A Note on the Witten‘s Invariants of Three Manifolds

The purpose of this note is to study the three manifold invariants Qp(ML).For lens spaces L(7,1)and (7,2),we compute Q5(L(7,1) and Q5(L(7.2)) concretely,which enables us to prove that the in-variants can distinguish homotopy equivalence manifolds.

Laminar Flow and Heat Transfer Characteristics in Jackets of Triangular Flow Channels

Laminar flow and heat transfer characteristics of jacketed vessel with triangular flow channels were numerically studied under hydrodynamically and thermally fully developed conditions. Constant heat flux at theheated wall was assumed. The numerical program code interms of vorticity, stream function, axial velocity com ponent and energy equations was written based on a finite volume method. Based on the numerical results, the flow and temperature field were given, and the effects of Dean and Prandtl numbers on flow and heat transfer were ex amined, and the correlations of flow resistance and mean Nusselt number were developed for the jacket. The results show that the structure of secondary flow is steady two vortices in the investigated range of dimensionless curvatureratio and Reynolds number. Two peaks of local Nusselt number increase significantly with Prandtl and Dean num ber increasing, but the local Nusselt numbers near two ends and at the center of the heated wall increase only slightly. The center and two ends of heated wall are the poor positions for heat transfer in the jacket. Compared with the outer half coil jacket at the same area of heated wall, curvature radius, Reynolds number and Prandtl number, e jacket of triangular flow chmnel has lower flow resistance and less mean Nusselt number.

Mixed convection characteristics in lid-driven cavity containing heated triangular block

This research work numerically analyzes 2D,steady state,mixed convective heat transfer for Newtonian fluids in lid driven square enclosure with centered triangular block（blockage—10%or 30%）maintained either at the constant wall temperature or constant heat flux thermal conditions.The fluid flow in the enclosure is initiated by top moving wall in＋x-direction,while all other walls are stationary.The top and bottom walls are thermally insulated.In particular,the governing field equations are solved for range of governing parameters such as,Reynolds number（1–1000）,Prandtl number（1–100）,and Grashof number

An Algorithm of Witten's Invariants of Some 3-manifolds

In this paper, I give out an algorithm of the witten' s invariants of the 3-manifolds obtained by surgery on the links whose corresponding graphs may be no longer trees.

On Stable Minimal Cylinders in 3-manifolds

The purpose of this paper is to improve an important Theorem of Fisher-ColbrieSchoen’s in [1].

The complex nonlinear systems with fractal as well as chaotic dynamics of annual runoff processes in the three headwaters of the Tarim River

This paper attempted to identify fractal and chaotic characteristics of the annual runoff processes in headwaters of the Tarim River.Methods of fractal analyses were used to explore several aspects of the temporal changes from 1957 to 2002.The main findings are as follows:(1) The annual runoff processes of the three headwaters of the Tarim River are com-plex nonlinear systems with fractal as well as chaotic dynamics.(2) The correlation dimensions of attractor derived from the time series of the annual runoff for the Hotan,Yarkand and Aksu rivers are all greater than 3.0 and non-integral,implying that all three rivers are chaotic dynamical systems that are sensitive to initial conditions,and the dynamic modeling of their annual runoff process requires at least four independent variables.(3) The time series of annual runoff in each river presents a long-term correlation characteristic.The Hurst exponent for the period of 1989 to 2002 suggests that we may expect to see an increasing trend in the annual runoff of the Aksu and Yarkand rivers in the years after 2002,but a decreasing tendency for the Hotan River in the same period.

Discrete element method analysis of lateral resistance of fouled ballast bed

The lateral resistance of sleeper plays an important role in ensuring the stability of a railway track, which may change in the operation of railway, due to the fouling in the ballast bed. In this work, discrete element method was adopted to investigate the effect of fouling on the lateral resistance of sleeper. The shape information of ballast was captured by method of three-dimensional vision reconstruction. In order to calibrate the mechanical parameters and verify the models, a lateral resistance field test was carried out by using a custom-made device. The contact force distributions in the different parts of sleeper as well as the interaction between ballast and sleeper were discussed in depth. The results show that fouling of ballast bed evidently reduces the lateral resistance of sleeper and the decreasing degree is also related to the fouled position of ballast bed, in the order of shoulder > bottom > side.Therefore, the effect of fouling, especially the fouling in the ballast shoulder, on the lateral resistance of sleeper, should be taken into account in ballast track maintenance work.

Numerical Simulation of Interaction Between Laminar Flow and Elastic Sheet

A numerical simulation of the interaction between laminar flow with low Reynolds number and a highly flexible elastic sheet is presented. The mathematical model for the simulation includes a three-dimensional finitevolume based fluid solver for incompressible viscous flow and a combined finite-discrete element method for the three-dimensional deformation of solid. An immersed boundary method is used to couple the simulation of fluid and solid. It is implemented through a set of immersed boundary points scattered on the solid surface. These points provide a deformable solid wall boundary for the fluid by adding body force to Navier-Stokes equations. The force from the fluid is also obtained for each point and then applied on the boundary nodes of the solid. The vortex-induced vibration of the highly flexible elastic sheet is simulated with the established mathematical model. The simulated results for both swing pattern and oscillation frequency of the elastic sheet in low Reynolds number flow agree well with experimental data.

3D hydrodynamics involving multiple eccentric impellers in unbaffled cylindrical tank

In this paper, the numerical predictions of 3D hydrodynamics and power consumption in a vessel stirred by mul- tiple eccentrically located impellers are presented. The vessel is a fiat-bottomed cylindrical one equipped with six-curved bladed impellers. Aqueous solutions of xanthan gum are used, which have a shear thinning behavior with yield stress. The influence of several parameters on the mixing efficiency has been investigated, namely： the stirring rate, fluid theology, impeller number and impeller clearance from the tank bottom. Our predicted results are compared with other experimental data and a satisfactory agreement is found.

Vibration and Instability of Third⁃Order Shear Deformable FGM Sandwich Cylindrical Shells Conveying Fluid

The vibration and instability of functionally graded material(FGM)sandwich cylindrical shells conveying fluid are investigated.The Navier-Stokes relation is used to describe the fluid pressure acting on the FGM sandwich shells.Based on the third-order shear deformation shell theory,the governing equations of the system are derived by using the Hamilton’s principle.To check the validity of the present analysis,the results are compared with those in previous studies for the special cases.Results manifest that the natural frequency of the fluid-conveying FGM sandwich shells increases with the rise of the core-to-thickness ratio and power-law exponent,while decreases with the rise of fluid density,radius-to-thickness ratio and length-to-radius ratio.The fluid-conveying FGM sandwich shells lose stability when the non-dimensional flow velocity falls in 2.1-2.5,which should be avoided in engineering application.

GIS-based Numerical Modelling of Debris Flow Motion across Three-dimensional Terrain

The objective of this study is to incorporate a numerical model with GIS to simulate the movement, erosion and deposition of debris flow across the three dimensional complex terrain. In light of the importance of erosion and deposition processes during debris flow movement, no entrainment assumption is unreasonable. The numerical model considering these processes is used for simulating debris flow. Raster grid networks of a digital elevation model in GIS provide a uniform grid system to describe complex topography. As the raster grid can be used as the finite difference mesh, the numerical model is solved numerically using the Leap-frog finite difference method. Finally, the simulation results can be displayed by GIS easily and used to debris flow evaluation. To illustrate this approach, the proposed methodology is applied to the Yohutagawa debris flow that occurred on 2oth October 2010, in Amami- Oshima area, Japan. The simulation results that reproduced the movement, erosion and deposition are in good agreement with the field investigation. The effectiveness of the dam in this real-ease is also verified by this approach. Comparison with the results were simulated by other models, shows that the present coupled model is more rational and effective.

A General Study on Equivalent Resistance of Three-Terminal Ladder Network

In view of the application importance of resistance network in modern science and technology, this paper presents the basic structure of a three terminals ladder shaped resistance network, for which, to study in- depth the equivalent resistance, carry out network analysis by applying virtual current method and construct a model of two elements three orders differential equation. Based on different marginal conditions, two general adaptive rules for the three-terminal ladder shaped inlet resistance, as well as two ultimate rules for the equiva- lent resistance of three-terminal infinite ladder shaped were given.

The Design of Desk Lamp on 3D Modeling based on Solidworks

This paper introduces the solidworks design flow, design methods and design principles of desk lamp. And we use the ergonomics principle, analysis the function of each part of the lamp, and analyses the design of lamp components, to meet user needs in the bedroom, do practical, human nature and convenient lamp. The application shows that the use of this software can make easier the realization of the design of the parts with complex shape, raise the design efficiency, optimize the product structure and save the cost.

On δ-reducible Heegaard Splittings

In this paper, we shall prove that any Heegaard splitting of a δ-reducible 3-manifold M, say M = W U V, can be obtained by doing connected sums, boundary connected sums and self-boundary connected sums from Heegaard splittings of n manifolds M1,..., Mn, where Mi is either a solid torus or an irreducible, δ-irreducible manifold.

Mbius geometry of three-dimensional Wintgen ideal submanifolds in S^5

Wintgen ideal submanifolds in space forms are those ones attaining equality at every point in the socalled DDVV inequality which relates the scalar curvature,the mean curvature and the normal scalar curvature.This property is conformal invariant;hence we study them in the framework of Mbius geometry,and restrict to three-dimensional Wintgen ideal submanifolds in S5.In particular,we give Mbius characterizations for minimal ones among them,which are also known as(3-dimensional)austere submanifolds(in 5-dimensional space forms).

Lattice Boltzmann simulation of flow around two,three and four circular cylinders in close proximity

Cross-flows around two,three and four circular cylinders in tandem,side-by-side,isosceles triangle and square arrangements are simulated using the incompressible lattice Boltzmann method with a second-order accurate curved boundary condition at Reynolds number 200 and the cylinder center-to-center transverse or/and longitudinal spacing 1.5D,where D is the identical circular cylinder diameter.The wake patterns,pressure and force distributions on the cylinders and mechanism of flow dynamics are investigated and compared among the four cases.The results also show that flows around the three or four cylinders significantly differ from those of the two cylinders in the tandem and side-by-side arrangements although there are some common features among the four cases due to their similarity of structures,which are interesting,complex and useful for practical applications.This study provides a useful database to validate the simplicity,accuracy and robustness of the Lattice Boltzmann method.

On embedding infinite cyclic covers in compact 3-manifolds

We show that there are knots whose infinite cyclic covers do not embed in any compact 3-manifold,answering a question of Jiang et al.(2006).

Modeling complex crack problems using the three-node triangular element fitted to numerical manifold method with continuous nodal stress

A three-node triangular element fitted to numerical manifold method with continuous nodal stress, called Trig_3-CNS(NMM)element, was recently proposed for linear elastic continuous problems and linear elastic simple crack problems. The Trig_3-CNS(NMM) element can be considered as a development of both the Trig_3-CNS element and the numerical manifold method(NMM).Inheriting all the advantages of Trig_3-CNS element, calculations using Trig_3-CNS(NMM) element can obtain higher accuracy than Trig_3 element without extra degrees of freedom(DOFs) and yield continuous nodal stress without stress smoothing. Inheriting all the advantages of NMM, Trig_3-CNS(NMM) element can conveniently treat crack problems without deploying conforming mathematical mesh. In this paper,complex problems such as a crucifix crack and a star-shaped crack with many branches are studied to exhibit the advantageous features of the Trig_3-CNS(NMM) element. Numerical results show that the Trig_3-CNS(NMM) element is prominent in modeling complex crack problems.

Bilateral self-amalgamation of a Heegaard splitting and Hempel distance

A closed orientable Haken 3-manifold containing a non separating incompressible closed surface has two canonical Heegaard splittings, which are called self-amalgamation and bilateral self-amalgamation.Heegaard distance introduced by Hempel is a useful index in studying Heegaard splitting. This paper studies the stabilization problem for the bilateral self-amalgamation, and proves that if the distance of bilateral selfamalgamation of a Heegaard splitting is at least 9, then it is unstabilized, weakly reducible and irreducible.

A Note on Heegaard Genus of Self-amalgamated 3-Manifold

Let M be a connected orientable compact irreducible 3-manifold. Suppose that αM consists of two homeomorphic surfaces F1 and F2, and both F1 and F2 are compressible in M. Suppose furthermore that g(M, F1) = g(M) + g(F1), where g(M, F1)is the Heegaard genus of M relative to F1. Let Mfbe the closed orientable 3-manifold obtained by identifying F1 and F2 using a homeomorphism f : F1 → F2. The authors show that if f is sufficiently complicated, then g(Mf) = g(M, αM) + 1.