log periodic crashes revisited we revisit the finding that crashes can be deterministic and governed by log periodic formulas d sornette a johansen significance of log periodic precursors to financial crashes quant finance 1 2001 452 471 d sornette w x zhou the us 2000 2002 market descent how much longer and deeper quant finance 2 2002 468 481 one and two harmonic equations are usually employed to fit daily data during bubble episodes but a three harmonics has been shown to fit anti bubbles a johansen d sornette financial anti bubbles log periodicity in gold and nikkei collapses int j mod phys c 10 1999 563 575 here we show that the three harmonic formula can work for bubble episodes as well as anti bubbles this is illustrated with daily data from the brazilian real us dollar exchange rate and we also show that the three harmonics can fit an intraday data set from that foreign exchange rate

evidence of log periodicity in corporate bond spreads abstract in this paper we looked for evidence of log periodicity in recent us corporate bond spreads while we found evidence of log period behavior in both aaa and baa spreads the evidence supporting log periodicity in high yield hy spreads was found to be weak we conducted additional tests to confirm our results bivariate spectral analysis and surrogate data analysis and confirmed log periodicity in aaa and baa spreads but not in hy spreads we attempt to explain this variance using the recently developed market microstructure models of farmer sornette ausloos and others

can log periodic power law structures arise from random fluctuations abstract recent research has established log periodic power law lppl patterns prior to the detonation of the german stock index dax bubble in 1998 the purpose of this article is to explore whether a langevin equation extracted from real world data can generate synthetic time series with comparable lppl structures to this end we first estimate the stochastic process underlying the dax log returns during the period from mid 1997 until end 2003 the employed data set contains about 3 93 10 6 intraday dax quotes at a sampling rate of 15 s our results indicate that the dax log returns can be described as a markov process as a consequence a langevin equation is derived based on this model equation we run extensive simulations in order to generate 100 synthetic dax trajectories each covering 3000 trading days we find lppl behavior in ten artificial time series moreover we can establish a link between lppl patterns and ensuing bubble bursts in seven synthetic 600 week windows however the lppl components in most synthetic trajectories differ fundamentally from those lppl structures that have previously been detected in real financial time series summarized this paper demonstrates that lppl structures are not necessarily the signature of imitative behavior among investors but can also stem from noise even though the likelihood of this is extremely low thus our findings confirm with high statistical confidence that the lppl structures in the dax development are rooted deeper than only in the random fluctuations of the german stock market

on the numerical solution of the equation frac partial 2 z partial x 2 frac partial 2 z partial y 2 left frac partial 2 z partial x partial y right 2 f and its discretizations i the equation indicated in the title is the simplest representative of the class of nonlinear equations of monge ampere type equations with such nonlinearities arise in dynamic meteorology geometric optics elasticity and differential geometry in some special cases heuristic procedures for numerical solution are available but in order for them to be successful a good initial guess is required for a bounded convex domain nonnegativef and dirichlet data we consider a special discretization of the equation based on its geometric interpretation for the discrete version of the problem we propose an iterative method that produces a monotonically convergent sequence no special information about an initial guess is required and to initiate the iterates a routine step is made the method is self correcting and is structurally suitable for a parallel computer the computer program modules and several examples are presented in two appendices

damped dynamic substructures the dynamic substructure method is extended to lightly or heavily damped systems both internal and external dampings are considered the damped dynamic flexibility associated with the slave co ordinates is first expanded in terms of the damped fixed interface natural modes and the condensed dynamic stiffness associated with the master co ordinates is formed subsequently the convergence of the condensed dynamic stiffness with respect to the damped natural modes can be improved by means of the static matrices since the dynamic stiffness method is equivalent to the modal synthesis method the component mode method and kron s method the theory presented here is readily applicable to these methods are restricted to symmetric damping matrices

generalised lagrangian solutions for atmospheric and oceanic flows atmospheric or oceanic flows strongly constrained by rotation and stratification can be described by a set of lagrangian partial differential equations called the semigeostrophic equations in these equations the trajectories must be determined implicitly generalised solutions of these equations are defined as a sequence of rearrangements of the fluid which need not be smooth these solutions are closely related to generalised solutions of the monge ampere equation existence and uniqueness of such solutions is proved the evolution is shown to be a sequence of minimum energy states of the fluid giving strong physical plausibility to the solutions

improved fixed interface method for modal synthesis the paper takes a fresh look at the important topic of substructure synthesis an improved technique for synthesizing the modes obtained with fixed interface coordinates is proposed the reduction in the size of a substructure model is achieved through a higher order mode superposition approach for the first time an analytical rationale for the popular craig bampton hurty method is presented it is shown that the craigbampton hurty method is a special case of the proposed method the proposed approach can be used to improve the accuracy of the calculated eigenproperties by utilizing the higher order component modes but without calculating additional normal modes of the substructures numerical results are presented to demonstrate the effectiveness of the proposed method with the increasing order of the mode superposition approach used the approach is expected to be efficient if the degrees of freedom at the interface of the substructures are relatively small compared with the degrees of freedom of the substructures

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

unconditional stability and error estimates of the modified characteristics fems for the time dependent incompressible mhd equations abstract this paper focuses on the unconditional stability and convergence of characteristics type methods for the time dependent incompressible mhd equations for this purpose we introduce a new characteristics time discrete system the optimal error estimates in l 2 and h 1 norms for the typical modified characteristics finite element method unconditionally can be deduced while the whole previous works require certain time step restrictions some numerical experiments document performance of the characteristics type methods for the time dependent incompressible mhd equations

oscillations in the tsallis income distribution abstract oscillations in the complementary cumulative distribution function ccdf of individual income data have been found in the data of various countries studied by different authors at different time periods but the dynamical origins of this behavior are currently unknown although these datasets can be fitted by different functions at different income ranges the tsallis distribution has recently been found capable of fitting the whole distribution by means of only two parameters this procedure showed clearly such oscillatory feature in the entire income range feature but made it particularly visible at the tail of the distribution although log periodic functions fitted to the data are capable of describing this behavior a different approach to naturally disclose such oscillatory characteristics is to allow the tsallis q parameter to become complex in this paper we use this idea in order to describe the behavior of the ccdf of the brazilian personal income recently studied empirically by soares et al 2016 typical elements of periodic motion such as amplitude and angular frequency coupled to this income analysis were obtained by means of this approach a highly non linear function for the ccdf was obtained through this methodology and a numerical test showed it capable of recovering the main oscillatory feature of the original ccdf of the personal income data of brazil

how to quantify deterministic and random influences on the statistics of the foreign exchange market it is shown that price changes of the u s dollar german mark exchange rates upon different delay times can be regarded as a stochastic marcovian process furthermore we show how kramers moyal coefficients can be estimated from the empirical data finally we present an explicit fokker planck equation which models very precisely the empirical probability distributions in particular their non gaussian heavy tails

log periodic self similarity an emerging financial law abstract a hypothesis that the financial log periodicity cascading self similarly through various time scales carries signatures of a law is pursued it is shown that the most significant historical financial events can be classified amazingly well using a single and unique value of the preferred scaling factor 2 which indicates that its real value should be close to this number this applies even to a declining decelerating log periodic phase crucial in this connection is identification of a super bubble bubble on bubble phenomenon identifying a potential universal preferred scaling factor as undertaken here may significantly improve the predictive power of the corresponding methodology several more specific related results include evidence that i the real end of the high technology bubble on the stock market started with a decelerating log periodic draw down in the beginning of september 2000 ii a parallel 2000 2002 decline seen in the standard poor s 500 from the log periodic perspective is already of the same significance as the one of the early 1930s and of the late 1970s iii all this points to a much more serious global crash in around 2025 of course from a level much higher at least one order of magnitude than in 2000

how many modes are acceptable and how do people improve the modes abstract in this paper an effective judgement and improvement regarding the truncation errors of the modal synthesis method are put forward the truncation errors estimated are considered as error forces and their reasonable bound is defined to determine the number of acceptable modes the initial eigenvalues and eigenvectors can be calculated from the modal synthesis method the renewed eigenvalues are obtained from rayleigh quotient and the renewed eigenvectors are obtained from an equation which is essentially the expression for error forces then the renewed eigenvectors are put into rayleigh quotient again as an iterative process until the permitted error forces or definite errors of eigenvalues are reached as a result the proposed approach is called an error force vector method obviously the process is also an accuracy improvement for eigenpair updating

markov analysis and kramers moyal expansion of nonstationary stochastic processes with application to the fluctuations in the oil price we describe a general method for analyzing a nonstationary stochastic process x t which unlike many of the previous analysis methods does not require x t to have any scaling feature the method is used to study the fluctuations in the daily price of oil it is shown that the returns time series y t ln x t 1 x t is a stationary and markov process characterized by a markov time scale t m the coefficients of the kramers moyal expansion for the probability density function p y tmid r y 0 t 0 are computed p y tmid r y 0 t 0 satisfies a fokker planck equation which is equivalent to a langevin equation for y t that provides quantitative predictions for the oil price over times that are of the order of t m also studied is the average frequency of positive slope crossings nu alpha p y i alpha y i 1 alpha for the returns where t alpha 1nu alpha is the average waiting time for observing y t alpha again

new improved series expansion for solving the moving oscillator problem a new method able to evaluate the dynamic stress response of an elastic beam subject to moving oscillators is presented the proposed procedure improves the convergence and accuracy of the conventional eigenfunction series expansion of beam response taking into account the gravitational inertial and damping effects due to the moving oscillators the improvement of the conventional solution is obtained by means of an extension to continuous systems of the dynamic correction method originally proposed for discretized structures the proposed method is able to account for the truncated higher order eigenfunctions by adding a pseudo static term to the conventional series expansion numerical results are presented to demonstrate the capability of the method to accurately determine the discontinuity and jump in bending moment and shear force distributions respectively

weak existence for the semigeostrophic equations formulated as a coupled monge ampere transport problem hoskins s semigeostrophic equations are reformulated as a coupled monge ampere transport problem b j hoskins quart j royal met soc 97 1971 pp 139 153 existence of global weak solutions is obtained for this formulation

a nonconforming morley finite element method for the fully nonlinear monge ampere equation in this paper we study finite element approximations of the viscosity solution of the fully nonlinear monge ampere equation det d 2 u f 0 using the well known nonconforming morley element our approach is based on the vanishing moment method which was recently proposed as a constructive way to approximate fully nonlinear second order equations by the author and feng j sci comput 38 1 74 98 2009 the vanishing moment method approximates the monge ampere equation by the fourth order quasilinear equation epsilon delta 2u epsilon rm det d 2u epsilon f with appropriate boundary conditions we develop a finite element scheme using the n dimensional morley element introduced in wang and xu numer math 103 155 169 2006 to approximate the regularized fourth order problem in two and three dimensions and then derive optimal order error estimates

a modified characteristic finite element method for a fully nonlinear formulation of the semigeostrophic flow equations this paper develops a fully discrete modified characteristic finite element method for a coupled system consisting of the fully nonlinear monge ampere equation and a transport equation the system is the eulerian formulation in the dual space for b j hoskins semigeostrophic flow equations which are widely used in meteorology to model frontogenesis to overcome the difficulty caused by the strong nonlinearity we first formulate at the differential level a vanishing moment approximation of the semigeostrophic flow equations a methodology recently proposed by the authors x feng and m neilan j sci comput 38 2009 pp 74 98 which involves approximating the fully nonlinear monge ampere equation by a family of fourth order quasilinear equations we then construct a fully discrete modified characteristic finite element method for the regularized problem it is shown that under certain mesh constraint the proposed numerical method converges with an optimal order rate of convergence in particular the obtained error bounds show explicit dependence on the regularization parameter varepsilon numerical tests are also presented to validate the theoretical results and to gauge the efficiency of the proposed fully discrete modified characteristic finite element method

component mode iteration for frequency calculations this paper presents component mode iteration a new method for carrying out a form of subspace iteration on the substructure level for structures composed of substructures in component mode iteration substructure displacement is in terms of component modes with the interface displacements as coefficients component modes are chosen such that exact compatibility is assured throughout the structure the reduced structure eigenvalue problem is assembled and solved for lower mode approximations the resulting estimates of lower mode interface motion are used to refine the component modes by approximating the dynamic response of substructures to interface motion with a vector iteration process for this step the substructures are independent and can be processed in parallel reassembly and solution of the reduced structure problem yields improved lower mode estimates further iterations of refinement and reassembly are performed until convergence takes place the initial component modes are generated automatically and at negligible expense and numerical examples show that convergence with these is excellent yielding very accurate results after only one iteration indeed convergence is shown to be significantly faster than in the standard subspace iteration method

improved estimation of fokker planck equations through optimization an improved method for the description of hierarchical complex systems by means of a fokker planck equation is presented in particular the limited memory broyden fletcher goldfarb shanno algorithm for constraint problems is used to minimize the distance between the numerical solutions of the fokker planck equation and the empirical probability density functions and thus to estimate properly the drift and diffusion term of the fokker planck equation the optimization routine is applied to a time series of velocity measurements obtained from a turbulent helium gas jet in order to demonstrate the benefits and to quantify the improvements of this optimization routine

unconditional stability and error estimates of modified characteristics fems for the navier stokes equations the paper is concerned with the unconditional stability and convergence of characteristics type methods for the time dependent navier stokes equations we present optimal error estimates in l 2 l2 and h 1 h1 norms for a typical modified characteristics finite element method unconditionally while all previous works require certain time step restrictions the analysis is based on an iterated characteristic time discrete system with which the error function is split into a temporal error and a spatial error with a rigorous analysis to the characteristic time discrete system we prove that the difference between the numerical solution and the solution of the time discrete system is tau independent where tau denotes the time stepsize thus numerical solution in w 1 infty w1 is bounded and optimal error estimates can be obtained in a traditional way numerical results confirm our analysis and show clearly the unconditional stability and convergence of the modified characteristics finite element method for the time dependent navier stokes equations the approach used in this paper can be easily extended to many other characteristics based methods

numerical methods for fully nonlinear elliptic equations of the monge ampere type abstract in this article we discuss the numerical solution of the dirichlet problem for the monge ampere equation in two dimensions the solution of closely related problems is also discussed these include a family of pucci s equations the equation prescribing the harmonic mean of the eigenvalues of the hessian of a smooth function of two variables and a minimization problem from nonlinear elasticity where the cost functional involves the determinant of the gradient of vector valued functions to solve the monge ampere equation we consider two methods the first one reduces the monge ampere equation to a saddle point problem for a well chosen augmented lagrangian to solve this saddle point problem we advocate an uzawa douglas rachford algorithm the second method combines nonlinear least squares and operator splitting this second method being simpler to implements we apply variants of it to the solution of the other problems for the space discretization we use mixed finite element approximations closely related to methods already used for the solution of linear and nonlinear bi harmonic problems through these approximations the solution of the above problems is essentially reduced to the solution of discrete poisson problems the methods discussed in this article are validated by the results of numerical experiments

estimation of kramers moyal coefficients at low sampling rates an optimization procedure for the estimation of kramers moyal coefficients from stationary one dimensional markovian time series data is presented the method takes advantage of a recently reported approach that allows one to calculate exact finite sampling interval effects by solving the adjoint fokker planck equation therefore it is well suited for the analysis of sparsely sampled time series the optimization can be performed either by making a parametric ansatz for drift and diffusion functions or parameter free we demonstrate the power of the method in several numerical examples with synthetic time series

analysis of the real estate market in las vegas bubble seasonal patterns and prediction of the csw indices we analyze 27 house price indices of las vegas from june 1983 to march 2005 corresponding to 27 different zip codes these analyses confirm the existence of a real estate bubble defined as a price acceleration faster than exponential which is found however to be confined to a rather limited time interval in the recent past from approximately 2003 to mid 2004 and has progressively transformed into a more normal growth rate comparable to pre bubble levels in 2005 there has been no bubble till 2002 except for a medium sized surge in 1990 in addition we have identified a strong yearly periodicity which provides a good potential for fine tuned prediction from month to month a monthly monitoring using a model that we have developed could confirm by testing the intra year structure if indeed the market has returned to normal or if more turbulence is expected ahead we predict the evolution of the indices one year ahead which is validated with new data up to september 2006 the present analysis demonstrates the existence of very significant variations at the local scale in the sense that the bubble in las vegas seems to have preceded the more global usa bubble and has ended approximately two years earlier mid 2004 for las vegas compared with mid 2006 for the whole of the usa

revisiting the moving force problem abstract the problem of the vibration of a beam subject to a travelling force is considered the purpose of the study is to develop simple tools for finding the maximum deflection of a beam for any given velocity of the travelling force it is shown that for given boundary conditions there exists a unique response velocity dependence function a technique to determine this function is suggested which is based on the assumption that the maximum beam response can be adequately approximated by means of the first beam mode to illustrate this the maximum response function is calculated analytically for a simply supported ss beam and constructed numerically for a clamped clamped beam the effect of the higher modes on the maximum response is investigated and the relative error of the one mode approximation for a ss beam is constructed the estimates obtained substantiate the assumption about adequacy of the one mode approximation in a wide range of velocities in particular the relative error in the neighborhood of the velocity that results in the largest response is less than one percent

a substructure approach for the midfrequency vibration of stochastic systems a novel substructure coupling technique based on the proper orthogonal decomposition method is presented for the midfrequency range vibration of linear dynamical systems with parameter uncertainty for a given frequency band the methodology permits the derivation of an adaptive basis for each subsystem and the construction of a reduced order model of the global structure the formulation is directed toward the efficient probabilistic characterization of model based predictions in the framework of a stochastic finite element method the efficiency of the substructure method has been contrasted both from the viewpoint of adopting free free and fixed fixed substructure proper orthogonal modes in order to arrive at a reduced subsystem model the distinction as well as similarity of the present methodology with the component mode synthesis is also pointed out the proper orthogonal modes are obtained from both frequency and time domain approaches and their suitability is discussed in relation to the behavior of a specific system the substructure approach elegantly integrates with a version of stochastic finite elements based on orthogonal decompositions and projections of stochastic processes

dynamic condensation with synthesis of substructure eigenproperties abstract for structures divided into substructures which are defined by their eigenproperties a combination of the node synthesis and condensation techniques is presented to obtain a first few important eigenproperties of the combined system accurately the accuracy of the calculated modal properties can be increased by iteratively solving an improved and condensed eigenvalue problem if desired the higher modal properties can also be obtained by solving a complementry eigenvalue problem associated with the higher modes the availability of the substructure modes in this approach also provides a very convenient and efficient means of selecting the most appropriate kept degrees of freedom in the condensed eigenvalue problem to expediate the convergence to the correct eigenproperties the methods for coupling of the substructure eigenproperties obtained with different boundary conditions at the interface are described several examples are presented to demonstrate the application of the approach

on three modal synthesis variants abstract a unified formulation of fixed free and loaded interface variants of modal synthesis is presented a formal proof of the exactness of all three variants in the absence of modal truncation is established the sensitivity of the truncation error to the different interface conditions is studied

mixed finite element methods for the fully nonlinear monge ampere equation based on the vanishing moment method this paper studies mixed finite element approximations of the viscosity solution to the dirichlet problem for the fully nonlinear monge ampere equation det d 2u 0 f 0 based on the vanishing moment method which was proposed recently by the authors in x feng and m neilan j scient comp doi 10 1007 s10915 008 9221 9 2008 in this approach the second order fully nonlinear monge ampere equation is approximated by the fourth order quasilinear equation varepsilon delta 2 u varepsilon det d 2u varepsilon f it was proved in x feng trans ams submitted that the solution u varepsilon converges to the unique convex viscosity solution u 0 of the dirichlet problem for the monge ampere equation this result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations a task which has been impracticable before the goal of this paper is threefold first we develop a family of hermann miyoshi type mixed finite element methods for approximating the solution u varepsilon of the regularized fourth order problem which computes simultaneously u varepsilon and the moment tensor sigma varepsilon d 2u varepsilon second we derive error estimates which track explicitly the dependence of the error constants on the parameter varepsilon for the errors u varepsilon u varepsilon h and sigma 0 sigma h varepsilon finally we present a detailed numerical study on the rates of convergence in terms of powers of varepsilon for the error u 0 u h varepsilon and sigma varepsilon sigma h varepsilon and numerically examine what is the best mesh size h in relation to varepsilon in order to achieve these rates due to the strong nonlinearity of the underlying equation the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work for the problem to overcome the difficulty we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations

discontinuous galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations this paper develops a discontinuous galerkin dg finite element differential calculus theory for approximating weak derivatives of sobolev functions and piecewise sobolev functions by introducing numerical one sided derivatives as building blocks various first and second order numerical operators such as the gradient divergence hessian and laplacian operator are defined and their corresponding calculus rules are established among the calculus rules are product and chain rules integration by parts formulas and the divergence theorem approximation properties and the relationship between the proposed dg finite element numerical derivatives and some well known finite difference numerical derivative formulas on cartesian grids are also established besides independent interest in numerical differentiation the primary motivation and goal of developing the dg finite element differential calculus is to solve partial differential equations it is shown that several existing finite element finite difference and discontinuous galerkin methods can be rewritten compactly using the proposed dg finite element differential calculus framework moreover new discontinuous galerkin methods for linear and nonlinear pdes are also obtained from the framework

evidence of markov properties of high frequency exchange rate data we present a stochastic analysis of a data set consisting of 106 quotes of the us dollar german mark exchange rate evidence is given that the price changes x upon different delay times can be described as a markov process evolving in thus the dependence of the probability density function pdf p x on the delay time can be described by a fokker planck equation a generalized diffusion equation for p x this equation is completely determined by two coefficients d1 x and d2 x drift and diffusion coefficient respectively we demonstrate how these coefficients can be estimated directly from the data without using any assumptions or models for the underlying stochastic process furthermore it is shown that the solutions of the resulting fokker planck equation describe the empirical pdfs correctly including the pronounced tails

renormalization group analysis of the 2000 2002 anti bubble in the us s p500 index explanation of the hierarchy of five crashes and prediction abstract we propose a straightforward extension of our previously proposed log periodic power law model of the anti bubble regime of the usa stock market since the summer of 2000 in terms of the renormalization group framework to model critical points using a previous work by gluzman and sornette phys rev e 65 2003 036142 on the classification of the class of weierstrass like functions we show that the five crashes that occurred since august 2000 can be accurately modeled by this approach in a fully consistent way with no additional parameters our theory suggests an overall consistent organization of the investors forming a collective network which interact to form the pessimistic bearish anti bubble regime with intermittent acceleration of the positive feedbacks of pessimistic sentiment leading to these crashes we develop retrospective predictions that confirm the existence of significant arbitrage opportunities for a trader using our model finally we offer a prediction for the unknown future of the us s p500 index extending over 2003 and 2004 that refines the previous prediction of sornette and zhou quant finance 2 2002 468

medium and small scale analysis of financial data abstract a stochastic analysis of financial data is presented in particular we investigate how the statistics of log returns change with different time delays the scale dependent behaviour of financial data can be divided into two regions the first time range the small timescale region in the range of seconds seems to be characterised by universal features the second time range the medium timescale range from several minutes upwards can be characterised by a cascade process which is given by a stochastic markov process in the scale a corresponding fokker planck equation can be extracted from given data and provides a non equilibrium thermodynamical description of the complexity of financial data

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem