On semi-infinite systems of linear inequalities

1971 ◽  
Vol 10 (2) ◽  
pp. 252-258 ◽  
Author(s):  
R. G. Jeroslow ◽  
K. O. Kortanek
Keyword(s):  
Linear Inequalities ◽  
Infinite Systems ◽  
Systems Of Linear Inequalities

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

2020 ◽  
Vol 52 (4) ◽  
pp. 1249-1283
Author(s):  
Masatoshi Kimura ◽  
Tetsuya Takine
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Continuous Time ◽  
State Transition ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Systems Of Linear Inequalities ◽  
Free Sets ◽  
Practical Implications

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

scholarly journals Lower Bounds for the Football Pool Problem for 7 and 8 Matches

10.37236/945 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wolfgang Haas
Keyword(s):  
Lower Bounds ◽  
Covering Radius ◽  
Linear Inequalities ◽  
Minimal Cardinality ◽  
Covering Code ◽  
Ternary Code ◽  
Systems Of Linear Inequalities

Let $k_3(n)$ denote the minimal cardinality of a ternary code of length $n$ and covering radius one. In this paper we show $k_3(7)\ge 156$ and $k_3(8)\ge 402$ improving on the best previously known bounds $k_3(7)\ge 153$ and $k_3(8)\ge 398$. The proofs are founded on a recent technique of the author for dealing with systems of linear inequalities satisfied by the number of elements of a covering code, that lie in $k$-dimensional subspaces of F${}_3^n$.

scholarly journals An algorithm for constructing the skeleton graph of degenerate systems of linear inequalities

PLoS ONE ◽  
2017 ◽  
Vol 12 (4) ◽  
pp. e0175819 ◽  
Author(s):  
José Manuel Méndez Martínez ◽  
Jesús Urías
Keyword(s):  
Linear Inequalities ◽  
Skeleton Graph ◽  
Systems Of Linear Inequalities
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