Nearest-neighbour systems: a lemma with application to Bartlett's global solutions

1972 ◽  
Vol 9 (02) ◽  
pp. 418-421 ◽  
Author(s):  
J. E. Besag
Keyword(s):  
Markov Model ◽  
Equilibrium Distribution ◽  
Global Solutions ◽  
Nearest Neighbour ◽  
Spatial Equilibrium ◽  
Dimensional Lattice

Bartlett has proposed two alternative forms of spatial-temporal Markov model, each leading to the standard spatial equilibrium distribution for nearest-neighbour systems on a multi-dimensional lattice. The validity of one of these forms is discussed with the aid of a lemma.

Nearest-neighbour systems: a lemma with application to Bartlett's global solutions

1972 ◽  
Vol 9 (2) ◽  
pp. 418-421 ◽  
Author(s):  
J. E. Besag
Keyword(s):  
Markov Model ◽  
Equilibrium Distribution ◽  
Global Solutions ◽  
Nearest Neighbour ◽  
Spatial Equilibrium ◽  
Dimensional Lattice

Bartlett has proposed two alternative forms of spatial-temporal Markov model, each leading to the standard spatial equilibrium distribution for nearest-neighbour systems on a multi-dimensional lattice. The validity of one of these forms is discussed with the aid of a lemma.

Physical nearest-neighbour models and non-linear time-series

1971 ◽  
Vol 8 (2) ◽  
pp. 222-232 ◽  
Author(s):  
M. S. Bartlett
Keyword(s):  
Equilibrium Distribution ◽  
Linear Models ◽  
Linear Time ◽  
Three Dimensional ◽  
Approximate Solutions ◽  
Nearest Neighbour ◽  
Spatial Equilibrium ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Non Linear

A general class of spatial-temporal Markov processes is defined leading to the standard spatial equilibrium distribution for nearest-neighbour models on a multi-dimensional lattice. Physical properties are obtainable from the marginal spatial spectral function. However, only the simplest one-dimensional case corresponds to a linear model with a readily derived spectrum. Non-linear models corresponding to two- and three-dimensional lattices are presented in their simplest terms, and a preliminary discussion of approximate solutions is included.

Physical nearest-neighbour models and non-linear time-series

1971 ◽  
Vol 8 (02) ◽  
pp. 222-232 ◽  
Author(s):  
M. S. Bartlett
Keyword(s):  
Equilibrium Distribution ◽  
Linear Models ◽  
Linear Time ◽  
Three Dimensional ◽  
Approximate Solutions ◽  
Nearest Neighbour ◽  
Spatial Equilibrium ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Non Linear

A general class of spatial-temporal Markov processes is defined leading to the standard spatial equilibrium distribution for nearest-neighbour models on a multi-dimensional lattice. Physical properties are obtainable from the marginal spatial spectral function. However, only the simplest one-dimensional case corresponds to a linear model with a readily derived spectrum. Non-linear models corresponding to two- and three-dimensional lattices are presented in their simplest terms, and a preliminary discussion of approximate solutions is included.

scholarly journals A Markov model for Selmer ranks in families of twists

2014 ◽  
Vol 150 (7) ◽  
pp. 1077-1106 ◽  
Author(s):  
Zev Klagsbrun ◽  
Barry Mazur ◽  
Karl Rubin
Keyword(s):  
Markov Process ◽  
Markov Model ◽  
Elliptic Curve ◽  
Arbitrary Number ◽  
Equilibrium Distribution ◽  
Two Dimensional ◽  
Absolute Galois Group ◽  
Quadratic Twists ◽  
The Family ◽  
Positive Integers

AbstractWe study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$ over an arbitrary number field $K$. Under the assumption that ${\rm Gal}(K(E[2])/K) \ {\cong }\ S_3$, we show that the density (counted in a nonstandard way) of twists with Selmer rank $r$ exists for all positive integers $r$, and is given via an equilibrium distribution, depending only on a single parameter (the ‘disparity’), of a certain Markov process that is itself independent of $E$ and $K$. More generally, our results also apply to $p$-Selmer ranks of twists of two-dimensional self-dual ${\bf F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.

Properties of a One-Dimensional Coulomb Gas

1981 ◽  
Vol 36 (12) ◽  
pp. 1319-1322
Author(s):  
Daniel M. Heffernan ◽  
Richard L. Liboff
Keyword(s):  
Equilibrium Distribution ◽  
Plasma Frequency ◽  
Particle Density ◽  
Coulomb Gas ◽  
One Dimension ◽  
Spatial Equilibrium ◽  
One Dimensional ◽  
Particle Vibrations ◽  
A Charge ◽  
Reduced Distributions

AbstractThe BBKGY equations for N identical, impenetrable, charged particles which move in one dimension and lie in a charge neutralizing background, are shown to separate into N uncoupled equations for the sequence of N reduced distributions. The potential relevant to any subgroup of s adjoining particles is that of an «-dimensional harmonic oscillator whose frequency is the plasma frequency of the aggregate. The «-particle spatial equilibrium distribution reveals that particle vibrations remain centered about fixed, uniformly distributed sites as σ/T goes from zero to infinity, where σ is particle density and T is temperature. Thus it is concluded that the system suffers no change in phase for all σ and T.

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