
Keyword(s):  
Hall Effect ◽  
Square Lattice

Proposal for dispersion compensating square-lattice photonic crystal fiber

2021 ◽  
Vol 17 (3) ◽  
pp. 160-164
Author(s):  
Ashraful Hossain Howlader ◽  
Md. Sherajul Islam ◽  
S. M. A. Razzak
Keyword(s):  
Photonic Crystal ◽  
Photonic Crystal Fiber ◽  
Square Lattice ◽  
Crystal Fiber

Double-core photonic crystal fiber with square lattice

2004 ◽  
Author(s):  
Ryszard Buczynski ◽  
Przemyslaw Szarniak ◽  
Dariusz Pysz ◽  
Ireneusz Kujawa ◽  
Ryszard Stepien ◽  
...  
Keyword(s):  
Photonic Crystal ◽  
Photonic Crystal Fiber ◽  
Square Lattice ◽  
Crystal Fiber

Two‐Dimensional V‐VI Binary Nanosheets with Square Lattice: A Theoretical Investigation

2021 ◽  
Author(s):  
Xin Qiao ◽  
Xiaodong Lv ◽  
Yinan Dong ◽  
Yanping Yang ◽  
Fengyu Li
Keyword(s):  
Theoretical Investigation ◽  
Square Lattice ◽  
Two Dimensional
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