Arbitrary Edge Research Articles

The Euler characteristic of the configuration space of planar spidery linkages

The topology of configuration spaces of planar mechanical linkages has been studied by many people. Among these linkages, polygonal and spidery ones are particularly important. We call the configuration space of a polygonal linkage a polygon space. The study of polygon spaces was originated by Hausmann [7], Kapovich and Millson [10] and Walker [15]. Their study has a long history and has begotten many references. For example, the homology groups of polygon spaces of arbitrary edge lengths have been determined by Farber and Schutz [6].

An Open Canvas—2D Materials with Defects, Disorder, and Functionality

CONSPECTUS: While some exceptional properties are unique to graphene only (its signature Dirac-cone gapless dispersion, carrier mobility, record strength), other features are common to other two-dimensional materials. The broader family "beyond graphene" offers greater choices to be explored and tailored for various applications. Transition metal dichalcogenides (TMDCs), hexagonal boron nitride (h-BN), and 2D layers of pure elements, like phosphorus or boron, can complement or even surpass graphene in many ways and uses, ranging from electronics and optoelectronics to catalysis and energy storage. Their availability greatly relies on chemical vapor deposition growth of large samples, which are highly polycrystalline and include interfaces such as edges, heterostructures, and grain boundaries, as well as dislocations and point defects. These imperfections do not always degrade the material properties, but they often bring new physics and even useful functionality. It turns particularly interesting in combination with the sheer openness of all 2D sheets, fully exposed to the environment, which, as we show herein, can change and tune the defect structures and consequently all their qualities, from electronic levels, conductivity, magnetism, and optics to structural mobility of dislocations and catalytic activities. In this Account, we review our progress in understanding of various defects. We begin by expressing the energy of an arbitrary graphene edge analytically, so that the environment is regarded by "chemical phase shift". This has profound implications for graphene and carbon nanotube growth. Generalization of this equation to heteroelemental BN gives a method to determine the energy for arbitrary edges of BN, depending on the partial chemical potentials. This facilitates the tuning of the morphology and electronic and magnetic properties of pure BN or hybrid BN|C systems. Applying a similar method to three-atomic-layer TMDCs reveals more diverse edge structures for thermodynamically stable flakes. Moreover, CVD samples show new types of edge reconstruction, providing insight into the nonequilibrium growth process. Combining dislocation theory with first-principles computations, we could predict the dislocation cores for BN and TMDC and reveal their variable chemical makeup. This lays the foundation for the unique sensitivity to ambient conditions. For example, partial occupation of the defect states for dislocations in TMDCs renders them intrinsically magnetic. The exchange coupling between electrons from neighboring dislocations in grain boundaries further makes them half-metallic, which may find its applications in spintronics. Finally, brief discussion of monoelemental 2D-layer phosphorus and especially the structures and growth routes of 2D boron shows how theoretical assessment can help the quest for new synthetic routes.

Modeling the mechanics and dynamics of arbitrary edge drills

This paper presents a new approach for modelling the cutting forces and chatter stability limits in drills with arbitrary lip geometry. The oblique cutting geometry at each point on the drill lip is modelled using parametric curve equations. The cutting force and process damping coefficients at different parts of the drill lip are identified empirically; the cutting force coefficients are identified from non-symmetric drilling tests, and the process damping coefficients are identified from chatter-free orthogonal turning tests. The presented approach provides a practical method for modelling the cutting forces and vibration stability without needing the detailed geometry of drill lips. The accuracy of presented model in predicting lateral and torsional-axial chatter stability limits is verified by conducting drilling tests using drills with various edge geometries.

Thermal buckling of temperature-dependent FGM conical shells with arbitrary edge supports

Bifurcation behavior of heated conical shell made of a through-the-thickness functionally graded material is investigated in the present research. Properties of the shell are obtained based on a power law form across the thickness. Temperature dependency of the constituents is also taken into account. The heat conduction equation of the shell is solved based on an iterative generalized differential quadrature method (GDQM). General nonlinear equilibrium equations and the associated boundary conditions are obtained using the virtual displacement principle in the Donnell sense. The prebuckling solution of the shell is obtained under the assumption of linear membrane deformations. The stability equations are extracted via the concept of the adjacent equilibrium criterion. A semi-analytical solution employing the GDQM and trigonometric expansion is implemented to solve the stability equations. Numerical results of the present research are compared and validated with the known available data through the open literature. Some parametric studies are conducted to investigate the influences of various involved parameters, such as the cone semi-vertex angle, boundary conditions, power law index of composition rule, length to thickness ratio, and the radius to thickness ratio.

Distortion of the hyperbolicity constant in minor graphs

The main aim in this work is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph G\\e (respectively, G/e) obtained from the graph G by deleting (respectively, contracting) an arbitrary edge e from it. A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from G by contracting some edges, deleting some edges, and deleting some isolated vertices. Since we deal with connected graphs, our work provides information about the hyperbolicity constant of minor graphs.

Coulomb edge effects in graphene nanoribbons

Coulomb effects in graphene nanoribbons with arbitrary edges are investigated with the use of a mean-field Hubbard model. It was recently shown that chiral ribbons with minimal edges, characterized by the translation vector (n,m), have a similar structure of bands localized around the Fermi energy as pure zigzag ribbons (n−m,0). Here we show that these flat bands in both ribbon cases differ in detail due to the perturbation induced by armchair edge nodes. For chiral ribbons the edge bands split at the zone boundary, where the corresponding bands of (n−m,0) zigzag ribbons are degenerate. Coulomb interactions enhance strongly this splitting and at the same time they bring spin into play. We modify each edge keeping global sublattice balance to find that spin degeneracy can be partially lifted. The breaking of spin-degeneracy depends on the asymmetry between the edges and in some cases leads to spin-polarized currents.

Operationally optimal vertex-based shape coding with arbitrary direction edge encoding structures

The intention of shape coding in the MPEG-4 is to improve the coding efficiency as well as to facilitate the object-oriented applications, such as shape-based object recognition and retrieval. These require both efficient shape compression and effective shape description. Although these two issues have been intensively investigated in data compression and pattern recognition fields separately, it remains an open problem when both objectives need to be considered together. To achieve high coding gain, the operational rate-distortion optimal framework can be applied, but the direction restriction of the traditional eight-direction edge encoding structure reduces its compression efficiency and description effectiveness. We present two arbitrary direction edge encoding structures to relax this direction restriction. They consist of a sector number, a short component, and a long component, which represent both the direction and the magnitude information of an encoding edge. Experiments on both shape coding and hand gesture recognition validate that our structures can reduce a large number of encoding vertices and save up to 48.9% bits. Besides, the object contours are effectively described and suitable for the object-oriented applications. © The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI. (DOI: 10.1117/1.JEI.23.4.043009)

New Bounds for Edge-Cover by Random Walk

We show that the expected time for a random walk on a (multi-)graphGto traverse allmedges ofG, and return to its starting point, is at most 2m2; if each edge must be traversed in both directions, the bound is 3m2. Both bounds are tight and may be applied to graphs with arbitrary edge lengths. This has interesting implications for Brownian motion on certain metric spaces, including some fractals.

Natural vibration analysis of stiffened panels with arbitrary edge constraints using the assumed mode method

Natural vibration analysis of stiffened panels represents an important issue in different kinds of engineering applications. In this article, a procedure for the vibration analysis of stiffened panels with arbitrary edge constraints is presented. It is based on the assumed mode method, where natural frequencies and modes are determined by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange’s equations of motion. The Mindlin thick plate theory is applied for a plate, while the effect of stiffeners having the properties of Timoshenko beams is accounted for by adding their strain and kinetic energies to the corresponding plate energies. The accuracy of the proposed procedure is justified by several numerical examples which include the natural vibration analysis of stiffened panels with different framing sizes, their lengths and orientations, plate thicknesses and different combinations of boundary conditions. A comparison of results with those obtained by the finite element method is provided, and good agreement is achieved.

A note on r-equitable k-colorings of trees

A graph G = (V;E) is r-equitably k-colorable if there exists a partition of V into k independent sets V1, V2,... Vk such that ||Vi|- |Vj|| ? r for all i,j ? {1,2... k}. In this note, we show that if two trees T1 and T2 of order at least two are r-equitably k-colorable for r ? 1 and k ? 3, then all trees obtained by adding an arbitrary edge between T1 and T2 are also r-equitably k-colorable.

Size-dependent vibrations of post-buckled functionally graded Mindlin rectangular microplates

In this paper, the free vibration behavior of post-buckled functionally graded (FG) Mindlin rectangular microplates are described based on the modified couple stress theory (MCST). This theory enables the consideration of the size-effect through introducing material length scale parameters. The FG microplates made of a mixture of metal and ceramic are considered whose volume fraction of components is expressed by a power law function. By means of Hamilton's principle, the nonlinear governing equations and associated boundary conditions are derived for FG micro-plates in the postbuckling domain. The governing equations and boundary conditions are then discretized by using the generalized differential quadrature (GDQ) method before solving numerically by the pseudo-arclength continuation technique. In the solution procedure, the postbuckling problem of microplates is investigated first. Afterwards, the free vibration of microplates around the buckled configuration is discussed. The effects of dimensionless length scale parameter, material gradient index and aspect ratio on the on the postbuckling path and frequency of FG microplates subject to arbitrary edge supports are thoroughly discussed.

Harmonic vectors and matrix tree theorems

The paper describes an explicit combinatorial formula for a harmonic vector for the Laplacian of a directed graph with arbitrary edge weights. This result was motivated by questions from mathematical economics, and the formula plays a crucial role in forthcoming joint work of the author on the emergence of prices and money in an exchange economy. It turns out that the formula is closely related to well-studied problems in graph theory, in particular to the so-called matrix tree theorem due to W. Tutte and independently to R. Bott and J. Mayberry. As a further application of our considerations, we obtain a short new proof of both the matrix tree theorem as well as its generalization due to S. Chaiken.

Approximate natural vibration analysis of rectangular plates with openings using assumed mode method

Natural vibration analysis of plates with openings of different shape represents an important issue in naval architecture and ocean engineering applications. In this paper, a procedure for vibration analysis of plates with openings and arbitrary edge constraints is presented. It is based on the assumed mode method, where natural frequencies and modes are determined by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange's equations of motion. The presented solution represents an extension of a procedure for natural vibration analysis of rectangular plates without openings, which has been recently presented in the literature. The effect of an opening is taken into account in an intuitive way, i.e. by subtracting its energy from the total plate energy without opening. Illustrative numerical examples include dynamic analysis of rectangular plates with rectangular, elliptic, circular as well as oval openings with various plate thicknesses and different combinations of boundary conditions. The results are compared with those obtained by the finite element method (FEM) as well as those available in the relevant literature, and very good agreement is achieved.

Edge-Compositions of 3D Surfaces

Origami-based design methods enable complex devices to be fabricated quickly in plane and then folded into their final 3D shapes. So far, these folded structures have been designed manually. This paper presents a geometric approach to automatic composition of folded surfaces, which will allow existing designs to be combined and complex functionality to be produced with minimal human input. We show that given two surfaces in 3D and their 2D unfoldings, a surface consisting of the two originals joined along an arbitrary edge can always be achieved by connecting the two original unfoldings with some additional linking material, and we provide a polynomial-time algorithm to generate this composite unfolding. The algorithm is verified using various surfaces, as well as a walking and gripping robot design.

Geometrically nonlinear rectangular simply supported plates subjected to a moving mass

The dynamic deformation of a geometrically nonlinear rectangular simply supported plate under a moving lumped mass is evaluated using mode expansion method. The governing differential equations of motion for a largely deformable rectangular plate are derived using Lagrange method based on appropriate in and out-of-plane spatial functions which satisfy the proposed boundary conditions. Although the proposed procedure is applicable for any arbitrary edge boundary conditions, only the simply supported plates are addressed in the present work. On the other hand, all inertial components of the moving mass are included in the derivation of the equations of motion. A numerical example is used to study the dynamic behavior of the plate, considering large deformations. The obtained results indicate that ignoring the geometric nonlinearity in determining the vertical deformations of plates under the effect of moving masses, especially those of considerable weight and velocity, results in excessively large linear amplitudes leading to an unfavorable conservative structural design.

Approximate natural vibration analysis of rectangular plates with openings using assumed mode method

Natural vibration analysis of plates with openings of different shape represents an important issue in naval architecture and ocean engineering applications. In this paper, a procedure for vibration analysis of plates with openings and arbitrary edge constraints is presented. It is based on the assumed mode method, where natural frequencies and modes are determined by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange's equations of motion. The presented solution represents an extension of a procedure for natural vibration analysis of rectangular plates without openings, which has been recently presented in the literature. The effect of an opening is taken into account in an intuitive way, i.e. by subtracting its energy from the total plate energy without opening. Illustrative numerical examples include dynamic analysis of rectangular plates with rectangular, elliptic, circular as well as oval openings with various plate thicknesses and different combinations of boundary conditions. The results are compared with those obtained by the finite element method (FEM) as well as those available in the relevant literature, and very good agreement is achieved.

Almost Spanning Subgraphs of Random Graphs After Adversarial Edge Removal

Let Δ ≥ 2 be a fixed integer. We show that the random graph${\mathcal{G}_{n,p}}$with$p\gg (\log n/n)^{1/\Delta}$is robust with respect to the containment of almost spanning bipartite graphsHwith maximum degree Δ and sublinear bandwidth in the following sense: asymptotically almost surely, if an adversary deletes arbitrary edges from${\mathcal{G}_{n,p}}$in such a way that each vertex loses less than half of its neighbours, then the resulting graph still contains a copy of all suchH.

Optimal identification of sets of edges using 2-factors

The problem of identifying a single arbitrary edge in a graph using edge-identifying codes was recently considered by Foucaud et al. (in press) [4]. In this paper, we focus on locating more than one edge. First, we classify the graphs in which this is possible. Furthermore, for such graphs, we give a simple characterization of edge-identifying codes. Using the characterization, we give various lower and upper bounds for edge-identifying codes in terms of the order and size of a graph. In particular, codes with the minimum cardinality are obtained with the aid of 2-factors. In addition, we consider how the cardinality of the smallest edge-identifying codes behaves when an edge is added to a graph.

A lower bound on the number of Hamiltonian cycles through a prescribed edge in a crossed cube

The crossed cube proposed by Efe is one of the most notable variations of the hypercube, but some properties of the former are superior to those of the latter. For example, the diameter of the crossed cube is almost the half that of the hypercube. In this paper, we consider the problem of embedding a Hamiltonian cycle passing through a prescribed edge in crossed cubes. A concept termed the cycle pattern is used to construct a linear algorithm for embedding a structural Hamiltonian cycle so the cycle passes through an arbitrary given edge in the crossed cube. Further, we give the necessary and sufficient conditions for determining what kind of permutation generates a Hamiltonian cycle pattern of the crossed cube. As a result, we obtain a lower bound for the number of Hamiltonian cycles through a given edge in an n-dimensional crossed cube. Our work extends some recently obtained results.

Edge-fault-tolerant hamiltonicity of augmented cubes

Abstract The augmented cube AQ n is a hypercube Q n with additional edges between vertices that differ in a suffix. We show that AQ n with f arbitrary faulty edges contains a copy of Q n with at most n 2 n − 1 f faulty edges. This allows to transfer properties of Q n with faulty edges to AQ n with (more) faulty edges. In particular, we show that if f ⩽ 3 n − 7 and each vertex of AQ n is incident with at least two non-faulty edges then AQ n contains a hamiltonian cycle consisting only of non-faulty edges.