EXACT SOLUTION OF THE ROW-CONVEX POLYGON GENERATING FUNCTION FOR THE CHECKERBOARD LATTICE

1991 ◽  
Vol 05 (15) ◽  
pp. 2551-2562 ◽  
Author(s):  
W.J. TZENG ◽  
K.Y. LIN
Keyword(s):  
Exact Solution ◽  
Generating Function ◽  
Convex Polygon ◽  
Square Lattice ◽  
Honeycomb Lattice ◽  
Rectangular Lattice ◽  
Convex Polygons ◽  
Recursion Relations ◽  
Special Cases ◽  
Special Case

We have studied the row-convex polygons on a general checkerboard lattice and derived the recursion relations for the four-variable generating function. Exact solution of the row-convex polygon generating function is obtained for a special case of the checkerboard lattice. Our result includes the square lattice, rectangular lattice and isotropic honeycomb lattice as special cases.

ON THE ROW-CONVEX POLYGON GENERATING FUNCTION FOR THE CHECKERBOARD LATTICE

1991 ◽  
Vol 05 (20) ◽  
pp. 3275-3285 ◽  
Author(s):  
K.Y. LIN ◽  
W.J. TZENG
Keyword(s):  
Exact Solution ◽  
Generating Function ◽  
Convex Polygon ◽  
Square Lattice ◽  
Rectangular Lattice ◽  
Convex Polygons ◽  
Special Cases

Exact solution for the most general four-variable generating function of the number of row-convex polygons on the checkerboard lattice is derived. Previous results for the square lattice, rectangular lattice, and honeycomb latticc are special cases of our solution.

PERIMETER AND AREA GENERATING FUNCTIONS OF THE STAIRCASE AND ROW-CONVEX POLYGONS ON THE RECTANGULAR LATTICE

1991 ◽  
Vol 05 (11) ◽  
pp. 1913-1925 ◽  
Author(s):  
K. Y. LIN ◽  
W. J. TZENG
Keyword(s):  
Perturbation Method ◽  
Generating Function ◽  
Generating Functions ◽  
Square Lattice ◽  
Rectangular Lattice ◽  
Convex Polygons

The two-variable perimeter and area generating functions derived recently by Brak and Guttmann for the staircase and row-convex polygons on the square lattice are generalized to the rectangular lattice. We consider the three-variable generating function [Formula: see text] where cn,m,r is the number of appropriate polygons with 2n horizontal steps, 2m vertical steps and area r. Two generating functions G(x, y, z) for the staircase and row-convex polygons are derived. We also calculate the generating functions for the first and second area-weighted moments by the perturbation method of Lin.

scholarly journals An Approximation Theorem for Vector Equilibrium Problems under Bounded Rationality

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 45
Author(s):  
Wensheng Jia ◽  
Xiaoling Qiu ◽  
Dingtao Peng
Keyword(s):  
Exact Solution ◽  
Bounded Rationality ◽  
Equilibrium Problems ◽  
Approximation Theorem ◽  
Optimization Problems ◽  
Vector Equilibrium Problems ◽  
Special Cases ◽  
Full Rationality ◽  
Vector Equilibrium ◽  
Special Case

In this paper, our purpose is to investigate the vector equilibrium problem of whether the approximate solution representing bounded rationality can converge to the exact solution representing complete rationality. An approximation theorem is proved for vector equilibrium problems under some general assumptions. It is also shown that the bounded rationality is an approximate way to achieve the full rationality. As a special case, we obtain some corollaries for scalar equilibrium problems. Moreover, we obtain a generic convergence theorem of the solutions of strictly-quasi-monotone vector equilibrium problems according to Baire’s theorems. As applications, we investigate vector variational inequality problems, vector optimization problems and Nash equilibrium problems of multi-objective games as special cases.

Exact solution of the row-convex polygon perimeter generating function

1990 ◽  
Vol 23 (12) ◽  
pp. 2319-2326 ◽  
Author(s):  
R Brak ◽  
A J Guttmann ◽  
I G Enting
Keyword(s):  
Exact Solution ◽  
Generating Function ◽  
Convex Polygon

ISING MODEL ON A SELF-DUAL 3-6 LATTICE

1990 ◽  
Vol 04 (01) ◽  
pp. 113-121 ◽  
Author(s):  
N. C. CHAO ◽  
FELIX J. LEE ◽  
K. Y. LIN
Keyword(s):  
Phase Transition ◽  
Critical Temperature ◽  
Ising Model ◽  
Antiferromagnetic Interaction ◽  
Interaction Parameters ◽  
Square Lattice ◽  
Rectangular Lattice ◽  
The One ◽  
Special Case

We consider the Ising model on a self-dual 3-6 lattice with six interaction parameters which includes the rectangular lattice as a special case. Depending on the signs and magnitudes of the interactions, the model has either a unique critical temperature or no phase transition. In the special case where all six interactions are the same, the unique critical temperature is identical with (different from) the one of the square lattice Ising model with the same ferromagnetic (antiferromagnetic) interaction.

scholarly journals On a Generalization of One-Dimensional Kinetics

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1264
Author(s):  
Vladimir V. Uchaikin ◽  
Renat T. Sibatov ◽  
Dmitry N. Bezbatko
Keyword(s):  
Exact Solution ◽  
Asymptotic Behavior ◽  
Constant Velocity ◽  
Telegraph Equation ◽  
Material Derivative ◽  
Symmetric Random Walk ◽  
One Dimensional ◽  
Special Cases ◽  
Fractional Telegraph Equation ◽  
Asymmetric Case

One-dimensional random walks with a constant velocity between scattering are considered. The exact solution is expressed in terms of multiple convolutions of path-distributions assumed to be different for positive and negative directions of the walk axis. Several special cases are considered when the convolutions are expressed in explicit form. As a particular case, the solution of A. S. Monin for a symmetric random walk with exponential path distribution and its generalization to the asymmetric case are obtained. Solution of fractional telegraph equation with the fractional material derivative is presented. Asymptotic behavior of its solution for an asymmetric case is provided.

AN APPROXIMATION ALGORITHM FOR LOCATING MAXIMAL DISKS WITHIN CONVEX POLYGONS

2011 ◽  
Vol 21 (06) ◽  
pp. 661-684
Author(s):  
HIROFUMI AOTA ◽  
TAKURO FUKUNAGA ◽  
HIROSHI NAGAMOCHI
Keyword(s):  
Approximation Algorithm ◽  
Convex Polygon ◽  
Computation Time ◽  
Approximation Ratio ◽  
Convex Polygons ◽  
The Given

This paper considers a problem of locating the given number of disks into a container so that the area covered by the disks is maximized. In the problem, the radii of the disks can be changed arbitrarily unless they overlap outside of the container, and the disks are allowed to overlap with each other. We present an approximation algorithm for this problem assuming that the container is a convex polygon. Our algorithm achieves approximation ratio (0.78 - ϵ) for any small ϵ > 0. Since the computation time of our algorithm depends on the number of corners of the convex polygon exponentially, we also give a heuristic to reduce the number of corners.

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