For vertex-reinforced random walks, we extend results previously obtained on ℤ by Volkov [Ann. Probab. 29 (2001) 66–91] and by Basdevant, Schapira and Singh [Ann. Probab. 42 (2014) 527–558], and on ...

If k > 0, the vertex is a minimum turning point If k < 0, the vertex is a maximum turning point We can identify these properties from a quadratics graph or equation.

Every link in ℝ3 can be represented by a one-vertex ribbon graph. We prove a Markov type theorem on this subset of link diagrams. Journal Information This monthly journal, begun in 1950, is devoted ...