high order compact computation and nonuniform grids for streamfunction vorticity equations abstract fourth order compact difference schemes for steady streamfunction vorticity formulation of 2d incompressible navier stokes equations on nonuniform grids are derived using maple software package especially we deduce fourth order compact difference scheme of the first partial derivative terms in order to resolve boundary layers grid transformation techniques are used which maps a nonuniform grid onto a uniform one for use with the fourth order compact difference scheme a krylov subspace iterative method with an ilut preconditioning technique is employed to solve the resulting linear system the proposed high accuracy computation method is applied to two model problems computational results are compared with results computed by other schemes in the literature

multiplicative functions on the lattice of non crossing partitions and free convolution the lattice of non crossing partitions was introduced by kreweras kre in 1972 and examined further by poupard pou and edelman edel ede2 in the last time there has been an increased interest in this lattice on one hand from the purely combinatorial point of view siu eds sire bss and on the other hand from a quantum probabilistic point of view gss spel spe2 spe3 we became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures the free convolution introduced by voiculescu voil voi2 voi3 this connection between the free convolution and the lattice of non crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions we shall work out this connection in the last part of this work the intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other motivated by voiculescu s formula for the r series of free convolution we examine multiplicative functions on the lattice of non crossing partitions and prove our main theorem which is much in the spirit of voiculescu s formula we shall then use this main theorem for deriving some known results on non crossing partitions in a unified way this part of our work is purely combinatorial the free convolution serves only as a motivation for our theorem in the last section we shall switch to the probabilistie side and show how the lattice of non crossing partitions determines the structure of the free convolution voiculescu s formula will then follow as an easy corollary of our main theorem in this respect we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools

asymmetric non linear forced vibrations of free edge circular plates part 1 theory in this article a detailed study of the forced asymmetric non linear vibrations of circular plates with a free edge is presented the dynamic analogue of the von karman equations is used to establish the governing equations the plate displacement at a given point is expanded on the linear natural modes the forcing is harmonic with a frequency close to the natural frequency omega n of one asymmetric mode of the plate thus the vibration is governed by the two degenerated modes corresponding to omega n which are one to one internally resonant an approximate analytical solution using the method of multiple scales is presented attention is focused on the case where one configuration which is not directly excited by the load gets energy through non linear coupling with the other configuration slight imperfections of the plate are taken into account experimental validations are presented in the second part of this paper

a new family of 5 5 cc 4oc schemes applicable for unsteady navier stokes equations in this paper a new family of implicit compact finite difference schemes for computation of unsteady convection diffusion equation with variable convection coefficient is proposed the schemes which are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter are then applied to unsteady navier stokes system the proposed schemes where transport variable and its first derivatives are carried as the unknowns combine virtues of compact discretization and pade scheme for spatial derivative these schemes which are based on a five point stencil with constant coefficients named as 5 5 constant coefficient 4th order compact 5 5 cc 4oc give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods these schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable they efficiently capture both transient and steady solutions of linear and nonlinear convection diffusion equations with dirichlet as well as neumann boundary conditions subsequently the proposed schemes are applied to problems governed by the incompressible navier stokes equations the results obtained are in excellent agreement with analytical and available numerical results in all cases establishing efficiency and accuracy of the proposed schemes

non linear vibrations of free edge thin spherical shells modal interaction rules and 1 1 2 internal resonance this paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement the analog for thin shallow shells of von karman s theory for large deflection of plates is used the validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution the specific case of a free edge is considered the governing partial differential equations are expanded onto the natural modes of vibration of the shell the problem is replaced by an infinite set of coupled second order differential equations with quadratic and cubic non linear terms analytical expressions of the non linear coefficients are derived and a number of them are found to vanish as a consequence of the symmetry of revolution of the structure then for all the possible internal resonances a number of rules are deduced thus predicting the activation of the energy exchanges between the involved modes finally a specific mode coupling due to a 1 1 2 internal resonance between two companion modes and an axisymmetric mode is studied 2004 elsevier ltd all rights reserved

nonlinear normal modes in an intrinsic theory of anisotropic beams nonlinear normal modes in nonlinear oscillations of beams are derived from intrinsic equations that is using velocities and strains as primary degrees of freedom displacements and rotations are thus not system states but are instead obtained using the propagation of the local beam material reference frames as in rigid body dynamics it is shown that the intrinsic variables sufce to describe the free vibrations of the beam the approach does not need assumptions in the material properties i e it is valid for general anisotropic behavior or the beam kinematics i e it is based on cosserat s exact geometrical description of the deformable curve furthermore the nonlinear modal equations in intrinsic coordinates are obtained from integrals involving only products of the mode shapes and known coecients using this

reduced order models for large amplitude vibrations of shells including in plane inertia non linear normal modes nnms are used in order to derive reduced order models for large amplitude geometrically non linear vibrations of thin shells the main objective of the paper is to compare the accuracy of different truncations using linear and non linear modes in order to predict the response of shells structures subjected to harmonic excitation for an exhaustive comparison three different shell problems have been selected i a doubly curved shallow shell simply supported on a rectangular base ii a circular cylindrical panel with simply supported in plane free edges and iii a simply supported closed circular cylindrical shell in each case the models are derived by using refined shell theories for expressing the strain displacement relationship as a consequence in plane inertia is retained in the formulation reduction to one or two nnms shows perfect results for vibration amplitude lower or equal to the thickness of the shell in the three cases and this limitation is extended to two times the thickness for two of the selected models 2008 elsevier b v all rights reserved

a hierarchy of plate models derived from nonlinear elasticity by gamma convergence we derive a hierarchy of plate models from three dimensional nonlinear elasticity by convergence what distinguishes the different limit models is the scaling of the elastic energy per unit volume h where h is the thickness of the plate this is in turn related to the strength of the applied force h membrane theory derived earlier by le dret and raoult corresponds to 0 nonlinear bending theory to 2 von karman theory to 3 4 and linearized vk theory to 3 intermediate values of lead to certain theories with constraints a key ingredient in the proof is a generalization to higher derivatives of our rigidity result 29 which states that for maps v 0 1 3 3 the l2 distance of v from a single rotation is bounded by a multiple of the l2 distance from the set so 3 of all rotations

essentially compact schemes for unsteady viscous incompressible flows a new fourth order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced the scheme is based on the vorticity stream function formulation it is essentially compact and has the nice features of a compact scheme with regard to the treatment of boundary conditions it is also very efficient at every time step or runge kutta stage only two poisson like equations have to be solved the poisson like equations are amenable to standard fast poisson solvers usually designed for second order schemes detailed comparison with the second order scheme shows the clear superiority of this new fourth order scheme in resolving both the boundary layers and the gross features of the flow this efficient fourth order scheme also made it possible to compute the driven cavity flow at reynolds number 106on a 10242grid at a reasonable cost fourth order convergence is proved under mild regularity requirements this is the first such result to our knowledge

an example of a generalized brownian motion we present an example of a generalized brownian motion it is given by creation and annihilation operators on a twisted fock space ofl2 these operators fulfill for a fixed 1 1 the relationsc f c g c g c f f g 1 f g l2 we show that the distribution of these operators with respect to the vacuum expectation is a generalized gaussian distribution in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula we also indicate that our brownian motion is one component of ann dimensional brownian motion which is invariant under the quantum groups u n of woronowicz with v2

analysis of a fourth order compact scheme for convection diffusion in 1984 gupta et al introduced a compact fourth order finite difference convection diffusion operator with some very favorable properties in particular this scheme does not seem to suffer excessively from spurious oscillatory behavior and it converges with standard methods such as gauss seidel or sor hence multigrid regardless of the diffusion this scheme has been rederived developed including some variations and applied in both convection diffusion and navier stokes equations by several authors accurate solutions to high reynolds number flow problems at relatively coarse resolutions have been reported these solutions were often compared to those obtained by lower order discretizations such as second order central differences and first order upstream discretizations the latter it was stated achieved far less accurate results due to the artificial viscosity which the compact scheme did not include we show here that while the compact scheme indeed does not suffer from a cross stream artificial viscosity as does the first order upstream scheme when the characteristic direction is not aligned with the grid it does include a streamwise artificial viscosity that is inversely proportional to the natural viscosity this term is not always benign 7 refs 1 fig 1 tab

from the classical to the generalized von karman and marguerre von karman equations in this work we describe and analyze two models that were recently proposed for modeling generalized von karman plates and generalized marguerre von karman shallow shells first we briefly review the classical von karman and marguerre von karman equations their physical meaning and their mathematical justification we then consider the more general situation where only a portion of the lateral face of a nonlinearly elastic plate or shallow shell is subjected to boundary conditions of von karman type while the remaining portion is free using techniques from formal asymptotic analysis we obtain in each case a two dimensional boundary value problem that is analogous to but is more general than the classical equations in particular it is remarkable that the boundary conditions for the airy function can still be determined on the entire boundary of the nonlinearly elastic plate or shallow shell solely from the data following recent joint works we then reduce these more general equations to a single cubic operator equation which generalizes an equation introduced by berger and fife and whose sole unknown is the vertical displacement of the shell we next adapt an elegant compactness method due to lions for establishing the existence of a solution to this operator equation

high accuracy solutions of incompressible navier stokes equations abstract in recent years we have developed high accuracy finite difference approximations for partial differential equations of elliptic type with particular emphasis on the convection diffusion equation these approximations are of compact type have a local truncation error of fourth order and allow the use of standard iterative schemes to solve the resulting systems of algebraic equations in this paper we extend these high accuracy approximations to the solution of navier stokes equations solutions are obtained for the model problem of driven cavity and are compared with solutions obtained using other approximations and those obtained by other authors it is discovered that the high order approximations do indeed produce high accuracy solutions and have a potential for use in solving important problems of viscous fluid flows

stochastic calculus with respect to free brownian motion and analysis on wigner space we define stochastic integrals with respect to free brownian motion and show that they satisfy burkholder gundy type inequalities in operator norm we prove also a version of ito s predictable representation theorem as well as product form and functional form of ito s formula finally we develop stochastic analysis on the free fock space in analogy with stochastic analysis on the wiener space

high order difference methods for heat equations in polar cylindrical coordinates abstract fourth order difference methods for the solution of poisson equations in cylindrical polar coordinates are proposed the same technique is then applied to obtain o k 2 h 4 two level unconditionally stable adi methods for the solution of the heat equation in two dimensional polar coordinates and three dimensional cylindrical coordinates numerical examples given here show that the methods developed here retain their order and accuracy everywhere including the region in the vicinity of the singularity r 0

connected lie groups and property rd for a locally compact group property rd gives a control on the convolution norm of any compactly supported measure in terms of the l 2 norm of its density and the diameter of its support we give a complete classification of those lie groups with property rd

q gaussian processes non commutative and classical aspects we examine for 1 q 1 q gaussian processes i e families of operators non commutative random variables where the a t fulfill the q commutation relations for some covariance function equipped with the vacuum expectation state we show that there is a q analogue of the gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q gaussian processes into corresponding and much simpler questions in the underlying hilbert space in particular we use this idea to show that a large class of q gaussian processes possesses a non commutative kind of markov property which ensures that there exist classical versions of these non commutative processes this answers an old question of frisch and bourret fb

resolvents of r diagonal operators we consider the resolvent a 1 of any diagonal operator a in a ii 1 factor our main theorem theorem 1 1 gives a universal asymptotic formula for the norm of such a resolvent en route to its proof we calculate the transform of the operator c 2 where c is voiculescu s circular operator and we give an asymptotic formula for the negative moments of a 2 for any diagonal a we use a mixture of complex analytic and combinatorial techniques each giving finer information where the other can give only coarse detail in particular we introduce partition structure diagrams in section 4 a new combinatorial structure arising in free probability

high accuracy iterative solution of convection diffusion equation with boundary layers on nonuniform grids abstract a fourth order compact finite difference scheme and a multigrid method are employed to solve the two dimensional convection diffusion equations with boundary layers the computational domain is first discretized on a nonuniform stretched grid to resolve the boundary layers a grid transformation technique is used to map the nonuniform grid to a uniform one the fourth order compact scheme is applied to the transformed uniform grid a multigrid method is used to solve the resulting linear system numerical experiments are used to show that a graded mesh and a grid transformation are necessary to compute high accuracy solutions for the convection diffusion problems with boundary layers and dicretized by the fourth order compact scheme

lattice boltzmann method on curvilinear coordinates system using an interpolation based strategy the lattice boltzmann method is extended to apply to general curvilinear coordinate systems as an example a cylindrical coordinate system is used to simulate two dimensional flow around a circular cylinder numerical simulations are carried out for impulsive initial conditions with reynolds numbers up to 104 the agreement of our results with previous computational and experimental results is satisfactory compared with previous lattice boltzmann simulations of the same problem our new approach greatly enhances the computational efficiency

commutators of free random variables let a be a unital c algebra given together with a specified state a c consider two selfadjoint elements a b of a which are free with respect to in the sense of the free probability theory of voiculescu let us denote c i ab ba where the i in front of the commutator is introduced to make c selfadjoint in this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b some properties of the corresponding operation on probability measures are also discussed the methods we use are combinatorial based on the description of freeness in terms of non crossing partitions an important ingredient is the notion of r diagonal pair introduced and studied in our previous paper 12

a fourth order compact adi method for solving two dimensional unsteady convection diffusion problems n this article an exponential high order compact ehoc alternating direction implicit adi method in which the crank nicolson scheme is used for the time discretization and an exponential fourth order compact difference formula for the steady state 1d convection diffusion problem is used for the spatial discretization is presented for the solution of the unsteady 2d convection diffusion problems the method is temporally second order accurate and spatially fourth order accurate which requires only a regular five point 2d stencil similar to that in the standard second order methods the resulting ehoc adi scheme in each adi solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal gaussian decomposition and may also be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time the unconditionally stable character of the method was verified by means of the discrete fourier or von neumann analysis numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order adi method of karaa and zhang and the spatial third order compact scheme of note and tan

on block circulant preconditioners for high order compact approximations of convection diffusion problems we study some properties of block circulant preconditioners for high order compact approximations of convection diffusion problems for two dimensional problems the approximation gives rise to a nine point discretisation matrix and in three dimensions we obtain a nineteen point matrix we derive analytical expressions for the eigenvalues of the block circulant preconditioner and this allows us to establish the invertibility of the preconditioner in both two and three dimensions the eigenspectra of the preconditioned matrix in the two dimensional case is described for different test cases our numerical results indicate that the block circulant preconditioning leads to significant reduction in iteration counts and comparisons between the high order compact and upwind discretisations are carried out for the unpreconditioned systems we observe fewer iteration counts for the hoc discretisation but for the preconditioned systems we find similar iteration counts for both finite difference approximations of constant coefficient two dimensional convection diffusion problems

a transformation free hoc scheme for incompressible viscous flows past an impulsively started circular cylinder in this paper we present a higher order compact scheme for the unsteady two dimensional 2d navier stokes equations on nonuniform polar grids specifically designed for the incompressible viscous flows past a circular cylinder the scheme is second order accurate in time and at least third order accurate in space the scheme very efficiently computes both unsteady and time marching steady state flow for a wide range of reynolds numbers re ranging from 10 to 9500 for the impulsively started cylinder the robustness of the scheme is highlighted when it accurately captures the vortex shedding for moderate re represented by the von karman street and the so called and s phenomena for higher re comparisons are made with established numerical and experimental results and excellent agreement is found in all the cases both qualitatively and quantitatively

limit distributions of random matrices abstract we study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation in particular we are concerned with the ensemble of symmetric blocks of independent hermitian random matrices which are asymptotically free asymptotically free from diagonal deterministic matrices and whose norms are uniformly bounded almost surely this class contains symmetric blocks of unitarily invariant hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability we show that the associated matricially free gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices including non hermitian gaussian matrices and matrices of wishart type

a high order compact finite difference scheme and precise integration method based on modified hopf cole transformation for numerical simulation of n dimensional burgers system abstract this paper modifies a n dimensional hopf cole transformation to the n dimensional burgers system we obtain the n dimensional heat conduction equation through the modification of the hopf cole transformation then the fourth order precise integration method pim in combination with a spatially global sixth order compact finite difference cfd scheme is presented to solve the equation with high accuracy moreover coupling with the strang splitting method the scheme is extended to multi dimensional two three dimensional burgers system numerical results show that the proposed method appreciably improves the computational accuracy compared with the existing numerical method moreover the two dimensional and three dimensional examples demonstrate excellent adaptability and the numerical simulation results also have very high accuracy in medium reynolds numbers

hypercontractivity in non commutative holomorphic spaces we prove an analog of janson s strong hypercontractivity inequality in a class of non commutative holomorphic algebras our setting is the q gaussian algebras q associated to the q fock spaces of bozejko kummerer and speicher for q 1 1 we construct subalgebras open image in new window a q segal bargmann transform and prove janson s strong hypercontractivity open image in new window for r an even integer

the compact and crank nicolson adi schemes for two dimensional semilinear multidelay parabolic equations this paper deals with numerical solutions of initial boundary value problems of the two dimensional semilinear multidelay parabolic equations two types of alternating direction implicit adi schemes are suggested the unique solvability convergence and unconditional stability of the schemes are analyzed and hence the corresponding criteria are established especially by using the discrete energy method it is proven that the compact adi scheme can attain second order accuracy in time and fourth order accuracy in space and the crank nicolson adi scheme has second order accuracy in both time and space numerical experiments are performed to verify the efficiency and accuracy of the both schemes

geometrically nonlinear flexural vibrations of plates in plane boundary conditions and some symmetry properties abstract this study is devoted to the derivation of some properties of the von karman equations for geometrically nonlinear models of plates with a boundary of arbitrary shape for applications to nonlinear vibration and buckling an intrinsic formulation of the local partial differential equations in terms of the transverse displacement and an airy stress function as unknowns is provided classical homogeneous boundary conditions with vanishing prescribed forces and displacements are derived in terms of the airy stress function in the case of a boundary of arbitrary geometry a special property of this operator crucial for some energy conserving numerical schemes and called triple self adjointness is derived in the case of an edge of arbitrary shape it is shown that this property takes a simple form for some classical boundary conditions so that the calculations in some practical cases are also simplified the applications of this work are either semi analytical methods of solution using an expansion of the solution onto an eigenmode basis of the associated linear problem or special energy conserving numerical methods

a rational high order compact difference method for the steady state stream functionvorticity formulation of the navierstokes equations a rational high order compact rhoc finite difference fd method on the nine point stencil is proposed for solving the steady state two dimensional navierstokes equations in the stream functionvorticity form the resulting system of algebra equations can be solved by using the point successive over or under relaxation sor iteration numerical experiments involving two linear and two nonlinear problems with their analytical solutions and two flow problems including the lid driven cavity and backward facing step flows are carried out to validate the performance of the newly proposed method numerical solutions of the driven cavity problem with different grid mesh sizes maximum being 513513 for reynolds numbers ranging from 0 to 17500 are obtained and compared with some of the accurate results available in the literature

strong haagerup inequalities for free r diagonal elements abstract in this paper we generalize haagerup s inequality u haagerup an example of a nonnuclear c algebra which has the metric approximation property invent math 50 1978 1979 279 293 on convolution norm in the free group to a very general context of r diagonal elements in a tracial von neumann algebra moreover we show that in this holomorphic setting the inequality is greatly improved from its original form we give combinatorial proofs of two important special cases of our main result and then generalize these techniques en route we prove a number of moment and cumulant estimates for r diagonal elements that are of independent interest finally we use our strong haagerup inequality to prove a strong ultracontractivity theorem generalizing and improving the one in p biane free hypercontractivity comm math phys 184 1997 457 474

simulation of flows around an impulsively started circular cylinder by taylor series expansion and least squares based lattice boltzmann method the two dimensional incompressible viscous flow past an impulsively started circular cylinder for a wide range of reynolds numbers re 20 9500 is studied computationally by using an explicit taylor series expansion and least squares based lattice boltzmann method the final equation for distribution function in our method is in an explicit form and essentially has no limitation on choice of mesh structure and lattice model it can be easily applied to simulation of flows with curved boundaries such as the problem considered in this work for the flow past an impulsively started circular cylinder numerical results obtained by present method agree very well with experimental data and computational results of navier stokes equations available in the literature

hardening softening behaviour in non linear oscillations of structural systems using non linear normal modes the definition of a non linear normal mode nnm as an invariant manifold in phase space is used in conservative cases it is shown that normal form theory allows one to compute all nnms as well as the attendant dynamics onto the manifolds in a single operation then a single mode motion is studied the aim of the present work is to show that too severe truncature using a single linear mode can lead to erroneous results using single non linear mode motion predicts the correct behaviour hence the nonlinear change of co ordinates allowing one to pass from the linear modal variables to the normal ones linked to the nnms defines a framework to properly truncate non linear vibration pdes two examples are studied a discrete system a mass connected to two springs and a continuous one a linear euler bernoulli beam resting on a non linear elastic foundation for the latter a comparison is given between the developed method and previously published results

a compact monotonic discretization scheme for solving second order vorticity velocity equations abstract this paper presents a numerical method for solving the steady state navier stokes equations for incompressible fluid flows using velocities and vorticity as working variables the method involves solving a second order differential equation for the velocity and a convection diffusion equation for the vorticity in cartesian grids the key to the success of the numerical simulation of this class of flow equations depends largely on proper simulation of vorticity transport equation subject to proper boundary vorticity in this paper we present a monotonic advection diffusion multi dimensional scheme and a theoretically rigorous implementation of vorticity boundary conditions while the derivation of the proposed integral vorticity boundary condition is more elaborate and is more difficult to solve than conventional local approaches the present approach offers significant advantages in this study both lid driven and backward facing step problems have been selected for comparison and validation purposes

convergence and performance of iterative methods for solving variable coefficient convection diffusion equation with a fourth order compact difference scheme abstract we conduct convergence analysis on some classical stationary iterative methods for solving the two dimensional variable coefficient convection diffusion equation discretized by a fourth order compact difference scheme several conditions are formulated under which the coefficient matrix is guaranteed to be an m matrix we further investigate the effect of different orderings of the grid points on the performance of some stationary iterative methods multigrid method and preconditioned gmres three sets of numerical experiments are conducted to study the convergence behaviors of these iterative methods under the influence of the flow directions the orderings of the grid points and the magnitude of the convection coefficients

an operator compact implicit method of exponential type a fourth order accurate tridiagonal operator compact implicit finite difference method for diffusion convection equations is developed the coefficients of the difference operator contain exponentials of the coefficients of the differential operator in this way the method avoids spurious oscillations when the cell reynolds number is large numerical results and comparisons to other methods are presented

a simple compact fourth order poisson solver on polar geometry we present a simple and efficient compact fourth order poisson solver in polar coordinates this solver relies on the truncated fourier series expansion where the differential equations of the fourier coefficients are solved by the compact fourth order finite difference scheme by shifting a grid a half mesh away from the origin and incorporating the symmetry constraint of fourier coefficients we can easily handle coordinate singularities without pole conditions the numerical evidence confirms fourth order accuracy for the problem on an annulus and third order accuracy for the problem on a disk in addition a simple and comparably accurate approximation for the derivatives of the solution is also presented

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

asymptotic numerical methods and pade approximants for non linear elastic structures in this paper we apply asymptotic numerical methods for computing non linear equilibrium paths of elastic beam plate and shell structures the non linear branches are sought in the form of asymptotic expansions and they are determined by solving numerically fem several linear problems with a single stiffness matrix a large number of terms of the series can be easily computed by using recurrence formulas in comparison with a more classical step by step procedure the method is rapid and automatic we show with some examples that the choice of the expansion s parameter and the use of pade approximants play an important role in the determination of the size of the domain of convergence

a high order difference method for the steady state navier stokes equations a stable high order difference method is developed for solving the heated cavity and the driven cavity problems governed by the steady state navier stokes equations the numerical results compared favorably with other methods coordinate transformations were also used to reflect the boundary layer phenonmena without requiring many mesh points the methods used in this paper are accurate stable and applicable to other flow problems governed by the navier stokes equations

segal bargmann transform functional calculus on matrix spaces and the theory of semi circular and circular systems letmd c be the space ofd dcomplex matrices and xdbe the subspace of hermitian matrices we study the segal bargmann transform of functions on xd with values inmd c which are given by functional calculus we show that whend the transform of such a map becomes close to the space of holomorphic functional calculus onmd c and that this yields in the limit an isometry between thel2space of wigner s semi circle distribution and the hardy space of the disk we relate this isometry to voiculescu s theory of circular and semi circular systems and we study its analogue when the segal bargmann transform is replaced by the hall transform on unitary groups of large dimensions

dynamics of the wave turbulence spectrum in vibrating plates a numerical investigation using a conservative finite difference scheme the dynamics of the local kinetic energy spectrum of an elastic plate vibrating in a wave turbulence wt regime is investigated with a finite difference energy conserving scheme the numerical method allows the simulation of pointwise forcing together with realistic boundary conditions a set up which is close to experimental conditions in the absence of damping the framework of non stationary wave turbulence is used numerical simulations show the presence of a front propagating to high frequencies leaving a steady spectrum in its wake self similar dynamics of the spectra are found with and without periodic external forcing for the periodic forcing the mean injected power is found to be constant and the frequency at the cascade front evolves linearly with time resulting in a increase of the total energy for the free turbulence the energy contained in the cascade remains constant while the frequency front increases as t 1 3 these self similar solutions are found to be in accordance with the kinetic equation derived from the von karman plate equations the effect of the pointwise forcing is observable and introduces a steeper slope at low frequencies as compared to the unforced case the presence of a realistic geometric imperfection of the plate is found to have no effect on the global properties of the spectra dynamics the steeper slope brought by the external forcing is shown to be still observable in a more realistic case where damping is added

a justification of nonlinear properly invariant plate theories a single asymptotic derivation of three classical nonlinear plate theories is presented in a setting which preserves the frame invariance properties of three dimensional finite elasticity by a successive scaling of the external loading on the three dimensional body the nonlinear membrane theory the nonlinear inextensional theory and the von karman equations are derived as the leading order terms in the asymptotic expansion of finite elasticity the governing equations of the nonlinear inextensional theory are of particular interest where 1 plane strain kinematics and plane stress constitutive equations are derived simultaneously from the asymptotic analysis 2 the theory can be phrased as a minimization problem over the space of isometric deformations of a surface and 3 the local equilibrium equations are identical to those arising in the one director cosserat shell model furthermore it can be concluded that with a regular single scale asymptotic expansion it is not possible to obtain a system of plate equations in which finite membrane strain and finite bending strain occur simultaneously in the leading order term of an asymptotic analysis

flow past an impulsively started circular cylinder using a higher order semicompact scheme in the present investigation an attempt has been made to solve the two dimensional incompressible viscous flow past an impulsively started circular cylinder for reynolds numbers ranging from 20 to 5000 using higher order semicompact scheme hosc unlike conventional higher order compact schemes the hosc scheme has been developed to handle the circular geometry of the chosen problem and the intensive algebraic manipulations have been reduced considerably by relaxing the compactness of the computational stencil for few terms but retained for most of the terms of the discretized equations for the flow past an impulsively started circular cylinder the results obtained at low and moderate reynolds numbers have been validated with the experimental and numerical observations available in the literature for high reynolds number flows the present scheme rightly captures the alpha phenomenon at re 1000 and both beta and alpha phenomena one after the other at re 5000

direct numerical simulation of turbulent flow separation from a wall mounted hump a high order accurate numerical method for solving the incompressible navier stokes equations in general orthogonal coordinates is presented the method is applied to a test case of the nasa langley computational fluid dynamics validation workshop 2004 a turbulent flow over a wall mounted hump geometry results of direct numerical simulations dns for the unforced flow as well as for a case with steady suction are presented and compared to the available experimental data the dns predictions are shown to agree well with the experiments except in the vicinity of the experimental reattachment locations the simulations predict slightly longer recirculation regions for both the unforced and the controlled case the results presented in this work suggest that with the rapidly increasing computational resources of modern supercomputers such as the cray x1 dns is becoming a viable alternative to the use of turbulence models for investigating complex turbulent flows at moderately high reynolds numbers

properties of free entropy related to polar decomposition the free entropies of non selfadjoint random variables and of unitary random variables are introduced and discussed by the methods of voiculescu s free analysis the additivity is shown to be equivalent to freeness the relation among and is investigated in the case when is the polar decomposition the subadditivity is proven and applications to some maximization problems for are given

a discontinuous petrov galerkin methodology for adaptive solutions to the incompressible navier stokes equations the discontinuous petrov galerkin methodology with optimal test functions dpg of demkowicz and gopalakrishnan 18 20 guarantees the optimality of the solution in an energy norm and provides several features facilitating adaptive schemes whereas bubnov galerkin methods use identical trial and test spaces petrov galerkin methods allow these function spaces to differ in dpg test functions are computed on the fly and are chosen to realize the supremum in the inf sup condition the method is equivalent to a minimum residual method for well posed problems with sufficiently regular solutions dpg can be shown to converge at optimal rates the inf sup constants governing the convergence are mesh independent and of the same order as those governing the continuous problem 48 dpg also provides an accurate mechanism for measuring the error and this can be used to drive adaptive mesh refinements we employ dpg to solve the steady incompressible navier stokes equations in two dimensions building on previous work on the stokes equations and focusing particularly on the usefulness of the approach for automatic adaptivity starting from a coarse mesh we apply our approach to a manufactured solution due to kovasznay as well as the lid driven cavity flow backward facing step and flow past a cylinder problems

a high order finite difference method with richardsonextrapolation for 3d convection diffusion equation in this paper we extend the sun and zhang s 24 work on high order finite difference method which is based on the richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case firstly we employ a fourth order compact difference scheme to get the fourth order accurate solution on the fine and the coarse grids then we use the richardson extrapolation technique by combining the two approximate solutions to get a sixth order accurate solution on coarse grid finally we apply an operator interpolation scheme to achieve the sixth order accurate solution on the fine grid during this process we use alternating direction implicit adi method to solve the resulting linear systems numerical experiments are conducted to verify the accuracy and effectiveness of the present method

compact h4 finite difference approximations to operators of navier stokes type a method of obtaining compact finite difference approximations of h4 accuracy to operators of navier stokes type is considered the basic procedure is developed for operators in one space dimension and subsequently applied to problems in more space dimensions and in time four illustrative numerical examples are given which indicate clearly in various cases that excellent accuracy may be obtained using the methods comparisons with previous results and with the results of h2 accurate computations are made

r diagonal dilation semigroups this paper addresses extensions of the complex ornstein uhlenbeck semigroup to operator algebras in free probability theory if a 1 a k are free fancyscript r diagonal operators in a ii1 factor then d t a i 1 cdots a i n e nt a i 1 cdots a i n defines a dilation semigroup on the non self adjoint operator algebra generated by a 1 a k we show that d t extends in two different ways to a semigroup of completely positive maps on the von neumann algebra generated by a 1 a k moreover we show that d t satisfies an optimal ultracontractive property d t colon l 2 to l infty sim t 1 for small t 0