equivalence relations for homology cylinders and the core of the casson invariant let r be a compact oriented surface of genus g with one boundary component homology cylinders over r form a monoid ic into which the torelli group i of r embeds by the mapping cylinder construction two homology cylinders m and m are said to be y k equivalent if m is obtained from m by twisting an arbitrary surface s in m with a homeomorphim belonging to the k th term of the lower central series of the torelli group of s the j k equivalence relation on ic is defined in a similar way using the k th term of the johnson filtration in this paper we characterize the y 3 equivalence with three classical invariants 1 the action on the third nilpotent quotient of the fundamental group of r 2 the quadratic part of the relative alexander polynomial and 3 a by product of the casson invariant similarly we show that the j 3 equivalence is classified by 1 and 2 we also prove that the core of the casson invariant originally defined by morita on the second term of the johnson filtration of i has a unique extension to the corresponding submonoid of ic that is preserved by y 3 equivalence and the mapping class group action

the generalized nonlinear initial boundary riemann problem for quasilinear hyperbolic systems of conservation laws abstract in this paper we study the nonlinear initial boundary riemann problem and the generalized nonlinear initial boundary riemann problem for quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions on the domain t x t 0 x 0 under the assumption that each positive eigenvalue is either linearly degenerate or genuinely nonlinear we get the existence and uniqueness of the self similar solution to the nonlinear initial boundary riemann problem and of the global piecewise c 1 solution containing only shocks and or contact discontinuities to the corresponding generalized nonlinear initial boundary riemann problem it shows that the self similar solution to the nonlinear initial boundary riemann problem possesses the global structural stability

improvement of the scaled corrector method for bifurcation analysis using symmetry exploiting block diagonalization in the nonlinear bifurcation analysis for large scaled structures the standard eigenanalysis of the tangent stiffness matrix yields important information but at the same time demands a large amount of computational cost the scaled corrector method was developed as a numerically efficient eigenanalysis free bifurcation analysis strategy which exploits byproducts of the numerical iteration for path tracing this method however has a problem in its accuracy especially when eigenvalues are nearly or exactly coincidental as a remedy for this we propose a new bifurcation analysis method through the implementation of bifurcation mechanism of a symmetric structure into the scaled corrector method the bifurcation mode is accurately approximated by decomposing a scaled corrector vector into a number of vectors by means of block diagonalization method in group theoretic bifurcation theory and in turn by choosing the predominant one among these vectors in order to demonstrate the usefulness of this method it is applied to the bifurcation analysis of reticulated regular hexagonal truss domes to compute accurately the locations of double bifurcation points and nearly coincidental bifurcation points and associated bifurcation modes

global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields abstract in this paper we study the global existence and the asymptotic behavior of classical solution of the cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields we prove that the global c 1 solution exists uniquely if the bv norm of the initial data is sufficiently small based on the existence result on the global classical solution we show that when the time t tends to the infinity the solution approaches a combination of c 1 traveling wave solutions finally we give an application to the equation for time like extremal surfaces in the minkowski space time r 1 n

global structure instability of riemann solutions of quasilinear hyperbolic systems of conservation laws rarefaction waves abstract this work is a continuation of our previous work kong j differential equations 188 2003 242 271 global structure stability of riemann solutions of quasilinear hyperbolic systems of conservation laws shocks and contact discontinuities in the present paper we prove the global structure instability of the lax s riemann solution u u x t containing rarefaction waves of general n n quasilinear hyperbolic system of conservation laws combining the results in kong 2003 we prove that the lax s riemann solution of general n n quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non degenerate shocks and contact discontinuities but no rarefaction waves and other weak discontinuities

global shock waves for the supersonic flow past a perturbed cone we prove the global existence of a shock wave for the stationary supersonic gas flow past an infinite curved and symmetric cone the flow is governed by the potential equation as well as the boundary conditions on the shock and the surface of the body it is shown that the solution to this problem exists globally in the whole space with a pointed shock attached at the tip of the cone and tends to a self similar solution under some suitable conditions our analysis is based on a global uniform weighted energy estimate for the linearized problem combining this with the local existence result of chen li 1 we establish the global existence and decay rate of the solution to the nonlinear problem

nonlinear stability of a self similar 3 dimensional gas flow we show that the 3 dimensional supersonic gas flow past an infinite cone is nonlinear staple upon the perturbation of the obstacle the perturbed flow exists globally in space and tends to the self similar flow downstream there is a thin layer of concentration of vorticities and entropy variation our analysis is based on an approximation scheme using local self similar solutions as building blocks this enables us to obtain global estimates of the nonlinear interactions of waves needed for the stability analysis

global structure instability of riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary abstract this work is a continuation of our previous work z q shao d x kong y c li shock reflection for general quasilinear hyperbolic systems of conservation laws nonlinear anal tma 66 1 2007 93 124 in this paper we study the global structure instability of the riemann solution u u x t containing shocks at least one rarefaction wave for general n n quasilinear hyperbolic systems of conservation laws in the presence of a boundary we prove the nonexistence of global piecewise c 1 solution to a class of the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws on the quarter plane our result indicates that this kind of riemann solution u u x t mentioned above for general n n quasilinear hyperbolic systems of conservation laws in the presence of a boundary is globally structurally unstable some applications to quasilinear hyperbolic systems of conservation laws arising from physics and mechanics are also given

global structure stability of riemann solutions for general hyperbolic systems of conservation laws in the presence of a boundary abstract this work is a continuation of our previous work z q shao global structure instability of riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary j math anal appl 330 1 2007 511 540 in the present paper we study the global structure stability of the riemann solution u u x t containing only shocks and contact discontinuities of general n n quasilinear hyperbolic systems of conservation laws in the presence of a boundary we prove the existence and uniqueness of global piecewise c 1 solution containing only shocks and contact discontinuities to a class of the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws in the half space t x t 0 x 0 our result indicates that this kind of riemann solution u u x t mentioned above for general n n quasilinear hyperbolic systems of conservation laws in a half space possesses a global structural stability some applications to quasilinear hyperbolic systems of conservation laws arising in physics and other disciplines are also given

integral homology 3 spheres and the johnson filtration the mapping class group of an oriented surface g 1 of genus g with one boundary component has a natural decreasing filtration mg 1 mg i 3 me i d m g 1 3 where m g 1 k is the kernel of the action of m g 1 on the k th nilpotent quotient of 1 g 1 using a tree lie algebra approximating the graded lie algebra k m g 1 k m g 1 k 1 we prove that any integral homology sphere of dimension 3 has for some g a heegaard decomposition of the form m hg lg h g where g 1 3 and lg is such that h g lg h g s 3 this proves a conjecture due to s morita and shows that the core of the casson invariant is indeed the casson invariant

transonic nozzle flows and free boundary problems for the full euler equations abstract we establish the existence and uniqueness of transonic flows with a transonic shock through a two dimensional nozzle of slowly varying cross sections the transonic flow is governed by the steady full euler equations given an incoming smooth flow that is close to a constant supersonic state i e smooth cauchy data at the entrance and the subsonic condition with nearly horizontal velocity at the exit of the nozzle we prove that there exists a transonic flow whose downstream smooth subsonic region is separated by a smooth transonic shock from the upstream supersonic flow this problem is approached by a one phase free boundary problem in which the transonic shock is formulated as a free boundary the full euler equations are decomposed into an elliptic equation and a system of transport equations for the free boundary problem an iteration scheme is developed and its fixed point is shown to exist which is a solution of the free boundary problem by combining some delicate estimates for the elliptic equation and the system of transport equations with the schauder fixed point argument the uniqueness of transonic nozzle flows is also established by employing the coordinate transformation of euler lagrange type and detailed estimates of the solutions

boundary conditions for nonlinear hyperbolic systems of conservation laws we propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws a first approach is based on the vanishing viscosity method and a second one is related to the riemann problem the equivalence between these two conditions is studied the latter formulation is extended to treat numerically physically relevant boundary conditions monodimensional experiments are presented

the dynamics of relativistic strings moving in the minkowski space mathbb r 1 n in this paper we investigate the dynamics of relativistic in particular closed strings moving in the minkowski space mathbb r 1 n n ge 2 we first derive a system with n nonlinear wave equations of born infeld type which governs the motion of the string this system can also be used to describe the extremal surfaces in mathbb r 1 n we then show that this system enjoys some interesting geometric properties based on this we give a sufficient and necessary condition for the global existence of extremal surfaces without space like point in mathbb r 1 n with given initial data this result corresponds to the global propagation of nonlinear waves for the system describing the motion of the string in mathbb r 1 n we also present an explicit exact representation of the general solution for such a system moreover a great deal of numerical analyses are investigated and the numerical results show that in phase space various topological singularities develop in finite time in the motion of the string finally some important discussions related to the theory of extremal surfaces of mixed type in mathbb r 1 n are given

global weakly discontinuous solutions for hyperbolic conservation laws in the presence of a boundary abstract this work is a continuation of our previous work in the present paper we study the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws with non linear boundary conditions in the half space t x t 0 x 0 under the assumption that each characteristic with positive velocity is linearly degenerate we prove the existence and uniqueness of global weakly discontinuous solution u u t x with small amplitude and this solution possesses a global structure similar to that of the self similar solution u u x t of the corresponding riemann problem some applications to quasilinear hyperbolic systems of conservation laws arising in physics and other disciplines particularly to the system describing the motion of the relativistic string in minkowski space r 1 n are also given

a sum formula for the casson walker invariant we study the following question how does the casson walker invariant of a rational homology 3 sphere obtained by gluing two pieces along a surface depend on the two pieces our partial answer may be stated as follows for a compact oriented 3 manifold a with boundary a the kernel l a of the map from h 1 a q to h 1 a q induced by the inclusion is called the lagrangian of a let be a closed oriented surface and let a a b and b be four rational homology handlebodies such that a a b and b are identified via orientation preserving homeomorphisms with assume that l a l a and l b l b inside h 1 q and also assume that l a and l b are transverse then we express in terms of the form induced on 3 l a by the algebraic intersection on h 2 a a paired to the analogous form on 3 l b via the intersection form of the simple formula that we obtain naturally extends to the extension of the casson walker invariant of the author it also extends to gluings along non connected surfaces