Markov chains in the 2‐d Ising model: Exact solutions for special cases

1979 ◽  
Vol 71 (2) ◽  
pp. 1024-1029 ◽  
Author(s):  
Douglas Poland
Keyword(s):  
Ising Model ◽  
Markov Chains ◽  
Exact Solutions ◽  
Special Cases

Exact solutions for Ising-model odd-number correlations on planar lattices

1991 ◽  
Vol 44 (6) ◽  
pp. 2595-2608 ◽  
Author(s):  
J. H. Barry ◽  
T. Tanaka ◽  
M. Khatun ◽  
C. H. Múnera
Keyword(s):  
Ising Model ◽  
Exact Solutions ◽  
Planar Lattices

Truncation approximations of invariant measures for Markov chains

1998 ◽  
Vol 35 (03) ◽  
pp. 517-536 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Markov Chains ◽  
Transition Matrix ◽  
Invariant Measures ◽  
Invariant Distribution ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Positive Recurrent

Let P be the transition matrix of a positive recurrent Markov chain on the integers, with invariant distribution π. If (n) P denotes the n x n ‘northwest truncation’ of P, it is known that approximations to π(j)/π(0) can be constructed from (n) P, but these are known to converge to the probability distribution itself in special cases only. We show that such convergence always occurs for three further general classes of chains, geometrically ergodic chains, stochastically monotone chains, and those dominated by stochastically monotone chains. We show that all ‘finite’ perturbations of stochastically monotone chains can be considered to be dominated by such chains, and thus the results hold for a much wider class than is first apparent. In the cases of uniformly ergodic chains, and chains dominated by irreducible stochastically monotone chains, we find practical bounds on the accuracy of the approximations.

On the Generalized Navier-Stokes Equations

2003 ◽  
Author(s):  
Moustafa El-Shahed ◽  
Ahmed Salem
Keyword(s):  
Exact Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Hankel Transforms ◽  
Fourier Sine ◽  
Special Cases

In this paper, we present a general Inodel of the classical Navier-Stokes equations. With the help of Laplace, Fourier Sine transforms, finite Fourier Sine transforms, and finite Hankel transforms, an exact solutions for three different special cases have been obtained.

The existence of moments for stationary Markov chains

1983 ◽  
Vol 20 (01) ◽  
pp. 191-196 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Time Dependent ◽  
General Function ◽  
Convergence Of Moments ◽  
Special Cases

We give conditions under which the stationary distribution π of a Markov chain admits moments of the general form ∫ f(x)π(dx), where f is a general function; specific examples include f(x) = xr and f(x) = esx . In general the time-dependent moments of the chain then converge to the stationary moments. We show that in special cases this convergence of moments occurs at a geometric rate. The results are applied to random walk on [0, ∞).

Two-dimensional quasi-stationary flows in gas dynamics

1968 ◽  
Vol 64 (4) ◽  
pp. 1099-1108 ◽  
Author(s):  
A. G. Mackie
Keyword(s):  
Differential Equations ◽  
Unsteady Flow ◽  
Exact Solutions ◽  
Gas Dynamics ◽  
Two Dimensional ◽  
Independent Variables ◽  
Stationary Flows ◽  
First Case ◽  
Special Cases ◽  
Flow Round

In this paper we are concerned with the two-dimensional, unsteady flow of an inviscid, polytropic gas whose adiabatic index γ lies between 1 and 3. We recall that comparatively early in the study of gas dynamics we encounter two exact solutions of gas dynamic problems. One, in one-dimensional unsteady flow, is the expansion of a semi-infinite column of gas which is initially at rest behind a piston which, at time t = 0, begins to move with constant speed away from the gas. The second, in two-dimensional, steady, supersonic flow, is the Prandtl–Meyer flow round a sharp convex corner. Both of those flows may be regarded as special cases of more general exact solutions which are obtained by the method of characteristics (see, for example, Courant and Friedrichs(1)). On the other hand, each may be obtained directly from the appropriate equations by making use of the fact that, in so far as neither problem contains any characteristic length parameter in its formulation, the principle of dynamic similarity can be used to reduce the system of partial differential equations to one of ordinary differential equations. In the first case the independent variables x and t occur only in the combination x/t and in the second the independent variables x and y occur only in the combination x/y. Interesting and instructive as the derivation of these solutions from such principles may be, it is an unfortunate fact that they are the only non-trivial solutions of the respective equations. This is not altogether surprising as the equations are ordinary with (in this case) a limited number of non-trivially distinct solutions.

Exact solutions for scalar field cosmology in f(R) gravity

2017 ◽  
Vol 32 (30) ◽  
pp. 1750164 ◽  
Author(s):  
S. D. Maharaj ◽  
R. Goswami ◽  
S. V. Chervon ◽  
A. V. Nikolaev
Keyword(s):  
Scalar Field ◽  
Exact Solutions ◽  
Scalar Curvature ◽  
Evolution Equations ◽  
Initial Conditions ◽  
De Sitter ◽  
Airy Functions ◽  
Special Cases ◽  
The Universe ◽  
Current Stage

We study scalar field FLRW cosmology in the content of f(R) gravity. Our consideration is restricted to the spatially flat Friedmann universe. We derived the general evolution equations of the model, and showed that the scalar field equation is automatically satisfied for any form of the f(R) function. We also derived representations for kinetic and potential energies, as well as for the acceleration in terms of the Hubble parameter and the form of the f(R) function. Next we found the exact cosmological solutions in modified gravity without specifying the f(R) function. With negligible acceleration of the scalar curvature, we found that the de Sitter inflationary solution is always attained. Also we obtained new solutions with special restrictions on the integration constants. These solutions contain oscillating, accelerating, decelerating and even contracting universes. For further investigation, we selected special cases which can be applied with early or late inflation. We also found exact solutions for the general case for the model with negligible acceleration of the scalar curvature in terms of special Airy functions. Using initial conditions which represent the universe at the present epoch, we determined the constants of integration. This allows for the comparison of the scale factor in the new solutions with that for current stage of the universe evolution in the [Formula: see text]CDM model.

scholarly journals STAR-MATRIX MODELS

1995 ◽  
Vol 10 (35) ◽  
pp. 2709-2725
Author(s):  
E. VINTELER
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Matrix Models ◽  
Critical Behavior ◽  
Gaussian Model ◽  
Explicit Formulas ◽  
Special Cases ◽  
Q Matrix

The star-matrix models are difficult to solve due to the multiple powers of the Vandermonde determinants in the partition function. We apply to these models a modified Q-matrix aprpoach and we get results consistent with those obtained by other methods. As examples we study the inhomogeneous Gaussian model on Bethe tree and matrix q-Potts-like model. For the last model in the special cases q=2 and q=3, we write down explicit formulas which determine the critical behavior of the system. For q=2 we argue that the critical behavior is indeed that of the Ising model on the ϕ3 lattice.

scholarly journals Fuzzy Constrained Probabilistic Inventory Models Depending on Trapezoidal Fuzzy Numbers

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Mona F. El-Wakeel ◽  
Kholood O. Al-yazidi
Keyword(s):  
Exact Solutions ◽  
Lead Time ◽  
Fuzzy Numbers ◽  
Total Cost ◽  
Continuous Review ◽  
Fuzzy Models ◽  
First Case ◽  
Proposed Model ◽  
Special Cases ◽  
Cost Components

We discussed two different cases of the probabilistic continuous review mixture shortage inventory model with varying and constrained expected order cost, when the lead time demand follows some different continuous distributions. The first case is when the total cost components are considered to be crisp values, and the other case is when the costs are considered as trapezoidal fuzzy number. Also, some special cases are deduced. To investigate the proposed model in the crisp case and the fuzzy case, illustrative numerical example is added. From the numerical results we will conclude that Uniform distribution is the best distribution to get the exact solutions, and the exact solutions for fuzzy models are considered more practical and close to the reality of life and get minimum expected total cost less than the crisp models.

Random Vibrations in Discrete Nonlinear Dynamic Systems

1968 ◽  
Vol 10 (2) ◽  
pp. 168-174 ◽  
Author(s):  
R. G. Bhandari ◽  
R. E. Sherrer
Keyword(s):  
Exact Solutions ◽  
Density Function ◽  
Gaussian White Noise ◽  
Hermite Polynomials ◽  
Planck Equation ◽  
Degree Of Freedom ◽  
Multiple Series ◽  
Special Cases ◽  
Fokker Planck Equations ◽  
Fokker Planck

A one-degree-of-freedom system and a two-degree-of-freedom system containing Dis-placement and velocity dependent nonlinearities subjected to stationary gaussian white noise excitation have been studied by the method of the Fokker-Planck equation. Non-linearities have been represented by suitable polynomials. The Fokker-Planck equations governing the stationary probability density function for these systems have been solved by representing the density function by a multiple series of Hermite polynomials. The constants in the series expansion were determined by Galerkin's method. Analysis is developed for the system containing nonlinearities described by suitable polynomials in velocity and displacement dependent forces. Comparisons were made between series and exact solutions for those special cases for which exact solutions are known.

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