Resolvability in circulant graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d(u, s). A k-partition Π = {S 1 , S 2 , . . . , S k } of V (G) is called a resolving partition if for every two distinct vertices u, v ∈ V (G) there is a set S i in Π such that d(u, Si )≠ d(v, Si ). The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn , an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i-j (mod n) ∈ C , where CZn has the property that C =-C and 0 ■ C. The circulant graph is denoted by Xn, Δ where Δ = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn, 3 with connection set C = {1, n/2 , n-1} and prove that dim(Xn, 3 ) is independent of choice of n by showing that dim(Xn, 3 ) ={3 for all n ≡ 0 (mod 4), 4 for all n ≡ 2 (mod 4). We also study the partition dimension of a family of circulant graphs Xn,4 with connection set C = {±1, ±2} and prove that pd(Xn, 4 ) is independent of choice of n and show that pd(X5,4 ) = 5 and pd(Xn,4 ) ={3 for all odd n ≥ 9, 4 for all even n ≥ 6 and n = 7.     

The Conditional Covering Problem on Unweighted Interval Graphs with Nonuniform Coverage Radius

Let G = (V, E) be an interval graph with n vertices and m edges. A positive integer R(x) is associated with every vertex x ? V{x\in V}. In the conditional covering problem, a vertex x ? V{x \in V} covers a vertex y ? V{y \in V} (x ≠ y) if d(x, y) ≤ R(x) where d(x, y) is the shortest distance between the vertices x and y. The conditional covering problem (CCP) finds a minimum cardinality vertex set C í V{C\subseteq V} so as to cover all the vertices of the graph and every vertex in C is also covered by another vertex of C. This problem is NP-complete for general graphs. In this paper, we propose an efficient algorithm to solve the CCP with nonuniform coverage radius in O(n ²) time, when G is an interval graph containing n vertices.     

On nonconvex bifunction variational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which is called the nonconvex bifunction variational inequality. We suggest and analyze some iterative methods for solving nonconvex bifunction variational inequalities using the auxiliary principle technique. We prove that the convergence of implicit method requires only pseudomonotonicity, which is weaker condition than monotonicity. Our proof of convergence is very simple. Results proved in this paper may stimulate further research in this dynamic field.     

Dynamic characteristics of nanoindentation in Ni:A molecular dynamics simulation study

In this work,three-dimensional molecular dynamics simulation is carried out to elucidate the nanoindentation behaviour of single crystal Ni.The substrate indenter system is modelled using hybrid interatomic potentials including the manybody potential(embedded atom method) and two-body Morse potential.The spherical indenter is chosen,and the simulation is performed for different loading rates from 10 m/s to 200 m/s.Results show that the maximum indentation load and hardness of the system increase with the increase of velocity.The effect of indenter size on the nanoindentation response is also analysed.It is found that the maximum indentation load is higher for the large indenter whereas the hardness is higher for the smaller indenter.Dynamic nanoindentation is carried out to investigate the behaviour of Ni substrate to multiple loading-unloading cycles.It is observed from the results that the increase in the number of loading unloading cycles reduces the maximum load and hardness of the Ni substrate.This is attributed to the decrease in recovery force due to defects and dislocations produced after each indentation cycle.     

Vibration characteristics of FGM circular cylindrical shells filled with fluid using wave propagation approach

The vibration characteristics of a functionally graded material circular cylindrical shell filled with fluid are examined with a wave propagation approach. The shell is filled with an incompressible non-viscous fluid. Axial modal dependence is approximated by exponential functions. A theoretical study of shell vibration frequencies is analyzed for simply supported-simply supported, clamped-simply supported, and clamped-clamped boundary conditions with the fluid effect. The validity and the accuracy of the present method are confirmed by comparing the present results with those available in the literature. Good agreement is observed between the two sets of results.     

Matching effect in superconducting NbN thin film with a square lattice of holes

A square hole array is fabricated over a micro-bridge of NbN thin film by electron beam lithography and reactive ion milling. Magneto-resistance is measured across the micro-bridge filled with a hole array near the superconducting transition temperature. It is found that magneto-resistance minima occur when the number of vortices is an integer multiple or a fractional multiple of the number of holes. The temperature and the current dependences of the matching effect are studied.     

Numerical investigations of asymmetric flow of a micropolar fluid between two porous disks

Numerical solution is presented for the two- dimensional flow of a micropolar fluid between two porous coaxial disks of different permeability for a range of Reynolds number Re （-300≤ Re 〈 0） and permeability parameter A （1.0≤A ≤2.0）. The main flow is superimposed by the injection at the surfaces of the two disks. Von Karman＇s similarity transformations are used to reduce the governing equations of motion to a set of non-linear coupled ordinary differential equations （ODEs） in dimensionless form. An algorithm based on the finite difference method is employed to solve these ODEs and Richardson＇s extrapolation is used to obtain higher order accuracy. The results indicate that the parameters Re and A have a strong influence on the velocity and microrotation profiles, shear stresses at the disks and the position of the viscous/shear layer. The micropolar material constants cl, c2, c3 have profound effect on microrotation as compared to their effect on streamwise and axial velocity profiles. The results of micropolar fluids are compared with the results for Newtonian fluids.     

Vibrations of FGM thin cylindrical shells with exponential volume fraction law

In this paper, the influence of an exponential volume fraction law on the vibration frequencies of thin functionally graded cylindrical shells is studied. Material properties in the shell thickness direction are graded in accordance with the exponential law. Expressions for the strain-displacement and curvature-displacement relationships are taken from Love＇s thin shell theory. The Rayleigh-Ritz approach is used to derive the shell eigenfrequency equation. Axial modal dependence is assumed in the characteristic beam functions. Natural frequencies of the shells are observed to be dependent on the constituent volume fractions. The results are compared with those available in the literature for the validity of the present methodology.     

Heat transfer in circular microchannels during volumetric heating with magnetic field

Convective heat transfer within circular microchannels in a rectangular solid substrate with heat generation due to imposed magnetic field was studied. A detailed parametric study was performed by varying Reynolds number, magnetic field strength, working fluid, and the diameter of the channel. It was found that the heat transfer coefficient decreases downstream along the channel. Nusselt number increased with Reynolds number. The tube diameter, properties of the working fluid, and magnetic field strength affected the temperature distribution and heat transfer rate at the solid-fluid interface.     

Upscaled Unstructured Computational Grids for Efficient Simulation of Flow in Fractured Porous Media

Discrete fracture modeling (DFM) is currently the most promising approach for modeling of naturally fractured reservoirs and simulation of multiphase fluid flow therein. In contrast with the classical double-porosity/double permeability models, in the DFM approach all the interactions and fluid flow in and between the fractures and within the matrix are modeled in a unified manner, using the same computational grid. There is no need for computing the shape factors, which are crucial to the accuracy of the double-porosity models. We have exploited this concept in order to develop a new method for the generation of unstructured computational grids. In the new approach the geological model (GM) of the reservoir is first generated, using square or cubic grid blocks. The GM is then upscaled using a method based on the multiresolution wavelet transformations that we recently developed. The upscaled grid contains a distribution of the square or cubic blocks of various sizes. A map of the blocks’ centers is then used with an optimized Delauney triangulation method and the advancing-front technique, in order to generate the final unstructured triangulated grid suitable for use in any general reservoir simulator with any number of fluid phases. The new method also includes an algorithm for generating fractures that, contrary to the previous methods, does not require modifying their paths due to the complexities that may arise in spatial distribution of the grid blocks. It also includes an effective partitioning of the simulation domain that results in large savings in the computation times. The speed-up in the computations with the new upscaled unstructured grid is about three orders of magnitude over that for the initial GM. Simulation of waterflooding indicates that the agreement between the results obtained with the GM and the upscaled unstructured grid is excellent. The method is equally applicable to the simulations of multiphase flow in unfractured, but highly heterogeneous, reservoirs.