The (vertex) path-table of a tree T contains quantitative information about the paths in T . The entry (i, j) of this table gives the number of paths of length j passing through vertex vi. The path-table is a slight variation of the notion of path layer matrix. In this survey we review some work done on the vertex path-table of a tree and also introduce the edge path-table. We show that in gene...

We reduce the problem of computing a rectilinear shortest path between two given points s and t in the splinegonal domain S to the problem of computing a rectilinear shortest path between two points in the polygonal domain. As part of this, we define a polygonal domain P from S and transform a rectilinear shortest path computed in P to a path between s and t amid splinegon obstacles in S. When ...

In this paper,we study a variant of thepath cover problem, namely, the k -fixed-endpoint path cover problem. Given a graph G and a subset T of k vertices of V (G), a k fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices ofT are all endpoints of the paths inP . The k -fixed-endpoint path cover problem is to f...

where α and β range over all continuous non-decreasing functions with α(0) = β(0) = 0 and α(1) = β(1) = 1, and d′ is a distance metric between points. The functions α and β are also refereed to as reparametrizations. We call (α, β) a matching between P and Q. The Fréchet distance is described intuitively by a man walking a dog on a leash. The man follows a curve (path), and the dog follows anot...

The Brownian frame process T B is defined as T B t := (Bt−1+u)0≤u≤1 , t ∈ [0, 1] , where B is a real-valued Brownian motion with parameter set [−1, 1]. This thesis investigates properties of the path-valued Brownian frame process relevant to establishing an integration theory based on the theory of rough paths ([Lyons, 1998]). The interest in studying this object comes from its connection with ...

The Feynman path integral in p-adic quantum mechanics is considered. The probability amplitude K p (x ′′ , t ′′ ; x ′ , t ′) for one-dimensional systems with quadratic actions is calculated in an exact form, which is the same as that in ordinary quantum mechanics.