a current density conservative scheme for incompressible mhd flows at a low magnetic reynolds number part i on a rectangular collocated grid system a consistent conservative and accurate scheme has been designed to calculate the current density and the lorentz force by solving the electrical potential equation for magnetohydrodynamics mhd at low magnetic reynolds numbers and high hartmann numbers on a finite volume structured collocated grid in this collocated grid velocity u pressure p and electrical potential f are located in the grid center while current fluxes are located on the cell faces the calculation of current fluxes on the cell faces is conducted using a conservative scheme which is consistent with the discretization scheme for the solution of electrical potential poisson equation a conservative interpolation is used to get the current density at the cell center which is used to conduct the calculation of lorentz force at the cell center for momentum equations we will show that both conservative and consistent are important properties of the scheme to get an accurate result for high hartmann number mhd flows with a strongly non uniform mesh employed to resolve the hartmann layers and side layers of hunt s conductive walls and shercliff s insulated walls a general second order projection method has been developed for the incompressible navier stokes equations with the lorentz force included this projection method can accurately balance the pressure term and the lorentz force for a fully developed core flow this method can also simplify the pressure boundary conditions for mhd flows

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

the topological trace formula the topological trace formula is a computation of the lefschetz number of a hecke correspondence c acting on the weighted cohomology groups defined in ghm of a locally symmetric space x it expresses this lefschetz number as a sum of contributions from fixed point components of c on the reductive borel serre compactifi cation of x the proof uses the lefschetz fixed point formula of gm2

hodge cohomology of gravitational instantons we study the space of l2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type these metrics generalize the geometric structures at infinity of several different well known classes of metrics including asymptotically locally euclidean manifolds the known types of gravitational instantons and also poincare metrics on q rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in kahler manifolds the answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold the l2 signature formula implied by our result is closely related to the one proved by dai 25 and more generally by vaillant 67 and identifies dai s invariant directly in terms of intersection cohomology of differing perversities this work is also closely related to a recent paper of carron 12 and the forthcoming paper of cheeger and dai 17 we apply our results to a number of examples gravitational instantons among them arising in predictions about l2 harmonic forms in duality theories in string theory

large eddy simulation in complex geometric configurations using boundary body forces anumericalmethodispresentedthatallowslargeeddysimulation les ofturbulente owsincomplexgeometric cone gurations with moving boundaries and that retains the advantages of solving the navier stokes equations on e xed orthogonal grids the boundary conditions are applied independently of the grid by assigning body forces over surfaces that need not coincide with coordinate lines the use of orthogonal nondeforming grids simplie es grid generation facilitates theimplementation of high order nondissipativediscretization schemes andminimizes the spatial and temporal variations in e lter width that complicate unstructured deforming grid les dynamic subgrid scaleturbulence models areparticularly appealing in combination with the body forceprocedure because the dynamic model accounts automatically for the presence of solid walls without requiring damping functions the method is validated by simulations of the turbulent e ow in a motored axisymmetric piston cylinder assembly for which detailed experimental measurements are available computed mean and rms velocity proe les show very good agreement with measured ensemble averages thepresent numerical code runs on small personal computerlike workstations for a comparable level of accuracy computational requirements memory and cpu time are at least a factor of 10 lower compared to published simulations for the same cone guration obtained using an unstructured boundary e tted deforming grid approach

an immersed boundary method with formal second order accuracy and reduced numerical viscosity a formally second order accurate immersed boundary method is presented and tested in this paper we apply this new scheme to simulate the flow past a circular cylinder and study the effect of numerical viscosity on the accuracy of the computation by comparing the numerical results with those of a first order method the numerical evidence shows that the new scheme has less numerical viscosity and is therefore a better choice for the simulation of high reynolds number flows with immersed boundaries

on the m function and borg marchenko theorems for vector valued sturm liouville equations we will consider a vector valued sturm liouville equation of the form r u pu qu lambdawu xis an element of 0 b with p 1 w qis an element ofl loc 1 0 b mxm being hermitian and under some additional conditions on p 1 and w we give an elementary deduction of the leading order term asymptotics for the titchmarsh weyl m function corresponding to this equation in the special case of p w i qis an element ofl 1 0 infinity mxm and the neumann boundary conditions at 0 we will also prove that m 1 root lambda i r i r 1 where r lim n infinity r n sigma n 1 infinity q n for recursively defined sequences r n and q n if qis an element ofl loc 1 0 b mxm 0 bless than or equal toinfinity the same formula is valid with an exponentially small error for large lambda it is clear that expansions of this type are helpful in finding representatives of the kdv invariants for p w i we prove that the spectral measure corresponding to the equation r u lambdau uniquely determines q as well as b and the boundary conditions at 0 and b we finally give a new proof of a local form of the borg marchenko theorem cf gesztesy and simon on local borg marchenko uniqueness results commun math phys 211 273 287 2000 chap 3 a theorem which is due to simon see simon a new approach to inverse spectral theory i fundamental formalism ann math 150 1 29 1999 in the scalar case for applications to physics it is worth mentioning that vector valued sturm liouville equations appear in some problems in magneto hydro dynamics

an immersed boundary finite volume method for simulations of flow in complex geometries abstract a new immersed boundary method for simulating flows over or inside complex geometries is developed by introducing a mass source sink as well as a momentum forcing the present method is based on a finite volume approach on a staggered mesh together with a fractional step method both momentum forcing and mass source are applied on the body surface or inside the body to satisfy the no slip boundary condition on the immersed boundary and also to satisfy the continuity for the cell containing the immersed boundary in the immersed boundary method the choice of an accurate interpolation scheme satisfying the no slip condition on the immersed boundary is important because the grid lines generally do not coincide with the immersed boundary therefore a stable second order interpolation scheme for evaluating the momentum forcing on the body surface or inside the body is proposed three different flow problems decaying vortices and flows over a cylinder and a sphere are simulated using the immersed boundary method proposed in this study and the results agree very well with previous numerical and experimental results verifying the accuracy of the present method

a consistent and conservative scheme for mhd flows with complex boundaries on an unstructured cartesian adaptive system the numerical simulation of magnetohydrodynamics mhd flows with complex boundaries has been a topic of great interest in the development of a fusion reactor blanket for the difficulty to accurately simulate the hartmann layers and side layers along arbitrary geometries an adaptive version of a consistent and conservative scheme has been developed for simulating the mhd flows besides the present study forms the first attempt to apply the cut cell approach for irregular wall bounded mhd flows which is more flexible and conveniently implemented under adaptive mesh refinement amr technique it employs a volume of fluid vof approach to represent the fluid conducting wall interface that makes it possible to solve the fluid solid coupling magnetic problems emphasizing at how electric field solver is implemented when conductivity is discontinuous in cut cell for the irregular cut cells the conservative interpolation technique is applied to calculate the lorentz force at cell center on the other hand it will be shown how consistent and conservative scheme is implemented on fine coarse mesh boundaries when using amr technique then the applied numerical schemes are validated by five test simulations and excellent agreement was obtained for all the cases considered simultaneously showed good consistency and conservative properties

immersed boundary method for the mhd flows of liquid metals wall bounded magnetohydrodynamic mhd hereafter flows are of great theoretical and practical interest even for laminar cases mhd simulations are associated with very high computational cost due to the resolution requirements for the hartmann and side layers developing in the presence of solid obstacles in the presence of turbulence these difficulties are further compounded thus mhd simulations in complex geometries are currently a challenge the immersed boundary ib hereafter method is a reliable numerical tool for efficient hydrodynamic field simulations in arbitrarily geometries but it has not yet been extended for mhd simulations the present study forms the first attempt to apply the ib methodology for the computation of both the hydrodynamic and mhd fields a consistent numerical methodology is presented that is appropriate for efficient 3d mhd simulations in geometrically complicated domains using cartesian flow solvers for that purpose a projection scheme for the electric current density is presented based on an electric potential correction algorithm a suitable forcing scheme for electric density currents in the vicinity of non conducting immersed surfaces is also proposed the proposed methodology has been first extensively tested for hartmann layers in fully developed and developing channel and duct flows at hartmann numbers ha 500 2000 in order to demonstrate the potential of the method the three dimensional mhd flow around a circular cylinder at reynolds number re 200 is also presented the effects of grid resolution and variable arrangement on the simulation accuracy and consistency were examined when compared with existing numerical or analytic solutions excellent agreement was found for all the cases considered the proposed projection and forcing schemes for current densities were found capable of satisfying the charge conservation law in the presence of immersed non conducting boundaries finally we show how the proposed methodology can be used to extend the applicability of existing flow solvers that use the ib concept with a staggered variable arrangement

a general reconstruction algorithm for simulating flows with complex 3d immersed boundaries on cartesian grids in the present note a general reconstruction algorithm for simulating incompressible flows with complex immersed boundaries on cartesian grids is presented in the proposed method an arbitrary three dimensional solid surface immersed in the fluid is discretized using an unstructured triangular mesh and all the cartesian grid nodes near the interface are identified then the solution at these nodes is reconstructed via linear interpolation along the local normal to the body in a way that the desired boundary conditions for both pressure and velocity fields are enforced the overall accuracy of the resulting solver is second order as it is demonstrated in two test cases involving laminar flow past a sphere

discrete series characters and the lefschetz formula for hecke operators this paper consists of three independent but related parts in the rst part xx1 6 we give a combinatorial formula for the constants appearing in the numerators of characters of stable discrete series representations of real groups see x3 as well as an analogous formula for individual discrete series representations see x6 moreover we give an explicit formula theorem 5 1 for certain stable virtual characters on real groups by theorem 5 2 these include the stable discrete series characters and thus we recover the results of x3 in a more natural way in the second part x7 we use the character formula given in theorem 5 1 to rewrite the lefschetz formula of gm for the local contribution at a single xed point component to the trace of a hecke operator on weighted cohomology in the same spirit as that of arthur s lefschetz formula a in terms of stable virtual characters on real groups see theorem 7 14 b we then sum the contributions of the various xed point components and show that in the case of middle weighted cohomology the resulting global lefschetz xed point formula agrees with arthur s lefschetz formula this gives a topological proof of arthur s formula

an embedded boundary formulation for large eddy simulation of turbulent flows interacting with moving boundaries a non boundary conforming formulation for simulating complex turbulent flows with dynamically moving boundaries on fixed cartesian grids is proposed the underlying finite difference solver for the filtered incompressible navier stokes equations is based on a second order fractional step method on a staggered grid to satisfy the boundary conditions on an arbitrary immersed interface the velocity field at the grid points near the interface is reconstructed using momentum forcing without smearing the sharp interface the concept of field extension is also introduced to treat the points emerging from a moving solid body to the fluid laminar flow cases and large eddy simulations les are presented to demonstrate the formal accuracy and range of applicability of the method in particular simulations of laminar flow induced by the harmonic in line oscillation of a circular cylinder in quiescent fluid and from a transversely oscillating cylinder in a free stream are presented and compared to reference simulations and experiments les of turbulent flow over a traveling wavy wall and transitional flow through a bileaflet prosthetic heart valve are also shown all results are in very good agreement with reference results in the literature

a ghost cell immersed boundary method for flow in complex geometry an efficient ghost cell immersed boundary method gcibm for simulating turbulent flows in complex geometries is presented a boundary condition is enforced through a ghost cell method the reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second order accuracy of the base solver both dirichlet and neumann boundary conditions can be treated the current ghost cell treatment is both suitable for staggered and non staggered cartesian grids the accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface numerical results are compared with experimental data and boundary fitted grid results the method is further extended to an existing ocean model mitgcm to simulate geophysical flow over a three dimensional bump the method is easily implemented as evidenced by our use of several existing codes

l2 cohomology of arithmetic varieties abstract the l2 cohomology of arithmetic quotients of bounded symmetric domains is studied we establish the conjecture of zucker equating the l2 cohomology of these spaces to the intersection cohomology of their baily borel compactifications

invariant submanifold flows given a lie group acting on a manifold our aim is to analyze the evolution of differential invariants under invariant submanifold flows the constructions are based on the equivariant method of moving frames and the induced invariant variational bicomplex applications to integrable soliton dynamics and to the evolution of differential invariant signatures used in equivalence problems and object recognition and symmetry detection in images are discussed

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

combined immersed boundary finite difference methods for three dimensional complex flow simulations a second order accurate highly efficient method is developed for simulating unsteady three dimensional incompressible flows in complex geometries this is achieved by using boundary body forces that allow the imposition of the boundary conditions on a given surface not coinciding with the computational grid the governing equations therefore can be discretized and solved on a regular mesh thus retaining the advantages and the efficiency of the standard solution procedures two different forcings are tested showing that while the quality of the results is essentially the same in both cases the efficiency of the calculation strongly depends on the particular expression a major issue is the interpolation of the forcing over the grid that determines the accuracy of the scheme this ranges from zeroth order for the most commonly used interpolations up to second order for an ad hoc velocity interpolation the present scheme has been used to simulate several flows whose results have been validated by experiments and other results available in the literature finally in the last example we show the flow inside an ic piston cylinder assembly at high reynolds number to our knowledge this is the first example in which the immersed boundary technique is applied to a full three dimensional complex flow with moving boundaries and with a reynolds number high enough to require a subgrid scale turbulence model

local contribution to the lefschetz fixed point formula for a class of self correspondencesc calledweakly hyperbolic we give a computable formula for the contribution of a fixed point component to the lefschetz number ofc the formula applies to lefschetz numbers of cohomology with coefficients in a constructible complex of sheaves such as intersection homology

a consistent and conservative immersed boundary method for mhd flows and moving boundary problems abstract a consistent and conservative immersed boundary method has been developed to accurately and efficiently solve two topics magnetohydrodynamics flows with a complex boundary and moving boundary problems based on a least square interpolation reconstruction method a unified form of boundary condition is described to handle a single domain situation while a mixed boundary condition is used for coupled conditions of multi domain in order to achieve the consistency with a desired wall boundary condition a so called approximation correction procedure is done and an imposing neumann boundary condition is discussed with two kinds of different schemes besides a consistent and conservative scheme is implemented to satisfy both the charge and mass conservation laws on cells around the immersed surface then a conservative interpolation is reconstructed for velocity in moving boundary problems at last the applied numerical method is validated by stationary and moving boundary cases and an excellent agreement is obtained with good accuracy efficiency and conservation especially the consistent and conservative immersed boundary method can obtain almost the same accurate results as those from the cut cell technique seo and mittal 2011 14 for a moving boundary problem by reducing the spurious pressure

monotone scheme and boundary conditions for finite volume simulation of magnetohydrodynamic internal flows at high hartmann number a monotone scheme for finite volume simulation of magnetohydrodynamic internal flows at high hartmann number is presented the numerical stability is analysed with respect to the electromagnetic force standard central finite differences applied to finite volumes can only be numerically stable if the vector products involved in this force are computed with a scheme using a fully staggered grid the electromagnetic quantities electric currents and electric potential must be shifted by half the grid size from the mechanical ones velocity and pressure an integral treatment of the boundary layers is used in conjunction with boundary conditions for electrically conducting walls the simulations are performed with inhomogeneous electrical conductivities of the walls and reach high hartmann numbers in three dimensional simulations even though a non adaptive grid is used

weighted cohomology of arithmetic groups let g be a semisimple algebraic group defined over the rational numbers k a maximal compact subgroup ofg g r and g q a neat arithmetic subgroup letx g k be the locally symmetric space associated to and e the local system on x constructed out of a finite dimensional irreducible algebraic representation e of g fix a maximally q split torus s in g s is assumed to be nontrivial so that x is noncompact let a s r 0 and by a slight abuse of notation let x a denote the group of rational characters of s choose a minimal rational parabolic subgroup p0 s the choice determines a notion of positivity in x a and a set of positive roots among the roots of s in g let 0 be the half sum of the positive roots weighted cohomology is an invariant of introduced by m goresky g harder and r macpherson 12 in the study 15 16 12 14 of the trace of hecke operators in the cohomology of for each p x a q there are weight profiles p and p upper and lower p see 1 5 and groups w ph i e and w h i e the notion of profile here differs from but is equivalent to that of 12 see remark 1 2 below if p q 2 0 then the profiles p and q are dual and the corresponding groups are in poincare duality for very positive p one gets the compactly supported cohomology h i c x e and dually for very negative p the full cohomology h i x e the definition of the weighted cohomology groups is somewhat involved so i will attempt to give some idea of the motivation behind it detailed definitions are recalled in section 1 associated to p and p are complexes of sheaves wpc e and wpc e on the reductive borel serre compactification x of x let p be a rational parabolic subgroup np its unipotent radical sp the split centre of p np and xp the p boundary component in x the idea behind the construction in 12 is as follows the stalk cohomology of the direct image ri e by i x x at x xp is

the l2 lefschetz numbers of hecke operators introduction 5 1 hecke operators and l2 cohomology 52 the spectral decomposition of cohomology 6 3 application of the trace formula 54 the functions pa t 5 5 proof of theorem 5 1 6 the main formula appendix bibliography