Vertex Boundary Research Articles

Sine-Gordon solitons in networks: Scattering and transmission at vertices

We consider the sine-Gordon equation on metric graphs with simple topologies and derive vertex boundary conditions from the fundamental conservation laws together with successive space-derivatives of sine-Gordon equation. We analytically obtain traveling-wave solutions in the form of standard sine-Gordon solitons such as kinks and antikinks for star and tree graphs. We show that for this case the sine-Gordon equation becomes completely integrable just as in case of a simple 1D chain. This simple analysis provides a cornerstone for the numerical solution of the general case, including a quantification of the vertex scattering. Applications of the obtained results to Josephson junction networks and DNA double helix are discussed.

Dirichlet PageRank and Ranking Algorithms Based on Trust and Distrust

Motivated by numerous models of representing trust and distrust within a network ranking system, we examine a quantitative vertex ranking with consideration of the influence of a subset of nodes. We propose and analyze a general ranking metric, called _Dirichlet PageRank_, which gives a ranking of vertices in a subset _S_ of nodes subject to some specified conditions on the vertex boundary of _S_. In addition to the usual Dirichlet boundary condition (which disregards the influence of nodes outside of _S_), we consider general boundary conditions allowing the presence of negative (distrustful) nodes or edges. We give an efficient approximation algorithm for computing Dirichlet PageRank vectors. Furthermore, we give several algorithms for solving various trust-based ranking problems using Dirichlet PageRank with general boundary conditions.

Vertex Isoperimetric Inequalities for a Family of Graphs on $\mathbb{Z}^k$

We consider the family of graphs whose vertex set is $\mathbb{Z}^k$ where two vertices are connected by an edge when their $\ell_\infty$-distance is 1. We prove the optimal vertex isoperimetric inequality for this family of graphs. That is, given a positive integer $n$, we find a set $A\subset \mathbb{Z}^k$ of size $n$ such that the number of vertices who share an edge with some vertex in $A$ is minimized. These sets of minimal boundary are nested, and the proof uses the technique of compression.We also show a method of calculating the vertex boundary for certain subsets in this family of graphs. This calculation and the isoperimetric inequality allow us to indirectly find the sets which minimize the function calculating the boundary.

Some Structural Properties of Acyclic Heaps of Pieces

We introduce the notions of boundary vertex, linear equivalence, and effective boundary vertex in the context of Viennot’s heaps of pieces. We prove that in the heap of a fully commutative element in a star reducible Coxeter group, every boundary vertex is linearly equivalent to an effective boundary vertex.

Bandwidth and pathwidth of three-dimensional grids

We study the bandwidth and the pathwidth of multi-dimensional grids. It can be shown for grids, that these two parameters are equal to a more basic graph parameter, the vertex boundary width. Using this fact, we determine the bandwidth and the pathwidth of three-dimensional grids, which were known only for the cubic case. As a by-product, we also determine the two parameters of multi-dimensional grids with relatively large maximum factors.

Spectral Filtering in Quantum Y-Junction

We examine scattering properties of singular vertex of degree n =2 and 3, taking advantage of a new form of representing the vertex boundary condition, which has been devised to approximate a singular vertex with finite potentials. We show that proper identification of δ and δ' components in the connection condition between outgoing lines enables the designing of quantum spectral branch-filters.

Improvement of the superconvergent patch recovery technique by the use of constraint equations: the SPR‐C technique

AbstractThe superconvergent patch recovery (SPR) technique is widely used in the evaluation of a recovered stress field σ* from the finite element solution σfe. Several modifications of the original SPR technique have been proposed. A new improvement of the SPR technique, called SPR‐C technique (Constrained SPR), is presented in this paper. This new technique proposes the use of the appropriate constraint equations in order to obtain stress interpolation polynomials in the patch σ that locally satisfy the equations that should be satisfied by the exact solution. As a result the evaluated expressions for σ will satisfy the internal equilibrium and compatibility equations in the whole patch and the boundary equilibrium equation at least in vertex boundary nodes and, under certain circumstances, along the whole boundary of the patch coinciding with the boundary of the domain. The results show that the use of this technique considerably improves the accuracy of the recovered stress field σ* and therefore the local effectivity of the ZZ error estimator. Copyright © 2006 John Wiley & Sons, Ltd.

Buckling of triangular plates with elastic edge constraints

This paper presents the first-known investigation on the buckling behavior of triangular plates with both translational and rotational elastic edge constraints. The energy functional of a general triangular plate with elastic edge supports is derived, and thep-Ritz method is utilized to derive the governing eigenvalue equation for the buckling problem. Convergence and comparison studies are carried out to verify the validity and accuracy of the solution method. Extensive buckling factors are presented for several selected isosceles and right-angled triangular plates of various edge support conditions and subjected to isotropic inplane compressive load. The influence of elastic edge supports on the buckling factors for triangular plates of various vertex angles (aspect ratios) and boundary conditions is examined.

Demiarcs, creaons and genons

Useful insights into the representation of natural systems can be gained by decomposing directed graphs (digraphs) into elementary components. Arcs of digraphs can be split into male demiarcs (outarcs) which leave vertices and female demiarcs (inarcs) which enter demiarcs. Likewise, a vertex can be split into an input perceiving side called the creaon and an output generating side called the genon. Digraphs can be regarded as being hierarchically organized because each vertex in a level-1 digraph can be expanded into a level-2 digraph. In general, each vertex of a level- i digraph can be expanded into a level-( i+1) digraph. Arcs of a level- i digraph can be regarded as bundles of level-( i + 1) arcs which are split at the vertex boundary. These elementary graphical components are shown to be useful for depicting input-output systems such as organisms, ecosystems and societies.