statistics of shocks in solutions of inviscid burgers equation the purpose of this paper is to analyze statistical properties of discontinuities of solutions of the inviscid burgers equation having a typical realizationb y of the brownian motion as an initial datum this case was proposed and studied numerically in the companion paper by she aurell and frisch the description of the statistics is given in terms of the behavior of the convex hull of the random process w y int limits 0 y b eta eta d eta the hausdorff dimension of the closed set of thosey where the convex hull coincides withw is also studied

a fast phase space method for computing creeping rays creeping rays can give an important contribution to the solution of medium to high frequency scattering problems they are generated at the shadow lines of the illuminated scatterer by grazing incident rays and propagate along geodesics on the scatterer surface continuously shedding diffracted rays in their tangential direction in this paper we show how the ray propagation problem can be formulated as a partial differential equation pde in a three dimensional phase space to solve the pde we use a fast marching method the pde solution contains information about all possible creeping rays this information includes the phase and amplitude of the field which are extracted by a fast post processing computationally the cost of solving the pde is less than tracing all rays individually by solving a system of ordinary differential equations we consider an application to mono static radar cross section problems where creeping rays from all illumination angles must be computed the numerical results of the fast phase space method and a comparison with the results of ray tracing are presented

operator algebras with contractive approximate identities weak compactness and the spectrum abstract we continue our study of operator algebras with contractive approximate identities cais by presenting a couple of interesting examples of operator algebras with cais which in particular answer questions raised in previous papers in this series for example about whether roughly speaking weak compactness of an operator algebra or the lack of it can be seen in the spectra of its elements

a field space based level set method for computing multi valued solutions to 1d euler poisson equations we present a field space based level set method for computing multi valued solutions to one dimensional euler poisson equations the system of these equations has many applications and in particular arises in semiclassical approximations of the schrodinger poisson equation the proposed approach involves an implicit eulerian formulation in an augmented space called field space which incorporates both velocity and electric fields into the configuration both velocity and electric fields are captured through common zeros of two level set functions which are governed by a field transport equation simultaneously we obtain a weighted density f by solving again the field transport equation but with initial density as starting data the averaged density is then resolved by the integration of the obtained f against the dirac delta function of two level set functions in the field space moreover we prove that such obtained averaged density is simply a linear superposition of all multi valued densities and the averaged field quantities are weighted superposition of corresponding multi valued ones computational results are presented and compared with some exact solutions which demonstrate the effectiveness of the proposed method

a local level set method for paraxial geometrical optics we propose a local level set method for constructing the geometrical optics term in the paraxial formulation for the high frequency asymptotics of two dimensional 2 d acoustic wave equations the geometrical optics term consists of two multivalued functions a travel time function satisfying the eikonal equation locally and an amplitude function solving a transport equation locally the multivalued travel times are obtained by solving a level set equation and a travel time equation with a forcing term the multivalued amplitudes are computed by a neweulerian formula based on the gradients of travel times and takeoff angles as a byproduct the method is also able to capture the caustic locations the proposed eulerian method has complexity of o n 2 rm log n rather than o n 4 as typically seen in the lagrangian ray tracing method several examples including the well known marmousi synthetic model illustrate the accuracy and efficiency of the eulerian method

best approximation in c algebras on developpe des methodes permettant d obtenir les meilleures approximations des ideaux de sous algebres et de sous espaces des c algebres

on language equations with invertible operations the paper studies language equations of the type x l r and l y r where l and r are given languages and is an invertible binary word language operation for most of the considered insertion and deletion operations the existence of both a solution and a singleton solution to these equations proves to be decidable for given regular l and r in case l is a context free language and r is a regular one the existence of a solution is generally undecidable the results can be extended to more complex linear equations systems of linear equations as well as for equations of higher degree

characterizations of ordered operator spaces abstract we demonstrate new abstract characterizations for unital and non unital operator spaces we characterize unital operator spaces in terms of the cone of accretive operators operators whose real part is positive we show that matrix norms and accretive cones are induced by gauges although inducing gauges are not unique in general finally we show that completely positive completely contractive linear maps on non unital operator spaces extend to any containing operator system if and only if the operator space is induced by a unique gauge

upwind finite difference calculation of traveltimes seismic traveltimes can be computed efficiently on a regular grid by an upwind finite difference method the method solves a conservation law that describes changes in the gradient components of the traveltime field the traveltime field itself is easily obtained from the solution of the conservation law by numerical integration the conservation law derives from the eikonal equation and its solution depicts the first arrival time field the upwind finite difference scheme can be implemented in fully vectorized form in contrast to a similar scheme proposed recently by vidale the resulting traveltime field is useful both in kirchhoff migration and modeling and in seismic tomography many reliable methods exist for the numerical solution of conservation laws which appear in fluid mechanics as statements of the conservation of mass momentum etc a first order upwind finite difference scheme proves accurate enough for seismic applications upwind schemes are stable because they mimic the behavior of flu

realizable high order finite volume schemes for quadrature based moment methods dilute gas particle flows can be described by a kinetic equation containing terms for spatial transport gravity fluid drag and particle particle collisions however direct numerical solution of kinetic equations is often infeasible because of the large number of independent variables an alternative is to reformulate the problem in terms of the moments of the velocity distribution recently a quadrature based moment method was derived for approximating solutions to kinetic equations the success of the new method is based on a moment inversion algorithm that is used to calculate non negative weights and abscissas from the moments the moment inversion algorithm does not work if the moments are non realizable which might lead to negative weights it has been recently shown 14 that realizability is guaranteed only with the 1st order finite volume scheme that has an inherent problem of excessive numerical diffusion the use of high order finite volume schemes may lead to non realizable moments in the present work realizability of the finite volume schemes in both space and time is discussed for the 1st time a generalized idea for developing realizable high order finite volume schemes for quadrature based moment methods is presented these finite volume schemes give remarkable improvement in the solutions for a certain class of problems it is also shown that the standard runge kutta time integration schemes do not guarantee realizability however realizability can be guaranteed if strong stability preserving ssp runge kutta schemes are used numerical results are presented on both cartesian and triangular meshes

the use of a legendre pseudospectral viscosity technique to solve a class of nonlinear dynamic hamilton jacobi equations in this research we study the problem of finding the approximate solution of a class of hamilton jacobi equations namely the eikonal equation we employ the legendre pseudospectral viscosity method to solve this problem this method basically consists of adding a spectral viscosity to the equation this spectral viscosity which is sufficiently small to retain the formal spectral accuracy is large enough to stabilize the numerical scheme several test problems are considered and the numerical results are given to show the efficiency of the proposed method

numerical solution of the high frequency asymptotic expansion for the scalar wave equation new numerical methods are derived for calculation of high frequency asymptotic expansion of the scalar wave equation the nonlinear partial differential equations defining the terms in the expansio

multi phase computations of the semiclassical limit of the schrodinger equation and related problems whitham vs wigner abstract we present and compare two different techniques to obtain the multi phase solutions for the schrodinger equation in the semiclassical limit the first is whitham s averaging method which gives the modulation equations governing the evolution of multi phase solutions the second is the wigner transform a convenient tool to derive the semiclassical limit equation in the phase space the vlasov equation for the linear schrodinger equation motivated by the linear superposition principle we derive and prove the multi phase ansatz for the wigner function by the stationary phase method and then use the ansatz to close the moment equations of the vlasov equation and obtain the multi phase equations in the physical space we show that the multi phase equations so derived agree with those derived by whitham s averaging method which can be proved using different arguments generic way of obtaining and computing the multi phase equations by the wigner function is given and kinetic schemes are introduced to solve the multi phase equations the numerical schemes based on multiphase closure of the liouville equation are purely eulerian and only operate in the physical space several numerical examples are given to explore the validity of this approach similar studies are conducted for the linearized korteweg de vries equation and the linear wave equation

on restricted perturbations in inverse images and a description of normalizer algebras in c algebras abstract let e and f be banach spaces and t a bounded linear map from e into f using a certain perturbation function t we find a useful sufficient criterion that t maps the closed unit ball onto a closed set applying this to the quotient map from a c algebra a onto its quotient banach space a l r by the sum l r of closed left and right ideals l and r of a we obtain that the closed unit ball of a maps onto the closed unit ball of a l r this gives an independent description of the images of the right normalizer algebra n r d and the normalizer algebra n d n l d n r d of a hereditary c subalgebra d of a by the quotient map from a onto a cl ad cl da we use this to prove a necessary and sufficient criterion for a c algebra b to be a c quotient algebra of a c subalgebra of a c algebra a the latter criterion will be applied in the situation where a equals the car algebra in some forthcoming papers cf section 6 for more details

a characterization of operator algebras abstract an operator algebra is a uniformly closed algebra of bounded operators on a hilbert space in this paper we give a characterization of unital operator algebras in terms of their matricial norm structure more precisely if a is an l matricially normed space and also an algebra with a completely contractive multiplication and an identity of norm 1 then there is a completely isometric isomorphism of a onto a unital operator algebra indeed the multiplication on a need not be assumed to be associative for this conclusion to follow examples are given to show that the condition on the identity is necessary it follows from the above that the quotient of an operator algebra by a closed two sided ideal with the natural matricial structure is again an operator algebra up to complete isometric isomorphism

aggregation rates in one dimensional stochastic systems with adhesion and gravitation we consider one dimensional systems of self gravitating sticky particles with random initial data and describe the process of aggregation in terms of the largest cluster size l n at any fixed time prior to the critical time the asymptotic behavior of l n is also analyzed for sequences of times tending to the critical time a phenomenon of phase transition shows up namely for small initial particle speeds cold gas l n has logarithmic order of growth while higher speeds warm gas yield polynomial rates for l n

optimizing schema languages for xml numerical constraints and interleaving the presence of a schema offers many advantages in processing translating querying and storage of xml data basic decision problems such as equivalence inclusion and nonemptiness of intersection of schemas form the basic building blocks for schema optimization and integration and algorithms for static analysis of transformations it is thereby paramount to establish the exact complexity of these problems most common schema languages for xml can be adequately modeled by some kind of grammar with regular expressions at right hand sides in this paper we observe that apart from the usual regular operators of union concatenation and kleene star schema languages also allow numerical occurrence constraints and interleaving operators although the expressiveness of these operators remains within the regular languages the presence or absence of these operators has a significant impact on the complexity of the basic decision problems we present a complete overview of the complexity of the basic decision problems for dtds xsds and relax ng with regular expressions incorporating numerical occurrence constraints and interleaving we also discuss chain regular expressions and the complexity of the schema simplification problem incorporating the new operators

computing multi valued physical observables for the high frequency limit of symmetric hyperbolic systems we develop a level set method for the computation of multi valued physical observables density velocity energy etc for the high frequency limit of symmetric hyperbolic systems in any number of space dimensions we take two approaches to derive the method the first one starts with a weakly coupled system of an eikonal equation for phase s and a transport equation for density r ts h x s 0 t x r xr n email protected email protected x email protected ph x xs 0 the main idea is to evolve the density near the n dimensional bi characteristic manifold of the eikonal hamiltonian jacobi equation which is identified as the common zeros of n level set functions in phase space x k r 2 n these level set functions are generated from solving the liouville equation with initial data chosen to embed the phase gradient simultaneously we track a new quantity email protected t x k det email protected by solving again the liouville equation near the obtained zero level set f 0 but with initial density as initial data the multi valued density and higher moments are thus resolved by integrating f along the bi characteristic manifold in the phase directions the second one uses the high frequency limit of symmetric hyperbolic systems derived by the wigner transform this gives rise to liouville equations in the phase space with measure valued solution in its initial data due to the linearity of the liouville equation we can decompose the density distribution into products of function each of which solves the liouville equation with l initial data on any bounded domain it yields higher order moments such as energy and energy flux the main advantages of these new approaches in contrast to the standard kinetic equation approach using the liouville equation with a dirac measure initial data include 1 the liouville equations are solved with l initial data and a singular integral involving the email protected function is evaluated only in the post processing step thus avoiding oscillations and excessive numerical smearing 2 a local level set method can be utilized to significantly reduce the computation in the phase space these methods can be used to compute all physical observables for multi dimensional problems our method applies to the wave fields corresponding to simple eigenvalues of the dispersion matrix one such example is the wave equation which will be studied numerically in this paper

a geometrical optics based numerical method for high frequency electromagnetic fields computations near fold caustics part i this paper presents an eulerian numerical method for the computation of a bi valued solution of hamilton jacobi type equation in a particular geometric setting more precisely we consider high frequency electromagnetic fields in the vicinity of fold caustics

finite difference solution of the eikonal equation along expanding wavefronts we show that a scheme to solve the 2 d eikonal equation by a finite difference method can violate causality for moderate to large velocity contrasts v2 v1 2 as an alternative we present a finite difference scheme in which the solution region progresses outward from an expanding wavefront rather than an expanding square and therefore honors causality our method appears to be stable and reasonably accurate for a variety of velocity models with moderate to large velocity contrasts the penalty is a large increase in computational cost and programming effort

kirchhoff migration using eikonal based computation of traveltime source derivatives abstractthe computational efficiency of kirchhoff type migration can be enhanced by using accurate traveltime interpolation algorithms we addressed the problem of interpolating between a sparse source sampling by using the derivative of traveltime with respect to the source location we adopted a first order partial differential equation that originates from differentiating the eikonal equation to compute the traveltime source derivatives efficiently and conveniently unlike methods that rely on finite difference estimations the accuracy of the eikonal based derivative did not depend on input source sampling for smooth velocity models the first order traveltime source derivatives enabled a cubic hermite traveltime interpolation that took into consideration the curvatures of local wavefronts and can be straightforwardly incorporated into kirchhoff antialiasing schemes we provided an implementation of the proposed method to first arrival traveltimes by modifying the fast marching eikonal solver severa

rapid calculation of shuffles of two words resume un mot c du monoide libre c d alphabet c est appele un melange de deux mots u et de c s il existe k 0 et des mots u1 uk 1 k de c tels que u u1 uk 1 k et c u1 1u2 2 uk k nous donnons ici un algorithme qui determine les melanges de deux mots sans omission ni repetition

a quadrature based third order moment method for dilute gas particle flows dilute gas particle flows can be described by a kinetic equation containing terms for spatial transport gravity fluid drag and particle particle collisions however the direct numerical solution of the kinetic equation is intractable for most applications due to the large number of independent variables a useful alternative is to reformulate the problem in terms of the moments of the velocity distribution function closure of the moment equations is challenging for flows away from the equilibrium maxwellian limit in this work a quadrature based third order moment closure is derived that can be applied to gas particle flows at any knudsen number a key component of quadrature based closures is the moment inversion algorithm used to find the weights and abscissas a robust inversion procedure is proposed for moments up to third order and tested for three example applications riemann shock problem impinging jets and vertical channel flow extension of the moment inversion algorithm to fifth or higher order is possible but left to future work the spatial fluxes in the moment equations are treated using a kinetic description and hence a gradient diffusion model is not used to close the fluxes because the quadrature based moment method employs the moment transport equations directly instead of a discretized form of the boltzmann equation the mass momentum and energy are conserved for arbitrary knudsen number including the euler limit while developed here for dilute gas particle flows quadrature based moment methods can in principle be applied to any application that can be modeled by a kinetic equation e g thermal and non isothermal flows currently treated using lattice boltzmann methods and examples are given from the literature

on numerical ranges and roots existence of the fractional powers is established in banach algebra setting in terms of the numerical ranges of elements involved the behavior of the spectra and for hermitian algebras satisfying some additional hypotheses the numerical range under taking these powers also is investigated 2003 elsevier science usa all rights reserved

gaussian beam methods for the schrodinger equation with discontinuous potentials we propose eulerian and lagrangian gaussian beam methods for the schrodinger equation with discontinuous potentials at the quantum barriers where the potential is discontinuous we derive suitable interface conditions to account for quantum scattering information these scattering interface conditions are then built into the numerical fluxes in the eulerian level set formulation of the gaussian beam methods and are also used in the lagrangian formulation including an interface condition for the hessian matrix we carry both 1d and 2d numerical examples to verify the accuracy of the method

computing multivalued physical observables for the semiclassical limit of the schrodinger equation we develop a level set method for the computation of multivalued physical observables density velocity etc for the semiclassical limit of the schrodinger equation this method uses an eulerian formulation and applies directly to arbitrary number of space dimensions the main idea is to evolve the density near an n dimensional manifold that is identified as the common zeros of n level set functions in phase space these level set functions are generated from solving the liouville equation with initial data chosen to embed the phase gradient simultaneously we track a new quantity f by solving again the liouville equation near the obtained zero level set but with initial density as initial data the multivalued density and higher moments are thus resolved by integrating f along the n dimensional manifold in the phase directions we show that this is equivalent to using the wigner approach but decomposing the velocity from the density each of which evolves by the same liouville equation the main advantages of this approach in contrast to the standard kinetic equation approach using the liouville equation with a dirac measure initial data include 1 the liouville equations are solved with l initial data and a singular integral involving the email protected function is evaluated only in the post processing step thus avoiding oscillations and excessive numerical smearing 2 a local level set method can be utilized to significantly reduce the computation in the phase space these advantages allow us to compute for the first time all physical observables for multidimensional problems in an eulerian framework

extensions of operator algebras i abstract we transcribe a portion of the theory of extensions of c algebras to general operator algebras we also include several new general facts about approximately unital ideals in operator algebras and the c algebras which they generate

a hamiltonian preserving scheme for the liouville equation of geometrical optics with partial transmissions and reflections we construct a class of hamiltonian preserving numerical schemes for the liouville equation of geometrical optics with partial transmissions and reflections this equation arises in the high frequency limit of the linear wave equation with a discontinuous index of refraction in our previous work hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds j comput phys 214 2006 pp 672 697 we introduced the hamiltonian preserving schemes for the same equation when only complete transmissions or reflections occur at the interfaces these schemes are extended in this paper to the general case of partial transmissions and reflections the key idea is to build into the numerical flux the behavior of waves at the interface namely partial transmissions and reflections that satisfy snell s law of refraction with the correct transmission and reflection coefficients this scheme allows a hyperbolic stability condition under which positivity and stabilities in both l 1 and l infty norms are established numerical experiments are carried out to study the numerical accuracy

concentration and cavitation in the vanishing pressure limit of solutions to the euler equations for nonisentropic fluids numerical simulations 2 d riemann problem in gas dynamics and formation of spiral in nonlinear problems in engineering and science numerical and analytical approach beijing 1991 science press beijing 1992 pp 167 179 discrete contin dyn syst 1 1995 555 584 6 2000 419 430 for the euler equations for gas dynamics in the regime of small pressure showed that for one case the particles seem to be more sticky and tend to concentrate near some shock locations and for the other case in the region of rarefaction waves the particles seem to be far apart and tend to form cavitation in the region in this paper we identify and analyze the phenomena of concentration and cavitation by studying the vanishing pressure limit of solutions of the full euler equations for nonisentropic compressible fluids with a scaled pressure it is rigorously shown that any riemann solution containing two shocks and possibly one contact discontinuity to the euler equations for nonisentropic fluids tends to a shock solution to the corresponding transport equations and the intermediate densities between the two shocks tend to a weighted measure that along with the two shocks and possibly contact discontinuity forms the shock as the pressure vanishes by contrast it is also shown that any riemann solution containing two rarefaction waves and possibly one contact discontinuity to the euler equations for nonisentropic fluids tends to a two contact discontinuity solution to the transport equations and the nonvacuum intermediate states between the two rarefaction waves tend to a vacuum state as the pressure vanishes some numerical results exhibiting the processes of concentration and cavitation are presented as the pressure decreases 2003 elsevier b v all rights reserved msc 35l65 35b30 76e19 35q35 35l67 35b25 65m06

algorithmic decomposition of shuffle on words we investigate shuffle decomposability into two words we give an algorithm which takes as input a dfa m under certain conditions and determines the unique candidate decomposition into words u and v such that l m uv if m is shuffle decomposable in time o u v even though this algorithm does not determine whether or not the dfa is shuffle decomposable the sublinear time complexity of only determining the two words under the assumption of decomposability is surprising given the complexity of shuffle and demonstrates an interesting property of the operation we also show that for given words u and v and a dfa m we can determine whether uv l m in polynomial time

shuffle factorization is unique we prove that given a finite set of words s there exists at most one normalized multiset p such that s is the shuffle of the words in p the multiset p is effectively computable

using k branch entropy solutions for multivalued geometric optics computations this paper is devoted to a numerical simulation of the classical wkb system arising in geometric optics expansions it contains the nonlinear eikonal equation and a linear conservation law whose coefficient can be discontinuous we address the problem of treating it in such a way that superimposed signals can be reproduced by means of the kinetic formulation of multibranch solutions originally due to brenier and corrias some existence and uniqueness results are given together with computational test cases of increasing difficulty displaying up to five multivaluations

conditional quadrature method of moments for kinetic equations kinetic equations arise in a wide variety of physical systems and efficient numerical methods are needed for their solution moment methods are an important class of approximate models derived from kinetic equations but require closure to truncate the moment set in quadrature based moment methods qbmm closure is achieved by inverting a finite set of moments to reconstruct a point distribution from which all unclosed moments e g spatial fluxes can be related to the finite moment set in this work a novel moment inversion algorithm based on 1 d adaptive quadrature of conditional velocity moments is introduced and shown to always yield realizable distribution functions i e non negative quadrature weights this conditional quadrature method of moments cqmom can be used to compute exact n point quadratures for multi valued solutions also known as the multi variate truncated moment problem and provides optimal approximations of continuous distributions in order to control numerical errors arising in volume averaging and spatial transport an adaptive 1 d quadrature algorithm is formulated for use with cqmom the use of adaptive cqmom in the context of qbmm for the solution of kinetic equations is illustrated by applying it to problems involving particle trajectory crossing i e collision less systems elastic and inelastic particle particle collisions and external forces i e fluid drag

eulerian droplet model delta shock waves and solution of the riemann problem abstract we study an eulerian droplet model which can be seen as the pressureless gas system with a source term a subsystem of this model and the inviscid burgers equation with source term the condition for loss of regularity of a solution to burgers equation with source term is established the same condition applies to the eulerian droplet model and its subsystem the riemann problem for the eulerian droplet model is constructively solved by going through the solution of the riemann problems for the inviscid burgers equation with a source term and the subsystem respectively under suitable generalized rankine hugoniot relations and entropy condition the existence of delta shock solution is established the existence of a solution to the generalized rankine hugoniot conditions is proven some numerical illustrations are presented

two dimensional riemann problems for zero pressure gas dynamics with three constant states abstract the riemann problems for two dimensional zero pressure gas dynamics are solved completely when the initial data take three constant states having discontinuities on x y positive and x negative axes with the help of characteristic analysis by studying interactions among delta shocks vacuums and contact discontinuities the riemann solutions constructed exhibit nine different explicit configurations the mach reflection like configurations appear in some solutions

multi phase computations in geometrical optics in this work we propose a new set of partial differential equations pdes which can be seen as a generalization of the classical eikonal and transport equations to allow for solutions with multiple phases the traditional geometrical optics pair of equations suffer from the fact that the class of physically relevant solutions is limited in particular it does not include solutions with multiple phases corresponding to crossing waves our objective has been to generalize these equations to accommodate solutions containing more than one phase the new equations are based on the same high frequency approximation of the scalar wave equation as the eikonal and the transport equations however they also incorporate a finite superposition principle the maximum allowed number of intersecting waves in the solution can be chosen arbitrarily but a higher number means that a larger system of pdes must be solved the pdes form a hyperbolic system of conservation laws with source terms although the equations are only weakly hyperbolic and thus not well posed in the strong sense several examples show the viability of solving the equations numerically the technique we use to capture multivalued solutions is based on a closure assumption for a system of equations representing the moments

convergence results for an inhomogeneous system arising in various high frequency approximations this paper is devoted to both theoretical and numerical study of a system involving an eikonal equation of hamilton jacobi type and a linear conservation law as it comes out of the geometrical optics expansion of the wave equation or the semiclassical limit for the schrodinger equation we first state an existence and uniqueness result in the framework of viscosity and duality solutions then we study the behavior of some classical numerical schemes on this problem and we give sufficient conditions to ensure convergence as an illustration some practical computations are provided

direct computation of multivalued phase space solutions for hamilton jacobi equations a method is presented for the exact and complete resolution of hamiltonian systems only using the corresponding hamilton jacobi equation we show that it possible to get the ode solution from the corresponding pde

on the uniqueness of shuffle on words and finite languages we investigate a special variant of the shuffle decomposition problem for regular languages namely when the given regular language is the shuffle of finite languages the shuffle decomposition into finite languages is in general not unique that is there are l 1 l 2 l 3 l 4 with but l 1 l 2 l 3 l 4 however if all four languages are singletons with at least two combined letters it follows by a result of berstel and boasson j berstel l boasson shuffle factorization is unique theoretical computer science 273 2002 47 67 that the solution is unique that is l 1 l 2 l 3 l 4 we further show that if l 1 and l 2 are arbitrary finite sets and l 3 and l 4 are singletons with at least two letters in each the solution is unique therefore shuffle decomposition of words is unique not only over words but over arbitrary sets this is strong as we cannot let all four be arbitrary finite sets hopefully the obtained results will help to better understand the very nature of the shuffle operation

the inviscid burgers equation with initial data of brownian type the solutions to burgers equation in the limit of vanishing viscosity are investigated when the initial velocity is a brownian motion or fractional brownian motion function i e a gaussian process with scaling exponent 0 h 1 typea or the derivative thereof with scaling exponent 1 h 0 typeb largesize numerical experiments are performed helped by the fact that the solution is essentially obtained by performing a legendre transform the main result is obtained for typea and concerns the lagrangian functionx a which gives the location at timet 1 of the fluid particle which started at the locationa it is found to be a complete devil s staircase the cumulative probability of lagrangian shock intervals a also the distribution of shock amplitudes follows a a h law for small a the remaining regular lagrangian locations form a cantor set of dimensionh in eulerian coordinates the shock locations are everywhere dense the scaling properties of various statistical quantities are also found heuristic interpretations are provided for some of these results rigorous results for the case of brownian motion are established in a companion paper by ya sinai for typeb initial velocities e g white noise there are very few small shocks and shock locations appear to be isolated finally it is shown that there are universality classes of random but smooth non scaling initial velocities such that the long time large scale behavior is after rescaling the same as for typea orb

facial structure in operator algebra theory we give a complete description of the closed faces in twelve kinds of convex sets that appear in operator algebra theory these consist of positive parts of unit balls for c algebras and their dual spaces and for von neumann algebras and their pre duals of self adjoint parts of unit balls in the same four classes and finally of general unit balls in the four classes all these faces are shown to be semi exposed and naturally paired with a polar face in the dual or pre dual space

hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds in this paper we construct two classes of hamiltonian preserving numerical schemes for a liouville equation with discontinuous local wave speed this equation arises in the phase space description of geometrical optics and has been the foundation of the recently developed level set methods for multivalued solution in geometrical optics we extend our previous work in s jin x wen hamiltonian preserving schemes for the liouville equation with discontinuous potentials commun math sci 3 2005 285 315 for the semiclassical limit of the schrodinger equation into this system the designing principle of the hamiltonian preservation by building in the particle behavior at the interface into the numerical flux is used here and as a consequence we obtain two classes of schemes that allow a hyperbolic stability condition when a plane wave hits a flat interface the hamiltonian preservation is shown to be equivalent to snell s law of refraction in the case when the ratio of wave length over the width of the interface goes to zero when both length scales go to zero positivity and stabilities in both l 1 and l norms are established for both schemes the approach also provides a selection criterion for a unique solution of the underlying linear hyperbolic equation with singular discontinuous and measure valued coefficients benchmark numerical examples are given with analytic solution constructed to study the numerical accuracy of these schemes

generalized variational principles global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics we study systems of conservation laws arising in two models of adhesion particle dynamics the first is the system of free particles which stick under collision the second is a system of gravitationally interacting particles which also stick under collision in both cases mass and momentum are conserved at the collisions so the dynamics is described by 2 2 systems of conservations laws we show that for these systems global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the lax oleinik variational principle for scalar conservation laws however this weak solution is not unique among weak solutions satisfying the standard entropy condition we also study a modified gravitational model in which instead of momentum some other weighted velocity is conserved at collisions for this model we prove both existence and uniqueness of global weak solutions we then study the qualitative behavior of the solutions with random initial data we show that for continuous but nowhere differentiable random initial velocities all masses immediately concentrate on points even though they were continuously distributed initially and the set of shock locations is dense

numerical approximations of one dimensional linear conservation equations with discontinuous coefficients conservative linear equations arise in many areas of application including continuum mechanics or high frequency geometrical optics approximations this kind of equations admits most of the time solutions which are only bounded measures in the space variable known as duality solutions in this paper we study the convergence of a class of finite differences numerical schemes and introduce an appropriate concept of consistency with the continuous problem some basic examples including computational results are also supplied

on the quest for positivity in operator algebras abstract we show that in every nonzero operator algebra with a contractive approximate identity or c a i there is a nonzero operator t such that i t 1 in fact there is a c a i consisting of operators t with i 2 t 1 so the numerical range of the elements of our contractive approximate identity is contained in the closed disk center 1 2 and radius 1 2 this is the necessarily weakened form of the result for c algebras where there is always a contractive approximate identity consisting of operators with 0 t 1 the numerical range is contained in the real interval 0 1 so if an operator algebra has a c a i it must have operators with a certain amount of positivity

delta shock waves with dirac delta function in both components for systems of conservation laws abstract we study a class of non strictly and weakly hyperbolic systems of conservation laws which contain the equations of geometrical optics as a prototype the riemann problems are constructively solved the riemann solutions include two kinds of interesting structures one involves a cavitation where both state variables tend to zero forming a singularity the other is a delta shock wave in which both state variables contain dirac delta function simultaneously the generalized rankine hugoniot relation and entropy condition are proposed to solve the delta shock wave moreover with the limiting viscosity approach we show all of the existence uniqueness and stability of solution involving the delta shock wave the generalized rankine hugoniot relation is also confirmed then our theory is successfully applied to two typical systems including the geometric optics equations finally we present the numerical results coinciding with the theoretical analysis

numerical approximations of pressureless and isothermal gas dynamics we study several schemes of first or second order accuracy based on kinetic approximations to solve pressureless and isothermal gas dynamics equations the pressureless gas system is weakly hyperbolic giving rise to the formation of density concentrations known as delta shocks for the isothermal gas system the infinite speed of expansion into vacuum leads to zero timestep in the godunov method based on exact riemann solver the schemes we consider are able to bypass these difficulties they are proved to satisfy positiveness of density and discrete entropy inequalities to capture the delta shocks and to treat data with vacuum

on peak phenomena for non commutative h a non commutative extension of amar and lederer s peak set result is given as its simple applications it is shown that any non commutative h algebra h m has unique predual and moreover some restriction in some of the results of blecher and labuschagne are removed making them hold in full generality

on properties of bond free dna languages the input data for dna computing must be encoded into the form of single or double dna strands as complementary parts of single strands can bind together forming a double stranded dna sequence one has to impose restrictions on these sets of dna words languages to prevent them from interacting in undesirable ways we recall a list of known properties of dna languages which are free of certain types of undesirable bonds then we introduce a general framework in which we can characterize each of these properties by a solution of a uniform formal language inequation this characterization allows us among others to construct i a uniform algorithm deciding in polynomial time whether a given dna language possesses any of the studied properties and ii in many cases also an algorithm deciding whether a given dna language is maximal with respect to the desired property

aspects of shuffle and deletion on trajectories word and language operations on trajectories provide a general framework for the study of properties of sequential insertion and deletion operations a trajectory gives a syntactical constraint on the scattered insertion deletion of a word into from another one with an intuitive geometrical interpretation moreover deletion on trajectories is an inverse of the shuffle on trajectories these operations are a natural generalization of many binary word operations like catenation quotient insertion deletion shuffle etc besides they were shown to be useful e g in concurrent processes modelling and recently in biocomputing area we begin with the study of algebraic properties of the deletion on trajectories then we focus on three standard decision problems concerning linear language equations with one variable involving the above mentioned operations we generalize previous results and obtain a sequence of new ones particularly we characterize the class of binary word operations for which the validity of such a language equation is un decidable for regular and context free operands

high frequency wave propagation by the segment projection method geometrical optics is a standard technique used for the approximation of high frequency wave propagation computational methods based on partial differential equations instead of the traditional ray tracing have recently been applied to geometrical optics these new methods have a number of advantages but typically exhibit difficulties with linear superposition of waves in this paper we introduce a new partial differential technique based on the segment projection method in phase space the superposition problem is perfectly resolved and so is the problem of computing amplitudes in the neighborhood of caustics the computational complexity is of the same order as that of ray tracing the new algorithm is described and a number of computational examples are given including a simulation of waveguides

operator algebras with contractive approximate identities ii abstract we make several contributions to our recent program investigating structural properties of algebras of operators on a hilbert space for example we make substantial contributions to the noncommutative peak interpolation program begun by hay and the first author hay and neal another sample result an operator algebra has a contractive approximate identity iff the linear span of the elements with positive real part is dense we also extend the theory of compact projections to the most general case despite the title our algebras are often allowed to have no approximate identity

on the complexity and decidability of some problems involving shuffle abstract the complexity and decidability of various decision problems involving the shuffle operation denoted by are studied the following three problems are all shown to be np complete given a nondeterministic finite automaton nfa m and two words u and v is l m u v is u v l m and is l m u v it is also shown that there is a polynomial time algorithm to determine for nfa s m 1 m 2 and a deterministic pushdown automaton m 3 whether l m 1 l m 2 l m 3 the same is true when m 1 m 2 m 3 are one way nondeterministic l reversal bounded k counter machines with m 3 being deterministic other decidability and complexity results are presented for testing whether given languages l 1 l 2 and r from various languages families satisfy l 1 l 2 r and r l 1 l 2 several closure results on shuffle are also shown

residually finite dimensional operator algebras abstract we study non selfadjoint operator algebras that can be entirely understood via their finite dimensional representations in contrast with the elementary matricial description of finite dimensional c algebras in the non selfadjoint setting we show that an additional level of flexibility must be allowed motivated by this peculiarity we consider a natural non selfadjoint notion of residual finite dimensionality we identify sufficient conditions for the tensor algebra of a c correspondence to enjoy this property to clarify the connection with the usual self adjoint notion we investigate the residual finite dimensionality of the minimal and maximal c covers associated to an operator algebra

on finite index indexed grammars and their restrictions abstract the family l ind lin of languages generated by linear indexed grammars has been studied in the literature it is known that the parikh image of every language in l ind lin is semi linear however there are bounded semi linear languages that are not in l ind lin here we look at larger families of restricted indexed languages and study their combinatorial and decidability properties and their relationships

shuffle languages are in p abstract in this paper we show that shuffle languages are contained in one way nspace log n thus in p we consider the class of shuffle languages which emerges from the class of finite languages through regular operations union concatenation kleene star and shuffle operations shuffle and shuffle closure for every shuffle expression e we construct a shuffle automaton which accepts the language generated by e and we show that the automaton can be simulated by a one way nondeterministic turing machine in logarithmic space

global solutions to the two dimensional riemann problem for a system of conservation laws we study the global solutions to the two dimensional riemann problem for a system of conservation laws the initial data are three constant states separated by three rays emanating from the origin under the assumption that each ray in the initial data outside of the origin projects exactly one planar contact discontinuity this problem is classified into five cases by the self similar transformation the reduced system changes type from being elliptic near the origin to being hyperbolic far away in self similar plane then in hyperbolic region applying the generalized characteristic analysis method a goursat problem is solved to describe the interactions of planar contact discontinuities while in elliptic region a boundary value problem arises it is proved that this boundary value problem admits a unique solution based on these preparations five explicit solutions and their corresponding criteria can be obtained in self similar plane