on spectral multigrid methods for the time dependent navier stokes equations a new splitting scheme is proposed for the numerical solution of the time dependent incompressible navier stokes equations by spectral methods a staggered grid is used for the pressure improved intermediate boundary conditions are employed in the split step for the velocity and spectral multigrid techniques are used for the solution of the implicit equations

a spectral algorithm for the stokes problem in vorticity vector potential formulation and cylindrical geometry abstract in the numerical computation of incompressible flows using the navier stokes equations the main task generally lies in the solution of a stokes type problem in this paper we focus on this key point when using i a spectral method ii the vorticity vector potential formulation of the 3d navier stokes equations and iii a cylindrical geometry at the end we conclude with a numerical test of our stokes solver in the framework of rayleigh benard natural convection

semilocal generic formal fibers abstract let t be a complete local ring and c a finite set of incomparable prime ideals of t we find necessary and sufficient conditions for t to be the completion of an integral domain whose generic formal fiber is semilocal with maximal ideals the elements of c in addition if the characteristic of t is zero we give necessary and sufficient conditions for t to be the completion of an excellent integral domain whose generic formal fiber is semilocal with maximal ideals the elements of c

analysis of the kleiser schumann method the kleiser schumann algorithm for the approximation of the stokes problem by fourier legendre polynomials is analized stability when the degree of the polynomials increases is established whereas error estimates in sobolev spaces are proven

mean field theory for usp 4 sime so 5 the simple lie algebra usp 4 so 5 is a ten dimensional algebra that contains the angular momentum algebra su 2 in nuclear structure physics the algebra so 5 describes beta rigid collective modes in the bohr mottelson and interacting boson models the so 5 dual space consists of density matrices which are defined by the expectations of so 5 generators a coadjoint orbit is a common level surface in the dual space of the two so 5 casimirs this paper develops mean field theory on any coadjoint orbit of so 5 densities when the densities of a set of quantum states lie on one orbit the system is said to have a weak dynamical symmetry a lax pair determines the dynamics of so 5 densities on each coadjoint orbit analytic solutions are reported for rotating so 5 densities in equilibrium for a particular energy function

mean field theory for u n dynamical groups algebraic mean field theory amft is a many body physics modeling tool which firstly is a generalization of hartree fock mean field theory and secondly an application of the orbit method from lie representation theory the amft ansatz is that the physical system enjoys a dynamical group which may be either a strong or a weak dynamical lie group g when g is a strong dynamical group the quantum states are by definition vectors in one irreducible unitary representation irrep space and amft is equivalent to the kirillov orbit method for deducing properties of a representation from a direct geometrical analysis of the associated integral co adjoint orbit amft can be the only tractable method for analyzing some complex many body systems when the dimension of the irrep space of the strong dynamical group is very large or infinite when g is a weak dynamical group the quantum states are not vectors in one irrep space but amft applies if the densities of the states lie on one non integral co adjoint orbit the computational simplicity of amft is the same for both strong and weak dynamical groups this paper formulates amft explicitly for unitary lie algebras and applies the general method to the lipkin meshkov glick 2 model and the elliott 3 model when the energy in the 3 theory is a rotational scalar function marsden weinstein reduction simplifies amft dynamics to a two dimensional phase space

an efficient method for the solution of the incompressible navier stokes equations in cylindrical geometries the article presents a fast pseudo spectral navier stokes solver for cylindrical geometries which is shown to possess exponential rate of decay of the error the formulation overcomes the issues related to the axis singularity by employing in the radial direction a special set of collocation points together with standard chebyshev polynomials a multi domain technique with patching interfaces yields significant improvements in the conditioning of the algebraic problems arising from the discretization procedure and allows for an enhanced near wall resolution of wall bounded shear flows the elliptic kernel enjoys the efficiency of an analytic expansion of the harmonic extension the method is tested by computing the formation of taylor vortices in a rotating couette flow for both axisymmetric and non axisymmetric configurations a direct numerical simulation of a turbulent pipe flow at moderate reynolds number demonstrates the effectiveness of the method in as much as the axis singularity is concerned results compare well with reference experimental and numerical data

solutions to a time dependent su 3 mean field hamiltonian the time dependent su 3 mean field equations are solved for a particular energy function relevant to nuclear structure the model energy in the su 3 enveloping algebra is the sum of two terms which are the squared length of the angular momentum vector and the cubic rotational scalar x3 tr lql the mean field solutions for this energy have constant intrinsic quadrupole moments q in the three dimensional space of all angular momentum components in the rotating principal axis frame a trajectory is defined by the intersection of a sphere and a hyperboloid this conclusion is similar to the classical rigid rotor for which a solution is the intersection of a sphere and the inertia ellipsoid

three dimensional fully spectral numerical method for mantle convection with depth dependent properties a semi implicit fully spectral collocation method for the simulation of three dimensional mantle convection with depth dependent thermodynamic and transport properties is presented the variable property navier stokes equation expressed in terms of the primitive variable velocity and pressure is solved with the mass continuity and temperature equations the periodic horizontal boundary conditions allow a fourier expansion for the two horizontal directions the stress free impermeable isothermal boundary conditions along with the depth dependent coefficients are handled with a chebyshev expansion in the vertical direction in the limit of an infinite prandtl number appropriate to mantle convection the inertial terms in the momentum equation are unimportant in this case an explicit solution of a poisson equation for pressure can be avoided instead a fourth order equation for vertical velocity can be solved simultaneous imposition of both impermeable and continuity boundary conditions during the vertical velocity evaluation is discussed the pressure distributions on the top and bottom bounding planes were determined by means of an influence matrix technique the numerical method employed here avoids time splitting errors and enforces velocity boundary conditions and continuity over the entire domain including the boundaries to machine accuracy strongy time dependent three dimensional solutions up to a surface rayleigh number of 1 107 have been obtained strong upwellings pulsating chaotically are formed by the collective merging of cylindrical plumes

a formulation of navier stokes problem in cylindrical coordinates applied to the 3d entry jet in a duct a pseudospectral formulation of the three dimensional navier stokes equations in the cylindrical system of co ordinates is presented which automatically includes the regularity conditions at the polar axis for the fourier harmonics the mathematical system of equations is numerically implemented using standard methods for spatial discretisation and time advancement the method presented here for incompressible flows can be easily extended to different set of equations and to different numerical schemes of various levels of accuracy the formulation is applied to the solution of incompressible three dimensional entry jets in a circular duct

fibers of flat morphisms and weierstrass preparation theorem abstract we characterize flat extensions of commutative rings satisfying the weierstrass preparation theorem using this characterization we prove a variant of the weierstrass preparation theorem for rings of functions on a normal curve over a complete local domain of dimension one this generalizes recent works of harbater hartmann and krashen with a different method of proof

mean field theory of the general collective motion algebra gcm 3 the lie algebra gcm 3 is a 15 dimensional semidirect sum that contains the algebra of the general linear group gl 3 r the six dimensional abelian ideal of gcm 3 is generated by the mass quadrupole and monopole tensors the bohr mottelson quantum model of nuclear rotational and vibrational states is a particular irreducible representation of gcm 3 the gcm 3 dual space consists of density matrices which are defined by the expectations of the gcm 3 generators a coadjoint orbit is a level surface in the dual space of the gcm 3 casimir which is the squared length of the conserved kelvin circulation vector this paper develops mean field theory on any coadjoint orbit of gcm 3 densities a lax pair determines the dynamics of gcm 3 densities on each coadjoint orbit this lax equation is equivalent to the riemann ellipsoid equations of motion pacs number 21 60 fw

implementation of arbitrary inner product in the global galerkin method for incompressible navier stokes equations the global galerkin or weighted residuals method applied to the incompressible navier stokes equations is considered the basis functions are assumed to be divergence free and satisfy all the boundary conditions the method is formulated for an arbitrary inner product so that the pressure cannot be eliminated by galerkin projections on a divergence free basis a proposed straightforward procedure for the elimination of the pressure reduces the problem to an ode system without algebraic constraints to illustrate the applicability and the robustness of the numerical approach and to show that numerical solutions with unit and non unit weight functions yield similar results the driving lid cavity and natural convection benchmark problems are solved using the unit and chebyshev weight functions further implications of the proposed galerkin formulation are discussed

applications of patching to quadratic forms and central simple algebras this paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields in particular we use a patching approach to reprove and generalize a recent result of parimala and suresh in preprint arxiv 0708 3128 2007 on the u invariant of p adic function fields p 2 the strategy relies on a local global principle for homogeneous spaces for rational algebraic groups combined with local computations

on arithmetic macaulayfication of noetherian rings the rees algebra is the homogeneous coordinate ring of a blowing up the present paper gives a necessary and sufficient condition for a noetherian local ring to have a cohen macaulay rees algebra a noetherian local ring has a cohen macaulay rees algebra if and only if it is unmixed and all the formal fibers of it are cohen macaulay as a consequence of it we characterize a homomorphic image of a cohen macaulay local ring for non local rings this paper gives only a sufficient condition by using it however we obtain the affirmative answer to sharp s conjecture that is a noetherian ring having a dualizing complex is a homomorphic image of a finite dimensional gorenstein ring

local rings with zero dimensional formal fibers abstract we study noetherian local rings whose all formal fibers are of dimension zero universal catenarity and going up property of the canonical map to the completion are considered we present several characterizations of these rings including a characterization of weierstrass preparation type a characterization of local rings with going up property by a strong form of lichtenbaum hartshorne theorem is obtained as an application we give an upper bound for dimension of formal fibers of a large class of algebras over these rings

divergence free velocity fields in nonperiodic geometries the influence matrix method of enforcing incompressibility in pseudospectral simulations of fluid dynamics as described by kleiser and schumann for channel flow is generalized to other geometries a formalism of projection and matrix operators is introduced in which the influence matrix method is shown to be an application of the classic sherman morrison woodbury formula of numerical linear algebra special attention is paid to the tau correction applications to cartesian geometries illustrate the concepts and highlight the role of symmetry a coded implementation in a cylindrical geometry requiring special treatment of coordinate singularities is used to investigate properties of the influence matrix and to provide estimates of timings

su 3 mean field hamiltonian the su 3 mean field approximation describes collective nuclear rotation in a density matrix formalism the densities q i l 2 are 3 3 hermitian matrices in the su 3 dual space where q is the expectation of the quadrupole moment and l is the expectation of the angular momentum the mean field approximation restricts these densities to a level surface of the su 3 casimirs each level surface is a coadjoint orbit of the canonical transformation group su 3 for each density the su 3 mean field hamiltonian h is an element of the su 3 lie algebra a model su 3 energy functional and the symplectic structure on the coadjoint orbit determine uniquely the su 3 mean field hamiltonian the densities in time dependent su 3 mean field theory obey the dynamical equation i h on a coadjoint orbit the cranked mean field hamiltonian is h h i where is the angular velocity of the rotating principal axis frame a rotating equilibrium density in the body fixed frame is a self consistent solution to the equation h 0

mean field theory for su 4 so 6 algebraic mean field theory constructs a group theoretical model of quantum systems that have a weak dynamical symmetry but may break dynamical symmetry the strong defining condition for dynamical symmetry is that states belong to one irreducible representation space weak dynamical symmetry demands that the densities corresponding to the states have a constant value for each casimir quantum phase transitions and other complex systems exhibit weak dynamical symmetry furthermore mean field theory often yields analytic formulae for expectations and energy spectra that are not feasible in representation theory this paper develops mean field theory on any coadjoint orbit of su 4 densities the simple lie algebra su 4 so 6 is a 15 dimensional algebra that contains the subalgebra usp 4 so 5 and the angular momentum algebra su 2 the su 4 dual space consists of density matrices which are defined by the expectations of the su 4 generators a coadjoint orbit is a common level surface in the dual space of the three su 4 casimirs a lax pair determines the dynamics of these densities on each coadjoint orbit analytic solutions are reported for rotating su 4 densities in equilibrium for a particular energy function

computation of natural convection in two dimensional cavities with chebyshev polynomials abstract a semi implicit spectral method for the computation of buoyancy induced two dimensional flow of a viscous incompressible fluid is presented the incompressible navier stokes equations expressed in terms of the primitive variables velocity and pressure and an additional equation for the temperature are considered all the variables are expanded in double truncated series of chebyshev polynomials the pressure distribution at the boundary of the computational domain is determined by means of an influence matrix technique in order to satisfy the incompressibility condition everywhere in the field the algorithm is applied to the classical problem of natural convection of a boussinesq fluid within a differentially heated vertical square cavity results show that for a prandtl number of 0 71 corresponding to air and up to rayleigh numbers of 106 very accurate solutions are obtained with a spatial resolution of 33 polynomials in each direction new interesting phenomena are shown for higher values of the rayleigh number

how can we speed up matrix multiplication due to the new algebraic methods of algorithm design recently it became possible to perform multiplication and inversion of n times n matrices using o n 2 496 rather than o n 3 arithmetical operations consequently algorithms for several other computational problems of linear algebra and combinatorics have been accelerated the major ideas and techniques that have led to that progress are surveyed

spectral method solution of the stokes equations on nonstaggered grids abstract the stokes equations are solved using spectral methods with staggered and nonstaggered grids numerous ways to avoid the problem of spurious pressure modes are presented including new techniques using the pseudospectral method and a method solving the weak form of the governing equations a variation on the spectral element method developed by patera the pseudospectral methods using nonstaggered grids are simpler to implement and have comparable or better accuracy than the staggered grid formulations three test cases are presented a formulation with an exact solution a formulation with homogeneous boundary conditions and the driven cavity problem the solution accuracy is shown to be greatly improved for the driven cavity problem when the analytical solution of the singular flow behavior in the upper corners is separated from the computational solution

a spectral collocation method for the navier stokes equations abstract a fourier chebyshev spectral method for the incompressible navier stokes equations is described it is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time in this fully spectral algorithm a preconditioned iterative technique is used for solving the implicit equations arising from semi implicit treatment of pressure mean advection and vertical diffusion terms the algorithm is tested by applying it to hydrodynamic stability problems in channel flow and in external boundary layers with both constant and variable viscosity

a spectral numerical method for the navier stokes equations with applications to taylor couette flow a new spectral method for solving the incompressible navier stokes equations in a plane channel and between concentric cylinders is presented the method uses spectral expansions which inherently satisfy the boundary conditions and the continuity equation and yield banded matrices which are efficiently solved at each time step in addition the number of dependent variables is reduced resulting in a reduction in computer memory requirements several test problems have been computed for the channel flow and for flow between concentric cylinders including taylor couette flow with axisymmetric taylor vortices and wavy vortices in all cases agreement with available experimental and theoretical results is very good

poloidal toroidal decomposition in a finite cylinder the navier stokes equations in a finite cylinder are written in terms of poloidal and toroidal potentials in order to impose incompressibility regularity of the solutions is ensured in several ways first the potentials are represented using a spectral basis which is analytic at the cylindrical axis second the non physical discontinuous boundary conditions at the cylindrical corners are smoothed using a polynomial approximation to a steep exponential profile third the nonlinear term is evaluated in such a way as to eliminate singularities the resulting pseudo spectral code is tested using exact polynomial solutions and the spectral convergence of the coefficients is demonstrated our solutions are shown to agree with exact polynomial solutions and with previous calculations of axisymmetric vortex breakdown and of onset of non axisymmetric helical spirals parallelization by azimuthal wavenumber is shown to be highly effective

integral conditions for the pressure in the computation of incompressible viscous flows abstract the problem of finding the correct conditions for the pressure in the time discretized navier stokes equations when the incompressibility constraint is replaced by a poisson equation for the pressure is critically examined it is shown that the pressure conditions required in a nonfractional step scheme to formulate the problem as a system of split second order equations are of an integral character and similar to the previously discovered integral conditions for the vorticity the novel integral conditions for the pressure are used to derive a finite element method which is very similar to that developed by glowinski and pironneau and is the finite element counterpart of the influence matrix method of kleiser and schumann

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

sp 3 r coadjoint orbit theory the algebraic mean field method is applied to the symplectic lie algebra sp 3 r that describes geometrical collective states in atomic nuclei the expectations of sp 3 r generators define a symplectic density matrix the mean field approximation restricts the densities to a manifold that is a coadjoint orbit of the transformation group sp 3 r and a level surface of the symplectic casimir functions compared to representation theory mean field theory is technically simpler but yields similar predictions for physical properties of collective states the critical points of the energy functional on a coadjoint orbit surface define rotational bands the deformation kinetic energy and kelvin circulation of principal axis symplectic rotors are determined as a function of the angular momentum illustrative applications of coadjoint orbit theory are reported for the yrast rotational bands of a light 20ne and a heavy 166er deformed isotope

spectral approximation of the periodic nonperiodic navier stokes equations in order to approximate the navier stokes equations with periodic boundary conditions in two directions and a no slip boundary condition in the third direction by spectral methods we justify by theoretical arguments an appropriate choice of discrete spaces for the velocity and the pressure the compatibility between these two spaces is checked via an infsup condition we analyze a spectral and a collocation pseudo spectral method for the stokes problem and a collocation pseudo spectral method for the navier stokes equations we derive error bounds of spectral type i e which behave likem wherem depends on the number of degrees of freedom of the method and represents the regularity of the data

spurious modes in spectral collocation methods with two non periodic directions abstract collocation implementation of the kleiser schumann s method in geometries with two non periodic directions is shown to suffer from three spurious modes line column and checkerboard contaminating the computed pressure field the corner spurious modes are also present but they do not affect evaluation of pressure related quantities a simple methodology in the inversion of the influence matrix will efficiently filter out these spurious modes spectral methods spurious modes

an adaptive pseudo spectral method for reaction diffusion problems the spectral interpolation error was considered for both the chebyshev pseudo spectral and galerkin approximations a family of functionals i sub r u with the property that the maximum norm of the error is bounded by i sub r u j sub r where r is an integer and j is the degree of the polynomial approximation was developed these functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize i sub r u the number of collocation points is then chosen to maintain a prescribed error bound the method is illustrated by various examples from combustion problems in one and two dimensions