a genuinely multidimensional convective pressure flux split riemann solver for euler equations a simple inexpensive genuinely multidimensional hll type riemann solver for euler equations of gasdynamics is proposed the genuinely multidimensional formulation is based on the work of balsara 17 in the present approach the missing contact wave in the hll type solver is restored using the hll cps formulation of mandal and panwar 5 a few two dimensional standard test cases are solved to demonstrate the efficacy of the proposed solver a very simple new genuinely multidimensional hll type riemann solver for euler equations is proposed due to the use of closed form expressions for the fluxes the present method is easier to apply the contact discontinuity is restored using the hll cps formulation without the need for independently specifying its direction of evolution numerical test results demonstrate the efficacy of the present method of its ability to capture accurate solutions

a second order kinetic scheme for gas dynamics on arbitrary grids cartesian meshes for domains with complicated boundaries give rise to cut cells with arbitrarily small volumes explicit integration schemes over such meshes have a time step restriction proportional to the smallest cell volume we present an implementation of the kinetic scheme for gas dynamics by perthame b perthame boltzmann type schemes for gas dynamics and the entropy property siam j num anal 27 1990 1405 1421 on arbitrary cartesian meshes the formulation allows a time step based on the underlying regular cell size and retains l 1 stability positivity and second order convergence numerical convergence studies on arbitrary grids are presented

non uniform convergence of finite volume schemes for riemann problems of ideal magnetohydrodynamics this paper presents riemann test problems for ideal magnetohydrodynamics mhd finite volume schemes the test problems place emphasis on the hyperbolic irregularities of the ideal mhd system namely the occurrence of intermediate shocks and non unique solutions we investigate numerical solutions for the test problems obtained by several commonly used methods roe hlle central scheme all methods turned out to show a non uniform convergence behavior which may be paraphrased as pseudo convergence initially the methods show convergence towards a wrong solution with irregular wave patterns only after heavy grid refinement the methods switch to converge to the true solution this behavior is most pronounced whenever the solution should be unique but its phase space trajectory lies in the vicinity of that of a non unique solution we show detailed grid convergence studies and empirical error analysis the results may be related to similar results for a 2 2 model system and to time dependent investigations of other authors the non uniform convergence is expected to be present for any diffusive finite volume method and question the reliability of coarse grid ideal mhd solutions also in higher space dimensions

an extended finite element formulation for contact in multi material arbitrary lagrangian eulerian calculations a contact method for the multi dimensional lagrangian step in multi material arbitrary lagrangian eulerian ale formulations is presented the extended finite element method x fem is used to create independent fields i e velocity strain rate force mass etc for each material in the problem the accelerations of the multi material nodes are obtained by coupling the material force and mass fields as a function of the prescribed contact similarly the velocities of the multi material nodes are recalculated using the conservation of momentum when the prescribed contact requires it the coupling procedures impose the same nodal velocity on the coupled materials in the direction normal to their interface during the time step update as a result the overlap of materials is prevented and unwanted separation does not occur three different contacts are treated perfectly bonded frictionless slip and slip with friction example impact problems are solved and the numerical solutions are presented

a robust hllc type riemann solver for strong shock it is well known that for the eulerian equations the numerical schemes that can accurately capture contact discontinuity usually suffer from some disastrous carbuncle phenomenon while some more dissipative schemes such as the hll scheme are free from this kind of shock instability hybrid schemes to combine a dissipative flux with a less dissipative flux can cure the shock instability but also may lead to other problems such as certain arbitrariness of choosing switching parameters or contact interface becoming smeared in order to overcome these drawbacks this paper proposes a simple and robust hllc type riemann solver for inviscid compressible gas flows which is capable of preserving sharp contact surface and is free from instability the main work is to construct a hll type riemann solver and a hllc type riemann solver by modifying the shear viscosity of the original hll and hllc methods both of the two new schemes are positively conservative under some typical wavespeed estimations moreover a linear matrix stability analysis for the proposed schemes is accomplished which illustrates the hllc type solver with shear viscosity is stable whereas the hll type solver with vorticity wave is unstable our arguments and numerical experiments demonstrate that the inadequate dissipation associated to the shear wave may be a unique reason to cause the instability

a two dimensional hlle riemann solver and associated godunov type difference scheme for gas dynamics abstract the approximate three state riemann solver hlle is formally extended to a nine state two dimensional solver the associated godunov scheme is outlined and applied to two test problems published by elsevier science ltd

asymptotic stability for a linear difference equation with variable delay we shall obtain sufficient conditions for the uniform stability and the global asymptotic stability of the linear difference equation with variable delay xn 1 xn pnxn kn 0 n n where pn is a sequence of nonnegative real numbers kn is a sequence of nonnegative integers and there exists a nonnegative integer k such that kn k for n n

complete proper minimal surfaces in convex bodies of mathbbr 3 consider a convex domain b of r 3 we prove that there exist complete minimal surfaces which are properly immersed in b we also demonstrate that if d and dare convex domains with d bounded and the closure of d contained in dthen any minimal disk whose boundary lies in the boundary of d can be approximated in any compact subdomain of d by a complete minimal disk which is proper in d we apply these results to study the so called type problem for a minimal surface we demonstrate that the interior of any convex region of r 3 is not a universal region for minimal surfaces in the sense explained by meeks and perez in 9

schemes to compute unsteady flashing flows some ways to compute flashing flows in variable cross section ducts are provided focusing on the homogeneous relaxation model the basic numerical method relies on a splitting technique that is consistent with the overall entropy inequality the cross section is assumed to he continuous and the finite volume approach is applied to approximate homogeneous equations several suitable schemes to account for complex equation of state are discussed namely the rusanov scheme an approximate form of the roe scheme and the volumes finis roe vfroe scheme with the help of nonconservative variables to evaluate respective accuracy the homogeneous euler equations are computed first and the l1 error norm of transient solutions of shock tube experiments are plotted it is shown that the rusanov scheme is indeed less accurate which balances its interesting properties inasmuch as it preserves the positivity of the mean density and the maximum principle for the vapor quality computations of real cases are presented which account for the mass transfer term and the time space dependent cross sections

capturing shock reflections godunov type schemes based on exact or approximate solutions to the riemann problem have proven to be an excellent tool to compute approximate solutions to hyperbolic systems of conservation laws however there are many instances in which a particular scheme produces inappropriate results in this paper we consider several situations in which roe s scheme gives incorrect results or blows up all together and we propose an alternative flux formula that produces numerical approximations in which the pathological behavior is either eliminated or reduced to computationally acceptable levels

removal of pseudo convergence in coplanar and near coplanar riemann problems of ideal magnetohydrodynamics solved using finite volume schemes numerical schemes for ideal magnetohydrodynamics mhd that are based on the standard finite volume method fvm exhibit pseudo convergence in which irregular structures no longer exist only after heavy grid refinement we describe a method for obtaining solutions for coplanar and near coplanar cases that consist of only regular structures independent of grid refinement the method referred to as compound wave modification cwm involves removing the flux associated with non regular structures and can be used for simulations in two and three dimensions because it does not require explicitly tracking an alfven wave for a near coplanar case and for grids with 213 points or less we find root square mean errors rmses that are as much as 6 times smaller for the coplanar case in which non regular structures will exist at all levels of grid refinement for standard fvms the rmse is as much as 25 times smaller

the calabi yau conjectures for embedded surfaces in this talk i will discuss the proof of the calabi yau conjectures for embedded surfaces this is joint work with bill minicozzi cm9 the calabi yau conjectures about surfaces date back to the 1960s much work has been done on them over the past four decades in particular examples of jorge xavier from 1980 and nadirashvili from 1996 showed that the immersed versions were false we will show here that for embedded surfaces i e injective immersions they are in fact true their original form was given in 1965 in ca where e calabi made the following two conjectures about minimal surfaces see also s s chern page 212 of ch conjecture 1 prove that a complete minimal hypersurface in r must be unbounded calabi continued it is known that there are no compact minimal submanifolds of r or of any simply connected complete riemannian manifold with sectional curvature 0 a more ambitious conjecture is conjecture 2 a complete minimal hypersurface in r has an unbounded projection in every n 2 dimensional flat subspace

a two dimensional version of the godunov scheme for scalar balance laws a godunov scheme is derived for two dimensional scalar conservation laws without or with source terms following ideas originally proposed by boukadida and leroux math comput 63 1994 pp 541 553 in the context of a staggered lax friedrichs scheme in both situations the numerical fluxes are obtained at each interface of a uniform cartesian computational grid just by means of the external waves involved in the entropy solution of the elementary two dimensional 2d riemann problems in particular all the wave interaction phenomena are overlooked this restriction of the wave pattern suffices for deriving the exact numerical fluxes of the staggered lax friedrichs scheme but it furnishes only an approximation for the godunov scheme we show that under convenient assumptions these flux functions are smooth and the resulting discretization process is stable under nearly optimal cfl restriction a well balanced extension is presented relying on the curl free component of the helmholtz decomposi

the existence of least area surfaces in 3 manifolds this paper presents a new and unified approach to the existence theorems for least area surfaces in 3 manifolds introduction a surface f smoothly embedded or immersed in a riemannian manifold m is minimal if it has mean curvature zero at all points it is a least area surface in a class of surfaces if it has finite area which realizes the infimum of all possible areas for surfaces in this class the connection between these two ideas is that a surface which is of least area in a reasonable class of surfaces must be minimal the converse is false minimal surfaces are in general only critical points for the area function there are close analogies between these two concepts and the theory of geodesics in a riemannian manifold minimal surfaces correspond to geodesics and least area surfaces correspond to geodesic arcs or closed geodesics which have shortest length in some class of paths any geodesic a in a riemannian manifold m has the property that it is locally shortest i e if p and q are nearby points on a then the subarc of a which joins p and q is the shortest path in m from p to q it can also be proved that minimal surfaces are locally of least area but the proof is difficult and involves substantial knowledge of the theory of partial differential equations there are now a large number of theorems asserting the existence of surfaces of least area in various classes surfaces of this type have become an important tool in 3 dimensional topology in this paper we present a new approach to the proofs of these existence theorems this yields a simplified and unified method for the proof of the existence of minimal surfaces in 3 dimensional riemannian manifolds in 1930 douglas do and rado ra independently showed that a simple closed curve in ri which bounds a disk of finite area bounds a disk of least possible area this result was extended by morrey moi in 1948 to a general class of riemannian manifolds the homogeneously regular manifolds a class which includes all closed manifolds work of osserman 0 and gulliver g later showed that least area disks in 3 manifolds were immersed in their interiors more recently there has been a series of new existence results for closed surfaces of least area in manifolds of any dimension it follows from work of sacks and uhlenbeck s ui that if m is closed and 7r2 m is nonzero then there is an essential map of the 2 sphere into m which has least area among all essential maps sacks and uhlenbeck s uii and schoen and yau s y independently showed that if f f m is a map of a closed orientable surface f not the 2 sphere into a closed riemannian manifold m such received by the editors june 23 1986 1980 mathematics subject classification 1985 revision primary 53a10 secondary 57m35 the first author was supported by nsf postdoctoral fellowship dms84 14097 1988 american mathematical society 0002 9947 88 1 00 25 per page

permanence contractivity and global stability in logistic equations with general delays abstract we obtain new conditions of the permanence and contractivity of solutions and the global asymptotic stability for the positive equilibrium x 1 a j 0 m b j of the following logistic equation with general delays 0 1 dx t dt x t r t 1 ax t j 0 m b j x j t t t 0 x t t 0 t t 0 and t 0 0 where r t is a nonnegative continuous function on t0 a b 0 or a b 0 and b j 0mmin 0 bj each j t is piecewise continuous on t0 j t t for 0 j m and t min 1 j m j t as t the results improve that of j w h so and j s yu hokkaido math j 24 1995 269 286 for a logistic equation with nonlinear delay terms a similar result is obtained

on positivity preserving high order discontinuous galerkin schemes for compressible navier stokes equations we construct a local lax friedrichs type positivity preserving flux for compressible navier stokes equations which can be easily extended to multiple dimensions for generic forms of equations of state shear stress tensor and heat flux with this positivity preserving flux any finite volume type schemes including discontinuous galerkin dg schemes with strong stability preserving runge kutta time discretizations satisfy a weak positivity property with a simple and efficient positivity preserving limiter high order explicit runge kutta dg schemes are rendered preserving the positivity of density and internal energy without losing local conservation or high order accuracy numerical tests suggest that the positivity preserving flux and the positivity preserving limiter do not induce excessive artificial viscosity and the high order positivity preserving dg schemes without other limiters can produce satisfying non oscillatory solutions when the nonlinear diffusion in compressible navier stokes equations is accurately resolved

a first eigenvalue estimate for minimal hypersurfaces on obtient une borne inferieure pour la premiere valeur propre d une hypersurface orientable compacte plongee de facon minimale dans une variete orientable compacte a courbure de ricci positive

on a nonlinear delay population model we investigate uniform permanence of positive solutions of nonlinear delay equations in several examples we give explicit estimates for lower and upper limit of solutions we give conditions which imply that all solutions are asymptotically equivalent the nonlinear delay differential equation x t r t g t x t h x t t 0 is considered sufficient conditions are established for the uniform permanence of the positive solutions of the equation in several particular cases explicit formulas are given for the upper and lower limit of the solutions in some special cases we give conditions which imply that all solutions have the same asymptotic behavior in particular when they converge to a periodic or constant steady state

an embedded genus one helicoid there exists a properly embedded minimal surface of genus one with one end the end is asymptotic to the end of the helicoid this genus one helicoid is constructed as the limit of a continuous one parameter family of screw motion invariant minimal surfaces also asymptotic to the helicoid that have genus equal to one in the quotient

an analysis of finite difference and finite volume formulations of conservation laws abstract finite difference and finite volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws a new coordinate free formulation of systems of conservation laws is developed which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system the analysis considers general types of equations potential euler and navier stokes three dimensional unsteady flows with time varying grids are described using a single consistent nomenclature for both formulations grid motion due to a non inertial reference frame as well as flow adaptation is covered in comparing the two formulations it is found useful to distinguish between differences in numerical methods and differences in grid definition the former plays a role for non cartesian grids and results in only cosmetic differences in the manner in which geometric terms are handled the differences in grid definition for the two formulations is found to be more important since it affects the manner in which boundary conditions zonal procedures and grid singularities are handled at computational boundaries the proper interpretation of strong and weak conservation law forms for quasi one dimensional and axisymmetric flows is brought out

three circles theorems for schrodinger operators on cylindrical ends and geometric applications we show that for a schrodinger operator with bounded potential on a manifold with cylindrical ends the space of solutions that grows at most exponentially at infinity is finite dimensional and for a dense set of potentials or equivalently for a surface for a fixed potential and a dense set of metrics the constant function 0 is the only solution that vanishes at infinity clearly for general potentials there can be many solutions that vanish at infinity one of the key ingredients in these results is a three circles inequality or log convexity inequality for the sobolev norm of a solution u to a schrodinger equation on a product n 0 t w heren is a closed manifold with a certain spectral gap examples of such n s are all round spheres s n for n 1 and all zoll surfaces finally we discuss some examples arising in geometry of such manifolds and schrodinger operators c

a finite volume hllc based scheme for compressible interfacial flows with surface tension shock waves are often used in experiments to create a shear flow across liquid droplets to study secondary atomization similar behavior occurs inside of supersonic combustors scramjets under startup conditions but it is challenging to study these conditions experimentally in order to investigate this phenomenon further a numerical approach is developed to simulate compressible multiphase flows under the effects of surface tension forces the flow field is solved via the compressible multicomponent euler equations i e the five equation model discretized with the finite volume method on a uniform cartesian grid the solver utilizes a total variation diminishing tvd third order rungekutta method for time marching and second order tvd spatial reconstruction surface tension is incorporated using the continuum surface force csf model fluxes are upwinded with a modified hartenlaxvan leer contact hllc approximate riemann solver an interface compression scheme is employed to counter numerical diffusion of the interface the present work includes modifications to both the hllc solver and the interface compression scheme to account for capillary force terms and the associated pressure jump across the gasliquid interface a simple method for numerically computing the interface curvature is developed and an acoustic scaling of the surface tension coefficient is proposed for the non dimensionalization of the model the model captures the surface tension induced pressure jump exactly if the exact curvature is known and is further verified with an oscillating elliptical droplet and mach 1 47 and 3 shock droplet interaction problems the general characteristics of secondary atomization at a range of weber numbers are also captured in a series of simulations

persistence and global stability in a population model a difference equation modelling the dynamics of a population undergoing a density dependent harvesting is considered a sufficient condition is established for all positive solutions of the corresponding discrete dynamic system to converge eventually to the positive equilibrium elementary methods of differential calculus are used the result of this article provides a generalization of a result known for a simpler special model with no harvesting

a simple and accurate riemann solver for isothermal mhd a new approximate riemann solver for the equations of magnetohydrodynamics mhd with an isothermal equation of state is presented the proposed method of solution draws on the recent work of miyoshi and kusano in the context of adiabatic mhd where an approximate solution to the riemann problem is sought in terms of an average constant velocity and total pressure across the riemann fan this allows the formation of four intermediate states enclosed by two outermost fast discontinuities and separated by two rotational waves and an entropy mode in the present work a corresponding derivation for the isothermal mhd equations is presented it is found that the absence of the entropy mode leads to a different formulation which is based on a three state representation rather than four numerical tests in one and two dimensions demonstrate that the new solver is robust and comparable in accuracy to the more expensive linearized solver of roe although considerably faster

a direct eulerian grp scheme for compressible fluid flows a direct eulerian generalized riemann problem grp scheme is derived for compressible fluid flows riemann invariants are introduced as the main ingredient to resolve the generalized riemann problem grp directly for the eulerian formulation the crucial auxiliary lagrangian scheme in the original grp scheme is not necessary in the present framework the delicate sonic cases can be easily treated and the extension to multidimensional cases is obtained using the dimensional splitting technique

wave structure similarity of the hllc and roe riemann solvers application to low mach number preconditioning the approximate riemann solver of roe and the solver of harten lax van leer hll and its variants such as the hllc solver are widely used as building blocks of finite volume godunov type methods for the solution of the euler equations of gas dynamics and related hyperbolic flow models the hllc solver has gained increasing popularity over the last two decades since it possesses some of the good properties of the roe solver and in addition satisfies important entropy and positivity conditions with no need of special fixes in this work we rewrite the classical hllc riemann solver in a novel form that highlights the formal mathematical similarity of its wave structure with the one of the roe solver this similarity might be useful to extend to the hllc method some numerical techniques devised specifically for the roe method as an example of application we illustrate the design of a turkel type low mach number preconditioning technique for the hllc scheme by exploiting methodologies proposed in the l

on godunov type methods near low densities abstract when the energy of a flow is largely kinetic many conservative differencing schemes may fail by predicting non physical states with negative density or internal energy we describe as positively conservative the subclass of schemes that by contrast always generate physical solutions from physical data and show that the godunov method is positively conservative it is also shown that no scheme whose interface flux derives from a linearised riemann solution can be positively conservative classes of data that will bring about the failure of such schemes are described however the harten lax van leer hlle scheme is positively conservative under certain conditions on the numerical wavespeeds and this observation allows the linearised schemes to be rescued by modifying the wavespeeds employed

embedded minimal disks proper versus nonproper global versus local we construct a sequence of compact embedded minimal disks in a ball in r 3 with boundaries in the boundary of the ball and where the curvatures blow up only at the center the sequence converges to a limit which is not smooth and not proper if instead the sequence of embedded disks had boundaries in a sequence of balls with radii tending to infinity then we have shown previously that any limit must be smooth and proper

the existence of embedded minimal surfaces and the problem of uniqueness it was in the early nineteen thirties that douglas and radd solved the plateau problem for a rectifiable jordan curve in euclidean space the minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk in 1948 morrey generalized the theorem of douglas and radd to minimal surfaces in a homogeneously regular riemannian manifold in both cases the solutions are defective as they may possess branch points it was not until 1968 that osserman 24 was able to prove that interior branch points do not exist on the douglas radd morrey solution osserman s theorem in this full generality was proved by alt 2 3 and gulliver 8 if the jordan curve is real analytic gulliver and lesley 9 showed that the solution surface is free of branch points even on its boundary hence at least in this case we know that the douglas radd morrey solution is an immersion and represents a classical surface however the douglas radd morrey solution may in general have selfintersections it was then a classical problem to find conditions on the jordan curve to guarantee that the solution is an embedded disk in 1932 radd was the first to produce a general condition of this type he proved that if the curve can be projected onto a convex curve in the plane then it bounds a unique embedded minimal disk which is a graph over this plane it was not until the seventies that there was further progress on the problem of embeddedness osserman had conjectured that if a jordan curve is extreme that is lies on the boundary of its convex hull then the douglas rad6 solution is embedded gulliver and spruck 10 proved this conjecture under the additional assumption that the absolute total curvature of the jordan curve is less than 4 z right after this work tomi and tromba 32 showed that an extreme curve must bound an embedded minimal disk which need not be stable in 1976 almgren and simon 1 established the existence for such a curve of an embedded stable minimal disk which need not be a douglas radd solution finally in the same year the authors 18 proved that any douglas rad6 solution must be embedded

the ghost fluid method for compressible gas water simulation an analysis is carried out for the ghost fluid method gfm based algorithm as applied to the gas water riemann problems which can be construed as two single medium gfm riemann problems it is found that the inability to provide correct and consistent riemann waves in the respective real fluids by these two gfm riemann problems may lead to inaccurate numerical results based on this finding two conditions are suggested and imposed for the ghost fluid status in order to ensure that correct and consistent riemann waves are provided in the respective real fluids during the numerical decomposition of the singularity using these two conditions to analyse some of the existing gfm based algorithms such as the original gfm j comput phys 152 1999 457 the new version gfm j comput phys 166 2001 1 j comput phys 175 2002 200 and the modified gfm mgfm j comput phys 190 2003 651 it is found that there are ranges of conditions for each type of solution where either the original gfm or the new version gfm or both are unable to provide correct or consistent riemann waves in one of the real fluids within these ranges examples can be found such that either the original gfm or the new version gfm or both are unable to provide accurate results the mgfm is also found to encounter difficulties when applied to nearly cavitating flow various examples are presented to demonstrate the conclusions obtained the mgfm with proposed modification when applied to nearly cavitating flow is then found to be quite robust and can provide relatively reasonable results

uniform stability of nonlinear difference systems abstract we offer sufficient conditions for the uniform stability and uniformly asymptotic stability of the zero solution of the nonlinear difference system formula where 0 1 f n s r sis the set of all functions n k 0 rfor somek n andxn sis defined byxn m x n m form n k 0

persistence contractivity and global stability in logistic equations with piecewise constant delays abstract we establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium n 1 a i 0 m b i of the following differential equation with piecewise constant arguments dn t dt n t r t 1 an t i 0 m b i n n i n t n 0 1 2 n 0 n 0 0 and n j n j 0 j 1 2 m where r t is a nonnegative continuous function on 0 r t 0 i 0mbi 0 bi 0 i 0 1 2 m and a i 0mbi 0 these new conditions depend on a b0 and i 1mbi and hence these are other type conditions than those given by so and yu hokkaido math j 24 1995 269 286 and others in particular in the case m 0 and r t r 0 we offer necessary and sufficient conditions for the persistence and contractivity of solutions we also investigate the following differential equation with nonlinear delay terms dx t dt x t r t 1 ax t g x t x t 1 x t m t 0 x k k 0 0 k m and 0 0 where r t is a nonnegative continuous function on 0 r t 0 1 ax g x x x 0 has a unique solution x 0 and g x0 x1 xm c1 0 0 0

non proper helicoid like limits of closed minimal surfaces in 3 manifolds we show that there exists a metric with positive scalar curvature on s 2 s 1 and a sequence of embedded minimal cylinders that converges to a minimal lamination that in a neighborhood of a strictly stable 2 sphere is smooth except at two helicoid like singular ities on the 2 sphere the construction is inspired by a recent example by d hoffman and b white

the space of embedded minimal surfaces of fixed genus in a 3 manifold iii planar domains this paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3 manifold in cm3 cm5 we describe the case where the surfaces are topologically disks on any fixed small scale although the focus of this paper general planar domains is more in line with cm6 we will prove a result here namely corollary iii 3 5 below which is needed in cm5 even for the case of disks roughly speaking there are two main themes in this paper the first is that stability leads to improved curvature estimates this allows us to find large graphical regions these graphical regions lead to two possibilities

the space of embedded minimal surfaces of fixed genus in a 3 manifold iv locally simply connected this paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3manifold the key is to understand the structure of an embedded minimal disk in a ball in r3 this was undertaken in cm3 cm4 and the global version of it will be completed here see the discussion around figure 12 for the local case and cm15 for some more details our main results are theorem 0 1 the lamination theorem and theorem 0 2 the one sided curvature estimate the lamination theorem is stated in the global case where the lamination is in fact a foliation the first four papers of this series show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multivalued graph this is done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase like the helicoid to prove that such a disk is a double spiral staircase we showed in the first three papers of the series that it is built out of n valued graphs where n is a fixed number in this paper we will deal with how the multi valued graphs fit together and in particular prove regularity of the set of points of large curvature the axis of the double spiral staircase the first theorem is the global version of the statement that every embedded minimal disk is a double spiral staircase

regularity of isoperimetric hypersurfaces in riemannian manifolds we add to the literature the well known fact that an isoperimetric hypersurface s of dimension at most six in a smooth riemannian manifold m is a smooth submanifold if the metric is merely lipschitz then s is still holder differentiable

asymptotic behavior of solutions of first order nonlinear delay difference equations we obtain sufficient conditions which ensure that all solutions of the first order non linear delay difference equation yn f n yn k 0 n n n0 tend to zero as n in e n n0 n0 n0 1 n0 2 n0 is a fixed nonnegative integer k is a nonnegative integer f n n0 r r is a real valued function and yn yn 1 yn

comparison of some flux corrected transport and total variation diminishing numerical schemes for hydrodynamic and magnetohydrodynamic problems two versions of flux corrected transport and two versions of total variation diminishing schemes are tested for several one and two dimensional hydrodynamic and magnetohydrodynamic problems two of the schemes ydfct and tvdlf are tested extensively for the first time the results give an insight into the limitations of the methods their relative strengths and weaknesses some subtle points of the algorithms and the effects of selecting different options for certain methods are emphasised

a robust and accurate led bgk solver on unstructured adaptive meshes starting from the bgk model of the boltzmann equation we develop a robust and accurate finite volume gas kinetic scheme on unstructured triangular meshes the proposed numerical approach is composed of two steps an initial reconstruction step and a gas evolution step in the initial reconstruction step an unstructured version of the local extremum diminishing interpolation is applied to the conservative variables and to compute left and right states along a node edge in the gas evolution step the local integral solution of the bgk model is used to compute numerical fluxes at a cell interface this approach provides an alternative to riemann solvers and yields numerical schemes which possess many desirable properties that may not be found in godunov type schemes a classich refinement adaptive procedure is implemented to increase the spatial resolution of high speed unsteady flow characteristics such as shock waves contact discontinuities or expansion waves with minimal computational costs and memory overheads this procedure involves mesh enrichment coarsening steps to either insert nodes on an edge center in high gradient regions or delete nodes in over resolved regions numerical results of several test cases for unsteady compressible inviscid flows are presented in order to verify the accuracy and robustness of the current numerical approach the computed results are compared with analytical solutions experimental data the results of structured mesh calculations and the results obtained by widely used flux splitting methods

the space of minimal embeddings of a surface into a three dimensional manifold of positive ricci curvature in this paper we obtain a curvature estimate for embedded minimal surfaces in a three dimensional manifold of positive ricci curvature in terms of the geometry of the ambient manifold and the genus of the minimal surface it should be mentioned that there are two main points in our result one is the absence of a stability assumption and the other is the requirement of being embedded most known curvature estimates require the stability assumption and once the stability assumption is dropped many of these known results cease to be valid see ss and an for another example of a curvature estimate without the assumption of stability the embeddedness condition is rather subtle because of the way it enters in our proof our proof depends on the eigenvalue estimate and the area bound due to the first author and wang cw which require embeddedness in an essential way present knowledge indicates that closed embedded minimal surfaces in s 3 are rare while immersed surfaces are more plentiful for example only a finite number of minimal embeddings of a given genus are known on the other hand otsuki o constructed infinitely many immersed minimal tori with arbitrarily large area we obtain the curvature estimate indirectly by proving the smooth compactness theorem theorem 1 theorem 1 has many interesting consequences it shows that the set of conformal structures that can be realized on a minimal embedding in s 3 is a compact subset of the moduli space this is in contrast with the result of bryant b who showed that every riemann surface is conformally and minimally immersed in s 4 in view of our compactness result and the scarcity of examples it is very tempting to conjecture that there are only finitely many embedded minimal surfaces up to rigid motion in s 3 for each fixed genus throughout this paper manifold means manifold without boundary unless explictly stated otherwise when we say a sequence m i of surfaces converges to a

the space of embedded minimal surfaces of fixed genus in a 3 manifold ii multi valued graphs in disks this paper is the second in a series where we give a description of the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3 manifold the key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in r 3 we show here that if the curvature of such a disk becomes large at some point then it contains an almost flat multi valued graph nearby that continues almost all the way to the boundary this will be proved by showing the existence of small multi valued graphs near points of large curvature and then using the extension result for multi valued graphs proved in the first paper in this series there are two local models for embedded minimal disks by an embedded disk we mean a smooth injective map from the closed unit ball in r 2 into r 3 one model is the plane or more generally a minimal graph the other is a piece of a helicoid in the first four papers of this series we will show that every embedded minimal disk is either a graph of a function or is a double spiral staircase like a helicoid recall that a double spiral staircase consists of two spiral staircases that spiral together around a common axis one inside the other this will be done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase to prove that it is a double spiral staircase we will first prove that it is built out of n valued graphs where n is a fixed number these n valued graphs are like a single spiral staircase connecting n floors the existence of the n valued graphs was initiated in the first paper and will be completed here the third and fourth papers of this series will deal with how the multi valued graphs fit together and in particular prove regularity of the set of points of large curvature the axis of the double spiral staircase

the space of embedded minimal surfaces of fixed genus in a 3 manifold i estimates off the axis for disks this paper is the first in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed riemannian 3 manifold the key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in r3 with the flat metric this study is undertaken here and completed in cm6 these local results are then applied in cm7 where we describe the general structure of fixed genus surfaces in 3 manifolds there are two local models for embedded minimal disks by an embedded disk we mean a smooth injective map from the closed unit ball in r2 into r3 one model is the plane or more generally a minimal graph the other is a piece of a helicoid in the first four papers of this series we will show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multi valued graph this will be done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase to prove that such a disk

arbitrary high order discontinuous galerkin schemes based on the grp method for compressible euler equations combined with the discontinuous galerkin dg framework the generalized riemann problem grp method is applied to design a grp dg scheme with high order accuracy for compressible euler equations since numerical fluxes with second order accuracy in time are derived by the grp method the reconstruction steps for physical variables in the new scheme are halved compared with the traditional runge kutta discontinuous galerkin rk dg scheme the numerical results are also improved due to more introduced physical information several numerical examples verify the validity of the proposed schemes

modeled boltzmann equation and its application to shock capturing simulation a modified equilibrium distribution function for the bhatnagar gross krook type modeled boltzmann equation has recently been proposed the function was deduced using acoustics scaling to normalize the equation and allowed a correct recovery of similarly normalized euler equations it is a combination of a maxwellian distribution plus three other terms that are moments of particle velocity the lattice counterpart of the modified equilibrium distribution function also led to an exact recovery of the euler equations therefore there is no mach number limitation in the entire approach this lattice counterpart was able to replicate aeroacoustics problems involving vorticity acoustic and entropy acoustic interactions correctly and the simulations were carried out using a finite difference lattice boltzmann method employing only a two dimensional nine velocity lattice thus formulated the numerical scheme has no arbitrary constants and all calculations were carried out using one single relaxation time and a set of constants derived from the analysis this paper investigates the validity and extent of the formulation to capture shocks and resolve contact discontinuity and expansion waves in one and two dimensional riemann problems the simulations are carried out using the same two dimensional nine velocity lattice and identical set of constants and relaxation time they are compared with theoretical results and those obtained by solving the euler equations directly using harten s first order numerical scheme good agreement is obtained for all test cases however the modified equilibrium distribution function is not suitable for shock structure simulation for that an exact recovery of the navier stokes equations is required

viscous regularization of the euler equations and entropy principles this paper investigates a general class of viscous regularizations of the compressible euler equations a unique regularization is identified that is compatible with all the generalized entropies la harten et al siam j numer anal 35 1998 pp 2117 2127 and satisfies the minimum entropy principle a connection with a recently proposed phenomenological model by h brenner phys a 370 2006 pp 190 224 is made