generalization of the darboux transformation and generalized harmonic oscillators the darbroux transformation is generalized for time dependent hamiltonian systems which include a term linear in momentum and a time dependent mass the formalism for the n fold application of the transformation is also established and these formalisms are applied for a general quadratic system a generalized harmonic oscillator and a quadratic system with an inverse square interaction up to n 2 among the new features found it is shown for the general quadratic system that the shape of potential difference between the original system and the transformed system could oscillate according to a classical solution which is related to the existence of coherent states in the system

darboux transformations and supersymmetry for the generalized schrodinger equations in 1 1 dimensions we study intertwining relations supersymmetry and darboux transformations for generalized schrodinger equations in 1 1 dimensions we obtain intertwiners in an explicit form by means of which we construct arbitrary order darboux transformations for our class of equations we develop a corresponding supersymmetric formulation and prove equivalence of the darboux transformations with the supersymmetry formalism finally we show that our darboux transformations can also be constructed by means of point transformations avoiding the use of intertwiners

renormalized field theory of collapsing directed randomly branched polymers we present a dynamical field theory for directed randomly branched polymers and in particular their collapse transition we develop a phenomenological model in the form of a stochastic response functional that allows us to address several interesting problems such as the scaling behavior of the swollen phase and the collapse transition for the swollen phase we find that by choosing model parameters appropriately our stochastic functional reduces to the one describing the relaxation dynamics near the yang lee singularity edge this corroborates that the scaling behavior of swollen branched polymers is governed by the yang lee universality class as has been known for a long time the main focus of our paper lies on the collapse transition of directed branched polymers we show to arbitrary order in renormalized perturbation theory with ensuremath epsilon expansion that this transition belongs to the same universality class as directed percolation

ϵ expansion for the critical exponents of a vector spin glass the epsilon expansion for the critical exponents of a vector spin glass is calculated to o epsilon 3 epsilon 6 d a general cubic field theory is used

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

wavelet based image estimation an empirical bayes approach using jeffrey s noninformative prior the sparseness and decorrelation properties of the discrete wavelet transform have been exploited to develop powerful denoising methods however most of these methods have free parameters which have to be adjusted or estimated in this paper we propose a wavelet based denoising technique without any free parameters it is in this sense a universal method our approach uses empirical bayes estimation based on a jeffreys noninformative prior it is a step toward objective bayesian wavelet based denoising the result is a remarkably simple fixed nonlinear shrinkage thresholding rule which performs better than other more computationally demanding methods

yang lee edge singularities from extended activity expansions of the dimer density for bipartite lattices of dimensionality 2 d 7 we have extended in most cases through 24th order the series expansions of the dimer density in powers of the activity in the case of bipartite hyper simple cubic and hyper body centered cubic lattices of dimensionalities 2 d 7 a numerical analysis of these data yields estimates of the exponents characterizing the yang lee edge singularities for lattice ferromagnetic spin models as d varies between the lower and the upper critical dimensionalities our results are consistent with but more extensive and sometimes more accurate than those obtained from the existing dimer series or from the estimates of related exponents for lattice animals branched polymers and fluids we mention also that it is possible to obtain estimates of the dimer constants from our series for the various lattices

logarithmic corrections for spin glasses percolation and lee yang singularities in six dimensions we study analytically the logarithmic corrections to the critical exponents of the critical behaviour of correlation length susceptibility and specific heat for the temperature and the finite size scaling behaviour for a generic theory at its upper critical dimension six we have also computed the leading correction to scaling as a function of the lattice size we distinguish the obtained formulae to the following special cases percolation lee yang singularities and m component spin glasses we have compared our results for the ising spin glass case with numerical simulations finding a very good agreement finally and using the results obtained for the lee yang singularities in six dimensions we have computed the logarithmic corrections to the singular part of the free energy for lattice animals in eight dimensions

critical exponents for the percolation problem and the yang lee edge singularity gives details of a calculation of critical exponents for a class of field theory models that have an interaction cubic in the fields the results have already been reported give the exponents to third order in epsilon where epsilon 6 d and d is the dimensionality of space the class of models includes the percolation problem and the yang lee edge singularity so the authors give explicit results for the exponents to order epsilon 3 in these cases by resummation methods based on the symptotic behaviour of the epsilon expansion they obtain numerical estimates for these exponents for a number of interesting values of d

wavelet threshold estimators for data with correlated noise wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level dependent soft threshold to the coefficients in the wavelet transform a variety of threshold choices is proposed including one based on an unbiased estimate of mean squared error the practical performance of the method is demonstrated on examples including data from a neurophysiological context the theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable bench mark that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity and more generally is obtained by the use of an oracle that provides information that is not actually available in the data it is shown that the level dependent threshold estimator performs well relative to the bench mark risk and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator the wavelet domain structure of both short and long range dependent noise is considered and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes adapting automatically to the regularity properties of the underlying model the proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem

empirical bayes selection of wavelet thresholds this paper explores a class of empirical bayes methods for level dependent threshold selection in wavelet shrinkage the prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy tailed density the mixing weight or sparsity parameter for each level of the transform is chosen by marginal maximum likelihood if estimation is carried out using the posterior median this is a random thresholding procedure the estimation can also be carried out using other thresholding rules with the same threshold details of the calculations needed for implementing the procedure are included in practice the estimates are quick to compute and there is software available simulations on the standard model functions show excellent performance and applications to data drawn from various fields of application are used to explore the practical performance of the approach by using a general result on the risk of the corresponding marginal maximum likelihood approach for a single sequence overall bounds on the risk of the method are found subject to membership of the unknown function in one of a wide range of besov classes covering also the case of f of bounded variation the rates obtained are optimal for any value of the parameter p in 0 simultaneously for a wide range of loss functions each dominating the lq norm of the th derivative with 0 and 0 q 2 attention is paid to the distinction between sampling the unknown function within white noise and sampling at discrete points and between placing constraints on the function itself and on the discrete wavelet transform of its sequence of values at the observation points results for all relevant combinations of these scenarios are obtained in some cases a key feature of the theory is a particular boundary corrected wavelet basis details of which are discussed overall the approach described seems so far unique in combining the properties of fast computation good theoretical properties and good performance in simulations and in practice a key feature appears to be that the estimate of sparsity adapts to three different zones of estimation first where the signal is not sparse enough for thresholding to be of benefit second where an appropriately chosen threshold results in substantially improved estimation and third where the signal is so sparse that the zero estimate gives the optimum accuracy rate

conductivity exponents from the analysis of series expansions for random resistor networks there has been considerable controversy in recent years over the value of the conductivity exponent t this exponent can be deduced from series expansions via the scaling relations t zeta d 2 nu where zeta is deduced from differences between the exponents of the resistive gamma r percolative gamma p and conductive gamma c susceptibility the author finds that allowance for non analytic confluent corrections to scaling and the use of recent pc estimates leads to estimates for gamma r gamma p and gamma c that are somewhat different to those of fisch and harris 1978 however the differences between these exponents do not change significantly moreover the change in accepted estimates of nu in the last five years cancels some of this remaining discrepancy and she concludes using the relation zeta gamma r gamma p that t 1 31 d 2 t 2 04 d 3 t 2 39 d 4 t 2 72 d 5 with an error of about or 0 10 in each case the d 2 estimate is in significantly better agreement with those of other methods than that of fisch and harris

empirical bayes approach to improve wavelet thresholding for image noise reduction wavelet threshold algorithms replace small magnitude wavelet coefficients with zero and keep or shrink the other coefficients this is basically a local procedure because wavelet coefficients characterize the local regularity of a function although a wavelet transform has decorrelating properties structures in images like edges are never decorrelated completely and these structures appear in the wavelet coefficients a classification based on a local criterion like coefficient magnitude is not the perfect method to distinguish important uncorrupted coefficients from coefficients dominated by noise we therefore introduce a geometrical prior model for configurations of important wavelet coefficients and combine this with local characterization of a classical threshold procedure into a bayesian framework the local characterization is incorporated into the conditional model whereas the prior model describes only configurations not coefficient values more precisely local characterization favors co

a semi local paradigm for wavelet denoising wavelet denoising methods have been proven useful for many one and two dimensional problems most existing methods can in principle be carried over to three dimensional problems such as the denoising of volumetric positron emission tomography pet images but they may not be sufficiently flexible in allowing some regions of an image to be denoised more aggressively than others in this paper we propose a semi local paradigm for wavelet denoising the semi local paradigm involves the division of an image into suitable blocks which are then individually denoised to denoise the blocks we use our modification of the generalized cross validation gcv technique of jansen and bultheel to choose thresholding parameters we also present risk estimators to guide some of the other choices involved in the implementation experiments with phantom pet images show that the semi local paradigm provides superior denoising compared to standard application of the gcv technique an asymptotic analysis demonstrates that under some regularity conditions semi local denoising is asymptotically consistent on the logarithmic scale the paper concludes with a discussion on the nature of semi local denoising and some topics for future research

amplitude ratios and β estimates from general dimension percolation moments low concentration series are generated for moments of the percolation cluster size distribution gamma j sj 1 s is the number of sites on a cluster for j 2 8 and general dimensionality d these diverge at pc as gamma j approximately aj pc p gamma j with gamma j gamma j gamma j 2 delta where delta gamma beta is the gap exponent the series yield new accurate values for delta and beta delta 2 23 or 0 05 2 10 or 0 04 2 03 or 0 05 and beta 0 44 or 0 15 0 66 or 0 09 0 83 or 0 08 at d 3 4 5 in addition ratios of the form ajak aman with j k m n are shown to be universal new values for some of these ratios are evaluated from the series from the epsilon expansion epsilon 6 d and exactly in d 1 and on the bethe lattice the results are in excellent agreement with each other for all dimensions results for different lattices at d 2 3 agree very well these amplitude ratios are much better behaved than other ratios considered in the past and should thus be more useful in characterising percolating systems

1 σ expansion for critical concentrations in the diluted spin glass the author studies the threshold concentrations pq for critical behaviour at zero temperature in the correlation functions 2q for an ising spin glass in which nearest neighbour interactions randomly assume the values j 0 and j with respective probabilities p 2 1 p and p 2 here q j sisj q av where denotes an average over the ground states for fixed configuration of j and av an average over all such configurations due to frustration effects pc p2 p4 p6 pinfinity where pc is the percolation threshold thus spin glass theory with only 2 critical at p p2 applies and the critical exponents along the t 0 axis are the same as thermal critical exponents for p 1 when the values j and j are replaced by distributions of narrow width the frustration is removed and percolation exponents are expected at p pc

high order behaviour of the critical exponents for the scalar phi3 model with imaginary coupling for real coupling constants the vertex functions for the phi 3 model have a real and imaginary part the author presents a method of renormalising both parts of the vertex functions for small coupling g0 a consequence of the method is that the renormalisation group functions develop a non perturbative imaginary part this leads to an imaginary part for the critical exponents valid for 6 d epsilon 0 where d is the dimension of the space a dispersion relation in epsilon then yields the high order behaviour of the critical exponents as a power series in epsilon in particular the values of the overall universal coefficients c of the factorial growth are obtained

inverse scattering for a schrodinger equation with energy dependent potential in this article the inverse scattering problem of reconstructing the energy dependent potential ie2 m2 p x q x of a schrodinger equation on the line from its reflection coefficients and bound state data i e poles of the transmission coefficients and associated norming constants is solved using the marchenko integral equation approach

regularization of wavelet approximations in this paper we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced the approach can apply readily to many other statistical contexts various new penalty functions are proposed the hard thresholding and soft thresholding estimators of donoho and johnstone are specific members of nonlinear regularized wavelet estimators they correspond to the lower and upper envelopes of a class of the penalized least squares estimators necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions the sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax to efficiently solve penalized least squares problems nonlinear regularized sobolev interpolators nrsi are proposed as initial estimators wh

series study of random percolation in three dimensions new high density series data for the mean number percolation probability and susceptibility of finite clusters are presented for bond and site percolation on four standard three dimensional lattices a pade approximant analysis of both high and low density series makes particular use of the rather precise unbiased estimates of the percolation threshold pc obtained recently by heermann and stauffer 1981 for the simple cubic lattice for this lattice the authors obtain the biased estimate gamma 1 73 or 0 03 for the site problem and a similar estimate but with larger uncertainties for the bond problem such a value is significantly larger than earlier series estimates assuming gamma to be universal they obtain precise although biased estimates of pc for both bond and site percolation on all four lattices using the bond estimates of pc they find an overall biased estimate of beta 0 454 or 0 008 for bond percolation on all three dimensional lattices the corresponding site problem requires further study scaling estimates of other critical exponents are alpha 0 64 or 0 05 delta 4 81 or 0 14 delta 2 18 or 0 04 nu 0 88 or 0 02 and nu 0 03 or 0 03

korteweg de vries modified korteweg de vries systems and darboux transforms in 1 1 and 2 1 dimensions the jaulent miodek hierarchy m jaulent and i miodek lett math phys 1 243 1976 and its two dimensional analog are considered from the point of view of darboux transforms structures a new form of those transforms as well as old fashioned forms are presented the determinant form of solitons is obtained the combination of different transforms in the 2 1 case allows the buildup of localized solutions in both spatial dimensions including the two dimensional korteweg de vries kdv equation

wavelet shrinkage using cross validation summary wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics this paper concentrates on the estimation of functions and images from noisy data by using wavelet shrinkage a modified form of twofold cross validation is introduced to choose a threshold for wavelet shrinkage estimators operating on data sets of length a power of 2 the cross validation algorithm is then extended to data sets of any length and to multidimensional data sets the algorithms are compared with established threshold choosers by using simulation an application to a real data set arising from anaesthesia is presented

multiple wavelet threshold estimation by generalized cross validation for images with correlated noise denoising algorithms based on wavelet thresholding replace small wavelet coefficients by zero and keep or shrink the coefficients with absolute value above the threshold the optimal threshold minimizes the error of the result as compared to the unknown exact data to estimate this optimal threshold we use generalized cross validation this procedure does not require an estimation for the noise energy originally this method assumes uncorrelated noise in this paper we describe how we can extend it to images with correlated noise

bond percolation and the yang lee edge singularity problems in three dimensions field theories with a trilinear interaction are used to describe both the random bond percolation problem and the yang lee edge singularity problems in three dimensions at criticality renormalisation group functions are calculated to order four for the exponents eta omega and the beta function and to order five for the remaining critical exponents estimations of the critical coupling constant and the exponents for percolation theory are found by summing conformal transformations of the borel transformed series

spreading of percolation in three and four dimensions the authors present monte carlo simulations of the spreading of bond percolation on three and four dimensional simple hyper cubic lattices the algorithm is the same as that applied previously in two dimensions with the spreading proceeding from a hyperplane they find the spreading dimension to be d 1 82 or 0 02 three dimensional and d 1 88 or 0 03 four dimensional also they obtain values of the percolation probability and of the static exponents with errors at least comparable to the best values from the literature

theory of minimum spanning trees ii exact graphical methods and perturbation expansion at the percolation threshold continuing the program begun by the authors in a previous paper we develop an exact low density expansion for the random minimum spanning tree mst on a finite graph and use it to develop a continuum perturbation expansion for the mst on critical percolation clusters in space dimension d the perturbation expansion is proved to be renormalizable in d 6 dimensions we consider the fractal dimension d sub p of paths on the latter mst our previous results lead us to predict that d sub p 2 for d d sub c 6 using a renormalization group approach we confirm the result for d 6 and calculate d sub p to first order in epsilon 6 d for d 6 using the connection with critical percolation with the result d sub p 2 epsilon 7 o epsilon sup 2

critical behaviour of an m vector spin glass for m a special case of the spin glass model is considered in which the number of spin components m becomes infinitely large the authors derive a field theory hamiltonian for this model and show that its upper critical dimensionality is eight the critical exponents for this theory are calculated in an expansion in epsilon 8 d to second order they notice that this second order expansion is identical to that for the p to infinity limit of the q3 model in epsilon 6 d

generalized schrodinger equations with quadratical energy dependence in the potential darboux transformations and application to the heun class we construct darboux transformations of arbitrary order for generalized schrodinger type equations the potentials of which are second degree polynomials in the energy our equations are allowed to contain first derivative terms with arbitrary coefficients such as they occur for example in position dependent mass scenarios our darboux transformations are shown to be applicable to equations from the heun class

a bochner type characterization theorem for exceptional orthogonal polynomials abstract it was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of darboux transformations in this paper we prove this conjecture which paves the road to a complete classification of all exceptional orthogonal polynomials in some sense this paper can be regarded as the extension of bochner s result for classical orthogonal polynomials to the exceptional class as a supplementary result we derive a canonical form for exceptional operators based on a bilinear formalism and prove that every exceptional operator has trivial monodromy at all primary poles

spanning forests and osp n 2m invariant σ models the present paper is part of our ongoing work on osp n 2m supersymmetric models their relation with the potts model at q 0 and spanning forests and the rigorous analytic continuation of the partition function as an entire function of n 2m a feature first predicted by parisi and sourlas in the 1970s here we accomplish two main steps first we analyze in detail the role of the ising variables that arise when the constraint in the osp 1 2 model is solved and we point out two situations in which the ising and forest variables decouple second we establish the analytic continuation for the osp n 2m model in some special cases when the underlying graph is a forest and for the nienhuis action on a cubic graph we also make progress in understanding the series parallel graphs

directed lattice animals and the lee yang edge singularity the exponents nu perpendicular to gamma of directed animals in d dimensions are shown to equal those for the lee yang problem in d 1 dimensions for d 2 the author obtains the exact results nu perpendicular to theta 1 2 and for d or 7 the scaling relation theta d 1 nu perpendicular to

adaptive wavelet estimation a block thresholding and oracle inequality approach we study wavelet function estimation via the approach of block thresh olding and ideal adaptation with oracle oracle inequalities are derived and serve as guides for the selection of smoothing parameters based on an oracle inequality and motivated by the data compression and localization properties of wavelets an adaptive wavelet estimator for nonparametric re gression is proposed and the optimality of the procedure is investigated we show that the estimator achieves simultaneously three objectives adaptiv ity spatial adaptivity and computational efficiency specifically it is proved that the estimator attains the exact optimal rates of convergence over a range of besov classes and the estimator achieves adaptive local minimax rate for estimating functions at a point the estimator is easy to imple ment at the computational cost of on simulation shows that the es timator has excellent numerical performance relative to more traditional wavelet estimators 1 introduction wavelet methods have demonstrated considerable suc cess in nonparametric function estimation in terms of spatial adaptivity com putational efficiency and asymptotic optimality in contrast to the traditional linear procedures wavelet methods achieve near optimal convergence rates over large function classes such as besov classes and enjoy excellent mean squared error properties when used to estimate functions that are spatially inhomogeneous for example as shown by donoho and johnstone 1998 wavelet methods can outperform optimal linear methods even at the level of convergence rate over certain besov classes standard wavelet methods achieve adaptivity through term by term thresholding of the empirical wavelet coefficients there each individual empirical wavelet coefficient is compared with a predetermined threshold a wavelet coefficient is retained if its magnitude is above the threshold level and is discarded otherwise a well known example of term by term thresholding is donoho and johnstone s visushrink donoho and johnstone 1994 visushrink is spatially adaptive and the estimator is within a log arithmic factor of the optimal convergence rate over a wide range of besov classes visushrink achieves a degree of tradeoff between variance and bias contributions to the mean squared error however the tradeoff is not optimal visushrink reconstruction is often over smoothed hall kerkyacharian and picard 1999 considered block thresholding for wavelet function estimation which thresholds empirical wavelet coefficients in

higher order darboux transformations and wronskian representations for schrodinger equations with quadratically energy dependent potentials we construct higher order darboux transformations for schrodinger equations with quadratically energy dependent potentials similar to the case of conventional darboux also known as supersymmetric transformations both the solutions and the potentials of transformed equations are expressed through wronskians we discuss properties of our darboux transformations and provide an application

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

block cluster theory of site percolation on the four and five dimensional ordinary hypercubic lattices site percolation on the ordinary hypercubic lattice in four and five dimensions is studied using block cluster theory the exact analytic renormalisation group equations of block cluster theory are obtained by a computer algorithm they are solved numerically and yield for the susceptibility exponent gamma 1 277 d 4 gamma 1 031 d 5 and for the correlation length exponent nu 0 566 d 5

recovering edges in ill posed inverse problems optimality of curvelet frames we consider a model problem of recovering a function f x 1 x 2 from noisy radon data the function f to be recovered is assumed smooth apart from a discontinuity along a c 2 curve that is an edge we use the continuum white noise model with noise level varepsilon traditional linear methods for solving such inverse problems behave poorly in the presence of edges qualitatively the reconstructions are blurred near the edges quantitatively they give in our model mean squared errors mses that tend to zero with noise level varepsilon only as o varepsilon 1 2 as varepsilon to 0 a recent innovation nonlinear shrinkage in the wavelet domain visually improves edge sharpness and improves mse convergence to o varepsilon 2 3 however as we show here this rate is not optimal in fact essentially optimal performance is obtained by deploying the recently introduced tight frames of curvelets in this setting curvelets are smooth highly anisotropic elements ideally suited for detecting and synthesizing curved edges to deploy them in the radon setting we construct a curvelet based biorthogonal decomposition of the radon operator and build curvelet shrinkage estimators based on thresholding of the noisy curvelet coefficients in effect the estimator detects edges at certain locations and orientations in the radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain we prove that the curvelet shrinkage can be tuned so that the estimator will attain within logarithmic factors the mse o varepsilon 4 5 as noise level varepsilon to 0 this rate of convergence holds uniformly over a class of functions which are c 2 except for discontinuities along c 2 curves and except for log terms is the minimax rate for that class our approach is an instance of a general strategy which should apply in other inverse problems we sketch a deconvolution example

critical and off critical conformal analysis of the ising quantum chain in an imaginary field the critical curve with central charge c 22 5 of the ising quantum chain in an imaginary longitudinal field is calculated by finite size extrapolation of the yang lee square root branch point position various power laws are found the field dependence of the off criticality mass scale and the four lowest off criticality scaling functions are calculated the latter turn out to be universal along the critical line and agree very well with the results of the truncated hilbert space method for the c 22 5 continuum theory the author obtains the universal bulk scaling coefficient in qualitative agreement with the thermodynamic bethe ansatz result the scattering phaseshift calculated by luscher s method 1988 reproduces the cardy mussardo minimal solution 1989 for the s matrix if a boundary defect is introduced the only non trivial primary field decouples so that the spectrum consists only of the tower of the unit operator in the antiferromagnetic regime with a non staggered imaginary field the author finds boundary dependent lines where the gap vanishes

averages of best wavelets basis estimates for denoising abstract donoho and johnstone introduced an adaptive algorithm that extends nonlinear thresholding denoising in a fixed orthonormal basis to a multiple basis setting in their work a search for an optimal basis from a large collection of orthonormal bases i e a library is introduced that technique gives the so called best ortho basis estimate in this paper we study the situation when many such libraries are available we propose an algorithm that exploits the availability of many best ortho basis approximations the algorithm uses a strengthening of the convexity of the l2 norm to produce an estimate which is an average of best ortho basis estimates conditions under which the proposed algorithm offers improvements and corresponding numerical examples are also described

renormalization group theory for cooling first order phase transitions in potts models we develop a dynamic field theoretic renormalization group rg theory for cooling first order phase transitions in the potts model it is suggested that the well known imaginary fixed points of the q state potts model for q 10 3 in the rg theory are the origin of the dynamic scaling found recently from numerical simulations apart from logarithmic corrections this indicates that the real and imaginary fixed points of the potts model are both physical and control the scalings of the continuous and discontinuous phase transitions respectively of the model our one loop results for the scaling exponents are already not far away from the numerical results further the scaling exponents depend on q only slightly consistent with the numerical results therefore the theory is believed to provide a natural explanation of the dynamic scaling including the scaling exponents and their scaling laws for various observables in the cooling first order phase transition of the potts model

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

running wave solutions of the two dimensional sine gordon equation after the lamb substitution the 2d sine gordon equation was solved three classes of solutions for this equation were found ad hoc the 2d sine gordon equation was reduced to an algebraic system consisting of three equations the solutions given of the 2d sine gordon equation are a generalization of the solutions of the 1d sine gordon equation they are also a generalization of the solutions of the equation phi xx yy sin phi x y

diffusion percolation i infinite time limit and bootstrap percolation several models for the dynamic growth of percolation clusters or diffusion percolation dp are introduced and analysed in these models a random walker an ant walks on percolation clusters which are occupied with initial site concentration pi the ant is allowed to step off such clusters and add new sites to them if certain conditions are met some of these models are shown to have a one to one correspondence with models of bootstrap percolation bp in which sites which do not have a required number of neighbours are successively culled two new percolation thresholds have been calculated for two diffusion percolation models on the square lattice

asymptotic behaviour of the perturbation expansion in n 0 field theories the author develops a method for studying the high order behaviour of the perturbation expansion in theories in which the number of field components n is taken to zero this procedure is illustrated on the field theory formulation of the percolation problem which can be considered as the n 0 limit of the n 1 state potts model the saddle points controlling the asymptotic behaviour are labelled by an integer r 1 2 n for positive integer n but after continuation to n 0 the dominant contribution effectively comes from the saddle point with r infinity the perturbation expansion is found to be oscillatory at large orders and its behaviour is calculated

comment on generalization of the darboux transformation and generalized harmonic oscillators the authors song and klauder 2003 j phys a math gen 36 8673 84 present a generalized darboux transformation applicable to hamiltonians with linear terms in the momentum we show here that this generalized darboux transformation is just the standard darboux transformation in different coordinates

simulations of lattice animals and trees the scaling behaviour of randomly branched polymers in a good solvent is studied in two to nine dimensions using as microscopic models lattice animals and lattice trees on simple hypercubic lattices as a stochastic sampling method we use a biased sequential sampling algorithm with re sampling similar to the pruned enriched rosenbluth method perm used extensively for linear polymers essentially we start simulating percolation clusters either site or bond re weigh them according to the animal tree ensemble and prune or branch the further growth according to a heuristic fitness function in contrast to previous applications of perm this fitness function is not the weight with which the actual configuration would contribute to the partition sum but is closely related to it we obtain high statistics of animals with up to several thousand sites in all dimension 2 d 9 in addition to the partition sum number of different animals we estimate gyration radii and numbers of perimeter sites in all dimensions we verify the parisi sourlas prediction and we verify all exactly known critical exponents in dimensions 2 3 4 and 8 in addition we present the hitherto most precise estimates for growth constants in d 3 for clusters with one site attached to an attractive surface we verify for d 3 the superuniversality of the cross over exponent at the adsorption transition predicted by janssen and lyssy but not for d 2 there we find 0 480 4 instead of the conjectured 1 2 finally we discuss the collapse of animals and trees arguing that our present version of the algorithm is also efficient for some of the models studied in this context but showing that it is not very efficient for the classical model for collapsing animals

ageing properties of critical systems in the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest ageing phenomena are observed during this everlasting non equilibrium evolution a simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high temperature phase to the critical point the aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case particular emphasis is put on the field theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities fluctuation dissipation relations are discussed and in particular the concept of fluctuation dissipation ratio fdr is reviewed emphasizing its connection with the definition of a possible effective temperature the renormalization group approach to critical dynamics is summarized and the scaling forms of the time dependent non equilibrium correlation and response functions of a generic observable are discussed from them the universality of the associated fdr follows as an amplitude ratio it is then possible to provide predictions for ageing quantities in a variety of different models in particular the results for models a b and c dynamics of the o n ginzburg landau hamiltonian and model a dynamics of the weakly dilute ising magnet and of the 3 theory are reviewed and compared with the available numerical results and exact solutions the effect of a planar surface on the ageing behaviour of model a dynamics is also addressed within the mean field approximation

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed