computation of steady and unsteady vortex dominated flows with shock waves the unsteady euler equations have been derived in the conservation form for the flow relative motion with respect to a rotating frame of reference the resulting equations are solved by using a central difference finite volume scheme with four state runge kutta time stepping for steady flow applications local time stepping is used and for unsteady applications the minimum global time stepping is used a three dimensional fully vectorized computer program has been developed and applied to steady and unsteady maneuvering delta wings the capability of the three dimensional program has been demonstrated for a rigid sharp edged delta wing undergoing uniform rolling in a conical flow and rolling oscillations in a locally conical flow

a family of symplectic integrators stability accuracy and molecular dynamics applications the following integration methods for special second order ordinary differential equations are studied leapfrog implicit midpoint trapezoid stormer verlet and cowell numerov we show that all are members or equivalent to members of a one parameter family of schemes some methods have more than one common form and we discuss a systematic enumeration of these forms we also present a stability and accuracy analysis based on the idea of modified equations and a proof of symplecticness it follows that cowell numerov and lim2 a method proposed by zhang and schlick are symplectic a different interpretation of the values used by these integrators leads to higher accuracy and better energy conservation hence we suggest that the straightforward analysis of energy conservation is misleading

symplectic surgery and the spinc dirac operator abstract letgbe a compact connected lie group and m a compact hamiltoniang space with moment mapj m g under the assumption that these data are pre quantizable one can construct an associated spinc dirac operator formula whose equivariant index yields a virtual representation ofg we prove a conjecture of guillemin and sternberg that if 0 is a regular value ofj the multiplicityn 0 of the trivial representation in the index space formula is equal to the index of the spinc dirac operator for the symplectic quotientm0 j 1 0 g this generalizes previous results for the case thatg tis abelian i e a torus

computation of vortex dominated flow for a delta wing undergoing pitching oscillation the conservative unsteady euler equations for the flow relative to a moving frame of reference are used to solve for the three dimensional steady and unsteady flows around a sharp edged delta wing the resulting equations are solved by using an implicit approximately factored finite volume scheme implicit second order and explicit second and fourth order dissipations are added to the scheme the boundary conditions are explicitly satisfied the grid is generated by locally using a modified joukowski transformation in crossflow planes at the grid chord stations the computational applications cover a steady flow around a delta wing whose results serve as the initial conditions for the unsteady flow around a pitching delta wing at a large mean angle of attack the steady results are compared with the experimental data and the unsteady results are compared with results of a flux difference splitting scheme

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

symplectic and poisson geometry on b manifolds abstract let m 2 n be a poisson manifold with poisson bivector field we say that m is b poisson if the map n m 2 n t m intersects the zero section transversally on a codimension one submanifold z m this paper will be a systematic investigation of such poisson manifolds in particular we will study in detail the structure of m in the neighborhood of z and using symplectic techniques define topological invariants which determine the structure up to isomorphism we also investigate a variant of de rham theory for these manifolds and its connection with poisson cohomology

influence of numerical dissipation on computational euler equations for vortex dominated flows etude d ecoulements supersoniques tourbillonnaires pour des ailes delta a aretes vives ou arrondies on developpe un schema de volumes finis

quantization of presymplectic manifolds and circle actions we prove several versions of quantization commutes with reduction for circle actions on manifolds that are not symplectic instead these manifolds possess a weaker structure such as a spinc structure our theorems work whenever the quantization data and the reduction data are compatible this condition always holds if we start from a presymplectic in particular symplectic manifold

eigenvalue distributions of reduced density matrices given a random quantum state of multiple distinguishable or indistinguishable particles we provide an effective method rooted in symplectic geometry to compute the joint probability distribution of the eigenvalues of its one body reduced density matrices as a corollary by taking the distribution s support which is a convex moment polytope we recover a complete solution to the one body quantum marginal problem we obtain the probability distribution by reducing to the corresponding distribution of diagonal entries i e to the quantitative version of a classical marginal problem which is then determined algorithmically this reduction applies more generally to symplectic geometry relating invariant measures for the coadjoint action of a compact lie group to their projections onto a cartan subalgebra and can also be quantized to provide an efficient algorithm for computing bounded height kronecker and plethysm coefficients

total pressure loss in vortical solutions of the conical euler equations a technique for the solution of the conically self similar form of the euler equations is described solutions for the flow past a flat plate delta wing at angle of attack are presented these solutions show strong leading edge vortices with large total pressure losses in the cores a study of the effects of various computational parameters on the total pressure loss is made an explanation for the cause of the total pressure loss is presented it is shown to be consistent with the results for both a quasi one dimensional model problem and the conically self similar flow past the flat plate delta wing

euler and navier stokes solutions for flow over a conical delta wing the simulation of vortical flow about a conical delta wing 14 1 elliptic cone by the euler equations is studied navier stokes solutions are generated for comparison with the inviscid results at the flight condition considered the viscous solutions reveal a large primary leading edge separation vortex and a smaller secondary vortex the viscous solutions are shown to be grid resolved on a coarse grid the euler solutions result in a primary separation vortex but no secondary vortex a comparison of pressure coefficient with the viscous solution shows surprising agreement the agreement is unfortunately fortuitous when the euler equations are solved on the fine viscous grid the large separation vortex is eliminated instead a cross flow shock appears shock curvature introduces sufficient vorticity to produce a small vortex downstream of the shock this inviscid result fundamentally different from the viscous case is a valid solution to the euler equations the coarse grid euler solution is characterized by the production of spurious vorticity and entropy due to numerical error at the leading edge it is neither a valid solution to the euler equations nor a reliable approximation to the real viscous flow the use of a kutta condition is investigated for both the fineand coarse grid euler solutions it is shown that the kutta condition fails to produce a grid independent simulation of the actual viscous flow

heckman kostant and steinberg formulas for symplectic manifolds abstract let m be a compact symplectic manifold on which a compact connected lie group k acts in a hamiltonian fashion in part i we derive a formula for j m which generalizes the heckman formula for co adjoint orbits here j is the moment map associated with the action of k on m and m the liouville measure on m in part ii we derive a quantum analogue of this formula which extends to the symplectic setting classical multiplicity formulas of kostant and steinberg

practical construction of modified hamiltonians one of the most fruitful ways to analyze the effects of discretization error in the numerical solution of a system of differential equations is to examine the modified equations which are equations that are exactly satisfied by the approximate discrete solution these do not actually exist in general but rather are defined by an asymptotic expansion in powers of the discretization parameter nonetheless if the expansion is suitably truncated the resulting modified equations have a solution which is remarkably close to the discrete solution in the case of a hamiltonian system of ordinary differential equations the modified equations are also hamiltonian if and only if the integrator is symplectic evidence for the existence of a hamiltonian for a particular calculation is obtained by calculating modified hamiltonians and monitoring how well they are conserved also energy drifts caused by numerical instability are better revealed by evaluating modified hamiltonians doing this calculation would normally be complicated and highly dependent on the details of the method even if differences are used to approximate derivatives a relatively simple procedure is presented here nearly independent of the internal structure of the integrator for obtaining highly accurate estimates for modified hamiltonians as a bonus of the method of construction the modified hamiltonians are exactly conserved by a numerical solution in the case of a quadratic hamiltonian

on geometric quantization of b symplectic manifolds abstract we study a notion of pre quantization for b symplectic manifolds we use it to construct a formal geometric quantization of b symplectic manifolds equipped with hamiltonian torus actions with nonzero modular weight we show that these quantizations are finite dimensional t modules

unsteady transonic airfoil computation using implicit euler scheme on body fixed grid the unsteady euler equations have been derived for the flow relative motion with respect to a frame of reference that is rigidly attached to the moving airfoil the resulting equations preserve the conservation form the grid is generated once by an elliptic solver without any need for dynamic grid computation an implicit approximately factored finite volume scheme has been developed and implemented through a fully vectorized computer program the scheme is based on the spatial approximate factorization of beam and warming implicit second order and explicit second and fourth order dissipations are added to the scheme the boundary conditions are explicitly satisfied the scheme is applied to steady and unsteady transonic rigid airfoil flows for forced harmonic airfoil motions periodic solutions are achieved within the third cycle of oscillation the results are in good agreement with the experimental data sonic flows past a naca 0015 airfoil at a constant pitch rate 18 the latter case is the same application considered in ref 16 the numerical solution has been obtained by using the flux vector splitting and the flux difference splitting meth ods extended for dynamic meshes 19 in ref 18 it is noted that different turbulence models give different normal force and pitching moment coefficients numerical solutions of the unsteady euler equations are less expensive than those of the unsteady navier stokes equations for the unsteady transonic flow the unsteady euler equations adequately model most of the real flow features with the exception of viscous effects whenever they are substantial the euler equations model shock waves and their motion entropy increase across shocks and entropy gradient and vorticity production and convection behind shocks as can be seen from crocco s theorem and the inviscid vorticity transport equa tion recently successful time accurate solutions of the un steady euler equations have been presented for pitching air foils19 and wings in transonic flows 20 and for the rolling oscillation of delta wings a vortex dominated flow problem in a locally conical supersonic flows 21 to the best of bur knowledge ref 21 is the first work done for unsteady vortex dominated flows with shock waves which is directly applica ble to maneuvering delta wings in this paper we present an implicit approximately factored finite volume scheme for the time accurate numerical solution of the unsteady euler equations of the flow relative motion with respect to an airfoil fixed frame of reference the scheme is applied to steady and unsteady transonic flows around a naca 0012 airfoil for the unsteady flows the airfoil is in pitching oscillation about a small and moderate mean angle of attack at a large amplitude the computational results are compared with the experimental data

on geometric quantization of b m symplectic manifolds we study the formal geometric quantization of b m symplectic manifolds equipped with hamiltonian actions of a torus t with nonzero leading modular weight the resulting virtual t modules are finite dimensional when m is odd as in 4 when m is even these virtual modules are not finite dimensional and we compute the asymptotics of the representations for large weight

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

torus fibrations and localization of index ii local index for acyclic compatible system we give a framework of localization for the index of a dirac type operator on an open manifold suppose the open manifold has a compact subset whose complement is covered by a family of finitely many open subsets each of which has a structure of the total space of a torus bundle under an acyclic condition we define the index of the dirac type operator by using the witten type deformation and show that the index has several properties such as excision property and a product formula in particular we show that the index is localized on the compact set

unsteady navier stokes computations past oscillating delta wing at high incidence the unsteady compressible thin layer navier stokes equations written in the moving frame of reference for the flow relative motion are solved for the unsteady supersonic flow around a round edged delta wing for supersonic flow a local conical flow solution has been obtained from the three dimensional equations pseudotime stepping is used for the steady flow problem while time accurate stepping is used for the unsteady flow problem the computational scheme is an implicit approximately factored finite volume scheme which uses implicit and explicit dissipation terms the scheme is verified for the steady flow solution the scheme is then applied to a delta wing undergoing rolling oscillation at a reduced frequency of 1 337 with 15 deg maximum amplitude about a mean angle of attack of 10 deg for a mach number of 2 and a reynolds number of 0 5 x 10 6

an analytic proof of the geometric quantization conjecture of guillemin sternberg we present a direct analytic approach to the guillemin sternberg conjecture gs that geometric quantization commutes with symplectic reduction which was proved recently by meinrenken m1 m2 and vergne v1 v2 et al besides providing a new proof of this conjecture our methods also lead immediately to further extensions in various contexts

the moduli space of fano manifolds with kahler ricci solitons abstract we construct a canonical hausdorff complex analytic moduli space of fano manifolds with kahler ricci solitons this enlarges the moduli space of fano manifolds with kahler einstein metrics we discover a moment map picture for kahler ricci solitons and give complex analytic charts on the topological space consisting of kahler ricci solitons by studying differential geometric aspects of this moment map some stacky words and arguments on gromov hausdorff convergence help to glue them together in the holomorphic manner

long time step methods for oscillatory differential equations considered are numerical integration schemes for nondissipative dynamical systems in which multiple time scales are present it is assumed that one can do an explicit separation of the rhs forces into fast forces and slow forces such that i the fast forces contain the high frequency part of the solution ii the fast forces are conservative and iii the reduced problem consisting only of the fast forces can be integrated much more cheaply than the full problem the fast forces are allowed to have low frequency components particular applications for which the schemes are intended include n body problems for which most of the forces are slow and nonlinear wave phenomena for which the fast forces can be propagated by spectral methods the assumption of cheap integration of fast forces implies that the overall cost of integration is primarily determined by the step size used to sample the slow forces a long time step method is one in which this step size exceeds half the period of the fastest normal mode present in the full system an existing method that comes close to qualifying is the impulse method also known as verlet i and r respa it is shown that it might fail though for a couple of reasons first it suffers a serious loss of accuracy if the step size is near a multiple of the period of a normal mode and second it is unstable if the step size is near a multiple of half the period of a normal mode proposed in this paper is a mollified impulse method having an error bound that is independent of the frequency of the fast forces it is also shown to possess superior stability properties theoretical results are supplemented by numerical experiments the method is efficient and reasonably easy to implement

a vector partition function for the multiplicities of slkc abstract we use gelfand tsetlin diagrams to write down the weight multiplicity function for the lie algebra sl k c type ak 1 as a single partition function this allows us to apply known results about partition functions to derive interesting properties of the weight diagrams we relate this description to that of the duistermaat heckman measure from symplectic geometry which gives a large scale limit way to look at multiplicity diagrams we also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors using symplectic geometry we prove that the partition of the permutahedron into domains of polynomiality of the duistermaat heckman function is the same as that for the weight multiplicity function and give an elementary proof of this for sl 4 c a3

multiplicities formula for geometric quantization part ii 1 introduction let p be a compact manifold let h be a compact lie group acting on the right on p we assume that the stabilizer of each element y e p is a finite subgroup of h the space m p h is an orbifold and every orbifold can be presented this way if h acts freely then m is a manifold if ae graded space of h invariant solutions of the h horizontal dirac operator on p twisted by the line bundle let m o 9 be the moment map for the g action assume 0 is a regular value of let mred be the reduced orbifold of m that is mred o g consider the reduced orbifold line bundle rd lu lto g on m oa in the ease where both g andh are torus we prove here the formula

explicit symplectic integrators using hessian vector products in 1991 rowlands proposed an effectively fourth order effectively two stage explicit symplectic integrator based on using a hessian vector product to modify the force evaluation in the leapfrog method and evidence indicates that for modest accuracy this method is highly competitive here we explore the possible existence of even more efficient fourth order explicit symplectic integrators also based on the use of hessian vector products and the concept of effective order first it is shown that the cost of a force evaluation plus a hessian vector product is less than twice the cost of the force alone for a sum of two body interactions then a new method is found that is generally better than both the method of rowlands and that of calvo according to both a theoretical measure of the error and limited numerical experiments the basic motivation behind the new method is quite simple do a hessian vector computation only every other step significantly cutting costs while only marginally increasing the error the idea of effective order means that we allow for both the possibility of preprocessing the initial values before application of the basic method and the possibility of postprocessing the values obtained by the basic method but at output points only for some applications processing is unnecessary but in any case processing has been shown to be possible at low additional cost the derivation of the new method illustrates how to simplify by the use of h series the determination of parameters for methods of increased effective accuracy