On the Second Smallest and the Largest Normalized Laplacian Eigenvalues of a Graph

Let G be a simple connected graph with order n.Let L(G)and Q(G)be the normalized Laplacian and normalized signless Laplacian matrices of G,respectively.Letλk(G)be the k-th smallest normalized Laplacian eigenvalue of G.Denote byρ(A)the spectral radius of the matrix A.In this paper,we study the behaviors ofλ2(G)andρ(L(G))when the graph is perturbed by three operations.We also study the properties ofρ(L(G))and X for the connected bipartite graphs,where X is a unit eigenvector of L(G)corresponding toρ(L(G)).Meanwhile we characterize all the simple connected graphs withρ(L(G))=ρ(Q(G)).

Influence of Screw Design on Burden Descending Velocity and Particle Segregation in COREX Shaft Furnace

COREX shaft furnace（SF）is an industrial system that employs screw feeders;thus,the burden descending velocity and particle segregation in the SF can be directly affected by the design of screw.A three-dimensional actual size model of COREX-3000 SF was established using the discrete element method.Four types of burdens,including pellet,ore,flux and coke,were considered in this model.With this consideration,the effect of screw design on solid flow was investigated.Results showed that,in the base case,burdens fell primarily down from the first flight of the screw.The burden descending velocities were nearly uniform in the peripheral direction and decreased along the radial direction.In addition,the normalized particle size increased in the center area and decreased in the wall area.Reducing the flight diameter of screw benefited an even flow pattern and restrained the rolling tendency of burden from the edge to center areas.An optimized case was also proposed,in which a uniform solid flow profile could be obtained and the evenness of descending velocity along the radius could be greatly improved.

Deformation Behavior of Medium-strength TA18 High-pressure Tubes During NC Bending with Different Bending Radii

To improve the forming quality and forming limit of the numerical control （NC） bending of high-pressure titanium alloy tubes, in this study, using three-dimensional （3D） finite element method, deformation behavior of medium-strength TA 18 high-pressure tubes during NC bending with different bending radii is investigated. The results show that the cross-sectional deformation and the wall thickness variation during NC bending of TA18 tubes using a small bending radius （less than 2 times of tube outside diameter） are clearly different from that using a normal bending radius （between 2 and 4 times of tube outside diameter）. For bending with a normal bending radius, with or without a mandrel, the distribution of the flattening in the bending area resembles a platform and an asymmetric parabola, respectively. For bending with a small bending radius, with or without a mandrel, the flattening both distributes like a parabola, but the former has a stable peak which deflects toward the initial bending section, and the latter has a more pronounced peak with a bending angle and deflects slightly toward the bending section. The wall thickness variations with a normal bending radius, with and without a mandrel, both resemble a platform when the bending angle exceeds a certain angle. For the bending with a small radius, the distribution of the wall thickness variation without a mandrel follows an approximate parabola which increases in value as the bending angle increases. If a mandrel is used, the thickening ratio increases from the initial bending section to the bending section.

Bending characteristics of dual-hole PM-PCF based on LP_(01) and LP_(11) modal interference

The bending characteristics of dual-hole polarization maintaining photonic crystal fiber(PM-PCF) are demonstrated in this paper. The modal interference is induced by the LP_(01) mode and LP_(11) mode propagating in a single PM-PCF with the same polarization direction. Simulation results demonstrate that the bending radius induces the phase difference between LP_(01) mode and LP_(11) mode, which leads to the change of light interference intensity on the fiber output facet. The relationship between bending radius and normalized interference intensity with three different bending angles is discussed, where the bending angle is defined as the angle between hole axis and the x axis. The bending sensitivity is obtained to be about-188.78/m around the bending radius of 1.5 cm with the bending angle of 90°. The bending characteristics can contribute for online bending detection in widespread areas.