approximation order of bivariate spline interpolation for arbitrary smoothness by using the algorithm of nurnberger and riessinger 1995 we construct hermite interpolation sets for spaces of bivariate splines sqr 1 of arbitrary smoothness defined on the uniform type triangulations it is shown that our hermite interpolation method yields optimal approximation order for q 3 5r 1 in order to prove this we use the concept of weak interpolation and arguments of birkhoff interpolation

can one make any crash prediction in finance using the local hurst exponent idea abstract we apply the hurst exponent idea for investigation of djia index time series data the behavior of the local hurst exponent prior to drastic changes in financial series signal is analyzed the optimal length of the time window over which this exponent can be calculated in order to make some meaningful predictions is discussed our prediction hypothesis is verified with examples of 1929 and 1987 crashes as well as with more recent phenomena in stock market from the period 1995 to 2003 some interesting agreements are found

approximated maximum likelihood estimation in multifractal random walks we present an approximated maximum likelihood method for the multifractal random walk processes of e bacry et al phys rev e 64 026103 2001 the likelihood is computed using a laplace approximation and a truncation in the dependency structure for the latent volatility the procedure is implemented as a package in the r computer language its performance is tested on synthetic data and compared to an inference approach based on the generalized method of moments the method is applied to estimate parameters for various financial stock indices

multifractal analysis of ssec in chinese stock market a different empirical result from heng seng index abstract in this paper high frequency per 5 min data of shanghai stock exchange composite index ssec from january 1999 to july 2001 is analyzed by multifractal we find that the correlation of the parameters of the multifractal spectra with the variation of daily return z in ssec is noticeably different from that in previous studies of heng seng index in hong kong stock market sun et al phys a 291 2001 553 562 sun et al phys a 301 2001 473 482 so we suppose that there may not be a universal rule for the dependence of the parameters of the multifractal spectra with daily return of a stock index then we construct a new measurement of market risk based on multifractal spectra and test its ability of predicting index fluctuations with a more thorough method than that in sun et al phys a 301 2001 473 482

time dependent hurst exponent in financial time series abstract we calculate the hurst exponent h t of several time series by dynamical implementation of a recently proposed scaling technique the detrending moving average dma in order to assess the accuracy of the technique we calculate the exponent h t for artificial series simulating monofractal brownian paths with assigned hurst exponents h we next calculate the exponent h t for the return of high frequency tick by tick sampled every minute series of the german market we find a much more pronounced time variability in the local scaling exponent of financial series compared to the artificial ones the dma algorithm allows the calculation of the exponent h t without any a priori assumption on the stochastic process and on the probability distribution function of the random variables as happens for example in the case of the kitagawa grid and the extended kalmann filtering methods the present technique examines the local scaling exponent h t around a given instant of time this is a significant advance with respect to the standard wavelet transform or to the higher order power spectrum technique which instead operate on the global properties of the series by legendre or fourier transform of qth order moments

on hurst exponent estimation under heavy tailed distributions abstract in this paper we show how the sampling properties of the hurst exponent methods of estimation change with the presence of heavy tails we run extensive monte carlo simulations to find out how rescaled range analysis r s multifractal detrended fluctuation analysis m f d f a detrending moving average d m a and generalized hurst exponent approach g h e estimate hurst exponent on independent series with different heavy tails for this purpose we generate independent random series from stable distribution with stability exponent changing from 1 1 heaviest tails to 2 gaussian normal distribution and we estimate the hurst exponent using the different methods r s and g h e prove to be robust to heavy tails in the underlying process g h e provides the lowest variance and bias in comparison to the other methods regardless the presence of heavy tails in data and sample size utilizing this result we apply a novel approach of the intraday time dependent hurst exponent and we estimate the hurst exponent on high frequency data for each trading day separately we obtain hurst exponents for s p500 index for the period beginning with year 1983 and ending by november 2009 and we discuss the surprising result which uncovers how the market s behavior changed over this long period

assessing market uncertainty by means of a time varying intermittency parameter for asset price fluctuations abstract maximum likelihood estimation techniques for multifractal processes are applied to high frequency data in order to quantify intermittency in the fluctuations of asset prices from time records as short as one month these methods permit extraction of a meaningful intermittency parameter characterising the degree of volatility clustering we can therefore study the time evolution of volatility clustering and test the statistical significance of this variability by analysing data from the oslo stock exchange and comparing the results with the investment grade spread we find that the estimates of are lower at times of high market uncertainty

optimal error estimates in jacobi weighted sobolev spaces for polynomial approximations on the triangle spectral approximations on the triangle by orthogonal polynomials are studied in this paper optimal error estimates in weighted semi norms for both the l2 and h 1 0 orthogonal polynomial projections are established by using the generalized koornwinder polynomials and the properties of the sturm liouville operator on the triangle these results are then applied to derive error estimates for the spectral galerkin method for second and fourth order equations on the triangle the generalized koornwinder polynomials and approximation results developed in this paper will be useful for many other applications involving spectral and spectral element approximations in triangular domains

a triangular spectral element method applications to the incompressible navier stokes equations abstract encouraged by the success of spectral elements methods in computational fluid dynamics and p type finite element methods in structural mechanics we wish to extend these ideas to solving high order polynomial approximations on triangular domains as the next generation of spectral element solvers we introduce here a complete formulation using a modal basis which has been implemented in a new code ne t r the new basis has the following properties jacobi polynomials of mixed weights semi orthogonality hierarchical structure generalized tensor warped product variable order and a new apex co ordinate system allowing automated integration with gaussian quadrature we have discussed the formulation using a matrix notation which allows for an easy interpretation of the forward and backward transformations we use this notation to formulate the linear advection and helmholtz equations in an efficient manner and show that we recover the following properties well conditioned matrices asymptotic operation count of o n 3 scaling o n 2 of the spectral radius of the weak convective operator and exponential convergence using polynomials up to 40 having constructed and numerically analysed these equations we are then able to solve the incompressible navier stokes equations using a high order splitting scheme we demonstrate a variety of results showing exponential convergence using both deformed and straight triangular subdomains for both the stokes and navier stokes problems

predictability of multifractal analysis of hang seng stock index in hong kong in this paper the daily hang seng index in hong kong stock market is studied by multifractal analysis the main parameter of multifractal spectra used is f which can be used to characterize the ratio of number of highest index moments to that of lowest ones the dependence of today s gain probability g and the day s index increase probability n with f of the previous 3 days are studied it is found that g and n can reach 70 80 at the large positive f region and can be very close to 20 at the big negative f region the predictability decreases with the increasing number of the previous days

application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes we consider the numerical solution of the shallow water equations on unstructured grids we focus on flows over wet areas the extension to the case of dry bed will be reported elsewhere the shallow water equations fall into the category of systems of conservation laws which can be symmetrized thanks to the existence of a mathematical entropy coinciding in this case with the total energy our aim is to show the application of a particular class of conservative residual distribution rd schemes to the discretization of the shallow water equations and to analyze their discrete accuracy and stability properties we give a review of conservative rd schemes showing relations between different approaches previously published and recall l stability and accuracy criteria characterizing the schemes in particular the accuracy of the rd method in presence of source terms is analyzed and conditions to construct rth order discretizations on irregular triangular grids are proved it is shown that the rd approach gives a natural way of obtaining high order discretizations which moreover preserves exactly the steady lake at rest solution independently on mesh topology nature of the variation of the bottom and polynomial order of interpolation used for the unknowns we also consider more general analytical solutions which are less investigated from the numerical view point on irregular grids linearity preserving rd schemes yield a truly second order approximation we also sketch a strategy to achieve discretizations which preserve exactly some of these solutions numerical results on steady and time dependent problems involving smooth and non smooth variations of the bottom topology show very promising features of the approach

multifractal analysis of hang seng index in hong kong stock market abstract in this paper the daily hang seng index in hong kong stock market is analyzed by multifractal the correlation of the parameters of the multifractal spectra with the variation of close return z is studied statistically it is found that the amount of the variation of return is correlated with the amount of of that day the increase or decrease of the return is related to the positive or negative value of f the gain probability and the increasing day s probability can be higher than 70 at the larger f region and be lower than 20 at the more negative region

a pseudo spectral scheme for the incompressible navier stokes equations using unstructured nodal elements a pseudo spectral scheme for solving the incompressible navier stokes equations using unstructured nodal triangles is proposed efficient algorithms are developed with numerical evidence that indicates optimal rates of convergence can be achieved navier stokes simulations of kovasznay shear layer roll up and row past a cylinder are included to show comparisons between the different nodal sets considered and an alternative modal approach c 2000 academic press

a new triangular and tetrahedral basis for high order hp finite element methods in this paper we describe the foundations of a new hierarchical modal basis suitable for high order hp finite element discretizations on unstructured meshes it is based on a generalized tensor product of mixed weight jacobi polynomials the generalized tensor product property leads to a low operation count with the use of sum factorization techniques variable p order expansions in each element are readily implemented which is a crucial property for efficient adaptive discretizations numerical examples demonstrate the exponential convergence for smooth solutions and the ability of this formulation to handle easily very complex two and three dmensional computational domains employing standard meshes

multifractality in stock indexes fact or fiction abstract multifractal analysis and extensive statistical tests are performed upon intraday minutely data within individual trading days for four stock market indexes including hsi szsc s p 500 and nasdaq to check whether the indexes instead of the returns possess multifractality we find that the mass exponent q is linear and the singularity q is close to 1 for all trading days and all indexes furthermore we find strong evidence showing that the scaling behaviors of the original data sets cannot be distinguished from those of shuffled time series hence the so called multifractality in the intraday stock market indexes is merely an illusion

on the constants in hp finite element trace inverse inequalities we derive inverse trace inequalities for hp finite elements utilizing orthogonal polynomials we show how to derive explicit expressions for the constants when considering triangular and tetrahedral elements we also discuss how to generalize this technique to the general d simplex

modeling electricity spot prices using mean reverting multifractal processes abstract we discuss stochastic modeling of volatility persistence and anti correlations in electricity spot prices and for this purpose we present two mean reverting versions of the multifractal random walk mrw in the first model the anti correlations are modeled in the same way as in an ornstein uhlenbeck process i e via a drift damping term and in the second model the anti correlations are included by letting the innovations in the mrw model be fractional gaussian noise with h 1 2 for both models we present approximate maximum likelihood methods and we apply these methods to estimate the parameters for the spot prices in the nordic electricity market the maximum likelihood estimates show that electricity spot prices are characterized by scaling exponents that are significantly different from the corresponding exponents in stock markets confirming the exceptional nature of the electricity market in order to compare the damped mrw model with the fractional mrw model we use ensemble simulations and wavelet based variograms and we observe that certain features of the spot prices are better described by the damped mrw model the characteristic correlation time is estimated to approximately half a year

spectral approximations on the triangle in this paper we describe a new family of polynomials which are eigenfunctions of a singular sturm liouville problem on the triangle t 2 x y x 0 y 0 x y 1 the polynomials are shown to be orthogonal over t 2 with respect to a unit weight function and may be used in approximations which are exponentially convergent for functions which are infinitely smooth in t 2 the zeros of the polynomials may be used in cubature formulae on t 2

a triangular spectral element method using fully tensorial rational basis functions a rational approximation in a triangle is proposed and analyzed in this paper the rational basis functions in the triangle are obtained from the polynomials in the reference square through a collapsed coordinate transform optimal error estimates for the l 2 and h 1 0 orthogonal projections are derived with upper bounds expressed in the original coordinates in the triangle it is shown that the rational approximation is as accurate as the polynomial approximation in the triangle illustrative numerical results which are in agreement with the theoretical estimates are presented

multifractal random walk we introduce a class of multifractal processes referred to as multifractal random walks mrws to our knowledge it is the first multifractal process with continuous dilation invariance properties and stationary increments mrws are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio the mrws are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments we briefly explain how in the same way one can build stationary multifractal processes or positive random measures

approximation of patches by cr finite elements of powell sabin type in this work we develop a method to extend a function that is defined in a finite set of disjoint patches to a bigger domain containing all of them the way to extend the function is by minimizing an energy functional which controls the closeness of the extended function to the original one over the patches as well as the smoothness of the final reconstructed function we show the existence and uniqueness of solution of this problem and we give a convergence result as well as several graphical and numerical examples

approximation by discrete variational splines in this paper we present a numerical approximation of curves and surfaces from a given scattered data set an approximating curve or surface problem is obtained by minimizing a quadratic functional in a parametric finite element space its solution is called a discrete smoothing variational spline the existence and uniqueness of this problem are shown as long as the convergence of the method is established finally some particular examples are given

dynamical generalized hurst exponent as a tool to monitor unstable periods in financial time series we investigate the use of the hurst exponent dynamically computed over a weighted moving time window to evaluate the level of stability instability of financial firms financial firms bailed out as a consequence of the 2007 2008 credit crisis show a neat increase with time of the generalized hurst exponent in the period preceding the unfolding of the crisis conversely firms belonging to other market sectors which suffered the least throughout the crisis show opposite behaviors we find that the multifractality of the bailed out firms increase at the crisis suggesting that the multi fractal properties of the time series are changing these findings suggest the possibility of using the scaling behavior as a tool to track the level of stability of a firm in this paper we introduce a method to compute the generalized hurst exponent which assigns larger weights to more recent events with respect to older ones in this way large fluctuations in the remote past are less likely to influence the recent past we also investigate the scaling associated with the tails of the log returns distributions and compare this scaling with the scaling associated with the hurst exponent observing that the processes underlying the price dynamics of these firms are truly multi scaling