We prove that knots with braid index three in the 3-sphere satisfy the Property P conjecture.

We study propagation property for one-dimensional nonlinear diffusion equations with convection-absorption, including the prototype model ∂t(u m)− ∂x(|∂xu|p−1∂xu)− μ|∂xu|q−1∂xu+ λu = 0, where m, p, q, k > 0, and n-dimensional simplified variant ∂t(u )−Δp+1u = 0, where Δp+1u = div (|∇u|p−1∇u). Among the conclusions, we make complete classification of the parameters in the first equation to disti...

We show that if a knot has a minimal spanning surface that admits certain Gabai disks, then this knot has Property P. As one of the applications we extend and simplify a recent result of Menasco and Zhang that closed 3-braid knots have Property P. Other applications are given.

Let G = (V,E) be a graph. A k-neighborhood in G is a set of vertices consisting of all the vertices at distance at most k from some vertex of G. The hypergraph on vertex set V which edge set consists of all the k-neighborhoods of G for all k is the neighborhood hypergraph of G. Our goal in this paper is to investigate the complexity of a graph in terms of its neighborhoods. Precisely, we define...

A property of graphs is a non-empty set of graphs. A property P is called hereditary if every subgraph of any graph with property P also has property P. Let P1, . . . ,Pn be properties of graphs. We say that a graph G has property P1◦ · · · ◦Pn if the vertex set of G can be partitioned into n sets V1, . . . , Vn such that the subgraph of G induced by Vi has property Pi; i = 1, . . . , n. A here...

We study the existence and uniqueness of the positive solutions of the problem (P): ∂tu − ∆u + u q = 0 (q > 1) in Ω × (0,∞), u = ∞ on ∂Ω × (0,∞) and u(., 0) ∈ L(Ω), when Ω is a bounded domain in R . We construct a maximal solution, prove that this maximal solution is a large solution whenever q < N/(N − 2) and it is unique if ∂Ω = ∂Ω c . If ∂Ω has the local graph property, we prove that there e...