three dimensional supersonic viscous flow over a cone at incidence abstract the viscous supersonic flow over a sharp cone at incidence is examined numerically with a coupled strongly implicit algorithm for the properties in the plane normal to the cone axis the navier stokes equations are considered in a boundary layer like or parabolized manner and global relaxation is considered for the pressure interaction it is shown that departure effects can be effectively eliminated by forward differencing for the axial pressure gradient moreover this approximation retains the implicit free pressure interaction required for geometries where axial flow separation is possible author

positivity and lower and upper solutions for fourth order boundary value problems abstract this paper is devoted to the study the boundary value problem u 4 t f t u t for all t i 0 1 u 0 u 1 u 0 u 1 0 we prove the existence of at least one two or three solutions in the presence of a pair of not necessarily ordered lower and upper solutions the proof follows from maximum principles related to the operator u 4 m u and amann s three solutions theorem

constant sign solutions of two point fourth order problems in this paper we characterize the sign of the green s function related to the fourth order linear operator u 4 mu coupled with the two point boundary conditions u 1 u 0 u 0 u 0 0 we obtain the exact values on the real parameter m for which the related green s function is negative in 0 1 i 0 1 such property is equivalent to the fact that the operator satisfies a maximum principle in the space of functions that fulfil the homogeneous boundary conditions when m 0 the best estimate follows from spectral theory when m 0 we obtain an estimation by studying the disconjugacy properties of the solutions of the homogeneous equation u 4 mu 0 the optimal value is attained by studying the exact expression of the green s function such study allow us to ensure that there is no real parameter m for which the green s function is positive on 0 1 i 0 1 moreover we obtain maximum principles of this operator when the solutions verify suitable non homogeneous boundary conditions we apply the obtained results by means of the method of lower and upper solutions to nonlinear problems coupled with these boundary conditions

the weber seifert dodecahedral space is non haken in this paper we settle thurston s old question of whether the weber seifert dodecahedral space is non haken a problem that has been a benchmark for progress in computational 3 manifold topology over recent decades we resolve this question by combining recent significant advances in normal surface enumeration new heuristic pruning techniques and a new theoretical test that extends the seminal work of jaco and oertel

the relation between flux vector splitting and parabolized schemes abstract the relationship between pns and thin layer navier stokes algorithms is used to develop traditional as well as new pns procedures the use of characteristics based flux vector splitting gives rise to a parabolized system that is based on the predominant physics of the flow while pressure gradient based flux vector splitting is shown to give the traditional parabolized scheme of vigneron comparisons with tlns results show the characteristics based pns system gives results that are at least as accurate as the more traditional pressure gradient split pns system the use of a safety factor in the pressure gradient splitting is shown to cause inaccuracies and should be avoided the interpretation of pns procedures as the first sweep of a tlns adi procedure also suggests an obvious global pressure iteration method that is mathematically well posed and hence leads to an efficient rapidly converging global iteration procedure

pl equivariant surgery and invariant decompositions of 3 manifolds using normal surface theory h j2 we introduce the notion of least weight normal surfaces the weight of a normal surface is a nonnegative integer invariant of the normal isotopy class of the surface if we focus on a particular class of normal surfaces and choose representatives which minimize the weight over the class then we have least weight normal surfaces it is remarkable how these least weight normal surfaces exhibit many of the same useful properties as least area minimal surfaces they provide a piecewise linear pl environment to obtain the recent topological results coming from the analysis and geometry of least area surfaces in an impressive series of papers meeks and yau m y i m y z m y 3 m y m y meeks simon and yau m s y scott s meeks and scott m s and freedman hass and scott f h s introduced the analysis and geometry of least area surfaces into the study of topological questions about 3 manifolds the consequences have led to the resolution of many outstanding problems in the topology of 3 manifolds since most of the problems being solved with these new techniques are topological in nature it has been felt that there should be a more topological approach to the proofs of these results besides the bulk of the work in the theory of

calculation of supersonic flows with strong viscous inviscid interaction a procedure for the calculation of strong viscous inviscid interactions in two dimension al laminar super sonic flows with and without separation is described the equations solved are the so called parabolized navier stokes equations the streamwise pressure gradient term is written as a combination of a forward and a backward difference to provide a path for upstream propagation of information global iteration is utilized to repeatedly update the pressure field from an initial guess until convergence is achieved the numerical scheme employed is a new alternating direction explicit ade procedure which is used as an alternative to the more difficult to program multigrid strategy to accelerate convergence results are presented for flows past two flat plate related bodies he calculation of supersonic flows in which rapid changes in geometry or boundary condition occur often requires techniques other than classical prandtl boundary layer theory or other solution methods suitable for weakly interacting flows the reason for this is that at high reynolds number rapid changes in geometry or boundary condition induce large changes in the properties of the bound ary layer over a very short length scale when this occurs the phenomenon of upstream influence wherein information is propagated in the upstream direction becomes important this type of flow is referred to as a strongly interacting flow methods that treat such flows as initial value problems are not suitable because they are boundary value in nature in the predominant flow direction in this paper a procedure for two dimensional supersonic strongly interacting flows is developed which utilizes the parabolized or reduced navier stokes equations we will employ the former term here for simplicity it will be as sumed that x and y are nearly streamwise and stream normal directions respectively with special treatment of the pressure gradient term dp dx in the streamwise momentum equation one can develop a procedure which is well suited for calculating weakly interacting flows 1 4 in order to calculate strongly interacting flows a means for propagating information in the upstream direction is required this can be satisfied by utilizing a forward difference for the stream wise pressure gradient in subsonic flows where upstream influence is important even if the viscous inviscid interaction is weak a forward difference of px has been shown by rubin5 to allow the calculation of such flows lin and rubin6 later successfully applied forward differencing of px to a supersonic flow with weak interaction in that study forward differencing was utilized throughout the flowfield including the supersonic flow in the present technique for ward differencing of px is applied only in the subsonic por

z 2 thurston norm and complexity of 3 manifolds a new lower bound on the complexity of a 3 manifold is given using the mathbb z 2 thurston norm this bound is shown to be sharp and the minimal triangulations realising it are characterised using normal surfaces consisting entirely of quadrilateral discs

positive solutions of fourth order problems with clamped beam boundary conditions abstract in this paper we make an exhaustive study of the fourth order linear operator u 4 m u coupled with the clamped beam conditions u 0 u 1 u 0 u 1 0 we obtain the exact values on the real parameter m for which this operator satisfies an anti maximum principle such a property is equivalent to the fact that the related green s function is nonnegative in 0 1 0 1 when m 0 we obtain the best estimate by means of the spectral theory and for m 0 we attain the optimal value by studying the oscillation properties of the solutions of the homogeneous equation u 4 m u 0 by using the method of lower and upper solutions we deduce the existence of solutions for nonlinear problems coupled with this boundary conditions

inflations of ideal triangulations abstract starting with an ideal triangulation of m the interior of a compact 3 manifold m with boundary no component of which is a 2 sphere we provide a construction called an inflation of the ideal triangulation to obtain a strongly related triangulation of m itself besides a step by step algorithm for such a construction we provide examples of an inflation of the two tetrahedra ideal triangulation of the complement of the figure eight knot in s 3 giving a minimal triangulation having ten tetrahedra of the figure eight knot exterior as another example we provide an inflation of the one tetrahedron gieseking manifold giving a minimal triangulation having seven tetrahedra of a non orientable compact 3 manifold with klein bottle boundary several applications of inflations are discussed

a monotone method for constructing extremal solutions to fourth order periodic boundary value problems in this paper we describe a constructive method which yields two monotone sequences that converge uniformly to extremal solutions to the periodic boundary value problemu 4 t f t u t u t email protected 0 email protected u 0 u email protected u 0 u email protected u 0 u email protected u 0 u email protected in the presence of an upper solution b and a lower solution a with b a

positive solutions of nonlinear fourth order boundary value problems with local and non local boundary conditions we establish new existence results for multiple positive solutions of fourth order nonlinear equations which model deflections of an elastic beam we consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non local boundary conditions with a unified approach our method is to show that each boundary value problem can be written as the same type of perturbed integral equation in the space c 0 1 involving a linear functional u but although we seek positive solutions the functional is not assumed to be positive for all positive u the results are new even for the classic boundary conditions of clamped or hinged ends when u 0 because we obtain sharp results for the existence of one positive solution for multiple solutions we seek optimal values of some of the constants that occur in the theory which allows us to impose weaker assumptions on the nonlinear term than in previous works our non local boundary conditions contain multi point problems as special cases and for the first time in fourth order problems we allow coefficients of both signs

optimizing the double description method for normal surface enumeration many key algorithms in 3 manifold topology involve the enumeration of normal surfaces which is based upon the double description method for finding the vertices of a convex polytope typically we are only interested in a small subset of these vertices many key algorithms in 3 manifold topology involve the enumeration of normal surfaces which is based upon the double description method for finding the vertices of a convex polytope typically we are only interested in a small subset of these vertices thus opening the way for substantial optimization here we give an account of the vertex enumeration problem as it applies to normal surfaces and present new optimizations that yield strong improvements in both running time and memory consumption the resulting algorithms are tested using the freely available software package regina

increasing the number of fibered faces of arithmetic hyperbolic 3 manifolds we exhibit a closed hyperbolic 3 manifold which satisfies a very strong form of thurston s virtual fibration conjecture in particular this manifold has finite covers which fiber over the circle in arbitrarily many ways more precisely it has a tower of finite covers where the number of fibered faces of the thurston norm ball goes to infinity in fact faster than any power of the logarithm of the degree of the cover and we give a more precise quantitative lower bound the example manifold m is arithmetic and the proof uses detailed number theoretic information at the level of the hecke eigenvalues to drive a geometric argument based on fried s dynamical characterization of the fibered faces the origin of the basic fibration m s 1 is the modular elliptic curve e x 0 49 which admits multiplication by the ring of integers of q 7 we first base change the holomorphic differential on e to a cusp form on gl 2 over k q 3 and then transfer over to a quaternion algebra d k ramified only at the primes above 7 the fundamental group of m is a quotient of the principal congruence subgroup of o d of level 7 to analyze the topological properties of m we use a new practical method for computing the thurston norm which is of independent interest we also give a noncompact finite volume hyperbolic 3 manifold with the same properties by using a direct topological argument

0 efficient triangulations of 3 manifolds efficient triangulations of 3 manifolds are defined and studied it is shown that any triangulation of a closed orientable irreducible 3 manifold m can be modified to a 0 efficient triangulation orm can be shown to be one of the manifolds s 3 rp 3 or l 3 1 similarly any triangulation of a com pact orientable irreducible irreducible 3 manifold can be modified to a 0 efficient triangulation the notion of a 0 efficient ideal tr is de fined it is shown if m is a compact orientable irreducible irreducible 3 manifold having no essential annuli and distinct from the 3 cell then m admits an ideal triangulation furthermore it is shown that any ideal trian gulation of such a 3 manifold can be modified to a 0 efficient ideal triangula tion a 0 efficient triangulation of a closed manifold has only one vertex or the manifold is s 3 and the triangulation has precisely two vertices 0 efficient tri angulations of 3 manifolds with boundary and distinct from the 3 cell have all their vertices in the boundary and then just one vertex in each bound ary component as tools we introduce the concepts of barrier surface and shrinking as well as the notion of crushing a triangulation along a normal sur face a number of applications are given including an algorithm to construct an irreducible decomposition of a closed orientable 3 manifold an algorithm to construct a maximal collection of pairwise disjoint normal 2 spheres in a closed 3 manifold an alternate algorithm for the 3 sphere recognition prob lem results on edges of low valence in minimal triangulations of 3 manifolds and a construction of irreducible knots in closed 3 manifolds

embeddability in the 3 sphere is decidable we show that the following algorithmic problem is decidable given a 2 dimensional simplicial complex can it be embedded topologically or equivalently piecewise linearly in r3 by a known reduction it suffices to decide the embeddability of a given triangulated 3 manifold x into the 3 sphere s3 the main step which allows us to simplify x and recurse is in proving that if x can be embedded in s3 then there is also an embedding in which x has a short meridian that is an essential curve in the boundary of x bounding a disk in s3 b x with length bounded by a computable function of the number of tetrahedra of x

volume optimization normal surfaces and thurston s equation on triangulated 3 manifolds we propose a finite dimensional variational principle on triangulated 3 manifolds so that its critical points are related to solutions to thurston s gluing equation and haken s normal surface equation the action functional is the volume

computing trisections of 4 manifolds we describe an algorithm to compute trisections of orientable four manifolds using arbitrary triangulations as input this results in explicit complexity bounds for the trisection genus of a 4 manifold in terms of the number of pentachora 4 simplices in a triangulation

time iterative solutions of viscous supersonic nozzle flows supersonic viscous flows in high expansion ratio rocket nozzles are studied numerically using the thin layer navier stokes equations a discretization scheme using upwind flux vector split differencing in the streamwise direction and central differencing in the cross stream direction is chosen of the three implicit approximate factorization schemes considered the diagonally dominant alternating direction and the pafabolized alternating direction methods are found to be much faster than standard alternating direction procedures the optimum courant numbers for these methods are about 500 and produce convergence rates that rival direct methods proper downstream boundary conditions for the subsonic portion of the outflow are shown to allow control over the exit boundary layer thickness including recirculating flows with re entry conditions at the exit plane representative nozzle flowflelds are shown for conical and contoured nozzles for different reynolds numbers wall temperatures and back pressure and for both laminar and turbulent flows excellent global mass conservation is demonstrated when the strongly conservative equations are used but the quasiconservative equations lead to conservation errors that are unacceptably large the computed results for a typical high reynolds number flow through the high expansion nozzle are compared to the method of characteristics moc prediction with good agreement

angle structures and normal surfaces let m be the interior of a compact 3 manifold with boundary and let t be an ideal triangulation of m this paper describes necessary and sufficient conditions for the existence of angle structures semi angle structures and generalised angle structures on m t respectively in terms of a generalised euler characteristic function on the solution space of the normal surface theory of m t this extends previous work of kang and rubinstein and is itself generalised to a more general setting for 3 dimensional pseudo manifolds

iterative space marching method for compressible sub trans and supersonic flows anewiterativefully coupledimplicitspace marching method isproposedforsolving thetwo dimensionalsteady eulerequationsforcompressiblee owsatmach numbers ranging from subsonic to supersonic aspecial treatment of the streamwise pressure gradient component permits us to calculate both supersonic e ow regions where the euler equations are hyperbolic and subsonic regions where the equations reveal elliptic properties to take into account the elliptic effects of subsonic and transonic e ows space marching sweeps are carried out iteratively a new parabolic pressure correction procedure is developed to accelerate the convergence rate this procedure can be applied for subsonic and transonic regimes and is consistent with the characteristic analysis of the euler equations ateach marching station a newton iterativetechniqueisused tosolvethenonlinearsystem ofequations in a fully coupled manner to resolve strong shocks and contact discontinuities as well as smooth e owe elds with high accuracy implicit symmetric second order total variation diminishing and upwind second order richardson schemes are employed to approximate the transverse and streamwise derivatives respectively the method is tested on theproblem ofthee owin a ductwith a circulararcbump fordifferentmach numberregimes numerical calculations show that the method is accurate is robust and can efe ciently be applied for calculating subsonic transonic and supersonic e ows without streamwise separation

the method of lower and upper solutions for second third fourth and higher order boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u n t t u t u i a u i b i r i 0 n 1 where is a caratheodory function we obtain necessary and sufficient conditions in to guarantee the existence of solutions between a lower solution and an upper solution for n 2 if n 3 either or and n 4 if furthermore we obtain sufficient conditions in for n 2 k 6 when