Solution of an Extremal Problem
| Autor: | G. Sh. Rubinshtein, Kazimierz Urbanik |
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| Rok vydání: | 1957 |
| Předmět: | |
| Zdroj: | Theory of Probability & Its Applications. 2:364-366 |
| ISSN: | 1095-7219 0040-585X |
| DOI: | 10.1137/1102025 |
| Popis: | Let N be a set of pairs of integers $\langle {i,j} \rangle ,i, j = 1,2, \cdots ,h$, and M a non-empty subset of N. We shall denote by $\mathcal{P}_M$ the class of all systems $P = \{ p_{ij} \} ,\langle {i,j} \rangle \in N$ satisfying the following conditions: \[ p_{ij} \geqq 0\quad {\text{for}}\quad \langle {i,j} \rangle \in N, \,p_{ij} = 0\quad {\text{for}}\quad\langle {i,j} \rangle \in N - M,\,\sum\limits_{\langle {i,j} \rangle \in N} {p_{ij} = 1.} \] Let \[ \Phi (P) = \sum\limits_{\langle {i,j} \rangle \in N} {p_{ij} \log } \frac{{p_{ij} }}{{\sum\limits_{k = 1}^n {p_{ik} } \sum\limits_{l = 1}^n {p_{ej} } }}. \] The following theorem is proved: \[ \mathop {\max }\limits_{P \in \mathcal{P}_M } \Phi (P) = \log r(M), \] where $r(M)$ is the greatest number of pairs $\langle {i_1 ,j_1 } \rangle ,\langle {i_2 ,j_2 } \rangle , \cdots ,\langle {i_s ,j_s } \rangle $ belonging to M, such that \[ i_k \ne i_l > i_k \ne j_l \quad \text{for}\quad k \ne l,\quad k,l = 1,2, \cdots ,s. \]This is an answer to a problem ra... |
| Databáze: | OpenAIRE |
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