We consider a bipartite version of the color degree matrix problem. A bipartite graph G(U, V, E) is half-regular if all vertices in U have the same degree. We give necessary and sufficient conditions for a bipartite degree matrix (also known as demand matrix) to be the color degree matrix of an edge-disjoint union of halfregular graphs. We also give necessary and sufficient perturbations to transform realizations of a half-regular degree matrix into each other. Based on these perturbations, a Markov chain Monte Carlo method is designed in which the inverse of the acceptance ratios are polynomial bounded.Realizations of a half-regular degree matrix are generalizations of Latin squares, and they also appear in applied neuroscience.
Let A be an associative unital algebra, B k its successive quotients of lower central series and N k the successive quotients of ideals generated by lower central series. The geometric and algebraic aspects of B k 's and N k 's have been of great interest since the pioneering work of [11]. In this paper, we will concentrate on the case where A is a noncommutative polynomial algebra over Z modulo a single homogeneous relation. Both the torsion part and the free part of B k 's and N k 's are explored. Many examples are demonstrated in detail, and several general theorems are proved. Finally we end up with Appendix A about the torsion subgroups of N k (A n (Z)) and some open problems.
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