numerical solutions to compressible flows in a nozzle with variable cross section compressible flows in a nozzle can be modeled by the gas dynamics equations in one dimensional space with source terms it turns out that along stationary waves the entropy is conserved investigating properties of the system leads us to the determination of stationary waves relying on this analysis we construct a numerical scheme which takes into account the use of stationary waves our scheme is shown to be capable of maintaining equilibrium states this demonstrates the efficiency of the new scheme over classical ones which usually give unsatisfactory results when reducing the refinement of the mesh size moreover our scheme converges much faster than the classical ones in most cases

a well balanced scheme for the shallow water equations with topography a non negativity preserving and well balanced scheme that exactly preserves all the smooth steady states of the shallow water system including the moving ones is proposed in addition the scheme must deal with vanishing water heights and transitions between wet and dry areas a godunov type method is derived by using a relevant average of the source terms within the scheme in order to enforce the required well balance property a second order well balanced muscl extension is also designed numerical experiments are carried out to check the properties of the scheme and assess the ability to exactly preserve all the steady states

the riemann problem for a nonisentropic fluid in a nozzle with discontinuous cross sectional area we present a full investigation of the riemann problem for a nonisentropic polytropic fluid in a nozzle with piecewise constant cross section first we introduce the concept of elementary waves which turn out to make up riemann solutions second we study a procedure to select admissible stationary waves relying on the monotone criterion by projecting all the wave curves in the p u plane we construct riemann solutions existence of riemann solutions can be obtained for large initial data furthermore we establish the uniqueness of riemann solutions in strictly hyperbolic domains our argument can lead to estimate regions where the riemann problem admits a unique solution

nonlinear resonance in systems of conservation laws the riemann problem for a general inhomogeneous system of conservation laws is solved in a neighborhood of a state at which one of the nonlinear waves in the problem takes on a zero speed the inhomogeneity is modeled by a linearly degenerate field the solution of the riemann problem determines the nature of wave interactions and thus the riemann problem serves as a canonical form for nonlinear systems of conservation laws generic conditions on the fluxes are stated and it is proved that under these conditions the solution of the riemann problem exists is unique and has a fixed structure this demonstrates that in the above sense resonant inhomogeneous systems generically have the same canonical form the wave curves for these systems are only lipschitz continuous in a neighborhood of the states where the wave speeds coincide and so in contrast to strictly hyperbolic systems the implicit function theorem cannot be applied directly to obtain existence and uniqueness here we show that existence

invariant region preserving dg methods for multi dimensional hyperbolic conservation law systems with an application to compressible euler equations abstract an invariant region preserving irp limiter for multi dimensional hyperbolic conservation law systems is introduced as long as the system admits a global invariant region which is a convex set in the phase space it is shown that the order of approximation accuracy is not destroyed by the irp limiter provided the cell average is away from the boundary of the convex set moreover this limiter is explicit and easy for computer implementation a generic algorithm incorporating the irp limiter is presented for high order finite volume type schemes for arbitrarily high order discontinuous galerkin dg schemes to hyperbolic conservation law systems sufficient conditions are obtained for cell averages to remain in the invariant region provided the projected one dimensional system shares the same invariant region as the full multi dimensional hyperbolic system does the general results are then applied to both one and two dimensional compressible euler equations so to obtain high order irp dg schemes numerical experiments are provided to validate the proven properties of the irp limiter and the performance of irp dg schemes for compressible euler equations

a helson szego theorem for subdiagonal subalgebras with applications to toeplitz operators abstract we formulate and establish a noncommutative version of the well known helson szego theorem about the angle between past and future for subdiagonal subalgebras we then proceed to use this theorem to characterise the symbols of invertible toeplitz operators on the noncommutative hardy spaces associated to subdiagonal subalgebras

a multiwave approximate riemann solver for ideal mhd based on relaxation ii numerical implementation with 3 and 5 waves in the first part of this work bouchut et al j comput phys 108 7 41 2007 we introduced an approximate riemann solver for one dimensional ideal mhd derived from a relaxation system we gave sufficient conditions for the solver to satisfy discrete entropy inequalities and to preserve positivity of density and internal energy in this paper we consider the practical implementation and derive explicit wave speed estimates satisfying the stability conditions of bouchut et al j comput phys 108 7 41 2007 we present a 3 wave solver that well resolves fast waves and material contacts and a 5 wave solver that accurately resolves the cases when two eigenvalues coincide a full 7 wave solver which is highly accurate on all types of waves will be described in a follow up paper we test the solvers on one dimensional shock tube data and smooth shear waves

a godunov type method for the shallow water equations with discontinuous topography in the resonant regime abstract we investigate the riemann problem for the shallow water equations with variable and possibly discontinuous topography and provide a complete description of the properties of its solutions existence uniqueness in the non resonant regime multiple solutions in the resonant regime this analysis leads us to a numerical algorithm that provides one with a riemann solver next we introduce a godunov type scheme based on this riemann solver which is well balanced and of quasi conservative form finally we present numerical experiments which demonstrate the convergence of the proposed scheme even in the resonance regime except in the limiting situation when riemann data precisely belong to the resonance hypersurface

why nonconservative schemes converge to wrong solutions error analysis this paper attempts to give a qualitative and quailtitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws we show that these schemes converge strongly in li norm to the solution of an inhomogeneous conservation law containing a borel measure source term moreover we analyze the properties of this borel measure and derive a sharp estimate for the l1 error between the limit function given by the scheme and the correct solution in general the measure source term is of the order of the entropy dissipation measure associated with the scheme in certain cases the error can be small for short times which makes it difficult to detect numerically but generically such an error will grow in time and this would lead to a large error for large time calculations finally we show that a local correction of any high order accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution

applications of the fuglede kadison determinant szego s theorem and outers for noncommutative we first use properties of the fuglede kadison determinant on l p m for a finite von neumann algebra m to give several useful variants of the noncommutative szego theorem for l p m including the one usually attributed to kolmogorov and krein as an application we solve the longstanding open problem concerning the noncommutative generalization to arveson s noncommutative h p spaces of the famous outer factorization of functions f with log f integrable using the fuglede kadison determinant we also generalize many other classical results concerning outer functions

why the muscl hancock scheme is l 1 stable the finite volume methods are one of the most popular numerical procedure to approximate the weak solutions of hyperbolic systems of conservation laws they are developed in the framework of first order numerical schemes several approaches are proposed to increase the order of accuracy the van leer methods are interesting ways one of them namely the muscl hancock scheme is full time and space second order accuracy in the present work we exhibit relevant conditions to ensure the l1 stability of the method a cfl like condition is established and a suitable limitation procedure for the gradient reconstruction is developed in order to satisfy the stability criterion in addition we show that the conservative variables are not useful within the gradient reconstruction and the procedure is extended in the framework of the primitive variables numerical experiments are performed to show the interest and the robustness of the method

factorization in analytic operator algebras abstract a constructive and unified approach is used to obtain the upper lower factorization of positive operators and the outer function factorization of positive operator valued functions on the circle for a projection nest e it is shown that every positive operator admits a canonical factorization c a a with a an outer operator if and only if e is well ordered with new methods we generalize the inner outer factorizations obtained by arveson for nests of order type z and the riesz factorization due to shields for trace class triangular operators weak factorization is obtained in noncommutative h1 spaces associated with general nest subalgebras of a semifinite factor characterizations of a nehari type are given for the associated hankel forms and hankel operators

a multiwave approximate riemann solver for ideal mhd based on relaxation i theoretical framework we present a relaxation system for ideal magnetohydrodynamics mhd that is an extension of the suliciu relaxation system for the euler equations of gas dynamics from it one can derive approximate riemann solvers with three or seven waves that generalize the hllc solver for gas dynamics under some subcharacteristic conditions the solvers satisfy discrete entropy inequalities and preserve positivity of density and internal energy the subcharacteristic conditions are nonlinear constraints on the relaxation parameters relating them to the initial states and the intermediate states of the approximate riemann solver itself the 7 wave version of the solver is able to resolve exactly all material and alfven isolated contact discontinuities practical considerations and numerical results will be provided in another paper

a fast and stable well balanced scheme with hydrostatic reconstruction for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom it is a hyperbolic system of conservation laws that approximately describes various geophysical flows such as rivers coastal areas and oceans when completed with a coriolis term or granular flows when completed with friction numerical approximate solutions to this system may be generated using conservative finite volume methods which are known to properly handle shocks and contact discontinuities however in general these schemes are known to be quite inaccurate for near steady states as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions this difficulty can be overcome by using the so called well balanced schemes we describe a general strategy based on a local hydrostatic reconstruction that allows us to derive a well balanced scheme from any given numerical flux for the homogeneous problem whenever the initial solver satisfies some classical stability properties it yields a simple and fast well balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality

the riemann problem for the one dimensional free surface shallow water equations with a bed step theoretical analysis and numerical simulations in this paper the solution of the riemann problem for the one dimensional free surface shallow water equations over a bed step is analyzed both from a theoretical and a numerical point of view particular attention has been paid to the wave that is generated at the location of the bed discontinuity starting from the classical shallow water equations considering the bed level as an additional variable and adding to the system an equation imposing its time invariance we show that this wave is a contact wave across which one of the riemann invariants namely the energy is not constant this is due to the fact that the relevant problem is nonconservative we demonstrate that in this type of system riemann invariants do not generally hold in contact waves furthermore we show that in this case the equations that link the flow variables across the contact wave are the generalized rankine hugoniot relations and we obtain these for the specific problem from the numerical point of view we present an accurate and efficient solver for the step riemann problem to be used in a finite volume godunov type framework through a two step predictor corrector procedure the solver is able to provide solutions with any desired accuracy the predictor step uses a well balanced generalized roe solver while the corrector step solves the exact nonlinear system of equations that consitutes the problem by means of an iterative procedure that starts from the predictor solution in order to show the effectiveness and the accuracy of the proposed approach we consider several step riemann problems and compare the exact solutions with the numerical results obtained by using a standard roe approach far from the step and the novel two step algorithm for the fluxes over the step achieving good results

exact solution of the riemann problem for the shallow water equations with discontinuous bottom geometry in this paper we present the exact solution of the riemann problem for the non linear shallow water equations with a step like bottom the solution has been obtained by solving an enlarged system that includes an additional equation for the bottom geometry and then using the principles of conservation of mass and momentum across the step the resulting solution is unique and satisfies the principle of dissipation of energy across the shock wave we provide examples of possible wave patterns numerical solution of a first order dissipative scheme as well as an implementation of our riemann solver in the second order upwind method are compared with the proposed exact riemann problem solution a practical implementation of the proposed exact riemann solver in the framework of a second order upwind tvd method is also illustrated

finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws we develop here a new class of finite volume schemes on unstructured meshes for scalar conservation laws with stiff source terms the schemes are of equilibrium type hence with uniform bounds on approximate solutions valid in cell entropy inequalities and exact for some equilibrium states convergence is investigated in the framework of kinetic schemes numerical tests show high computational efficiency and a significant advantage over standard cell centered discretization of source terms equilibrium type schemes produce accurate results even on test problems for which the standard approach fails for some numerical tests they exhibit exponential type convergence rate in two of our numerical tests an equilibrium type scheme with 441 nodes on a triangular mesh is more accurate than a standard scheme with 50002 grid points

a well balanced flux vector splitting scheme designed for hyperbolic systems of conservation laws with source terms abstract we propose a way to construct robust numerical schemes for the computations of numerical solutions of one and two dimensional hyperbolic systems of balance laws in order to reduce the computational cost we selected the family of flux vector splitting schemes we reformulate the source terms as nonconservative products and treat them directly in the definition of the numerical fluxes by means of generalized jump relations this is applied to a 1d shallow water system with topography and to a 2d simplified model of two phase flows with damping effects numerical results and comparisons with a classical centered discretizations scheme are supplied

on the solution to the riemann problem for the compressible duct flow the quasi one dimensional euler equations in a duct of variable cross section form probably one of the simplest nonconservative systems we consider the riemann problem for it and discuss its properties in particular for some initial conditions the solution to the riemann problem appears to be nonunique in order to rule out the nonphysical solutions we provide two dimensional computations of the euler equations in a duct of corresponding geometry and compare them with the one dimensional 1d results then the physically relevant 1d solutions satisfy a kind of entropy rate admissibility criterion

the riemann problem for the baer nunziato two phase flow model we consider the riemann problem for the two phase flow model proposed by baer and nunziato int j multiphase flows 12 1986 861 it describes the flame spread and the deflagration to detonation transition ddt in gaspermeable reactive granular materials the model is given by a non strictly hyperbolic non conservative system of partial differential equations we investigate the structure of the riemann problem and construct the exact solution for it we establish that the solution across one wave is not unique and propose to use the evolutionarity criterion to single out the admissible solution due to special structure of the riemann problem for the baer nunziato model we are able to introduce a notion of a weak solution for it finally we propose a number of test cases based on the exact solution to the riemann problem for the baer nunziato model

numerical approximations of the 10 moment gaussian closure we propose a numerical scheme to approximate the weak solutions of the 10 moment gaussian closure the moment gaussian closure for gas dynamics is governed by a conservative hyperbolic system supplemented by entropy inequalities whose solutions satisfy positiveness of density and tensorial pressure we consider a suliciu type relaxation numerical scheme to approximate the solutions these methods are proved to satisfy all the expected positiveness properties and all the discrete entropy inequalities the scheme is illustrated by several numerical experiments

well balanced high order extensions of godunov s method for semilinear balance laws this paper is concerned with the development of well balanced high order numerical schemes for systems of balance laws with a linear flux function whose coefficients may be variable first well balanced first order numerical schemes are obtained based on the use of exact solvers of riemann problems that include both the flux and the source terms godunov s methods so obtained are extended to higher order schemes by using a technique of reconstruction of states the main contribution of this paper is to introduce a reconstruction technique that preserves the well balanced property of godunov s methods some numerical experiments are presented to verify in practice the properties of the developed numerical schemes

a variant of van leer s method for multidimensional systems of conservation laws abstract we present a new variant of van leer s construction of upwind finite volume schemes for hyperbolic systems of conservation laws fluxes are computed with second order accuracy using an interpolation rather than a slope reconstruction a correction of the interpolated values is necessary and performed globally on each cell by a conservation argument it can be used on a rectangular or triangles based dual grid to obtain a genuinely multidimensional scheme one of our main concerns in this construction is to prove that the second order reconstruction combined with a boltzmann solver gives nonnegative values of the pressure and density for gas dynamics even on an unstructured mesh this allows us to derive a rigorous cfl condition thus our approach is very robust

efficient well balanced hydrostatic upwind schemes for shallow water equations the proposed work concerns the numerical approximations of the shallow water equations with varying topography the main objective is to introduce an easy and systematic technique to enforce the well balance property and to make the scheme able to deal with dry areas to access such an issue the derived numerical method is obtained by involving the free surface instead of the water height and this produces the scheme well balanced independently from the numerical flux function associated with the homogeneous problem as a consequence we obtain an easy well balanced scheme which preserves non negative water height when compared with the well known hydrostatic reconstruction the presented topography discretization does not involve any max function known to introduce some numerical errors as soon as the topography admits very strong variations or discontinuities a second order muscl accurate reconstruction is adopted the proposed hydrostatic upwind scheme is next extended for considering 2d simulations performed over unstructured meshes several 1d and 2d numerical experiments are performed to exhibit the relevance of the scheme

subdiagonal algebras with beurling type invariant subspaces abstract let a be a maximal subdiagonal algebra in a finite von neumann algebra m if every right invariant subspace of a in the noncommutative hardy space h 2 is of beurling type then we say a is of type 1 we determine generators of these algebras and consider a riesz type factorization theorem for the noncommutative h 1 space we show that the right analytic toeplitz algebra on the noncommutative hardy space h p associated with a type 1 subdiagonal algebra with multiplicity 1 is hereditary reflexive

equilibrium schemes for scalar conservation laws with stiff sources we consider a simple model case of stiff source terms in hyperbolic conservation laws namely the case of scalar conservation laws with a zeroth order source with low regularity it is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced cfl condition and refined meshes required because of the lack of accuracy on equilibrium states the source term should be taken into account in the upwinding and discretized at the nodes of the grid in order to solve numerically the problem we introduce a so called equilibrium schemes with the properties that i the maximum principle holds true ii discrete entropy inequalities are satisfied iii steady state solutions of the problem are maintained one of the difficulties in studying the convergence is that there are no bv estimates for this problem we therefore introduce a kinetic interpretation of upwinding taking into account the source terms based on the kinetic formulation we give a new convergence proof that only uses property ii in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions the computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that in comparison with an usual upwind scheme the corresponding equilibrium version is far more accurate furthermore numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients

robustness of muscl schemes for 2d unstructured meshes we consider second order accuracy muscl schemes to approximate the solutions of hyperbolic system of conservation laws in the context of the 2d unstructured grids we propose a limitation procedure on the gradient reconstruction to enforce several stability properties we establish that the muscl scheme preserves the invariant domains and satisfy a set of entropy inequalities a conservation assumption is not useful in the present work to define the piecewise linear approximations and the proposed limitation can be understood as a systematic correction of the standard gradient reconstruction procedure the numerical method is applied to the compressible euler equations the gradient reconstruction is performed using the characteristic variables several numerical tests exhibit stability and robustness of the scheme

a fully well balanced positive and entropy satisfying godunov type method for the shallow water equations this work is devoted to the derivation of a fully well balanced numerical scheme for the well known shallow water model during the last two decades several well balanced strategies have been introduced with a special attention to the exact capture of the stationary states associated with the so called lake at rest by fully well balanced we mean here that the proposed godunov type method is also able to preserve stationary states with non zero velocity the numerical procedure is shown to preserve the positiveness of the water height and satisfies a discrete entropy inequality

a multi well balanced scheme for the shallow water mhd system with topography the shallow water magnetohydrodynamic system involves several families of physically relevant steady states in this paper we design a well balanced numerical scheme for the one dimensional shallow water magnetohydrodynamic system with topography that resolves exactly a large range of steady states two variants are proposed with slightly different families of preserved steady states they are obtained by a generalized hydrostatic reconstruction algorithm involving the magnetic field and with a cutoff parameter to remove singularities the solver is positive in height and semi discrete entropy satisfying which ensures the robustness of the method

relaxation approximation of the euler equations the aim of this paper is to show how solutions to the one dimensional compressible euler equations can be approximated by solutions to an enlarged hyperbolic system with a strong relaxation term the enlarged hyperbolic system is linearly degenerate and is therefore suitable to build an efficient approximate riemann solver from a theoretical point of view the convergence of solutions to the enlarged system towards solutions to the euler equations is proved for local in time smooth solutions we also show that arbitrarily large shock waves for the euler equations admit smooth shock profiles for the enlarged relaxation system in the end we illustrate these results of convergence by proposing a numerical procedure to solve the enlarged hyperbolic system we test it on various cases

a subsonic well balanced reconstruction scheme for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom in past years efficient well balanced methods have been proposed in order to well resolve solutions close to steady states at rest here we describe a strategy based on a local subsonic steady state reconstruction that allows one to derive a subsonic well balanced scheme preserving exactly all the subsonic steady states it generalizes the now well known hydrostatic solver and like the latter it preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality an application to the euler poisson system is proposed

higher composition laws ii on cubic analogues of gauss composition in our first article 2 we developed a new view of gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms moreover we showed that the groups arising from these composition laws were closely related to the class groups of orders in quadratic number fields while the spaces underlying those composition laws were closely related to certain exceptional lie groups in this paper our aim is to develop analogous laws of composition on certain spaces of forms so that the resulting groups yield information on the class groups of orders in cubic fields that is we wish to obtain genuine cubic analogues of gauss composition the fundamental object in our treatment of quadratic composition 2 was the space of 2 x 2 x 2 cubes of integers in particular gauss composition arose from the three different ways of slicing a cube a into two 2 x 2 matrices mi ni i 1 2 3 each such pair mi ni gives rise to a binary quadratic form qa x y qi x y defined by qi x y det mix niy the cube law of 2 declares that as a ranges over all cubes the sum of qi q2 q3 is zero it was shown in 2 that the cube law gives a law of addition on binary quadratic forms that is equivalent to gauss composition various other invariant theoretic constructions using the space of 2 x 2 x 2 cubes led to several new composition laws on other spaces of forms furthermore we showed that each of these composition laws gave rise to groups that are closely related to the class groups of orders in quadratic fields based on the quadratic case described above our first inclination for the cubic case might be to examine 3 x 3 x 3 cubes of integers a 3 x 3 x 3 cube c can be sliced in three different ways into three 3 x 3 matrices li mi ni i 1 2 3 we may therefore obtain from c three ternary cubic forms fl x y z f2 x y z f3 x y z defined by

on power bounded operators in finite von neumann algebras abstract it is proved that any power bounded operator of class c 1 in a finite von neumann algebra is conjugate to a unitary this solves a conjecture stated by i kovacs in 1970 an important ingredient of the proof is the study of completely positive projections on some operator space

reliability of first order numerical schemes for solving shallow water system over abrupt topography we compare some first order well balanced numerical schemes for shallow water system with special interest in applications where there are abrupt variations of the topography we show that the space step required to obtain a prescribed error depends on the method moreover the solutions given by the numerical scheme can be significantly different if not enough space resolution is used we shall pay special attention to the well known hydrostatic reconstruction technique where it is shown that the effect of large bottom discontinuities might be missed and a modification is proposed to avoid this problem

gauss composition for p1 and the universal jacobian of the hurwitz space of double covers abstract in this paper we give an explicit description of the moduli space of line bundles on hyperelliptic curves including singular curves we study the universal jacobian j 2 g n of degree n line bundles over the hurwitz stack of double covers of p 1 by a curve of genus g our main results are the construction of a smooth irreducible universally closed but not separated moduli compactification j bd 2 g n of j 2 g n whose points we describe simply and explicitly as sections of certain vector bundles on p 1 a description of the global geometry and moduli properties of these stacks and a computation of the picard groups of j bd 2 g n and j 2 g n in the cases when n g is even an important ingredient of our work is the parametrization of line bundles on double covers by binary quadratic forms this parametrization generalizes the classical number theoretic correspondence between ideal classes of quadratic rings and integral binary quadratic forms which in particular gives the group law on integral binary quadratic forms first discovered by gauss

a well balanced scheme for the numerical processing of source terms in hyperbolic equations in a variety of physical problems one encounters source terms that are balanced by internal forces and this balance supports multiple steady state solutions that are stable typical of these are gravity driven flows such as those described by the shallow water equations over a nonuniform ocean bottom 1 10 qquad h t hu x 0 text and hu t left hu 2 frac gh 2 2 right x ga x x h 0 many classic numerical schemes cannot maintain these steady solutions or achieve them in the long time limit with an acceptable level of accuracy because they do not preserve the proper balance between the source terms and internal forces we propose here a numerical scheme adapted to a scalar conservation law that preserves this balance and that can it is hoped be extended to more general hyperbolic systems the proof of convergence of this scheme toward the entropy solution is given and some numerical tests are reported

proof of the conjecture of a grothendieck on the fuglede kadison determinant let m be a von neumann algebra with a faithful semifinite trace for compact elements x m grothendieck introduced in seminaire bourbaki 1955 a function 1 x generalizing the fuglede kadison determinant and conjectured that 1 x y 1 x 1 y in this paper the inequality is proved as a corollary a direct proof of the clarkson inequalities is obtained

why many theories of shock waves are necessary kinetic functions equivalent equations and fourth order models we consider several systems of nonlinear hyperbolic conservation laws describing the dynamics of nonlinear waves in presence of phase transition phenomena these models admit under compressive shock waves which are not uniquely determined by a standard entropy criterion but must be characterized by a kinetic relation building on earlier work by lefloch and collaborators we investigate the numerical approximation of these models by high order finite difference schemes and uncover several new features of the kinetic function associated with physically motivated second and third order regularization terms especially viscosity and capillarity terms on one hand the role of the equivalent equation associated with a finite difference scheme is discussed we conjecture here and demonstrate numerically that the numerical kinetic function associated with a scheme approaches the analytic kinetic function associated with the given model especially since its equivalent equation approaches the regularized model at a higher order on the other hand we demonstrate numerically that a kinetic function can be associated with the thin liquid film model and the generalized camassa holm model finally we investigate to what extent a kinetic function can be associated with the equations of van der waals fluids whose flux function admits two inflection points

exactly well balanced discontinuous galerkin methods for the shallow water equations with moving water equilibrium hyperbolic conservation laws with source terms often admit steady state solutions where the fluxes and source terms balance each other to capture this balance and near equilibrium solutions well balanced methods have been introduced and performed well in many numerical tests shallow water equations have been extensively investigated as a prototype example in this paper we develop well balanced discontinuous galerkin methods for the shallow water system which preserve not only the still water at rest steady state but also the more general moving water equilibrium the key idea is the recovery of well balanced states a special source term approximation and the approximation of the numerical fluxes based on a generalized hydrostatic reconstruction we also study the extension of the positivity preserving limiter presented in 40 in this framework numerical examples are provided at the end to verify the well balanced property and good resolution for smooth and discontinuous solutions

invariant domain preserving discretization independent schemes and convex limiting for hyperbolic systems abstract we introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving the method is discretization independent provided elementary symmetry and skew symmetry properties are satisfied by the scheme the method is formally first order accurate in space then we introduce a series of higher order methods when these methods violate the invariant domain properties they are corrected by a limiting technique that we call convex limiting after limiting the resulting methods satisfy all the invariant domain properties that are imposed by the user see theorem 7 24 and is formally high order accurate the two key novelties are that i limiting is done by enforcing bounds on quasiconcave functionals ii the bounds that are enforced on the solution at each time step are necessarily satisfied by the low order approximation

factorization in subdiagonal algebras let m be a finite von neumann algebra and let a be a maximal subdiagonal algebra of m with respect to a faithful normal conditional expectation we show that ifsis an invertible operator in m then there exists an isometrywin m such that boths 1wandw sbelong to a we also give several characterizations of a maximal subdiagonal algebra a such that every invertible operatorsin m can be factored asua whereuis a unitary operator inmand bothaanda 1are in a

a multiphase godunov method for compressible multifluid and multiphase flows we propose a new model and a solution method for two phase compressible flows the model involves six equations obtained from conservation principles applied to each phase completed by a seventh equation for the evolution of the volume fraction this equation is necessary to close the overall system the model is valid for fluid mixtures as well as for pure fluids the system of partial differential equations is hyperbolic hyperbolicity is obtained because each phase is considered to be compressible two difficulties arise for the solution one of the equations is written in non conservative form non conservative terms exist in the momentum and energy equations we propose robust and accurate discretisation of these terms the method solves the same system at each mesh point with the same algorithm it allows the simulation of interface problems between pure fluids as well as multiphase mixtures several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible the method provides reliable results is able to compute strong shock waves and deals with complex equations of state

why many theories of shock waves are necessary convergence error in formally path consistent schemes we are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics and for solutions containing shock waves we investigate the convergence of finite difference schemes applied to such systems according to dal maso lefloch and murat s theory a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space in the present paper we consider schemes that are formally consistent with a given family of paths and we investigate their limiting behavior as the mesh is refined we first generalize to systems a property established earlier by hou and lefloch for scalar conservation laws and we prove that nonconservative schemes generate at the level of the limiting hyperbolic system an convergence error source term which provided the total variation of the approximations remains uniformly bounded is a locally bounded measure this convergence error measure is supported on the shock trajectories and as we demonstrate here is usually small in the special case that the scheme converges in the sense of graphs a rather strong convergence property often violated in practice then this measure source term vanishes we also discuss the role of the equivalent equation associated with a difference scheme here the distinction between scalar equations and systems appears most clearly since for systems the equivalent equation of a scheme that is formally path consistent depends upon the prescribed family of paths the core of this paper is devoted to investigate numerically the approximation of several simplified or full hyperbolic models arising in fluid dynamics this leads us to the conclusion that for systems having nonconservative products associated with linearly degenerate characteristic fields the convergence error vanishes for more general models this measure is evaluated very accurately especially by plotting the shock curves associated with each scheme under consideration as we demonstrate plotting the shock curves provide a convenient approach for evaluating the range of validity of a given scheme

noncommutative hardy space associated with semi finite subdiagonal algebras abstract let m be a von neumann algebra with a faithful normal semi finite trace and let a be maximal subdiagonal algebra of m then for 0 p we define the noncommutative h p space we obtain that the conjugation and herglotz maps are bounded linear maps from l p m into l p m for 1 p and continuous map from l 1 m into l 1 m we also give the dual space of h p a for 1 p and extend pisier s interpolation theorem to this case

convergence of the 2 2 godunov method for a general resonant nonlinear balance law we introduce a generalized solution of the riemann problem for a general resonant nonlinear balance law and we prove the convergence of the 2 x 2 godunov numerical method based on these solutions in particular we obtain generic conditions that guarantee a canonical structure for the elementary waves in the solution of the riemann problem and an interesting multiplicity of time asymptotic wave patterns is observed and characterized

a riemann problem in gas dynamics with bifurcation abstract we construct the solution of the riemann problem for 2 2 isothermal gas dynamics in a duct with discontinuous diameter besides shocks and rarefaction there are standing waves the solution exists globally it is obtained as an asymptotic solution for an appropriate cauchy problem with continuous data in certain cases bifurcation occurs and there are three solutions one of which is unstable this is an example of a riemann problem whose solution depends discontinuously on the initial data

a well balanced scheme for the shallow water equations with topography or manning friction we consider the shallow water equations with manning friction or topography as well as a combination of both these source terms the main purpose of this work concerns the derivation of a non negativity preserving and well balanced scheme that approximates solutions of the system and preserves the associated steady states including the moving ones in addition the scheme has to deal with vanishing water heights and transitions between wet and dry areas to address such issues a particular attention is paid to the study of the steady states related to the friction source term then a godunov type scheme is obtained by using a relevant average of the source terms in order to enforce the required well balance property an implicit treatment of both topography and friction source terms is also exhibited to improve the scheme while dealing with vanishing water heights a second order well balanced muscl extension is designed as well as an extension for the two dimensional case numerical experiments are performed in order to highlight the properties of the scheme

relaxation schemes for nonlinear kinetic equations a class of numerical schemes for nonlinear kinetic equations of boltzmann type is described following wild s approach the solution is represented as a power series with parameter depending exponentially on the knudsen number this permits us to derive accurate and stable time discretizations for all ranges of the mean free path these schemes preserve the main physical properties positivity conservation of mass momentum and energy moreover for some particular models the entropy property is also shown to hold

higher composition laws i a new view on gauss composition and quadratic generalizations two centuries ago in his celebrated work disquisitiones arithmeticae of 1801 gauss laid down the beautiful law of composition of integral binary quadratic forms which would play such a critical role in number theory in the decades to follow even today two centuries later this law of composition still remains one of the primary tools for understanding and computing with the class groups of quadratic orders it is hence only natural to ask whether higher analogues of this composition law exist that could shed light on the structure of other algebraic number rings and fields this article forms the first of a series of four articles in which our aim is precisely to develop such higher composition laws in fact we show that gauss s law of composition is only one of at least fourteen composition laws of its kind which yield information on number rings and their class groups in this paper we begin by deriving a general law of composition on 2 2 2 cubes of integers from which we are able to obtain gauss s composition law on binary quadratic forms as a simple special case in a manner reminiscent of the group law on plane elliptic curves we also obtain from this composition law on 2 2 2 cubes four further new laws of composition these laws of composition are defined on 1 binary cubic forms 2 pairs of binary quadratic forms 3 pairs of quaternary alternating 2 forms and 4 senary six variable alternating 3 forms more precisely gauss s theorem states that the set of sl2 z equivalence classes of primitive binary quadratic forms of a given discriminant d has an inherent group structure the five other spaces of forms mentioned above including the space of 2 2 2 cubes also possess natural actions by special linear groups over z and certain products thereof we prove that just like gauss s space of binary quadratic forms each of these group actions has the following remarkable properties first each of these six spaces possesses only a single polynomial invariant for the corresponding group action which we call the discriminant this discriminant invariant is found to take only values that

logmodularity and isometries of operator algebras we generalize some facts about function algebras to operator algebras using the noncommutative shilov boundary or c envelope first considered by arveson in the first part we study and characterize complete isometries between operator algebras in the second part we introduce and study a notion of logmodularity for operator algebras we also give a result on conditional expectations many miscellaneous applications are provided

analysis and approximation of conservation laws with source terms we consider a conservation law of the form cl i ut f u x ax where a cdot is a bounded piecewise smooth source term and f an even convex function we first characterize the solution to the riemann problem through a new lax type formula then we prove that for a cdot fixed the semigroup associated with cl is an l1 contraction and we obtain an existence theorem for weak solutions to cl we conclude by constructing godunov type difference schemes and proving that these schemes are l infty stable and have stable steady solutions similar in structure to those of cl some numerical tests are reported

interpolation and φ moment inequalities of noncommutative martingales this paper is devoted to the study of moment inequalities for noncommutative martingales in particular we prove the noncommutative moment analogues of martingale transformations stein s inequalities khintchine s inequalities for rademacher s random variables and burkholder gundy s inequalities the key ingredient is a noncommutative version of marcinkiewicz type interpolation theorem for orlicz spaces which we establish in this paper

on positivity preserving finite volume schemes for euler equations we consider the positivity preserving property of first and higher order finite volume schemes for one and two dimensional euler equations of gas dynamics a general framework is established which shows the positivity of density and pressure whenever the underlying one dimensional first order building block based on an exact or approximate riemann solver and the reconstruction are both positivity preserving appropriate limitation to achieve a high order positivity preserving reconstruction is described

noncommutative kothe duality using techniques drawn from the classical theory of rearrangement invariant banach function spaces we develop a duality theory in the sense of kothe for symmetric banach spaces of measurable operators affiliated with a semifinite von neumann algebra equipped with a distinguished trace a principal result of the paper is the identification of the kothe dual of a given banach space of measurable operators in terms of normality

factorization problems for nests factorization methods and characterizations of the universal factorization property let m be a von neumann algebra and let be a nest in m we consider the problem of factoring an invertible element s of m as s wa where w is a unitary element in m and both a and a 1 are elements of m which also belong to alg it is known that this is not always possible however using a variant of the lu decomposition for matrices we show that if is an injective nest and s is an invertible operator then there exists an isometry w such that both s 1w and w s belong to alg we characterize when an invertible operator factors with respect to an injective nest we also prove a result which simultaneously generalizes results of arveson and gohberg and krein finally we give a number of characterizations of those nests within a factor such that every invertible operator factors with respect to the nest

the resonant cases and the riemann problem for a model of two phase flows abstract the resonant cases in the riemann problem for a model of two phase flows with resonance are investigated which complete the construction of riemann solutions the model is given by a nonconservative hyperbolic system of balance laws the phase decomposition method is developed to construct solutions in each phase in a relatively separated way but still glued to each other via the contact discontinuities composite wave curves can be built and intersections of wave curves in each phase plane help to determine various configurations of riemann solutions in particular solutions containing multiple waves propagating with the same shock speed are constructed

high order positivity preserving kinetic schemes for the compressible euler equations we present a new class of high order kinetic flux splitting schemes for the compressible euler equations and we prove that these schemes are positivity preserving i e rho and t remain geq 0 the first order kinetic scheme is based on the maxwellian equilibrium function and was initially proposed by pullin j comput phys 34 1980 pp 231 244 our higher order extension can be seen as a variant of the corrected antidiffusive flux approach the necessity of a limitation on the antidiffusive correction appears naturally in order to satisfy the constraint of positivity

nest algebras and similarity transformations in recent years the theory of algebras of operators on hilbert space has been stimulated by developments in the theory of quasitriangularity andersen 1 has shown that up to unitary equivalence there is only one continuous quasitrianguilar algebra we use this to provide the following answer to a question posed by j r ringrose approximately 20 years ago similar continuous nests on separable hilbert space can fail to be unitarily equivalent theorem 2 2 a consequence is the existence of a nonhyperintransitive compact operator corollary 2 3 which answers a question of kadison and singer 12 and of gohberg and krein 11 we extend our initial theorem to show that arbitrary continuous nests are similar theorem 2 10 and that every maximal nest is similar to a multiplicity one nest theorem 2 11 a consequence is that every compact operator is similar to a hyperintransitive compact operator corollary 2 12 the similarity transformation can be induced by an arbitrarily small compact perturbation of a unitary operator the methods of section 2 apply only to the continuous parts of nests for general results the atomic core part must be dealt with in section 3 different methods are developed for this purpose again utilizing andersen s results these are used in section 4 to prove that a complete nest x admits an arveson factorization for every positive invertible operator if and only if x is countable as a family of subspaces theorem 4 7 a consequence is that if x is an uncountable nest with atomic core then some similarity transformation of x has a continuous part this could not be deduced from section 2 these methods also yield a weak factorization result which concludes the paper it should be noted that a negative resolution to the ringrose question was conjectured in recent years by several mathematicians including w arveson and j ringrose also many of the results presented in this paper were announced in an a m s bulletin article 16 finally we wish to thank the referee for suggesting that the original manuscript could be condensed and improved

from picard groups of hyperelliptic curves to class groups of quadratic fields let c be a hyperelliptic curve defined over mathbb q whose weierstrass points are defined over extensions of mathbb q of degree at most three and at least one of them is rational generalizing a result of r soleng in the case of elliptic curves we prove that any line bundle of degree 0 on c which is not torsion can be specialised into ideal classes of imaginary quadratic fields whose order can be made arbitrarily large this gives a positive answer for such curves to a question by agboola and pappas

fuglede kadison determinant theme and variations abstract we review the definition of determinants for finite von neumann algebras due to fuglede and kadison fuglede b kadison r 1952 ann math 55 520 530 and a generalization for appropriate groups of invertible elements in banach algebras from a paper by skandalis and the author 1984 after some discussion of k theory and whitehead torsion we indicate the relevance of these determinants to the study of torsion in topology contents are as follows 1 the classical setting 2 on von neumann algebras and traces 3 fuglede kadison determinant for finite von neumann algebras 4 motivating question 5 brief reminder of and bott periodicity 6 revisiting the fuglede kadison and other determinants 7 on whitehead torsion 8 a few lines on torsion