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

a bound on the total variation of the conserved quantities for solutions of a general resonant nonlinear balance law we introduce a new potential interaction functional and use it to define a new glimm type functional that bounds the total variation of the conserved quantities at time t 0 by the total variation at time t 0 in glimm approximate solutions of a general resonant nonlinear balance law

on computing structural changes in evolving surfaces and their appearance as a surface undergoes a one parameter family of deformations its shape and its appearance change smoothly except at certain critical parameter values where abrupt structural changes occur this paper considers the case of surfaces defined as the zero set of smooth density functions undergoing a gaussian diffusion process and addresses the problem of computing the critical parameter values corresponding to structural changes in the parabolic curves of a surface and in its aspect graph an algorithm based on homotopy continuation and curve tracing is proposed in the case of polynomial density functions whose zero set is an algebraic surface it has been implemented and examples are presented

suppression of oscillations in godunov s method for a resonant non strictly hyperbolic system we prove the stability of the 2 x 2 godunov numerical method in a resonant nonlinear system of conservation laws the system we study provides one of the simplest settings in which wave speeds can coincide in a nonlinear problem namely an inhomogeneous equation in which the inhomogeneity is treated as an unknown variable ut f a u x o at o this is a model for resonance in more complicated systems such as transonic flow in a variable area duct and certain resonant problems in multiphase flow and elasticity we show that the total variation of the conserved quantities can grow at most linearly in weak solutions generated by the 2 2 godunov method under the assumption that a x satisfies the threshold smoothness condition that the total variation of a x is finite we show by counterexample that the condition is sharp in the sense that there is no bound on the growth rate based on the c norm of this variable this is the most complicated setting in which the stability of the 2 2 godunov method has been demonstrated and our results provide the first such result for a numerical method that is essentially based on the solution of the riemann problem for a resonant non strictly hyperbolic system the solution of the riemann problem is more interesting when wave speeds coincide because the coordinate system of riemann invariants is singular and there exists a multiplicity of possible time asymptotic wave patterns for solutions as a consequence numerical methods that are based on the solution of the riemann problem can introduce spurious oscillations in the approximate solutions indeed counterexamples show that the total variation in u of the waves in the riemann problem step of the 2 x 2 godunov method can tend to infinity as ax 0 even when the initial total variation in u is zero thus our results verify that the averaging step in the godunov method wipes out the numerical oscillations that can occur in the riemann problem solution step of the method we interpret this as demonstrating that numerical methods based on the riemann problem are viable in this non strictly hyperbolic setting

optimal results on tv bounds for scalar conservation laws with discontinuous flux abstract this paper is concerned with the total variation of the solution of scalar conservation law with discontinuous flux in one space dimension one of the main unsettled questions concerning conservation law with discontinuous flux was the boundedness of the total variation of the solution near interface in 1 it has been shown by a counter example at t 1 that the total variation of the solution blows up near interface but in that example the solution become of bounded variation after time t 1 so the natural question is what happens to the bv ness of the solution for large time here we give a complete picture of the bounded variation of the solution for all time for a uniform convex flux with only l data we obtain a natural smoothing effect in bv for all time t t 0 also we give a counter example even for a bv data to show that the assumptions which have been made are optimal

correlations between eigenvalues of large random matrices with independent entries we derive the connected correlation functions for eigenvalues of large hermitian random matrices with independently distributed elements using both a diagrammatic and a renormalization group rg inspired approach with the diagrammatic method we obtain a general form for the one two and three point connected green functions for this class of ensembles when matrix elements are identically distributed and then discuss the derivation of higher order functions by the same approach using the rg approach we rederive the one and two point green functions and show they are unchanged by choosing certain ensembles with nonidentically distributed elements throughout we compare the green functions we obtain to those from the class of ensembles with unitary invariant distributions and discuss universality in both ensemble classes textcopyright 1996 the american physical society

universal shocks in the wishart random matrix ensemble we show that the derivative of the logarithm of the average characteristic polynomial of a diffusing wishart matrix obeys an exact partial differential equation valid for an arbitrary value of n the size of the matrix in the large n limit this equation generalizes the simple inviscid burgers equation that has been obtained earlier for hermitian or unitary matrices the solution through the method of characteristics presents singularities that we relate to the precursors of shock formation in the burgers equation the finite n effects appear as a viscosity term in the burgers equation using a scaling analysis of the complete equation for the characteristic polynomial in the vicinity of the shocks we recover in a simple way the universal bessel oscillations so called hard edge singularities familiar in random matrix theory

stability of a resonant system of conservation laws modeling polymer flow with gravitation abstract we prove l 1 uniqueness and stability for a resonant 2 2 system of conservation laws that arise as a model for two phase polymer flow in porous media the analysis uses the equivalence of the eulerian and lagrangian formulation of this system and the results are first established for an auxiliary scalar equation our methods are based on front tracking approximations for the auxiliary equation and the kru kov entropy condition for scalar conservation laws

convergence of a difference scheme for conservation laws with a discontinuous flux convergence is established for a scalar finite difference scheme based on the godunov or engquist osher eo flux for scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient other works in this direction have established convergence for methods employing the solution of 2 2 riemann problems the algorithm discussed here uses only scalar riemann solvers satisfaction of a set of kruzkov type entropy inequalities is established for the limit solution from which geometric entropy conditions follow assuming a piecewise constant coefficient it is shown that these conditions imply l1 contractiveness for piecewise c1 solutions thus extending a well known theorem

a model of continuous sedimentation of flocculated suspensions in clarifier thickener units the chief purpose of this paper is to formulate and partly analyze a new mathematical model for continuous sedimentation consolidation processes of flocculated suspensions in clarifier thickener units this model appears in two variants for cylindrical and variable cross sectional area units respectively models 1 and 2 in both cases the governing equation is a scalar strongly degenerate parabolic equation in which both the convective and diffusion fluxes depend on parameters that are discontinuous functions of the depth variable the initial value problem for this equation is analyzed for model 1 we introduce a simple finite difference scheme and prove its convergence to a weak solution that satisfies an entropy condition a limited analysis of steady states as desired stationary modes of operation is performed numerical examples illustrate that the model realistically describes the dynamics of flocculated suspensions in clarifier thickeners

the immersed interface method for nonlinear differential equations with discontinuous coefficients and singular sources we extend the immersed interface method of leveque and li to find numerical solutions of one dimensional parabolic partial differential equations of the form u t beta x t u x x lambda x t u x kappa u u x f x where beta u beta u x and f may have known discontinuities at a known location x alpha at each time step a large weakly nonlinear system is set up using a difference scheme which is standard away from x alpha and which is derived for grid points near alpha by solving small linear systems which are determined from the jumps at x alpha the time stepping is done with a crank nicholson scheme and the nonlinear systems are solved with a levenberg marquardt method as an example we consider the flow of cars on a one lane highway with an entrance or exit where traffic is treated as a continuous fluid numerical examples show that we can compute solutions to these equations with second order accuracy

an engquist osher type scheme for conservation laws with discontinuous flux adapted to flux connections we consider scalar conservation laws with the spatially varying flux h x f u 1 h x g u where h x is the heaviside function and f and g are smooth nonlinear functions adimurthi mishra and veerappa gowda j hyperbolic differ equ 2 2005 pp 783 837 pointed out that such a conservation law admits many l 1 contraction semigroups one for each so called connection a b here we define entropy solutions of type a b involving kruzkov type entropy inequalities that can be adapted to any fixed connection a b it is proved that these entropy inequalities imply the l 1 contraction property for l infty solutions in contrast to the piecewise smooth setting of adimurthi mishra and veerappa gowda for a fixed connection these entropy inequalities include a single adapted entropy of the type used by audusse and perthame proc roy soc edinburgh sect a 135 2005 pp 253 265 we prove convergence of a new difference scheme that approximates entropy solutions of type a b for any connection a b if a few parameters are varied the scheme relies on a modification of the standard engquist osher flux is simple as no 2 times2 riemann solver is involved and is designed such that the steady state solution connecting a to b is preserved in contrast to most analyses of similar problems our convergence proof is not based on the singular mapping or compensated compactness methods but on standard spatial variation estimates away from the flux discontinuity some numerical examples are presented

growth motion and 1 parameter families of symmetry sets associated to every plane curve there is the locus of centres of circles bitangent to that curve the so called symmetry set of the curve we can view this set as the spine of our curve which can be recovered by taking the envelope of circles of varying radii along this spine varying the symmetry set in some isotopy while keeping the radius function fixed may be viewed as crudely modelling motion of the original curve viewed as a biological object fixing the symmetry set and varying the radius function can be considered to model growth crudely in this paper we shall describe the generic changes in the curves which take place in the process of growth and motion and outline the corresponding results for spheres centred on a space curve we also use the idea of a stratified morse function to describe the generic changes which occur in one parameter families of full bifurcation sets in the plane applying this to the bifurcation set of distance squared functions we find all the transitions of a symmetry set and evolute which occur in a generic isotopy of a plane curve

convergence of godunov type methods for a conservation law with a spatially varying discontinuous flux function we deal with single conservation laws with a spatially varying and possibly discontinuous coefficient this equation includes as a special case sin gle conservation laws with conservative and possibly singular source terms we extend the framework of optimal entropy solutions for these classes of equations based on a two step approach in the first step an interface connection vector is used to define infinite classes of entropy solutions we show that each of these classes of solutions is stable in l1 this allows for the possibility of choosing one of these classes of solutions based on the physics of the problem in the second step we define optimal entropy solutions based on the solution of a certain optimization problem at the discontinuities of the coefficient this method leads to optimal entropy solutions that are consistent with physically observed solutions in two phase flows in heterogeneous porous media another central aim of this paper is to develop suitable numerical schemes for these equations we develop and analyze a set of godunov type finite volume methods that are based on exact solutions of the corresponding riemann problem numerical experiments are shown comparing the performance of these schemes on a set of test problems

solutions of hamilton jacobi equations and scalar conservation laws with discontinuous space time dependence abstract we establish a unique stable solution to the hamilton jacobi equation u t h k x t u x 0 x t 0 with lipschitz initial condition where k x t is allowed to be discontinuous in the x t plane along a finite number of possibly intersecting curves parameterized by t we assume that for fixed k h k p is convex in p and lim p h k p p the solution is determined by showing that if k is made smooth by convolving k in the x direction with the standard mollifier then the control theory representation of the viscosity solution to the resulting hamilton jacobi equation must converge uniformly as the mollification decreases to a lipschitz continuous solution with an explicit control theory representation this also defines the unique stable solution to the corresponding scalar conservation law u t f k x t u x 0 x t 0 with k discontinuous

central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions in this paper two existing one dimensional mathematical models one for continuous sedimentation of monodisperse suspensions and one for settling of polydisperse suspensions are combined into a model of continuous separation of polydisperse mixtures this model can be written as a first order system of conservation laws for the local concentrations of each particle species with a flux vector that depends discontinuously on the space variable this application motivates the extension of the kurganov tadmor central difference scheme to systems with discontinuous flux the new central schemes are based on discretizing an enlarged system in which the discontinuous coefficients are viewed as additional conservation laws these additional conservation laws can either be discretized and the evolution of the discontinuity parameters is calculated in each time step or solved exactly that is the discontinuity parameters are kept constant with respect to time numerical examples and an l1 error study show that the kurganov tadmor scheme with first order in time discretization produces spurious oscillations whereas its semi discrete version discretized by a second order runge kutta scheme produces good results the scheme with discontinuity parameters kept constant is slightly more accurate than when these are evolved numerical examples illustrate the application to separation of polydisperse suspensions

random tensor theory extending random matrix theory to mixtures of random product states we consider a problem in random matrix theory that is inspired by quantum information theory determining the largest eigenvalue of a sum of p random product states in mathbb c d otimes k where k and p d k are fixed while d when k 1 the marcenko pastur law determines up to small corrections not only the largest eigenvalue 1 sqrt p d k 2 but the smallest eigenvalue min 0 1 sqrt p d k 2 and the spectral density in between we use the method of moments to show that for k 1 the largest eigenvalue is still approximately 1 sqrt p d k 2 and the spectral density approaches that of the marcenko pastur law generalizing the random matrix theory result to the random tensor case our bound on the largest eigenvalue has implications both for sampling from a particular heavy tailed distribution and for a recently proposed quantum data hiding and correlation locking scheme due to leung and winter

godunov type approximation for a general resonant balance law with large data abstract we consider the cauchy problem for the 2 2 nonstrictly hyperbolic system a t 0 u t f a u x g a u a x 0 a u t 0 a 0 u 0 for possibly large discontinuous and resonant data the generalized solution to the riemann problem is introduced interaction estimates are carried out using an original change of variables and the convergence of godunov approximations is shown uniqueness is addressed relying on a suitable extension of kru kov s techniques

a difference scheme for conservation laws with a discontinuous flux the nonconvex case in a previous work by the author convergence was established for a simple difference scheme approximating a scalar conservation law where the flux was concave and had a discontinuous spatially varying coefficient j d towers siam j numer anal 38 2000 pp 681 698 the main result of this paper is an extension of that convergence theorem to the situation where the flux may have any finite number of critical points additionally spatially varying source terms are allowed the spatially varying numerical flux is also shown to satisfy maximum and minimum principles and to be total variation decreasing tvd in time

a theory of l 1 dissipative solvers for scalar conservation laws with discontinuous flux we propose a general framework for the study of l1 contractive semigroups of solutions to conservation laws with discontinuous flux u t mathfrak f x u x 0 qquad mathfrak f x u left begin array ll f l u x 0 end array right quad quad quad rm cl where the fluxes fl fr are mainly assumed to be continuous developing the ideas of a number of preceding works baiti and jenssen in j differ equ 140 1 161 185 1997 towers in siam j numer anal 38 2 681 698 2000 towers in siam j numer anal 39 4 1197 1218 2001 towers et al in skr k nor vidensk selsk 3 1 49 2003 adimurthi et al in j math kyoto university 43 1 27 70 2003 adimurthi et al in j hyperbolic differ equ 2 4 783 837 2005 audusse and perthame in proc roy soc edinburgh a 135 2 253 265 2005 garavello et al in netw heterog media 2 159 179 2007 burger et al in siam j numer anal 47 1684 1712 2009 we claim that the whole admissibility issue is reduced to the selection of a family of elementary solutions which are piecewise constant weak solutions of the form c x c l11 left x 0 right we refer to such a family as a germ it is well known that cl admits many different l1 contractive semigroups some of which reflect different physical applications we revisit a number of the existing admissibility or entropy conditions and identify the germs that underly these conditions we devote specific attention to the vanishing viscosity germ which is a way of expressing the condition of diehl j hyperbolic differ equ 6 1 127 159 2009 for any given germ we formulate germ based admissibility conditions in the form of a trace condition on the flux discontinuity line x 0 in the spirit of vol pert math ussr sbornik 2 2 225 267 1967 and in the form of a family of global entropy inequalities following kruzhkov math ussr sbornik 10 2 217 243 1970 and carrillo arch ration mech anal 147 4 269 361 1999 we characterize those germs that lead to the l1 contraction property for the associated admissible solutions our approach offers a streamlined and unifying perspective on many of the known entropy conditions making it possible to recover earlier uniqueness results under weaker conditions than before and to provide new results for other less studied problems several strategies for proving the existence of admissible solutions are discussed and existence results are given for fluxes satisfying some additional conditions these are based on convergence results either for the vanishing viscosity method with standard viscosity or with specific viscosities adapted to the choice of a germ or for specific germ adapted finite volume schemes

random matrix ensembles with an effective extensive external charge recent theoretical studies of chaotic scattering have encountered ensembles of random matrices in which the eigenvalue probability density function contains a one body factor with an exponent proportional to the number of eigenvalues two such ensembles have been encountered an ensemble of unitary matrices specified by the so called poisson kernel and the laguerre ensemble of positive definite matrices here we consider various properties of these ensembles jack polynomial theory is used to prove a reproducing property of the poisson kernel and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the dyson circular ensemble for the laguerre ensemble the scaled global density is calculated exactly for all even values of the parameter while for random matrices with unitary symmetry the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class

signal and noise in correlation matrix using random matrix technique we determine an exact relation between the eigenvalue spectrum of the covariance matrix and of its estimator this relation can be used in practice to compute eigenvalue invariants of the covariance correlation matrix results can be applied in various problems where one experimentally estimates correlations in a system with many degrees of freedom like for instance those in statistical physics lattice measurements of field theory genetics quantitative finance and other applications of multivariate statistics

diffusions on symmetric spaces of type a iii and random matrix theories for rectangular matrices isotropic diffusion processes on cosets and and their zero curvature limit are studied from a unified viewpoint as indicated in our previous works the projection of the fokker planck equation onto the maximal commutative subgroup of these cosets can be described by using the radial part of the laplace beltrami operator by taking the zero curvature limit an integral which is a natural extension of the itzykson zuber integral to the rectangular matrices is explicitly evaluated the probability density function obtained from the diffusion on is studied in detail which can be applied to the quantum transport problem the explicit expressions for the probability density function in the metallic and insulating regimes are obtained for the metallic regime the integral representation for the hypergeometric function is used and the results are exact furthermore by using the orthogonal polynomial method n point correlation functions are obtained exactly for arbitrary n

well balanced schemes for conservation laws with source terms based on a local discontinuous flux formulation we propose and analyze a finite volume scheme of the godunov type for conservation laws with source terms that preserve discrete steady states the scheme works in the resonant regime as well as for problems with discontinuous flux moreover an additional modification of the scheme is not required to resolve transients and solutions of non linear algebraic equations are not involved our well balanced scheme is based on modifying the flux function locally to account for the source term and to use a numerical scheme especially designed for conservation laws with discontinuous flux due to the difficulty of obtaining bv estimates we use the compensated compactness method to prove that the scheme converges to the unique entropy solution as the discretization parameter tends to zero we include numerical experiments in order to show the features of the scheme and how it compares with a wellbalanced scheme from the literature

correlation functions in disordered systems recently we found that the correlation between the eigenvalues of random hermitian matrices exhibits universal behavior here we study this universal behavior and develop a diagrammatic approach which enables us to extend our previous work to the case in which the random matrix evolves in time or varies as some external parameters vary we compute the current current correlation function discuss various generalizations and compare our work with the work of other authors we study the distribution of eigenvalues of hamiltonians consisting of a sum of a deterministic term and a random term the correlation between the eigenvalues when the deterministic term is varied is calculated

ridges crests and sub parabolic lines of evolving surfaces the ridge lines on a surface can be defined either via contact of the surface with spheres or via extrema of principal curvatures along lines of curvature certain subsets of ridge lines called crest lines have been singled out by some authors for medical imaging applications there is a related concept of sub parabolic line on a surface also defined via extrema of principal curvatures in this paper we study in detail the structure of the ridge lines crest lines and sub parabolic lines on a generic surface and on a surface which is evolving in a generic one parameter family the mathematical details of this study are in bruce et al 1994c

renormalizing rectangles and other topics in random matrix theory we consider random hermitian matrices made of complex or realm n rectangular blocks where the blocks are drawn from various ensembles these matrices haven pairs of opposite real nonvanishing eigenvalues as well asm n zero eigenvalues form n these zero eigenvalues are kinematical in the sense that they are independent of randomness we study the eigenvalue distribution of these matrices to leading order in the large n m limit in which the rectangularity r m n is held fixed we apply a variety of methods in our study we study gaussian ensembles by a simple diagrammatic method by the dyson gas approach and by a generalization of the kazakov method these methods make use of the invariance of such ensembles under the action of symmetry groups the more complicated wigner ensemble which does not enjoy such symmetry properties is studied by large n renormalization techniques in addition to the kinematical function spike in the eigenvalue density which corresponds to zero eigenvalues we find for both types of ensembles that if r 1 is held fixed asn then nonzero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin this separation arises because the nonzero eigenvalues are repelled macroscopically from the origin finally we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes a behavior governed by airy functions asr 1 the lobes come closer and the airy oscillatory behavior near the endpoints that are close to zero breaks down we interpret this breakdown as a signal thatr 1 drives a crossover to the oscillation governed by bessel functions near the origin for matrices made of square blocks

discontinuous conservation laws for production networks with finite buffers in this study we use discontinuous conservation laws to model production networks with finite buffers a network formulation is characterized by the interaction between several production units described by discontinuous conservation laws under the assumption that no parts are lost therefore we discuss appropriate riemann problems and coupling conditions which are derived from a regularized model to solve the discontinuous problem in an accurate and fast way we introduce a novel algorithm based on different settings the new method is compared numerically with standard approaches

non hermitian random matrix theory summation of planar diagrams the single ring theorem and the disc annulus phase transition i review aspects of work done in collaboration with a zee and r scalettar 1997 nucl phys b 504 579 1997 nucl phys b 501 643 2001 j math phys 42 5718 on complex non hermitian random matrices i open by explaining why the bag of tools used regularly in analysing hermitian random matrices cannot be applied directly to analyse non hermitian matrices and then introduce the method of hermitization which solves this problem then for rotationally invariant ensembles i derive a master equation for the average density of eigenvalues in the complex plane in the limit of infinitely large matrices this is achieved by resumming all the planar diagrams which appear in the perturbative expansion of the hermitized green function remarkably this resummation can be carried out explicitly for any rotationally invariant ensemble i prove that in the limit of infinitely large matrices the shape of the eigenvalue distribution is either a disc or an annulus this is the celebrated single ring theorem which of these shapes is realized is determined by the parameters coupling constants which determine the ensemble by varying these parameters a phase transition may occur between the two possible shapes i briefly discuss the universal features of this transition as the analysis of this problem relies heavily on summation of planar feynman diagrams i make a special effort at presenting a pedagogical exposition of the diagrammatic method which some readers may find useful

targeting functional motifs of a protein family the structural organization of a protein family is investigated by devising a method based on the random matrix theory rmt which uses the physiochemical properties of the amino acid with multiple sequence alignment a graphical method to represent protein sequences using physiochemical properties is devised that gives a fast easy and informative way of comparing the evolutionary distances between protein sequences a correlation matrix associated with each property is calculated where the noise reduction and information filtering is done using rmt involving an ensemble of wishart matrices the analysis of the eigenvalue statistics of the correlation matrix for the lactamase family shows the universal features as observed in the gaussian orthogonal ensemble goe the property based approach captures the short as well as the long range correlation approximately following goe between the eigenvalues whereas the previous approach treating amino acids as characters gives the usual short range correlations while the long range correlations are the same as that of an uncorrelated series the distribution of the eigenvector components for the eigenvalues outside the bulk rmt bound deviates significantly from rmt observations and contains important information about the system the information content of each eigenvector of the correlation matrix is quantified by introducing an entropic estimate which shows that for the lactamase family the smallest eigenvectors low eigenmodes are highly localized as well as informative these small eigenvectors when processed gives clusters involving positions that have well defined biological and structural importance matching with experiments the approach is crucial for the recognition of structural motifs as shown in lactamase and other families and selectively identifies the important positions for targets to deactivate activate the enzymatic actions

semidiscrete central upwind scheme for conservation laws with a discontinuous flux function in space abstract in this paper a modified semidiscrete central upwind scheme is derived for the scalar conservation laws with a discontinuous flux function in space the new scheme is based on dealing with the phase transition at the stationary discontinuity where the unknown variable function is not continuous but the flux function is continuous the main advantages of the new scheme are the same as them of the original semidiscrete central upwind scheme numerical results are displayed to illustrate the efficiency of the methods

high resolution numerical approximation of traffic flow problems with variable lanes and free flow velocities this paper develops macroscopic traffic flow models for a highway section with variable lanes and free flow velocities that involve spatially varying flux functions to address this complex physical property we develop a riemann solver that derives the exact flux values at the interface of the riemann problem based on this solver we formulate godunov type numerical schemes to solve the traffic flow models numerical examples that simulate the traffic flow around a bottleneck that arises from a drop in traffic capacity on the highway section are given to illustrate the efficiency of these schemes

convergence via oslc of the godunov scheme for a scalar conservation law with time and space flux discontinuities this paper deals with a godunov scheme as applied to a scalar conservation law whose flux has discontinuities in both space and time we extend the definition of vanishing viscosity solution of karlsen and towers j hyperbolic differ equ 14 671 702 2017 which applies to a flux with a spatial discontinuity in order to accommodate the addition of temporal flux discontinuities and prove that this extended definition implies uniqueness we prove convergence of the godunov approximations to the unique vanishing viscosity solution as the mesh size converges to zero thus establishing well posedness for the problem the novel aspect of this paper is the use of a discrete one sided lipschitz condition oslc in the discontinuous flux setting in the classical setting where flux discontinuities are not present the oslc is well known to produce an immediate regularizing effect with a local spatial variation bound resulting at any positive time we show that the oslc also produces a regularizing effect at any finite distance from the spatial flux discontinuity this regularizing effect is not materially affected by temporal flux discontinuities when combined with a cantor diagonal argument these regularizing effects imply convergence of the godunov approximations with this new method it is possible to forgo certain assumptions about the flux that seem to be required when using two commonly used convergence methods

the dflu flux for systems of conservation laws the dflu numerical flux was introduced in order to solve hyperbolic scalar conservation laws with a flux function discontinuous in space we show how this flux can be used to solve a certain class of systems of conservation laws such as systems modeling polymer flooding in oil reservoir engineering furthermore these results are extended to the case where the flux function is discontinuous in the space variable such a situation arises for example while dealing with oil reservoirs which are heterogeneous numerical experiments are presented to illustrate the efficiency of this new scheme compared to other standard schemes like upstream mobility lax friedrichs and force schemes

stability of conservation laws with discontinuous coefficients we prove l1 contractivity of weak solutions to a conservation law with a flux function that may depend discontinuously on the space variable furthermore we show that the l1 difference between solutions to conservation laws with different flux functions is bounded by the total variation with respect to the space variable of the difference between the flux functions

localized short range correlations in the spectrum of the equal time correlation matrix we suggest a procedure to identify those parts of the spectrum of the equal time correlation matrix mathbf c where relevant information about correlations of a multivariate time series is induced using an ensemble average over each of the distances between eigenvalues all nearest neighbor distributions can be calculated individually we present numerical examples where a information about cross correlations is found in the so called bulk of eigenvalues which generally is thought to contain only random correlations and where b the information extracted from the lower edge of the spectrum of mathbf c is statistically more significant than that extracted from the upper edge we apply the analysis to electroencephalographic recordings with epileptic events

an engquist osher type finite difference scheme with a discontinuous flux function in space abstract an engquist osher type finite difference scheme is derived for dealing with scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient the new monotone difference scheme is based on introducing a new interface numerical flux function which is called a generalized engquist osher flux by means of this scheme the existence and uniqueness of weak solutions to the scalar conservation laws are obtained and the convergence theorem is established some numerical examples are presented and the corresponding numerical results are displayed to illustrate the efficiency of the methods

convergence of an engquist osher scheme for a multi dimensional triangular system of conservation laws we consider a multi dimensional triangular system of conservation laws this system arises as a model of three phase flow in porous media and includes multi dimensional conservation laws with discontinuous coefficients as a special case the system is neither strictly hyperbolic nor symmetric we propose an engquist osher type scheme for this system and show that the approximate solutions generated by the scheme converge to a weak solution numerical examples are also presented

correlation and network analysis of global financial indices random matrix theory rmt and network methods are applied to investigate the correlation and network properties of 20 financial indices the results are compared before and during the financial crisis of 2008 in the rmt method the components of eigenvectors corresponding to the second largest eigenvalue form two clusters of indices in the positive and negative directions the components of these two clusters switch in opposite directions during the crisis the network analysis uses the fruchterman reingold layout to find clusters in the network of indices at different thresholds at a threshold of 0 6 before the crisis financial indices corresponding to the americas europe and asia pacific form separate clusters on the other hand during the crisis at the same threshold the american and european indices combine together to form a strongly linked cluster while the asia pacific indices form a separate weakly linked cluster if the value of the threshold is further increased to 0 9 then the european indices france germany and the united kingdom are found to be the most tightly linked indices the structure of the minimum spanning tree of financial indices is more starlike before the crisis and it changes to become more chainlike during the crisis the average linkage hierarchical clustering algorithm is used to find a clearer cluster structure in the network of financial indices the cophenetic correlation coefficients are calculated and found to increase significantly which indicates that the hierarchy increases during the financial crisis these results show that there is substantial change in the structure of the organization of financial indices during a financial crisis

a relaxation scheme for conservation laws with a discontinuous coefficient lwe study a relaxation scheme of the jin and xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k x if k e bv we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution the murat tartar compensated compactness method is used to establish convergence we present numerical experiments with the relaxation scheme and comparisons are made with a front tracking scheme based on an exact 2 x 2 riemann solver

convergence of finite volume schemes for triangular systems of conservation laws we consider non strictly hyperbolic systems of conservation laws in triangular form which arise in applications like three phase flows in porous media we device simple and efficient finite volume schemes of godunov type for these systems that exploit the triangular structure we prove that the finite volume schemes converge to weak solutions as the discretization parameters tend to zero some numerical examples are presented one of which is related to flows in porous media

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

the riemann problem near a hyperbolic singularity the classification of solutions of quadratic riemann problems 1 the purpose of this paper is to classify the solutions of riemann problems near a hyperbolic singularity in a nonlinear system of conservation laws hyperbolic singularities play the role in the theory of riemann problems that rest points play in the theory of ordinary differential equations indeed generically only a finite number of structures can appear in a neighborhood of such a singularity in this the first of three papers the program of classification is discussed in general and the simplest structure that occurs is characterized

godunov type numerical methods for a model of granular flow we propose and analyze finite volume godunov type methods based on discontinuous flux for a 2 2 system of non linear partial differential equations proposed by hadeler and kuttler to model the dynamics of growing sandpiles generated by a vertical source on a flat bounded rectangular table the scheme is made well balanced by modifying the flux function locally by including source term as a part of the convection term its extension to multi dimensions is not straightforward for which an approach has been introduced here based on transport rays this approach is compared with another approach for inclusion of source term which uses the idea of inverting the divergence operator relying on the curl free component of the helmholtz decomposition of the source term numerical experiments are presented to illustrate the efficiency of the proposed scheme for both unsteady and steady state calculations and to make comparisons with the previously studied finite difference and semi lagrangian approaches by falcone and finzi vita

explicit hopf lax type formulas for hamilton jacobi equations and conservation laws with discontinuous coefficients we deal with a hamilton jacobi equation with a hamiltonian that is discontinuous in the space variable this is closely related to a conservation law with discontinuous flux recently an entropy framework for single conservation laws with discontinuous flux has been developed which is based on the existence of infinitely many stable semigroups of entropy solutions based on an interface connection in this paper we characterize these infinite classes of solutions in terms of explicit hopf lax type formulas which are obtained from the viscosity solutions of the corresponding hamilton jacobi equation with discontinuous hamiltonian this also allows us to extend the framework of infinitely many classes of solutions to the hamilton jacobi equation and obtain an alternative representation of the entropy solutions for the conservation law we have considered the case where both the hamiltonians are convex concave furthermore we also deal with the less explored case of sign changing coefficients in which one of the hamiltonians is convex and the other concave in fact in convex concave case we cannot expect always an existence of a solution satisfying rankine hugoniot condition across the interface therefore the concept of generalised rankine hugoniot condition is introduced and prove existence and uniqueness

the solution of nonstrictly hyperbolic conservation laws may be hard to compute some simple examples of nonstrictly hyperbolic conservation laws are presented that admit certain instability properties with respect to l 1 perturbations of the initial data one of the examples shows that a piecewise constant approximation generates an approximate solution with erroneous qualitative behaviour another example shows that the approximate solution is not well defined for both of these examples a classical stable entropy solution can be given by an explicit formula these properties may cause difficulties for numerical methods based on the solution of riemann problems

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

well posedness in bv t and convergence of a difference scheme for continuous sedimentation in ideal clarifier thickener units we consider a scalar conservation law modeling the settling of particles in an ideal clarifier thickener unit the conservation law has a nonconvex flux which is spatially dependent on two discontinuous parameters we suggest to use a kru kov type notion of entropy solution for this conservation law and prove uniqueness l1 stability of the entropy solution in the bvt class functions w x t with tw being a finite measure the existence of a bvt entropy solution is established by proving convergence of a simple upwind finite difference scheme of the engquist osher type a few numerical examples are also presented

parabolic curves of evolving surfaces in this article we show how certain geometric structures which are also associated with a smooth surface evolve as the shape of the surface changes in a 1 parameter family we concentrate on the parabolic set and its image under the gauss map but the same techniques also classify the changes in the dual of the surface all these have significance for computer vision for example through their connection with specularities and apparent contours with the aid of our complete classification which includes all the phenomena associated with multi contact tangent planes as well as those associated with parabolic sets we re examine examples given by j koenderink in his book 1990 under the title of morphological scripts we also explain some of the connections between parabolic sets and ridges of a surface where principal curvatures achieve turning values along lines of curvature the point of view taken is the analysis of the contact between surfaces and their tangent planes a systematic investigation of this yields the results using singularity theory the mathematical details are suppressed here and appear in bruce et al 1993

conservation law with the flux function discontinuous in the space variable ii we deal with a single conservation law with discontinuous convex concave type fluxes which arise while considering sign changing flux coefficients the main difficulty is that a weak solution may not exist as the rankine hugoniot condition at the interface may not be satisfied for certain choice of the initial data we develop the concept of generalized entropy solutions for such equations by replacing the rankine hugoniot condition by a generalized rankine hugoniot condition the uniqueness of solutions is shown by proving that the generalized entropy solutions form a contractive semi group in l 1 existence follows by showing that a godunov type finite difference scheme converges to the generalized entropy solution the scheme is based on solutions of the associated riemann problem and is neither consistent nor conservative the analysis developed here enables to treat the cases of fluxes having at most one extrema in the domain of definition completely numerical results reporting the performance of the scheme are presented

detection and characterization of changes of the correlation structure in multivariate time series we propose a method based on the equal time correlation matrix as a sensitive detector for phase shape correlations in multivariate data sets the key point of the method is that changes of the degree of synchronization between time series provoke level repulsions between eigenstates at both edges of the spectrum of the correlation matrix consequently detailed information about the correlation structure of the multivariate data set is imprinted into the dynamics of the eigenvalues and into the structure of the corresponding eigenvectors the performance of the technique is demonstrated by application to n f tori autoregressive models and coupled chaotic systems the high sensitivity the comparatively small computational effort and the excellent time resolution of the method recommend it for application to the analysis of complex spatially extended nonstationary systems

spectrum of the product of independent random gaussian matrices we show that the eigenvalue density of a product x x1x2 xm of m independent nxn gaussian random matrices in the limit n infinity is rotationally symmetric in the complex plane and is given by a simple expression rho z z 1 mpisigma 2 m z 2 2 m for z sigma the parameter sigma corresponds to the radius of the circular support and is related to the amplitude of the gaussian fluctuations this form of the eigenvalue density is highly universal it is identical for products of gaussian hermitian non hermitian and real or complex random matrices it does not change even if the matrices in the product are taken from different gaussian ensembles we present a self contained derivation of this result using a planar diagrammatic technique additionally we conjecture that this distribution also holds for any matrices whose elements are independent centered random variables with a finite variance or even more generally for matrices which fulfill pastur lindeberg s condition we provide a numerical evidence supporting this conjecture

viscosity solutions and convergence of monotone schemes for synthetic aperture radar shape from shading equations with discontinuous intensities the shape from shading sfs equation relating u y r the unknown angular height of a surface to i y r the known synthetic aperture radar sar intensity data from the surface is i frac u r 2 sqrt 1 u r 2 u y 2 where y and r are axial and radial cylindrical coordinates unlike the more common eikonal sfs equation which relates surface height in cartesian coordinates to optical photographic intensity data the above radar equation can be transformed into hamilton jacobi cauchy form ur g i uy 0 we explore the case where i is a discontinuous function which occurs commonly in radar data by considering sequences of continuous intensity functions that converge to i we obtain corresponding sequences of viscosity solutions we prove that these sequences must converge we also establish conditions that guarantee that these sequences converge to a common limit which we define as the solution to the radar equation finally we establish and demonstrate that when this common limit exists monot

the l 1 error estimates for a hamiltonian preserving scheme for the liouville equation with piecewise constant potentials we study the l 1 error of a hamiltonian preserving scheme developed in s jin and x wen commun math sci 3 2005 pp 285 315 for the liouville equation with a piecewise constant potential in one space dimension this problem has important applications in computations of the semiclassical limit of the linear schrodinger equation through barriers and of the high frequency waves through interfaces we use the l 1 error estimates established in x wen and s jin j comput math 26 2008 pp 1 22 x wen convergence of an immersed interface upwind scheme for linear advection equations with piecewise constant coefficients ii some related binomial coefficient inequalities preprint 2008 for the immersed interface upwind scheme to the linear advection equations with piecewise constant coefficients we prove that the scheme with the dirichlet incoming boundary conditions is l 1 convergent for a class of bounded initial data and derive the half order l 1 error bounds with explicit coefficients the initial conditions can be satisfied by applying the decomposition technique proposed in s jin h l liu s osher and r tsai j comput phys 205 2005 pp 222 241 for solving the liouville equation with measure valued initial data which arises in the semiclassical limit of the linear schrodinger equation

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

a comparison of convergence rates for godunov s method and glimm s method in resonant nonlinear systems of conservation laws we obtain time independent bounds on derivatives prove convergence and establish a rate of convergence for godunov s numerical method as applied to the initial value problem for a resonant inhomogeneous conservation law which is treated as a 2 times 2 nonstrictly hyperbolic system we compare the results with a corresponding analysis of glimm s method and see that our analysis gives equivalent sharp convergence rates in the strictly hyperbolic setting but an improvement is seen in godunov s method over glimm s method in the nonstrictly hyperbolic resonant regime the 2 times 2 glimm and godunov methods are the only methods for which we can obtain time independent bounds on derivatives these bounds represent a purely nonlinear phenomenon because there are no corresponding time independent bounds for the linearized equations which blow up at a linear rate in time