Let D = H \ ∪k=1Ck be a standard slit domain where H is the upper half plane and Ck, 1 ≤ k ≤ N , are mutually disjoint horizontal line segments in H. Given a Jordan arc γ ⊂ D starting at ∂H, let gt be the unique conformal map from D \ γ[0, t] onto a standard slit domain Dt satisfying the hydrodynamic normalization. We prove that gt satisfies an ODE with the kernel on its righthand side being th...

Let G and H be connected graphs. The tensor product G + H is a graph with vertex set V(G+H) = V (G) X V(H) and edge set E(G + H) ={(a , b)(x , y)| ax ∈ E(G) & by ∈ E(H)}. The graph H is called the strongly triangular if for every vertex u and v there exists a vertex w adjacent to both of them. In this article the tensor product of G + H under some distancebased topological indices are investiga...

We analyze the optimal value function v associated with a general parametric optimization problem via theory of viscosity solutions. The novelty is that we obtain regularity properties by showing it solution to set first-order equations. As consequence, in Banach spaces, provide sufficient conditions for local and global Lipschitz v. also derive, finite dimensions, optimality through comparison...