characteristic submanifold theory and toroidal dehn filling the exceptional dehn filling conjecture of the second author concerning the re lationship between exceptional slopes on the boundary of a hyperbolic knot manifold m has been verified in all cases other than small seifert filling slopes in this paper we verify it whenis a small seifert filling slope andis a toroidal filling slope in the generic case where m admits no punctured torus fibre or semi fibre and there is no incompressible torus in m which intersects m in one or two components under these hypotheses we show that 5 our proof is based on an analysis of the relationship between the topology of m the combinatorics of the intersection graph of an immersed disk or torus in m and the two sequences of characteristic subsurfaces associated to an essential punctured torus properly embedded in m

poincare bendixson theory for leaves of codimension one the level of a local minimal set of a c2 codimension one foliation of a compact manifold is a nonnegative integer defined inductively level zero corresponding to the minimal sets in the usual sense each leaf of a local minimal set at level k is at level k the authors develop a theory of local minimal sets level and how leaves at level k asymptotically approach leaves at lower level this last generalizes the classical poincare bendixson theorem and provides information relating growth topological type and level e g if l is a totally proper leaf at level k then l has exactly polynomial growth of degree k and topological type k

convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping we consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time asymptotically the hyperbolic model may be viewed as isentropic euler equations with friction term added to the momentum equation to model gas flow through a porous media in this case our result justifies darcy s law time asymptotically our model may also be viewed as an elastic model with damping

lifting representations to covering groups the primary purpose of this note is to prove that if a discrete subgroup z of z s c has no 2 torsion then it lifts to s c i e there is a homomorphism z s c such that the composition with the natural projection s c z s c is the identity on k note that the absence of 2 torsion is necessary since cyclic subgroups of even order do not lift this extends a result of km 2 for which z was required to be a function group our methods are simple and straightforward requiring only elementary covering space theory

on culler shalen seminorms and dehn filling if f is a finitely generated discrete group and g a complex algebraic lie group the g character variety of r is an affine algebraic variety whose points correspond to characters of representations of r with values in g marc culler and peter shalen developed the theory of sl2 c character varieties of finitely generated groups and applied their results to study the topology of 3 dimensional manifolds in the papers 6 7 8 consider the exterior m of a hyperbolic knot lying in a closed connected orientable 3 manifold the mostow rigidity theorem implies that the holonomy representation p iri m isom h3 psl2 c is unique up to conjugation and taking complex conjugates the orientability of m can be used to show 1 lifts to a representation p e hom ri m sl2 c whose character determines an essentially unique point of xp of x iri m the sl2 c character variety of irl m culler and shalen 8 proved that the component x0 of x x1ri m which contains x is a curve one of their major contributions was to show how x0 determines a norm on h1 wm r which encodes many topological properties of m in particular it provides information on the dehn fillings of m their construction may be applied to arbitrary curves in the sl2 c character variety of a connected compact orientable irreducible 3manifold whose boundary is a torus though in this generality one can only guarantee that it will define a seminorm the first half of this paper is devoted to the development of the general theory of culler shalen seminorms defined for curves of psl2 c characters by working over psl2 c we obtain a theory that is more generally applicable than its sl2 c counterpart while being only mildly more difficult to set up in the second half of this paper we apply the theory of culler shalen seminorms to study the dehn filling operation in particular we examine the relationship between fillings which yield manifolds having a positive dimen

convergence to nonlinear diffusion waves for solutions of euler equations with time depending damping abstract in this paper we are concerned with the asymptotic behavior of solutions to the system of euler equations with time depending damping in particular include the constant coefficient damping we rigorously prove that the solutions time asymptotically converge to the diffusion wave whose profile is self similar solution to the corresponding parabolic equation which justifies darcy s law compared with previous results about euler equations with constant coefficient damping obtained by hsiao and liu 1992 2 and nishihara 1996 9 we obtain a general result when the initial perturbation belongs to the same space i e h 3 r h 2 r our proof is based on the classical energy method

annuli in generalized heegaard splittings and degeneration of tunnel number abstract we analyze how a family of essential annuli in a compact 3 manifold will induce from a strongly irreducible generalized heegaard splitting of the ambient manifold generalized heegaard splittings of the complementary components there are specific applications to the subadditivity of tunnel number of knots improving somewhat bounds of kowng kw for example in the absence of 2 bridge summands the tunnel number of the sum of n knots is no less than frac 2 5 the sum of the tunnel numbers

monopoles and lens space surgeries monopole floer homology is used to prove that real projective three space cannot be obtained from dehn surgery on a nontrivial knot in the three sphere to obtain this result we use a surgery long exact sequence for monopole floer homology together with a nonvanishing theorem which shows that monopole floer homology detects the unknot in addition we apply these techniques to give information about knots which admit lens space surgeries and to exhibit families of three manifolds which do not admit taut foliations

free abelian covers short loops stable length and systolic inequalities we explore the geometry of the abel jacobi map f from a closed orientable riemannian manifold x to its jacobi torus h 1 x mathbb r h 1 x mathbb z mathbb r applying m gromov s filling inequality to the typical fiber of f we prove an interpolating inequality for two flavors of shortest length invariants of loops the procedure works provided the lift of the fiber is non trivial in the homology of the maximal free abelian cover overline x classified by f we show that the finite dimensionality of the rational homology of overline x is a sufficient condition for the homological non triviality of the fiber when applied to nilmanifolds our fiberwise inequality typically gives stronger information than the filling inequality for x itself in dimension 3 we present a sufficient non vanishing condition in terms of massey products this condition holds for certain manifolds that do not fiber over their jacobi torus such as 0 framed surgeries on suitable links our systolic inequality applies to surface bundles over the circle provided the algebraic monodromy has 1 dimensional coinvariants even though the massey product invariant vanishes for some of these bundles

convergence to strong nonlinear diffusion waves for solutions to p system with damping on quadrant in this paper we consider the so called p system with linear damping on quadrant we show that for a certain class of given large initial data v0 x u0 x the corresponding initial boundary value problem admits a unique global smooth solution v x t u x t and such a solution tends time asymptotically at the lp 2 p optimal decay rates to the corresponding nonlinear diffusion wave v x t u x t which satisfies 1 9 provided the corresponding prescribed initial error function v0 x u0 x lies in h3 r l1 r h2 r l1 r

foliations with good geometry the goal of this article is to show that there is a large class of closed hyperbolic 3 manifolds admitting codimension one foliations with good large scale geometric properties we obtain results in two directions first there is an internal result a possibly singular foliation in a manifold is quasi isometric if when lifted to the universal cover distance along leaves is efficienit up to a bounded multiplicative distortion in measuring distance in the universal cover this means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight this is the best geometric behavior we previously proved that nonsingular codimension one foliations in closed hyperbolic 3 manifolds can never be quasi isometric in this article we produce a large class of singular quasi isometric codimension one foliations in closed hyperbolic 3 manifolds the foliations are stable and unstable foliations of pseudo anosov flows our second result is an external result relating nonsingular foliations in hyperbolic 3 manifolds with their limit sets in the universal cover that is showing that leaves in the universal cover have good asymptotic behavior let g be a reebless finite depth foliation in a closed hyperbolic 3 manifold then g is not quasi isometric but suppose that g is transverse to a quasigeodesic pseudo anosov flow with quasi isometric stable and unstable foliations which are given by the internal result we then show that the lifts of leaves of g to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis the main tools used to prove these results are first a link between geometric properties of stable unstable foliations of pseudo anosov flows and the topology of these foliations in the tuniversal cover and second a topological theory of the joint structure of the pseudo anosov foliations in the universal cover reebless codimension one foliations are extremely useful for understanding the topology of 3 manifolds for instance they imply that the manifold is irreducible ro its universal cover is homeomorphic to r3 pa leaves are 7rl injective no and transversals are never null homotopic no hence they reflect topological properties of the manifold as for which manifolds have reebless foliations gabai gal ga2 ga3 proved that any compact oriented irreducible 3 manifold with nonzero first betti number has many reebless finite depth foliations roughly a

on the geometry of the slice of trace free sl_2 mathbb c characters of a knot group let k be a knot in an integral homology 3 sphere with exterior e k and let b 2 denote the two fold branched cover of branched along k we construct a map from the slice of trace free rm sl 2 mathbb c characters of 1 e k to the rm sl 2 mathbb c character variety of 1 b 2 when this map is surjective it describes the slice as the two fold branched cover over the rm sl 2 mathbb c character variety of b 2 with branched locus given by the abelian characters whose preimage is precisely the set of metabelian characters we show that each metabelian character can be represented as the character of a binary dihedral representation of 1 e k the map is shown to be surjective for all 2 bridge knots and all pretzel knots of type p q r an extension of this framework to n fold branched covers is also described

dehn fillings of knot manifolds containing essential once punctured tori in this paper we study exceptional dehn fillings on hyperbolic knot manifolds which contain an essential once punctured torus let m be such a knot manifold and let be the boundary slope of such an essential once punctured torus we prove that if dehn filling m with slope produces a seifert fibred manifold then 5 furthermore we classify the triples m when 4 more precisely when 5 then m is the unique manifold w h 3 2 obtained by dehn filling one boundary component of the whitehead link exterior with slope 3 2 and is the pair of slopes 5 0 further 4 if and only if m is the triple w h 2n 1 n 4 0 for some integer n with n 1 combining this with known results we classify all hyperbolic knot manifolds m and pairs of slopes on m where is the boundary slope of an essential once punctured torus in m and is an exceptional filling slope of distance 4 or more from refined results in the special case of hyperbolic genus one knot exteriors in s3 are also given

so 2n c character varieties are not varieties of characters abstract we prove that for n 2 and all groups of corank 2 the coordinate rings of s o 2 n c character varieties of are not generated by trace functions nor generalized trace functions sending s o 2 n c to t r for an arbitrary representation of s o 2 n c and any furthermore we give examples of non conjugate completely reducible representations undistinguishable by generalized trace functions hence s o 2 n c character varieties are not varieties of characters however we also prove that any generic s o 2 n c representation of a free group can be distinguished from all non equivalent representations by trace functions and by a single generalized trace function

finite dehn surgery on knots let k be a knot with a closed tubular neighbourhood n k in a connected orientable closed 3 manifold w such that the exterior of k m w intn k is irreducible we consider the problem of which dehn surgeries on k or equivalently which dehn fillings on m can produce 3 manifolds with finite fundamental group for convenience a surgery is called a g surgery if the resultant 3 manifold has fundamental group g if g is cyclic or finite the surgery is also called a cyclic surgery or a finite surgery similar terminology will be used for dehn fillings the manifold obtained by the dehn filling on m along m with slope r is denoted by m r let r1 r2 denote the minimal geometric intersection number the distance between two slopes r1 and r2 on m according to t1 m belongs to one of the following three mutually exclusive categories i m is a seifert fibred space admitting no essential torus ii m is a hyperbolic manifold i e intm admits a complete hyperbolic metric of finite volume iii m contains an essential torus it is a remarkable result the so called cyclic surgery theorem cgls that if m is not seifert fibred then all cyclic surgery slopes of k have mutual distance no larger than 1 and consequently there are at most 3 cyclic surgeries on k in this paper we consider finite dehn surgery on k and we prove for instance that if m is not a manifold of type i and is not a union along a torus of the twisted i bundle over the klein bottle and a cabled space then there are at most 6 finite and cyclic surgeries on k of maximal mutual distance 5 henceforth we shall use finite cyclic to mean either finite or infinite cyclic it turns out to be convenient to consider the three cases mentioned above separately in case i it is well known that one can completely classify the finite cyclic surgeries on k considering the torus knots for instance one sees that there exist infinitely many knots whose exteriors are of type i each of which admit an infinity of distinct finite cyclic or non cyclic surgery slopes our contributions deal with the cases ii and iii for the former we obtain

deformation spaces associated to compact hyperbolic manifolds in this paper we take a first step toward understanding representations of cocompact lattices in so n 1 into arbitrary lie groups by studying the deformations of rational representations see proposition 5 1 for a rather general existence result this proposition has a number of algebraic applications for example we remark that such deformations show that the margulis super rigidity theorem see 30 cannot be extended to the rank 1 case we remark also that if so n 1 is one of the standard arithmetic examples described in section 7 then has a faithful representation in so n 1 the galois conjugate of the uniformization representation and proposition 5 1 may be used to deform the direct sum of and the trivial representation in so n 2 thereby constructing non trivial families of irreducible orthogonal representations of however most of this paper is devoted to studying certain spaces of representations which are of interest in differential geometry in a sense which we now explain

pointwise estimates and lp convergence rates to diffusion waves for p system with damping abstract by introducing a new approximate green function we obtain the pointwise estimates on the solutions of euler equations with linear frictional damping from which we can deduce the optimal l p 1 p convergence rates to the nonlinear diffusion waves the pointwise estimates and l p 1 p convergence rates given in this paper are new

approximation by maximal cusps in boundaries of deformation spaces of kleinian groups let m be a compact oriented irreducible atoroidal 3 manifold with nonempty boundary let cc0 m denote the space of convex cocompact kleinian groups uniformizing m we show that any kleinian group in the boundary of cc0 m whose limit set is the whole sphere can be approximated by maximal cusps density of maximal cusps on the boundary of schottky space is derived as a corollary we further show that maximal cusps are dense in the boundary of the quasiconformal deformation space of any geometrically finite hyperbolic 3 manifold with connected conformal boundary

alternating knots with unknotting number one abstract we show that an alternating knot with unknotting number one has an unknotting crossing in any alternating diagram we also prove that an alternating knot has unknotting number one if and only if its branched double cover arises as half integer surgery on a knot in s 3 thus establishing a converse to the montesinos trick along the way we reprove a characterisation of almost alternating diagrams of the unknot originally due to tsukamoto these results are established using the obstruction to unknotting number one developed by greene

commensurability of 1 cusped hyperbolic 3 manifolds we give examples of non fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres in fact we give many examples of knot complements in homology spheres where every commensurable knot complement in a homology sphere has non monic alexander polynomial

non standard components of the character variety for a family of montesinos knots we show that there are montesinos knots with n 1 geq 4 tangles whose character varieties contain arbitrarily many irreducible components of dimension d for any 1 le d le n 2 moreover these irreducible components can be chosen so that the trace of the meridian is non constant

characteristic subsurfaces and dehn filling let m be a simple knot manifold using the characteristic submanifold theory and the combinatorics of graphs in surfaces we develop a method for bounding the distance between the boundary slope of an essential surface in m which is not a fiber or a semi fiber and the boundary slope of a certain type of singular surface applications include bounds on the distances between exceptional dehn surgery slopes it is shown that if the fundamental group of m has no non abelian free subgroup and if m is a reducible manifold which is not homeomorphic to s 1 s 2 or p 3 p 3 then 5 under the same condition on m it is shown that if m is seifert fibered then 6 moreover in the latter situation character variety techniques are used to characterize the topological types of m and m in case the bound of 6 is attained

on character varieties sets of discrete characters and nonzero degree maps a it knot manifold is a compact connected irreducible orientable 3 manifold whose boundary is an incompressible torus we first investigate virtual epimorphisms between the fundamental groups of small knot manifolds and prove minimality results for small knot manifolds with respect to nonzero degree maps these results are applied later in the paper where we fix a small knot manifold m and investigate various sets of characters of representations rho pi 1 m to rm psl 2 bbb c whose images are discrete we show that the topology of these sets is intimately related to the algebraic structure of the rm psl 2 bbb c character variety of m as well as dominations of manifolds by m and its dehn fillings we apply our results to the following question of shicheng wang it are nonzero degree maps between infinitely many distinct dehn fillings of two hyperbolic knot manifolds m and n induced by a nonzero degree map m to n we show that the answer is yes generically using this we show that if a small mathcal h minimal hyperbolic knot manifold admits non homeomorphic mathcal h minimal dehn fillings it admits infinitely many such fillings we also construct the first infinite families of small closed connected orientable manifolds which are minimal in the sense that they do not admit nonzero degree maps other than homotopy equivalences to any aspherical manifold

the maximal number of exceptional dehn surgeries thurston s hyperbolic dehn surgery theorem is one of the most important results in 3 manifold theory and it has stimulated an enormous amount of research if m is a compact orientable hyperbolic 3 manifold with boundary a single torus then the theorem asserts that for all but finitely many slopes s on m the manifold m s obtained by dehn filling along s also admits a hyperbolic structure the slopes s where m s is not hyperbolic are known as exceptional a major open question has been what is the maximal number of exceptional slopes on such a manifold m when m is the exterior of the figure eight knot the number of exceptional slopes is 10 and this was conjectured by gordon in 19 to be an upper bound that holds for all m in this paper we prove this conjecture

detection of essential surfaces in 3 manifolds with sl 2 trees according to bass serre theory se given a field f with a discrete valuation v f z there is a canonical way to construct a simplicial tree t tf v on which sl2 f acts simplicially without inversion when a group has a representation into the group sl2 f then there is an induced action of on the tree via the representation if further is the fundamental group of a compact 3 manifoldm and the action of on the tree t is nontrivial meaning that there is no point on t fixed by every element of then the action induces a splitting of m along an essential surface in the following way let m p m be the universal covering there is a 1 m equivariant map f m t which is transverse to the set e of midpoints of edges in t such that p f 1 e is an essential surface in m we say such an essential surface is associated to or detected by an sl2 tree a natural question is which essential surfaces in a compact 3 manifold can be associated to sl2 trees and how do they depend on the choice of the field and discrete valuation this is the main issue we are going to address in this paper any 3 manifold mentioned in this paper is automatically assumed orientable and connected by an essential surface in a compact 3 manifold m we mean an orientable properly embedded incompressible surface each component of which is neither boundary parallel in m nor bounds a 3 ball when it is a 2sphere also recall that a discrete valuation on a field f is a homomorphism v from the multiplicative group f onto the group of integers z such that v a b min v a v b for any a b f with a b 0 let v f z be a discrete valuation the set ov 0 a f v a 0 s schanuel x zhang department of mathematics suny at buffalo buffalo ny 14260 2900 usa e mail schanuel buffalo edu xinzhang math buffalo edu

lp convergence rate to nonlinear diffusion waves for p system with damping abstract in this paper we study the p system with frictional damping and show that the solutions time asymptotically tend to the nonlinear diffusion waves governed by the classical darcy s law by introducing an approximate green function we obtain the optimal l p 2 p convergence rate of the solution which is a perturbation of the nonlinear diffusion wave to the hyperbolic system

convergence to strong nonlinear diffusion waves for solutions of p system with damping this paper is concerned with the p system with frictional damping and our main purpose is two fold first we show that for a certain class of given large initial data nu 0 x u 0 x the cauchy problem 1 1 1 2 admits a unique global smooth solution nu t x u t x and such a solution tends time asymptotically at the optimal l p 2 less than or equal to p less than or equal to infinity decay rates to the corresponding nonlinear diffusion wave over bar t x u over bar t x governed by the classical darcy s law provided that the corresponding prescribed initial error function v 0 x u 0 x ties in h 3 x h 2 r boolean and l 1 x l 1 r compared with former results in this direction obtained by l hsiao and t p liu 1992 comm math phys 143 599 605 k nishihara 1996 j differential equations 131 171 188 and k nishihara w k wang and t yang 2000 j differential equations 161 191 218 our main novelty lies in the facts that the nonlinear diffusion wave over bar t x u over bar t x need not to be weak and v 0 x u 0 x can be chosen arbitrarily large secondly we show that the nonlinear diffusion waves over bar t x u over bar t x are nonlinear stable provided that the strength or the nonlinear diffusion waves is weak and that the initial disturbance v 0 x u 0 x satisfies the assumption that parallel tov 0xx x parallel to l infinity parallel tou 0x x parallel to l infinity is sufficiently small we also show that the smallness assumption imposed oil the strength of the diffusion waves is a necessary condition to guarantee the nonlinear stability result and compared with the corresponding results obtained by l hsiao and t p liu 1992 comm math phys 143 599 605 the smallness conditions we imposed on the initial disturbance are much more weaker c 2001 academic press

some relaxation problems in optimal control theory usually the space u r is a space lp q endowed with its weak topology the space y g is a sobolev space w y q endowed with its weak topology or with the l q topology in the one dimensional case and condition ii is guaranteed by some boundedness or growth hypotheses on j and a when condition i is not fulfilled in general the existence of a solution of 1 1 may fail nevertheless it may be interesting to study the asymptotic 212 0022 247x 87 3 00

asymptotic behavior of the solutions to the damped compressible euler equations with vacuum abstract the asymptotic behavior of solutions of the damped compressible euler equations is conjectured to obey to the famous porous media equations pmes the previous works on this topic concern the case away from vacuum where the system is strictly hyperbolic in present paper we prove that the l entropy weak solution with vacuum obtained by the compensated compactness theory converges strongly in l loc p 1 p space to the unique similarity solution of the related pme as time goes to infinity

convergence rate for compressible euler equations with damping and vacuum we study the asymptotic behavior of l weak entropy solutions to the compressible euler equations with damping and vacuum previous works on this topic are mainly concerned with the case away from the vacuum and small initial data in the present paper we prove that the entropy weak solution strongly converges to the similarity solution of the porous media equations in lp r 2 p with decay rates the initial data can contain vacuum and can be arbitrary large a new approach is introduced to control the singularity near vacuum for the desired estimates

global existence of solutions to 1 d euler equations with time dependent damping abstract in this paper we study the 1 d isentropic euler equations with time dependent damping our damping decays at a speed of order 1 with respect to time which is a little weaker than the linear one under our assumption we will prove the global existence of solutions to the euler system and obtain l 2 and l estimates for the solutions our approach is based on some detailed energy estimates

heegaard floer homology and fibred 3 manifolds given a closed 3 manifold y we show that the heegaard floer homology determines whether y fibres over the circle with a fibre of negative euler characteristic this is an analogue of an earlier result about knots proved by ghiggini and the author

dehn surgery on knots in d dehn considered the following method for constructing 3 manifolds remove a solid torus neighborhood n k of some knot x in the 3 sphere s and sew it back differently in particular he showed that taking x to be the trefoil one could obtain infinitely many non simply connected homology spheres o in this way let mx s n k then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus omk that bounds a meridional disk in the re attached solid torus we denote the resulting closed oriented 3 manifold by mk and say that it is obtained by r dehn surgery on x more generally one can consider the manifolds ml obtained by r dehn surgery on a fc component link l xi u u kk in s where r r it turns out that every closed oriented 3 manifold can be constructed in this way wal lie thus a good understanding of dehn surgery might lead to progress on general questions about the structure of 3 manifolds starting with the case of knots it is natural to extend the context a little and consider the manifolds m r obtained by attaching a solid torus v to an arbitrary compact oriented irreducible every 2 sphere bounds a 3 ball 3 manifold m with dm an incompressible torus where r is the isotopy class slope on dm of the boundary of a meridional disk of v we say that m r is the result of r dehn filling on m an observed feature of this construction is that

plane curves associated to character varieties of 3 manifolds consider a compact 3 manifold m with boundary consisting of a single torus the papers cs1 cs2 cgls discuss the variety of characters of sl2 c representations of zl m and some of the ways in which the topological structure of m is reflected in the algebraic geometry of the character variety we will describe in this paper a certain affine algebraic curve dm which is naturally associated to the character variety of m a basis b 9 1 for the peripheral subgroup of m determines an embedding p8 of dm into c x c with coordinates l and m the closure in c 2 of ps dm is a plane algebraic curve and therefore is defined by a polynomial am n0 m that after certain normalizations is uniquely determined up to multiplication by constants the polynomial am b is effectively computable and is an invariant of the manifold m together with the choice of basis b the results in this paper describe how geometric properties of the character variety and hence topological properties of m are reflected by the polynomial am b in the case that m is the complement of a knot k in a homology 3 sphere we may take the basis b to consist of the longitude and meridian of k with the usual orientation conventions this basis is well defined modulo the involution which inverts both the longitude and meridian it will follow from the construction of dm that if the basis b is obtained from b by inverting both generators then the regular maps pb and pb have the same image thus ar am s am b is an invariant of the knot the polynomial am b displays in a striking way information about the incompressible surfaces in m this involves the newton polygon of am b

asymptotic behavior of solutions for p system with relaxation in this paper we consider the asymptotic behavior of solutions for the cauchy problem for p system with relaxationvt ux 0 ut p v x 1e f v u e with initial data v u x 0 v 0 x u 0 x v u v 0 asx i we are interested to show the solutions of e i tend also to the equilibrium rarefaction waves and the traveling waves even if the limits v u of the initial data at x do not satisfy the equilibrium equation i e u f v when the limits of the initial data at infinity satisfy equilibrium states liu 9 studied the stability of rarefaction waves and traveling waves for the general 2 2 hyperbolic conservation laws with relaxation

asymptotic behavior of solutions to bipolar euler poisson equations with time dependent damping abstract in this paper we study the one dimensional euler poisson equations of bipolar hydrodynamic model for semiconductor device with time dependent damping effect j 1 t for 1 1 where the damping effect is time gradually degenerate for 0 and time gradually enhancing for 0 such a damping effect makes the hydrodynamic system possess the nonlinear diffusion phenomena time asymptotic weakly or strongly based on technical observation and by using the time weighted energy method where the weights are artfully chosen we prove that the system admits a unique global smooth solution which time asymptotically converges to the corresponding diffusion wave when the initial perturbation around the diffusion wave is small enough the convergence rates are specified in the algebraic forms o t 3 4 1 and o t 1 according to different values of in 1 1 7 and 1 7 1 respectively where 1 7 is the critical point and the convergence rate at the critical point is o t 6 7 ln t all these convergence rates obtained in different cases are optimal in the sense when the initial perturbations are l 2 integrable particularly when 1 7 the convergence rate is the fastest namely the asymptotic profile of the original system at the critical point is the best

dehn fillings of large hyperbolic 3 manifolds let m be a compact connected orientable hyperbolic 3 manifold whose boundary is a torus and which contains an essential closed surface s i t is conjectured that 5 is an upper bound for the distance between two slopes on m whose associated fillings are not hyperbolic manifolds in this paper we verify the conjecture when the first betti number of m is at least 2 by showing that given a pseudo anosov mapping class f of a surface and an essential simple closed curve in the surface then 5 is an upper bound for the diameter of the set of integers n for which the composition of f with the n th power of a dehn twist along is not pseudo anosov this sharpens an inequality of albert fathi for large manifolds m of first betti number 1 we obtain partial results set

unknotting tunnels and seifert surfaces let k be a knot with an unknotting tunnel gamma and suppose that k is not a 2 bridge knot there is an invariant rho p q in mathbb q 2 mathbb z with p odd defined for the pair k gamma the invariant rho has interesting geometric properties it is often straightforward to calculate for example for k a torus knot and gamma an annulus spanning arc rho k gamma 1 although rho is defined abstractly it is naturally revealed when k cup gamma is put in thin position if rho neq 1 then there is a minimal genus seifert surface f for k such that the tunnel gamma can be slid and isotoped to lie on f one consequence is that if rho k gamma neq 1 then mathrm genus k 1 this confirms a conjecture of goda and teragaito for pairs k gamma with rho k gamma neq 1

nonlinear diffusion waves for hyperbolic p system with nonlinear damping abstract this paper is concerned with the p system of hyperbolic conservation laws with nonlinear damping when the constant states are small the solutions of the cauchy problem for the damped p system globally exist and converge to their corresponding nonlinear diffusion waves which are the solutions of the corresponding nonlinear parabolic equation given by the darcy s law the optimal convergence rates are also obtained in order to overcome the difficulty caused by the nonlinear damping a couple of correction functions have been technically constructed the approach adopted is the elementary energy method together with the technique of approximating green function on the other hand when the constant states are large the solutions of the cauchy problem for the p system will blow up at a finite time

lower semicontinuity of integral functionals with nonconvex integrands by relaxation compactification a new approach to the lower semicontinuity of integral functionals is presented by a topological embedding of the control and state spaces in the hilbert cube and a simultaneous relaxation of the control functions a powerful approach emerges whose main features include i a generalized convexity condition is imposed upon the integrand of which the classical convexity condition is a special case ii in the embedded setting the integrand can be supposed lipschitz continuous in control and state arguments without loss of generality iii convergence in measure of the trajectories metamorphoses into l 1 norm convergence in the embedded setting

global and blow up solutions for compressible euler equations with time dependent damping abstract this paper deals with the cauchy problem for the compressible euler equations with time dependent damping where the time vanishing damping in the form of 1 t makes some fantastic variety of the dynamic system for 0 1 and 0 or 1 but 2 the solutions are proved to exist globally in time when the derivatives of the initial data are small but the initial data themselves can be arbitrarily large this is the so called challenging case of global solutions with large initial data while when the initial riemann invariants are monotonic and their derivatives with absolute value are large at least at one point then the solutions are still bounded but their derivatives will blow up at finite time somewhat like the singularity formed by shock waves for 1 and 0 or 1 but 0 1 the derivatives of solutions will blow up even for all initial data including the interesting case of blow up solutions with small initial data here the initial riemann invariants are monotonic in fact such a blow up phenomenon is determined by the mechanism of the dynamic system itself in order to prove the global existence of solutions with large initial data we introduce a new energy functional related to the riemann invariants which crucially enables us to build up the maximum principle for the corresponding riemann invariants and the uniform boundedness for the local solutions finally numerical simulations in different cases are carried out which further confirm our theoretical results

on character varieties of two bridge knot groups we find explicit models for the psl2 c and sl2 c character varieties of the fundamental groups of complements in s of an infinite family of two bridge knots that contains the twist knots we compute the genus of the components of these character varieties and deduce upper bounds on the degree of the associated trace fields we also show that these knot complements are fibered if and only if they are commensurable to a fibered knot complement in a z 2z homology sphere resolving a conjecture of hoste and shanahan

knot floer homology detects genus one fibred knots ozsv ath and szab o conjectured that knot floer homology detects fibred knots we propose a strategy to approach this conjecture based on contact topology and gabai s theory of sutured manifold decomposition we implement this strategy for genus one knots obtaining as a corollary that if rational surgery on a knot k gives the poincar e homology sphere sigma 2 3 5 then k is the left handed trefoil knot

a proof of the finite filling conjecture let m be a compact connected orientable hyperbolic 3 manifold whose boundary is a torus we show that there are at most five slopes on m whose associated dehn fillings have either a finite or an infinite cyclic fundamental group furthermore the distance between two slopes yielding such manifolds is no more than three and there is at most one pair of slopes which realize the distance three each of these bounds is realized when m is taken to be the exterior of the figure 8 sister knot

a general approach to lower semicontinuity and lower closure in optimal control theory a self contained approach to lower semicontinuity and lower closure evolves from an extension of relaxed control theory which is based on a central relative weak compactness criterion called tightness and relaxation in all but one variable two lower closure results for outer integral functionals with variable abstract time domain are developed the first of these has a convexity condition for the integrand and generalizes all similar results in the literature the second lower closure result is of a new kind among other things it implies a quite general version of fatou s lemma in several dimensions