For two vertices u and v in a graph G = (V, E), the detour distance D(u, v) is the length of a longest u–v path in G. A u–v path of length D(u, v) is called a u–v detour. A set S ⊆ V is called an edge detour set if every edge in G lies on a detour joining a pair of vertices of S. The edge detour number dn1(G) of G is the minimum order of its edge detour sets and any edge detour set of order dn1...

Abstract Garnet–kyanite–staurolite assemblages with large, late porphyroblasts of amphibole form garbenschists in Ordovician volcaniclastic rocks lying immediately south the Pearya terrane on northernmost Ellesmere Island, Canada. The schist, which together carbonate olistoliths makes up Petersen Bay Assemblage (PBA), displays a series parallel isograds that mark an increase metamorphic grade o...

In this work we study the problem of efficiently isolating witnesses for the complexity classes NL and LogCFL, which are two well-studied complexity classes contained in P. We prove that if there is a L/poly randomized procedure with success probability at least 2/3 for isolating an s-t path in a given directed graph with a source sink pair (s, t) then NL is contained in L/poly. By isolating a ...

An interior path-following algorithm is proposed for solving the nonlinear saddle point problem minimax c T x + (x) + b T y ? (y) ? y T Ax subject to (x; y) 2 X Y R n R m ; where (x) and (y) are smooth convex functions and X and Y are boxes (hyper-rectangles). This problem is closely related to models in stochastic programming and optimal control studied by Rockafellar and Wets. Existence condi...

Transition path sampling is a methodology which overcomes both the long timescale problem and the lack of prior knowledge about transition mechanisms. Here we briefly review the basic principles of transition path sampling, illustrate its application using autoionization in liquid water, and emphasize the capabilities and limitations of the methodology. 1 Why transition path sampling is needed ...

We consider the problem of finding minimum-link rectilinear paths in rectilinear polygonal domains in the plane. A path or a polygon is rectilinear if all its edges are axis-parallel. Given a set P of h pairwise-disjoint rectilinear polygonal obstacles with a total of n vertices in the plane, a minimumlink rectilinear path between two points is a rectilinear path that avoids all obstacles with ...

This main purpose of this paper is to develop an algorithm to find the shortest path on a network in which the weights of the edges are represented by bipolar neutrosophic numbers. Finally, a numerical example has been provided for illustrating the proposed approach. Introduction Smarandache [1, 2] introduced neutrosophic set and neutrosophic logic by considering the nonstandard analysis. The c...

Let us collect before the proof some notations and elementary facts to be used later. Given a simplicial complex J, by the notations Hn(T) and Zn(J) will be meant respectively the nth homology group and the group of M-cycles of T, formed by using finite chains with integral coefficients. Let P be a connected polyhedron and 5 be the w-sphere zZt-o xi = 1 in Euclidean (w + 1)-space (n^2). Let <p ...

An eclogite and its surrounding gneiss from the Malpica-Tuy zone, northwestern Spain, were studied. The eclogite contains besides garnet and omphacite, barroisitic amphibole, quartz, phengite with Si contents between 3.27 and 3.45 per formula unit (pfu), rutile, and accessory phases such as talc and (clino)zoisite-epidote. Garnet exhibits a chemical zonation with Alm59Prp13Grs24Sps4, Alm57Prp13...

Let D be a digraph and let λ ( ) denote the number of vertices in longest path . For pair vertex-disjoint induced subdigraphs A B , we say that is partition if V ∪ = The Path Partition Conjecture (PPC) states for every digraph, integer q with 1 ≤ − there exists such T vertex set { u … t } i ∈ [ ] H j : n composition Q arc p We acyclic (semicomplete, respectively) respectively). In this paper, i...