Hierarchy paths in the Hatcher-Thurston complex

Inspired by works of Masur-Minsky and Mahan Mj, we observe that the Hatcher-Thurston complex of a surface F is an interpolating complex between the pants complex and the curve complex, then we give a hierarchical structure for the Hatcher-Thurston complex. By this hierarchical structure, we show a distance formula in the Hatcher-Thurston complex related to subsurfaces of positive genera. As a corollary, we show that the Hatcher-Thurston complex has one end for surface with genus at least three, the proof runs the same line as a result of Masur-Schleimer, the key tool is the distance formula.     

On U-orthodox semigroups

As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semi-groups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in HU on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup WU and a V-ample semigroup ...     

Convergence of Finslerian metrics under Ricci flow

In this work,we study the convergence of evolving Finslerian metrics first in a general flow and next under Finslerian Ricci flow.More intuitively it is proved that a family of Finslerian metrics g(t)which are solutions to the Finslerian Ricci flow converges in C~∞ to a smooth limit Finslerian metric as t approaches the finite time T.As a consequence of this result one can show that in a compact Finsler manifold the curvature tensor along the Ricci flow blows up in a short time.     

MODELING,SIMULATION,AND OPTIMIZATION OF SURFACE ACOUSTIC WAVE DRIVEN MICROFLUIDIC BIOCHIPS

We will be concerned with the mathematical modeling, numerical simulation, and shape optimization of micro fluidic biochips that are used for various biomedical applications. A particular feature is that the fluid flow in the fluidic network on top of the biochips is in- duced by surface acoustic waves generated by interdigital transducers. We are thus faced with a multiphysics problem that will be modeled by coupling the equations of piezoelectricity with the compressible Navier-Stokes equations. Moreover, the fluid flow exhibits a multiscale character that will be taken care of by a homogenization approach. We will discuss and analyze the mathematical models and deal with their numerical solution by space-time discretizations featuring appropriate finite element approximations with respect to hierarchies of simplicial triangulations of the underlying computational domains. Simulation results will be given for the propagation of the surface acoustic waves on top of the piezoelectric substrate and for the induced fluid flow in the microchannels of the fluidic network. The performance of the operational behavior of the biochips can be significantly improved by shape optimization. In particular, for such purposes we present a multilevel interior point method relying on a predictor-corrector strategy with an adaptive choice of the continuation steplength along the barrier path. As a specific example, we will consider the shape optimization of pressure driven capillary barriers between microchannels and reservoirs.     

Orthogonal separable Hamiltonian systems on T~2

In this paper we characterize the Liouvillian integrable orthogonal separable Hamiltonian systems on T~2 for a given metric,and prove that the Hamiltonian flow on any compact level hypersurface has zero topological entropy.Furthermore,by examples we show that the integrable Hamiltonian systems on T~2 can have complicated dynamical phenomena.For instance they can have several families of invariant tori,each family is bounded by the homoclinic-loop-like cylinders and heteroclinic-loop-like cylinders.As we know,it is the first concrete example to present the families of invariant tori at the same time appearing in such a complicated way.     

Laguerre Geometry of Surfaces in R^3

Let f ： M → R3 be an oriented surface with non-degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L（f） = f（H2 - K）/KdM, is an invariant under the Laguerre transformations. The critical surfaces of the functional L are called Laguerre minimal surfaces. In this paper we study the Laguerre minimal surfaces in R^3 by using the Laguerre Gauss map. It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map. In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map. We show also that round spheres are the only compact Laguerre minimal surfaces in R^3. And we give a classification theorem of surfaces in R^3 with vanishing Laguerre form.     

On the nonexistence of a global nontrivial subsonic solution in a 3D unbounded angular domain

In this paper, under some assumptions on the flow with a low Mach number, we study the nonexistence of a global nontrivial subsonic solution in an unbounded domain Ω which is one part of a 3D ramp. The flow is assumed to be steady, isentropic and irrotational, namely, the movement of the flow is described by the potential equation. By establishing a fundamental a priori estimate on the solution of a second order linear elliptic equation in Ω with Neumann boundary conditions on Ω and Dirichlet boundary value at some point of Ω, we show that there is no global nontrivial subsonic flow with a low Mach number in such a domain Ω.     

Evolution of hypersurfaces by the mean curvature minus an external force field

In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force.     

THE SURFACE AREA PRESERVING MEAN CURVATURE FLOW IN QUASI-FUCHSIAN MANIFOLDS

In this paper,we consider the surface area preserving mean curvature flow in quasi-Fuchsian 3-manifolds.We show that the flow exists for all times and converges exponentially to a smooth surface of constant mean curvature with the same surface area as the initial surface.     

Laguerre minimal surfaces in ℝ3

Laguerre geometry of surfaces in R^3 is given in the book of Blaschke, and has been studied by Musso and Nicolodi, Palmer, Li and Wang and other authors. In this paper we study Laguerre minimal surface in 3-dimensional Euclidean space R^3. We show that any Laguerre minimal surface in R^3 can be constructed by using at most two holomorphic functions. We show also that any Laguerre minimal surface in R^3 with constant Laguerre curvature is Laguerre equivalent to a surface with vanishing mean curvature in the 3-dimensional degenerate space R0^3.     

Normal scalar curvature and a pinching theorem in S~m×R and H~m×R

We obtain an inequality in Sm×R and Hm×R which is similar to DDVV conjecture.As an application,we show that a minimal submanifold in H m×R with nonnegative scalar curvature must be a surface of the type γ×R,where γ is a geodesic in H m.In addition,we get a pinching theorem in Sm×R.     

Energy identity of the heat flow of H-systems at finite singular time

It is well known that there exists a global solution to the heat flow of H-systems.If the solution satisfies a certain energy inequality,it is global regular with at most finitely many singularities. Under the same energy inequality,we can show the energy identity of the heat flow of H-systems at finite singular time.The most interesting thing in our proof is that we find the singular points can only occur in the interior of the set in some sense.     

A THEORETICAL APPROACH TO THERMAL PROPERTY OF ARRAY OF CYLINDERS EMBEDDED IN HOMOGENEOUS MATRIX

We investigate in this article the thermal conductivity of array of cylinders embedded in a homogeneous matrix. Using Green's function, we confirm that the method invented by Rayleigh can be generalized to deal with thermal property of these systems. A technique for calculating effective thermal conductivities of these systems is proposed. As an example, we consider a system with square symmetry, and a neat formula for effective thermal conductivity is derived. We show that the method also includes the proof of Keller theorem.     

Schroedinger Flows on Compact Hermitian Symmetric Spaces and Related Problems

In this note,we prove that the Schroedinger flow of maps from a closed riemann surface into a compact irreducible Hermitian symmetic space admits a global weak solution.Also,we show the existence of weak solutions to the initial value problem of Heisenberg model with Lie algebra values,which is closely related to the Schroedinger flow on compact Hermitian symmetric spaces.     

Expansion of frames to tight frames

We show that every Bessel sequence （and therefore every frame） in a separable Hilbert space can be expanded to a tight frame by adding some elements. The proof is based on a recent generalization of the frame concept, the g-frame, which illustrates that g-frames could be useful in the study of frame theory. As an application, we prove that any Gabor frame can be expanded to a tight frame by adding one window function.     

THE GLOBAL ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE 3D FLOW AROUND A SUBMERGED BODY

We consider the numerical approximations of the three-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface in a channel. One vertical side is introduced as the up-stream artificial boundary and two vertical sides are introduced as the downstream arti-ficial boundaries. On the artificial boundaries, a sequence of high-order global artificial boundary conditions are given. Then the original problem is reduced to a problem defined on a finite computational domain, which is equivalent to a variational problem. After solving the variational problem by the finite element method, we obtain the numerical approximation of the original problem. The numerical examples show that the artificial boundary conditions given in this paper are very effective.     

Reflected BSDEs with random default time and related mixed optimal stopping-control problems

In this paper we study the one-dimensional reflected backward stochastic differential equations which are driven by Brownian motion as well as a mutually independent martingale appearing in a defaultable setting. Using a penalization method, we prove the existence and uniqueness of the solutions to these equations. As an application, we show that under proper assumptions the solution of the reflected equation is the value of the related mixed optimal stopping-control problem.     

Semipositivity of generic ordinary fibration over a curve

Let f : X → B be a generic ordinary proper fibration over a complete curve in positive characteristics, we prove that the dual of higher direct image sheaf R~1 f_*O_X is nef. As a corollary, we show that f_*ω_(S/B) is nef, if f : S → B is a fibration from a surface to a curve with generic ordinary fibres.     

几乎交错环链补中的不可压缩曲面

In this paper,we discuss mainly the properties of incompressible pairwies incom- prcssiblc surfaccs in almost altcrnating link complcmcnts. Lct L bca almost link and lct F be an incompressible palrwise incompressible surface in S~3-L.First,we give the properties that the surface F intersects with 2-spheres in S~3-L.The intersection consisting of a collection of circles and saddle-shaped discs is called a topological graph.One can compute the Euler Characteristic number of the surface by calculating the characteristic number of the graph.Next,we prove that if the graph is special simple,then the genus of the surface is zero.     

Convergence of Yang-Mills-Higgs flow for twist Higgs pairs on Riemann surfaces

We consider the gradient flow of the Yang-Mills-Higgs functional of twist Higgs pairs on a Hermitian vector bundle(E,H)over Riemann surface X.It is already known the gradient flow with initial data(A0,φ0)converges to a critical point(A∞,φ∞).Using a modified Chern-Weil type inequality,we prove that the limiting twist Higgs bundle(E,d′′A∞,φ∞)coincides with the graded twist Higgs bundle defined by the HarderNarasimhan-Seshadri filtration of the initial twist Higgs bundle(E,d′′A0,φ0),generalizing Wilkin’s results for untwist Higgs bundle.