most recent 30 from http://mathoverflow.net 2013-05-24T18:23:55Z http://mathoverflow.net/feeds/question/75095 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75095/recurrence-relation-with-hadamard-product Paolo 2011-09-10T13:03:15Z 2011-09-11T00:08:45Z

<p>I've been drawn to a problem that requires ascertaining the existence of fixed points in the following recurrence relation, any ideas would be much appreciated. I seek neccessary conditions on $A,B$ such that a fixed point exists. </p> <p>For <code>$A,B \in \{0,1\}^{mn}$</code> and $U,L,R,D$, the permutation matrices that shift rows up, left, right, and down respectively (rows and columns both cycling), i.e. <code>$[UA]_{1j} = [A]_{mj}$</code> etc; </p> <p>and $\circ_{i=0}^{k}A = A\circ A\circ A ...$, where $\circ$ denotes the hadamard product, excuse the awful notation. </p> <p>$A_{n+1} = UA_n + (I-U)A_n\circ\sum_{i=1}^{m}\circ_{k=1}^{i}(D^kB_n)$</p> <p>$B_{n+1} = RB_n + (I-R)B_n\circ\sum_{i=1}^{m}\circ_{k=1}^{i}(L^kA_n)$</p>