finite rank sums of products of toeplitz and hankel operators abstract on the dirichlet space of the unit disk we consider operators which are finite sums of toeplitz products hankel products or products of toeplitz and hankel operators we then give characterizations of when such operators have finite rank on the dirichlet space

a characterization of compact perturbations of toeplitz operators let x be a bounded operator on the hardy space h 2 of the unit circle it has been a longstanding problem to determine whether the condition that t u xt u x is compact for every inner function u implies that x is a compact perturbation of a toeplitz operator we show that the answer is affirmative

dynamic properties in a family of competitive growing models the properties of a wide variety of growing models generically called x rd involving the deposition of particles according to competitive processes such that a particle is attached to the aggregate with probability p following the mechanisms of a generic model x that provides the correlations and at random random deposition rd with probability 1 p are studied by means of numerical simulations and analytic developments the study comprises the following x models ballistic deposition random deposition with surface relaxation das sarma tamboronea kim kosterlitz lai das sarma wolf villain large curvature and three additional models that are variants of the ballistic deposition model it is shown that after a growing regime the interface width becomes saturated at a crossover time tx2 that by fixing the sample size scales with p according to tx2 p proportional variant p y p 0 where is an exponent also the interface width at saturation wsat scales as wsat p proportional variant p delta p 0 where delta is another exponent it is proved that in any dimension the exponents delta and y obey the following relationship delta y beta rd where beta rd 1 2 is the growing exponent for rd furthermore both exponents exhibit universality in the p 0 limit by mapping the behavior of the average height difference of two neighboring sites in discrete models of type x rd and two kinds of random walks we have determined the exact value of the exponent delta when the height difference between two neighbouring sites corresponds to a random walk that after walking steps returns to a distance from its initial position that is proportional to the maximum distance reached random walk of type a one has delta 1 2 on the other hand when the height difference between two neighboring sites corresponds to a random walk that after steps moves steps towards the initial position random walk of type b one has delta 1 finally by linking four well established universality classes namely edwards wilkinson kardar parisi zhang linear molecular beam epitaxy mbe and nonlinear mbe with the properties of type a and b of random walks eight different stochastic equations for all the competitive models studied are derived

non uniform convergence of finite volume schemes for riemann problems of ideal magnetohydrodynamics this paper presents riemann test problems for ideal magnetohydrodynamics mhd finite volume schemes the test problems place emphasis on the hyperbolic irregularities of the ideal mhd system namely the occurrence of intermediate shocks and non unique solutions we investigate numerical solutions for the test problems obtained by several commonly used methods roe hlle central scheme all methods turned out to show a non uniform convergence behavior which may be paraphrased as pseudo convergence initially the methods show convergence towards a wrong solution with irregular wave patterns only after heavy grid refinement the methods switch to converge to the true solution this behavior is most pronounced whenever the solution should be unique but its phase space trajectory lies in the vicinity of that of a non unique solution we show detailed grid convergence studies and empirical error analysis the results may be related to similar results for a 2 2 model system and to time dependent investigations of other authors the non uniform convergence is expected to be present for any diffusive finite volume method and question the reliability of coarse grid ideal mhd solutions also in higher space dimensions

a robust hllc type riemann solver for strong shock it is well known that for the eulerian equations the numerical schemes that can accurately capture contact discontinuity usually suffer from some disastrous carbuncle phenomenon while some more dissipative schemes such as the hll scheme are free from this kind of shock instability hybrid schemes to combine a dissipative flux with a less dissipative flux can cure the shock instability but also may lead to other problems such as certain arbitrariness of choosing switching parameters or contact interface becoming smeared in order to overcome these drawbacks this paper proposes a simple and robust hllc type riemann solver for inviscid compressible gas flows which is capable of preserving sharp contact surface and is free from instability the main work is to construct a hll type riemann solver and a hllc type riemann solver by modifying the shear viscosity of the original hll and hllc methods both of the two new schemes are positively conservative under some typical wavespeed estimations moreover a linear matrix stability analysis for the proposed schemes is accomplished which illustrates the hllc type solver with shear viscosity is stable whereas the hll type solver with vorticity wave is unstable our arguments and numerical experiments demonstrate that the inadequate dissipation associated to the shear wave may be a unique reason to cause the instability

schemes to compute unsteady flashing flows some ways to compute flashing flows in variable cross section ducts are provided focusing on the homogeneous relaxation model the basic numerical method relies on a splitting technique that is consistent with the overall entropy inequality the cross section is assumed to he continuous and the finite volume approach is applied to approximate homogeneous equations several suitable schemes to account for complex equation of state are discussed namely the rusanov scheme an approximate form of the roe scheme and the volumes finis roe vfroe scheme with the help of nonconservative variables to evaluate respective accuracy the homogeneous euler equations are computed first and the l1 error norm of transient solutions of shock tube experiments are plotted it is shown that the rusanov scheme is indeed less accurate which balances its interesting properties inasmuch as it preserves the positivity of the mean density and the maximum principle for the vapor quality computations of real cases are presented which account for the mass transfer term and the time space dependent cross sections

capturing shock reflections godunov type schemes based on exact or approximate solutions to the riemann problem have proven to be an excellent tool to compute approximate solutions to hyperbolic systems of conservation laws however there are many instances in which a particular scheme produces inappropriate results in this paper we consider several situations in which roe s scheme gives incorrect results or blows up all together and we propose an alternative flux formula that produces numerical approximations in which the pathological behavior is either eliminated or reduced to computationally acceptable levels

commuting toeplitz operators with pluriharmonic symbols let da z denote the lebesgue volume measure on the open unit ball bn f cn normalized so that the measure of bn equals 1 the 13ergman space l2 bn is the eiilbert space consisting of holomorphic functions on bn that are also in l2 bn da for f in l90 bn the toeplitz operator tf with symbol f is desned by t h p fh where p is the orthogonal projection from l2 bnnda onto ol2tbesda n called the bergman projection let dsr be the surface area measure on the unit sphere sn for 1 p 007 h s4 is the banach space of holomorphic functions on bn with the norm defined by

numerical analysis of shock wave reflection transition in steady flows different aspects of the transition between regular and mach reflections of strong shock waves in steady flows are numerically studied two approaches kinetic the direct simulation monte carlo method and continuum euler equations are used to investigate the hysteresis phenomenon in the flow about two symmetrical wedges in two and three dimensional statements the dependence of the final shock wave configuration on initial conditions the transition from regular to mach reflection by means of flow perturbations and three dimensional effects are examined the three dimensionality of the flow is shown to increase the angles of transition from regular to mach reflection and back and to decrease the mach stem height

finite rank toeplitz products with harmonic symbols abstract on the bergman space of the unit ball of c n we study the finite rank problem for toeplitz products with harmonic symbols we first solve the problem with two factors in case symbols have local continuous extension property up to the boundary also in case symbols have additional lipschitz continuity up to some part of the boundary we solve the problem for multiple products with number of factors depending on the dimension n analogous theorems on the polydisk are also obtained

an analysis of finite difference and finite volume formulations of conservation laws abstract finite difference and finite volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws a new coordinate free formulation of systems of conservation laws is developed which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system the analysis considers general types of equations potential euler and navier stokes three dimensional unsteady flows with time varying grids are described using a single consistent nomenclature for both formulations grid motion due to a non inertial reference frame as well as flow adaptation is covered in comparing the two formulations it is found useful to distinguish between differences in numerical methods and differences in grid definition the former plays a role for non cartesian grids and results in only cosmetic differences in the manner in which geometric terms are handled the differences in grid definition for the two formulations is found to be more important since it affects the manner in which boundary conditions zonal procedures and grid singularities are handled at computational boundaries the proper interpretation of strong and weak conservation law forms for quasi one dimensional and axisymmetric flows is brought out

toeplitz products with pluriharmonic symbols on the hardy space over the ball on the hardy space over the unit ball in cn we consider operators which have the form of a finite sum of products of several toeplitz operators we study characterizing problems of when such an operator is compact or of finite rank some of our results show higher dimensional phenomena

a finite volume hllc based scheme for compressible interfacial flows with surface tension shock waves are often used in experiments to create a shear flow across liquid droplets to study secondary atomization similar behavior occurs inside of supersonic combustors scramjets under startup conditions but it is challenging to study these conditions experimentally in order to investigate this phenomenon further a numerical approach is developed to simulate compressible multiphase flows under the effects of surface tension forces the flow field is solved via the compressible multicomponent euler equations i e the five equation model discretized with the finite volume method on a uniform cartesian grid the solver utilizes a total variation diminishing tvd third order rungekutta method for time marching and second order tvd spatial reconstruction surface tension is incorporated using the continuum surface force csf model fluxes are upwinded with a modified hartenlaxvan leer contact hllc approximate riemann solver an interface compression scheme is employed to counter numerical diffusion of the interface the present work includes modifications to both the hllc solver and the interface compression scheme to account for capillary force terms and the associated pressure jump across the gasliquid interface a simple method for numerically computing the interface curvature is developed and an acoustic scaling of the surface tension coefficient is proposed for the non dimensionalization of the model the model captures the surface tension induced pressure jump exactly if the exact curvature is known and is further verified with an oscillating elliptical droplet and mach 1 47 and 3 shock droplet interaction problems the general characteristics of secondary atomization at a range of weber numbers are also captured in a series of simulations

dynamic scaling in thin film growth with irreversible step edge attachment we study dynamic scaling in a model with collective diffusion cd of isolated atoms in terraces and irreversible aggregation at step edges simulations are performed in two dimensional substrates with several diffusion to deposition ratios r identical with d f data collapse of scaled roughness distributions confirms that this model is in the class of the fourth order nonlinear growth equation by villain lai and das sarma vlds with negligible finite size effects while estimates of scaling exponents show some discrepancies this result is consistent with the prediction of a recent renormalization group approach and improves previous numerical works on related models the roughness follows dynamic scaling as w lalpha r1 2f xi l with correlation length xi rt 1 z where z is the dynamic exponent we also propose a limited mobility lm model where the incident atom executes up to g steps before a new atom is adsorbed and irreversibly aggregates at step edges this model is also shown to belong to the vlds class the size of the plateaus in the film surface increases as g1 2 and the lateral correlation scales as g1 2t1 z the time evolution of the roughness reproduces that of the cd model if an equivalent parameter g approximately r2 z is chosen this suggests the possibility of using lm models with tunable diffusion length to simulate processes with simultaneous diffusion of many atoms a scaling approach is used to justify exponent values and dynamic relations for both models including the significant decrease of surface roughness as r or g increases

toeplitz operators on hardy sobolev spaces abstract in this paper we construct compact toeplitz operators and trace class toeplitz operators with unbounded symbols on hardy sobolev spaces we also characterize the compactness of finite sums of products of two toeplitz operators on these spaces then we study the fredholmness of the toeplitz operators and calculate the essential norm of them in addition we consider the c algebra generated by toeplitz operators

wave structure similarity of the hllc and roe riemann solvers application to low mach number preconditioning the approximate riemann solver of roe and the solver of harten lax van leer hll and its variants such as the hllc solver are widely used as building blocks of finite volume godunov type methods for the solution of the euler equations of gas dynamics and related hyperbolic flow models the hllc solver has gained increasing popularity over the last two decades since it possesses some of the good properties of the roe solver and in addition satisfies important entropy and positivity conditions with no need of special fixes in this work we rewrite the classical hllc riemann solver in a novel form that highlights the formal mathematical similarity of its wave structure with the one of the roe solver this similarity might be useful to extend to the hllc method some numerical techniques devised specifically for the roe method as an example of application we illustrate the design of a turkel type low mach number preconditioning technique for the hllc scheme by exploiting methodologies proposed in the l

the finite sum of finite products of toeplitz operators on the polydisk abstract a limit theorem is established for a finite sum of finite products of toeplitz operators on the hardy space of the polydisk as a consequence we show that the product of six toeplitz operators with pluriharmonic symbols is compact iff the product equals zero iff one of these toeplitz operators equals zero

zero products of toeplitz operators we prove that the product of finitely many toeplitz operators oil the hardy space is zero if and only if at least one of the operators is zero we use some new vector valued techniques that not only lead to a vector valued version of this result but also appear to be needed in the scalar case

on godunov type methods near low densities abstract when the energy of a flow is largely kinetic many conservative differencing schemes may fail by predicting non physical states with negative density or internal energy we describe as positively conservative the subclass of schemes that by contrast always generate physical solutions from physical data and show that the godunov method is positively conservative it is also shown that no scheme whose interface flux derives from a linearised riemann solution can be positively conservative classes of data that will bring about the failure of such schemes are described however the harten lax van leer hlle scheme is positively conservative under certain conditions on the numerical wavespeeds and this observation allows the linearised schemes to be rescued by modifying the wavespeeds employed

zero products of toeplitz operators with harmonic symbols on the bergman space of the unit ball in cn we solve the zero product problem for two toeplitz operators with harmonic symbols that have continuous extensions to some part of the boundary in the case where symbols have lipschitz continuous extensions to the boundary we solve the zero product problem for multiple products with the number of factors depending on the dimension n of the underlying space the number of factors is n 3 we also prove a local version of this result but with loss of a factor

the ghost fluid method for compressible gas water simulation an analysis is carried out for the ghost fluid method gfm based algorithm as applied to the gas water riemann problems which can be construed as two single medium gfm riemann problems it is found that the inability to provide correct and consistent riemann waves in the respective real fluids by these two gfm riemann problems may lead to inaccurate numerical results based on this finding two conditions are suggested and imposed for the ghost fluid status in order to ensure that correct and consistent riemann waves are provided in the respective real fluids during the numerical decomposition of the singularity using these two conditions to analyse some of the existing gfm based algorithms such as the original gfm j comput phys 152 1999 457 the new version gfm j comput phys 166 2001 1 j comput phys 175 2002 200 and the modified gfm mgfm j comput phys 190 2003 651 it is found that there are ranges of conditions for each type of solution where either the original gfm or the new version gfm or both are unable to provide correct or consistent riemann waves in one of the real fluids within these ranges examples can be found such that either the original gfm or the new version gfm or both are unable to provide accurate results the mgfm is also found to encounter difficulties when applied to nearly cavitating flow various examples are presented to demonstrate the conclusions obtained the mgfm with proposed modification when applied to nearly cavitating flow is then found to be quite robust and can provide relatively reasonable results

dependence of steady mach reflections on the reflecting wedge trailing edge angle two simultaneous numerical investigations using different codes were conducted the numerical method in the first study was the lcpfct algorithm which is a one dimensional flux corrected transport algorithm for solving generalized continuity equations with fourth order phase accuracy and minimum residual diffusion a computer code based on a w modified godunov s scheme was developped and used in the second study to complement these numerical studies an experimental study was also conducted

comparison of some flux corrected transport and total variation diminishing numerical schemes for hydrodynamic and magnetohydrodynamic problems two versions of flux corrected transport and two versions of total variation diminishing schemes are tested for several one and two dimensional hydrodynamic and magnetohydrodynamic problems two of the schemes ydfct and tvdlf are tested extensively for the first time the results give an insight into the limitations of the methods their relative strengths and weaknesses some subtle points of the algorithms and the effects of selecting different options for certain methods are emphasised

stochastic equation for conserved growth in a restricted solid on solid model we apply the master equation method naturally extended for the nonlocal growth process to directly deriving the continuum stochastic equation for a conserved growth model with a restricted solid on solid condition the villain lai das sarma growth equation we obtain for the model is consistent with the result of recent numerical simulations furthermore we find that only the relaxation of the deposited particles up to the nearest neighborhood for n51 condition or the next nearest neighborhood for n 1 condition determines the scaling property and the universality class of the model and the higher neighbor hopping processes play no essential role the choice of the regularization scheme in the derivation procedure is also discussed s1063 651x 98 07604 1

a robust and accurate led bgk solver on unstructured adaptive meshes starting from the bgk model of the boltzmann equation we develop a robust and accurate finite volume gas kinetic scheme on unstructured triangular meshes the proposed numerical approach is composed of two steps an initial reconstruction step and a gas evolution step in the initial reconstruction step an unstructured version of the local extremum diminishing interpolation is applied to the conservative variables and to compute left and right states along a node edge in the gas evolution step the local integral solution of the bgk model is used to compute numerical fluxes at a cell interface this approach provides an alternative to riemann solvers and yields numerical schemes which possess many desirable properties that may not be found in godunov type schemes a classich refinement adaptive procedure is implemented to increase the spatial resolution of high speed unsteady flow characteristics such as shock waves contact discontinuities or expansion waves with minimal computational costs and memory overheads this procedure involves mesh enrichment coarsening steps to either insert nodes on an edge center in high gradient regions or delete nodes in over resolved regions numerical results of several test cases for unsteady compressible inviscid flows are presented in order to verify the accuracy and robustness of the current numerical approach the computed results are compared with analytical solutions experimental data the results of structured mesh calculations and the results obtained by widely used flux splitting methods

the distribution function inequality for a finite sum of finite products of toeplitz operators abstract a generalized area function associated with a finite sum of finite products of toeplitz operators is introduced a distribution function inequality is established for the generalized area function by using the distribution function inequality we characterize when a finite sum of finite products of toeplitz operators on the hardy space is a compact perturbation of a toeplitz operator

modeled boltzmann equation and its application to shock capturing simulation a modified equilibrium distribution function for the bhatnagar gross krook type modeled boltzmann equation has recently been proposed the function was deduced using acoustics scaling to normalize the equation and allowed a correct recovery of similarly normalized euler equations it is a combination of a maxwellian distribution plus three other terms that are moments of particle velocity the lattice counterpart of the modified equilibrium distribution function also led to an exact recovery of the euler equations therefore there is no mach number limitation in the entire approach this lattice counterpart was able to replicate aeroacoustics problems involving vorticity acoustic and entropy acoustic interactions correctly and the simulations were carried out using a finite difference lattice boltzmann method employing only a two dimensional nine velocity lattice thus formulated the numerical scheme has no arbitrary constants and all calculations were carried out using one single relaxation time and a set of constants derived from the analysis this paper investigates the validity and extent of the formulation to capture shocks and resolve contact discontinuity and expansion waves in one and two dimensional riemann problems the simulations are carried out using the same two dimensional nine velocity lattice and identical set of constants and relaxation time they are compared with theoretical results and those obtained by solving the euler equations directly using harten s first order numerical scheme good agreement is obtained for all test cases however the modified equilibrium distribution function is not suitable for shock structure simulation for that an exact recovery of the navier stokes equations is required

compact products of toeplitz operators on the dirichlet space of the unit ball abstract we consider on the dirichlet space of the unit ball operators which have the form of a finite sum of products of several toeplitz operators and then give a characterization for which such an operator is compact we show as an application that for n 2 there is no nontrivial compact product of several toeplitz operators with pluriharmonic symbols which is a higher dimensional phenomenon whose one variable analogue is false

products of several toeplitz operators in this paper we give a necessary and sufficient condition for ta1ta2 tan ta1a2 an where the tai s are toeplitz operators on the hardy space of the unit disk we then show that ta1ta2ta3ta4ta5ta6 0 if and only if one of ai is identically zero a criteria for the compactness of ta1ta2 tan ta1a2 an is also given