maximum entropy and the nearly black object summary maximum entropy me inversion is a non linear inversion technique for inverse problems where the object to be recovered is known to be positive it has been applied in areas ranging from radio astronomy to various forms of spectroscopy sometimes with dramatic success in some cases me has attained an order of magnitude finer resolution and or an order of magnitude smaller noise level than that obtainable by standard linear methods the dramatic successes all seem to occur in cases where the object to be recovered is nearly black essentially zero in the vast majority of samples we show that near blackness is required both for signal to noise enhancements and for superresolution however other methods in particular minimum 1 norm reconstruction may exploit near blackness to an even greater extent

phase correlation of foreign exchange time series abstract correlation of foreign exchange rates in currency markets is investigated based on the empirical data of usd dem and usd jpy exchange rates for a period from february 1 1986 to december 31 1996 the return of exchange time series is first decomposed into a number of intrinsic mode functions imfs by the empirical mode decomposition method the instantaneous phases of the resultant imfs calculated by the hilbert transform are then used to characterize the behaviors of pricing transmissions and the correlation is probed by measuring the phase differences between two imfs in the same order from the distribution of phase differences our results show explicitly that the correlations are stronger in daily time scale than in longer time scales the demonstration for the correlations in periods of 1986 1989 and 1990 1993 indicates two exchange rates in the former period were more correlated than in the latter period the result is consistent with the observations from the cross correlation calculation

a confidence limit for the empirical mode decomposition and hilbert spectral analysis the confidence limit is a standard measure of the accuracy of the result in any statistical analysis most of the confidence limits are derived as follows the data are first divided into subsections and then under the ergodic assumption the temporal mean is substituted for the ensemble mean next the confidence limit is defined as a range of standard deviations from this mean however such a confidence limit is valid only for linear and stationary processes furthermore in order for the ergodic assumption to be valid the subsections have to be statistically independent for non stationary and nonlinear processes such an analysis is no longer valid the confidence limit of the method here termed emd hsa for empirical mode decomposition hilbert spectral analysis is introduced by using various adjustable stopping criteria in the sifting processes of the emd step to generate a sample set of intrinsic mode functions imfs the emd technique acts as a pre processor for hsa on the original data producing a set of components imfs from the original data that equal the original data when added back together each imf represents a scale in the data from smallest to largest the ensemble mean and standard deviation of the imf sample sets obtained with different stopping criteria are calculated and these form a simple random sample set the confidence limit for emd hsa is then defined as a range of standard deviations from the ensemble mean without evoking the ergodic assumption subdivision of the data stream into short sections is unnecessary hence the results and the confidence limit retain the full frequency resolution of the full dataset this new confidence limit can be applied to the analysis of nonlinear and non stationary processes by these new techniques data from length of day measurements and a particularly violent recent earthquake are used to demonstrate how the confidence limit is obtained and applied by providing a confidence limit for this new approach a stable range of stopping criteria for the decomposition or sifting phase emd has been established making the results of the final processing with hsa and the entire emd hsa method more definitive

partially finite convex programming part ii explicit lattice models in part i of this work we derived a duality theorem for partially finite convex programs problems for which the standard slater condition fails almost invariably our result depended on a constraint qualification involving the notion ofquasi relative interior the derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function the differentiability of various convex functions in lattices was considered at the end of part i in part ii we shall apply our results to a number of more concrete problems including variants of semi infinite linear programming l1 approximation constrained approximation and interpolation spectral estimation semi infinite transportation problems and the generalized market area problem of lowe and hurter 1976 as in part i we shall use lattice notation extensively but as we illustrated there in concrete examples lattice theoretic ideas can be avoided if preferred by direct calculation

partially finite programming in l_1 and the existence of maximum entropy estimates best entropy estimation is a technique that has been widely applied in many areas of science it consists of estimating an unknown density from some of its moments by maximizing some measure of the entropy of the estimate this problem can be modelled as a partially finite convex program with an integrable function as the variable a complete duality and existence theory is developed for this problem and for an associated extended problem which allows singular measure theoretic solutions this theory explains the appearance of singular components observed in the literature when the burg entropy is used it also provides a unified treatment of existence conditions when the burg boltzmann shannon or some other entropy is used as the objective some examples are discussed

exact boundary conditions for the initial value problem of convex conservation laws the initial value problem of convex conservation laws which includes the famous burgers inviscid equation plays an important rule not only in theoretical analysis for conservation laws but also in numerical computations for various numerical methods for example the initial value problem of the burgers equation is one of the most popular benchmarks in testing various numerical methods but in all the numerical tests the initial data have to be assumed that they are either periodic or having a compact support so that periodic boundary conditions at the periodic boundaries or two constant boundary conditions at two far apart spatial artificial boundaries can be used in practical computations in this paper for the initial value problem with any initial data we propose exact boundary conditions at two spatial artificial boundaries which contain a finite computational domain by using the lax s exact formulas for the convex conservation laws the well posedness of the initial boundary problem is discussed and the finite difference schemes applied to the artificial boundary problems are described numerical tests with the proposed artificial boundary conditions are carried out by using the lax friedrichs monotone difference schemes

dual methods in entropy maximization application to some problems in crystallography this paper is devoted to some infinite dimensional optimization problems with finitely many constraints these deal with entropy maximization and this paper is particularly concerned with those or

optimal l 1 rate of convergence for the viscosity method and monotone scheme to piecewise constant solutions with shocks we derive optimal error bounds for the viscosity method and monotone difference schemes to an initial value problem of scalar conservation laws with initial data being a finite number of piecewise constants subject to the initial discontinuities satisfying the entropy conditions it is known that the entropy solution of the problem is piecewise constant with a finite number of interacting shocks satisfying the entropy conditions a rigorous analysis shows that both the viscosity method and monotone schemes to approach the initial value problem have uniform l 1 error bounds of o epsilon and o delta x for the time infty t 0 respectively where epsilon and delta x are their corresponding viscosity coefficient and discrete mesh length the results are improvements over the half order rates of l 1 convergence numerical experiments for the lax friedrichs scheme are presented and numerical results justify the theoretical analysis

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

existence uniqueness and a constructive solution algorithm for a class of finite markov moment problems we consider a class of finite markov moment problems with an arbitrary number of positive and negative branches we show criteria for the existence and uniqueness of solutions and we characterize in detail the nonunique solution families moreover we present a constructive algorithm to solve the moment problems numerically and prove that the algorithm computes the right solution

error estimate and the geometric corrector for the upwind finite volume method applied to the linear advection equation this paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions we introduce what we call the geometric corrector which is a sequence associated with every finite volume mesh in mathbf r nd and every nonvanishing vector mathbf a of mathbf r nd first we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size then an order one error estimate for the finite volume scheme occurs afterwards we prove that this norm is indeed bounded by the mesh size in several cases including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimensions computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases such as the one with independent refined meshes

superresolution in the markov moment problem abstract suppose that bar x is a characteristic function and a 1 a 2 a n are weight functions on a finite measure space recent work of gamboa and gassiat observed conditions guaranteeing that x bar x 1 is small whenever 0 x 1 a e and the moment errors int x bar x a i are small i 1 2 n using concise and elementary techniques we obtain similar results under very mild assumptions we also provide precise error bounds

statistical properties of the volatility of price fluctuations we study the statistical properties of volatility measured by locally averaging over a time window t the absolute value of price changes over a short time interval ensuremath delta t we analyze the s 500 stock index for the 13 year period jan 1984 to dec 1996 we find that the cumulative distribution of the volatility is consistent with a power law asymptotic behavior characterized by an exponent ensuremath mu ensuremath approx 3 similar to what is found for the distribution of price changes the volatility distribution retains the same functional form for a range of values of t further we study the volatility correlations by using the power spectrum analysis both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents also both methods show the presence of a crossover at approximately 1 5 days in addition we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 u s companies over the two year period jan 1994 to dec 1995

existence of discrete shock profiles of a class of monotonicity preserving schemes for conservation laws when shock speed s times t x is rational the existence of solutions of shock profile equations on bounded intervals for monotonicity preserving schemes with continuous numerical flux is proved a sufficient condition under which the above solutions can be extended to j implying the existence of discrete shock profiles of numerical schemes is provided a class of monotonicity preserving schemes including all monotonicity preserving schemes with c 1 numerical flux functions the second order upwinding flux based muscl scheme the second order flux based muscl scheme with lax friedrichs splitting and the godunov scheme for scalar conservation laws are found to satisfy this condition thus the existence of discrete shock profiles for these schemes is established when s t x is rational

dynamic fluctuations of cross correlations in multi time scale abstract with the empirical mode decomposition we investigate the dynamic fluctuations of cross correlations in multi time scale in this approach the time series of price returns in financial markets is decomposed into a small number of intrinsic mode functions and the corresponding time series of phases can be obtained with the hilbert transform through the empirical mode decomposition we uncover that the cross correlations between stocks change significantly for different time scales and exhibit singular behaviors during a financial crisis with the phase correlation analysis we observe that the cross correlations between market indices vary also for different time scales dividing the total time interval into two sub intervals by the year 2007 we detect that the market indices are much more correlated after 2007

vanishing curvature viscosity for front propagation in this paper we study the front propagation with constant speed and small curvature viscosity we first investigate two related problems of conservation laws one of which is on the nonlinear viscosity methods for the conservation laws and the other one is on the structure of solutions to conservation laws with l 1 initial data we show that the nonlinear viscosity methods approaching the piecewise smooth solutions with finitely many discontinuity for convex conservation laws have the first order rate of l 1 convergence the solutions of conservation laws with l 1 initial data are shown to be bounded after t 0 if all singular points of initial data are from shocks these results suggest that the front propagation with constant speed and a small curvature viscosity will approach the front movements with a constant speed as the small parameter goes to zero after the front breaks down the cusps will disappear promptly and corners will be formed 2000 academic press

pointwise error estimates for scalar conservation laws with piecewise smooth solutions we introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation and in fact more general approximations to scalar conservation laws with piecewise smooth solutions to this end we derive a transport inequality for an appropriately weighted error function the key ingredient in our approach is a one sided interpolation inequality between classical l1 error estimates and lip stability bounds the one sided interpolation interesting for its own sake enables us to convert a global l1 result into a nonoptimal local estimate this in turn provides the necessary bounds on the coefficients of the above mentioned transport inequality estimates on the weighted error then follow from the maximum principle and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation unlike previous works in this direction our method can deal with finitely many waves with possible collisions moreover in our approach one does not follow the characteristics but instead makes use of the energy method and hence this approach could be extended to other types of approximate solutions

on the regularity of approximate solutions to conservation laws with piecewise smooth solutions in this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted w1 1 space convergence rate for the derivative of the approximate solutions is established under the assumption that a weak pointwise error estimate is given in other words we are able to convert weak pointwise error estimates to optimal error bounds in a weighted w1 1 space for convex conservation laws the assumption of a weak pointwise error estimate is verified by tadmor siam j numer anal 28 1991 pp 891 906 therefore one immediate application of our w1 1 convergence theory is that for convex conservation laws we indeed have w1 1 error bounds for the approximate solutions to conservation laws furthermore the co en pointwise error estimates of tadmor and tang siam j numer anal 36 1999 pp 1739 1758 are recovered by the use of the w1 1 convergence result

on the piecewise smooth solutions to non homogeneous scalar conservation laws abstract we study the structure and smoothness of non homogeneous convex conservation laws the question regarding the number of smoothness pieces is addressed it is shown that under certain conditions on the initial data the entropy solution has only a finite number of discontinuous curves we also obtain some global estimates on derivatives of the piecewise smooth entropy solution along the generalized characteristics these estimates play important roles in obtaining the optimal rate of convergence for various approximation methods to conservation laws

modified equation for adaptive monotone difference schemes and its convergent analysis a modified parabolic equation for adaptive monotone difference schemes based on equal arclength mesh applied to the linear convection equation is derived and its convergence analysis shows that solutions of the modified equation approach a discontinuous piecewise smooth solution of the linear convection equation at order one rate in the l 1 norm it is well known that solutions of the monotone schemes with uniform meshes and their modified equation approach the same discontinuous solution at a half order rate in the l 1 norm therefore the convergence analysis for the modified equation provided in this work demonstrates theoretically that the monotone schemes with adaptive grids can improve the solution accuracy numerical experiments also confirm the theoretical conclusions