Q-COLOR PROBLEM IN D DIMENSIONS

1988 ◽  
Vol 02 (06) ◽  
pp. 1503-1511 ◽  
Author(s):  
C. Y. PAN ◽  
X. Y. CHEN
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Cell Growth ◽  
Recursion Formula ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Four Dimensions ◽  
Color Problem ◽  
Hypercubic Lattice ◽  
Growth Method

Two methods are introduced to deal with the general q-color problem on a d-dimensional hypercubic lattice: one is the cell-growth method which gives an approximative solution of the problem in two and three dimensions, another is to combine the cell-growth method with the Monte Carlo simulation which leads to a recursion formula for the problem in d-dimensions. The results of both methods are in excellent agreement with Lieb's exact solution for the case of q = 3 in two dimensions and support Mattis' conjecture in the case of q > d, improve it in the case of 2 < q ≲ d. The validity of the recursion formula is supported by a Monte Carlo simulation up to four-dimensions.

FINITE TEMPERATURE ENTROPY OF THE ANTIFERROMAGNETIC POTTS MODEL

1988 ◽  
Vol 02 (06) ◽  
pp. 1495-1501 ◽  
Author(s):  
X. Y. CHEN ◽  
C. Y. PAN
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Potts Model ◽  
Finite Temperature ◽  
Three Dimensions ◽  
Universal Relation ◽  
Zero Temperature ◽  
Disorder Transition ◽  
Color Problem ◽  
Antiferromagnetic Potts Model

Monte Carlo simulation is used to deal with the finite temperature entropy of the q-state antiferromagnetic Potts model which is the extension of the general q-color problem (at zero temperature). The finite temperature entropy of the model in two and three dimensions is obtained which is consistent with the zero temperature results. A possible universal relation of the model to determine when the order-disorder transition happens is proposed.

NUMERICAL STUDY OF THE Q-COLOR PROBLEM ON A LATTICE

1987 ◽  
Vol 01 (01) ◽  
pp. 111-119 ◽  
Author(s):  
XIYAO CHEN ◽  
C.Y. PAN
Keyword(s):  
Numerical Study ◽  
Strong Support ◽  
Finite Size ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Zero Temperature ◽  
Color Problem ◽  
Exact Answer ◽  
Hypercubic Lattice ◽  
Good Agreement

By using the finite size extrapolation method and combined with a Monte Carlo simulation we have calculated the zero temperature entropy of the q-state Potts Antiferromagnet in two and three dimensions which is identical to the q-color problem in two and three dimensions. The model is laid on a hypercubic lattice. When q=3 (in two dimensions) the result is in good agreement with Lieb’s exact answer. When q>3 (in two and three dimensions) the results are in strong support of Mattis’ recent conjecture for the q-color problem. This method can also treat the d>3 cases without serious difficulties.

scholarly journals Bias Corrections for Probit and Logit Models with Two-way Fixed Effects

2017 ◽  
Vol 17 (3) ◽  
pp. 517-545 ◽  
Author(s):  
Mario Cruz-Gonzalez ◽  
Iván Fernández-Val ◽  
Martin Weidner
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Panel Data ◽  
Fixed Effects ◽  
Two Dimensions ◽  
Panel Data Models ◽  
Logit Models ◽  
Unobserved Effects ◽  
Parameter Problem ◽  
Bias Corrections

In this article, we present the user-written commands probitfe and logitfe, which fit probit and logit panel-data models with individual and time unobserved effects. Fixed-effects panel-data methods that estimate the unobserved effects can be severely biased because of the incidental parameter problem (Neyman and Scott, 1948, Econometrica 16: 1–32). We tackle this problem using the analytical and jackknife bias corrections derived in Fernández-Val and Weidner (2016, Journal of Econometrics 192: 291–312) for panels where the two dimensions ( N and T) are moderately large. We illustrate the commands with an empirical application to international trade and a Monte Carlo simulation calibrated to this application.

Probability Methods for Detecting Randomness in Small Sample Spatial Distributions

1994 ◽  
Vol 78 (3) ◽  
pp. 715-720 ◽  
Author(s):  
Frank O'brien
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Statistical Methods ◽  
Two Dimensions ◽  
Small Sample ◽  
Spatial Distributions ◽  
Theoretical Assumptions ◽  
Simulation Results ◽  
Probability Methods ◽  
Spatial Randomness

Several probability and statistical methods are discussed for detecting spatial randomness in two dimensions. One method is derived and proposed for its ease of application. Monte Carlo simulation results are presented in support of the theoretical assumptions of the proposed method.

Elimination of an isolated pore: Effect of grain size

1995 ◽  
Vol 10 (4) ◽  
pp. 1000-1015 ◽  
Author(s):  
Wan Y. Shih ◽  
Wei-Heng Shih ◽  
Ilhan A. Aksay
Keyword(s):  
Grain Size ◽  
Monte Carlo ◽  
Sample Size ◽  
Scaling Laws ◽  
Pore Diameter ◽  
Scaling Law ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Scaling Analysis ◽  
Scaling Relationships

The effect of grain size on the elimination of an isolated pore was investigated both by the Monte Carlo simulations and by a scaling analysis. The Monte Carlo statistical mechanics model for sintering was constructed by mapping microstructures onto domains of vectors of different orientations as grains and domains of vacancies as pores. The most distinctive feature of the simulations is that we allow the vacancies to move. By incorporating the outer surfaces of the sample in the simulations, sintering takes place via vacancy diffusion from the pores to the outer sample surfaces. The simulations were performed in two dimensions. The results showed that the model is capable of displaying various sintering phenomena such as evaporation and condensation, rounding of a sharp corner, pore coalescence, thermal etching, neck formation, grain growth, and growth of large pores. For the elimination of an isolated pore, the most salient result is that the scaling law of the pore elimination time tp with respect to the pore diameter dp changes as pore size changes from larger than the grains to smaller than the grains. For example, in sample-size-fixed simulations, tp ∼ d3p for dp < G and tp ∼ d2p for dp > G with the crossover pore diameter dc increasing linearly with G where G is the average grain diameter. For sample-size-scaled simulations, tp ∼ d4p for dp < G and tp ∼ d3p for dp > G. That tp has different scaling laws in different grain-size regimes is a result of grain boundaries serving as diffusion channels in a fine-grain microstructure such as those considered in the simulations. A scaling analysis is provided to explain the scaling relationships among tp, dp, and G obtained in the simulations. The scaling analysis also shows that these scaling relationships are independent of the dimensionality. Thus, the results of the two-dimensional simulations should also apply in three dimensions.

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