the hyperbolic class of quadratic time frequency representations i constant q warping the hyperbolic paradigm properties and members the time frequency tf version of the wavelet transform and the affine quadratic bilinear tf representations can be used for a tf analysis with constant q characteristic the paper considers a new approach to constant q tf analysis a specific tf warping transform is applied to cohen s class of quadratic tf representations which results in a new class of quadratic tf representations with constant q characteristic the new class is related to a hyperbolic tf geometry and is thus called the hyperbolic class hc two prominent tf representations previously considered in the literature the bertrand p sub 0 distribution and the altes marinovic q distribution are members of the new hc the authors show that any hyperbolic tf representation is related to both the wideband ambiguity function and a hyperbolic ambiguity function it is also shown that the hc is the class of all quadratic tf representations which are invariant to hyperbolic time shifts and tf scalings operations which are important in the analysis of doppler invariant signals and self similar random processes the paper discusses the definition of the hc via constant q warping some signal theoretic fundamentals of the hyperbolic tf geometry and the description of the hc by 2d kernel functions several members of the hc are considered and a list of desirable properties of hyperbolic tf representations is given together with the associated kernel constraints

a new implementation of the mellin transform and its application to radar classification of ships a modified mellin transform for digital implementation is developed and applied to range radar profiles of naval vessels the scale invariance property of the mellin transform provides a means for extracting features from the profiles which are insensitive to the aspect angle of the radar past implementations of the mellin transform based on the fft have required exponential sampling interpolation and the computation of a correction term all of which introduce errors into the transform in addition exponential sampling results in a factor of ln n increase in the number of data points an alternate implementation developed in the paper utilizes a direct expansion of the mellin integral definition this direct mellin transform dmt eliminates the implementation problems associated with the fft approach and does not increase the number of samples a scale and translation invariant transform is developed from a modification of the dmt the mdmt applied to the fft of the radar profiles results in the desired insensitivity without having the low pass filtering characteristic that exists in other fourier mellin implementations

high order symplectic integration methods for finite element solutions to time dependent maxwell equations in this paper we motivate the use of high order integration methods for finite element solutions of the time dependent maxwell equations in particular we present a symplectic algorithm for the integration of the coupled first order maxwell equations for computing the time dependent electric and magnetic fields symplectic methods have the benefit a conserving total electromagnetic field energy and are therefore preferred over dissipative methods such as traditional runge kutta in applications that require high accuracy and energy conservation over long periods of time integration we show that in the context or symplectic methods several popular schemes can be elegantly cast in a single algorithm we conclude with some numerical examples which demonstrate the superior performance of high order time integration methods

nested iteration and first order system least squares for incompressible resistive magnetohydrodynamics this paper develops a nested iteration algorithm to solve time dependent nonlinear systems of partial differential equations for each time step newton s method is used to form approximate solutions from a sequence of nested spaces where the resolution of the approximations increases as the algorithm progresses nested iteration results in most of the iterations being performed on coarser grids where minimal work is needed to reduce error to the level of discretization error the approximate solution on a given coarse grid is interpolated to a refined grid and is used as an initial guess for the problem posed there the approximation is then already close enough to the solution on the current grid that a minimal amount of work is needed to solve the refined problem due to the rapid convergence of newton s method near a solution the paper develops an algorithm that attempts to optimize accuracy per computational cost on each grid so that essentially no unnecessary work is done on any grid the nested iteration algorithm is then applied to a reduced two dimensional model of the incompressible resistive magnetohydrodynamic mhd equations using this algorithm on the mhd equations in the context of a first order system least squares finite element discretization and algebraic multigrid to solve the linearized systems instabilities in a model tokamak fusion reactor are simulated numerical results show that this highly complex nonlinear problem is solved in an equivalent of 30 80 fine grid relaxation sweeps per time step

scale invariances and lamperti transformations for stochastic processes scale invariant processes and hereafter processes with broken versions of this symmetry are studied by means of the lamperti transformation a one to one transformation linking stationary and self similar processes a general overview of the use of the transformation and of the stationary generators it builds is given for modelling and analysis of scale invariance we put an emphasis on generalizations to non strictly scale invariant situations the examples of discrete scale invariance and finite size scale invariance are developed by means of the lamperti transformation framework and some specific examples of processes with these generalized symmetries are given

form invariant linear filtering theory and applications the form invariant filters are by definition those shift variant filters such that their output for any given input turns out to be linearly scaled implying that its form does not change whenever the input is linearly scaled in this paper the most general classes of 1 d and 2 d linear form invariant filters are derived and their properties are discussed together with their implementation techniques two main implementation approaches are considered one based on the mellin transform the other on a combination of coordinate mappings and shift invariant filtering the paper also discusses the related works of other authors covering quite different fields such as optical pattern recognition image restoration and image reconstruction from projections radar signal processing etc it is shown that the mathematics of form invariant filtering provides a common framework if not a powerful unified approach to the many signal processing techniques spread in the above mentioned works and apparently different application areas the paper ends with a processing example showing the usefulness of form invariant filtering in a pattern recognition problem that is in the area where the most promising applications of such a filtering are foreseen

a theory for invariant object recognition in the frontoparallel plane we suggest a new computational theory for position size and orientation invariant recognition of planar gray scale forms in frontoparallel view the theory employs a spatial frequency representation which approximates the complex logarithmic conformally mapped wigner distribution of a gray scale image our discussion emphasizes the theoretical significance method of computation and practical utility of this powerful new representation for image pattern recognition

a fast correlation method for scale and translation invariant pattern recognition a size and position invariant description of an image function can be obtained via the absolute value of the mellin transform of its fourier amplitude spectrum if the transform is implemented on a digital computer via a discrete fourier mellin transform these exact invariances are not preserved due to sampling and border effects in this paper these effects are discussed and an alternative correlation method is proposed the method consists of calculating the normalized absolute magnitude of the discrete fourier transform dft of the image function which gives invariance to translation and multiplicative amplitude changes and a subsequent logarithmic distortion in x and y direction which converts scaling to translation two such transforms are compared by calculating the normalized euclidean distances between both for all possible relative shifts along the main diagonal if for some shift the distance has a minimum below a similarity threshold the underlying image functions will probably differ only by translation and scaling the magnitude of this shift is related to the scale factor between the objects good separation between similar and nonsimilar objects is possible if two size criteria imposed by the dft are met the total object size must not exceed n 4 n is the number of image points in each dimension and object details have to be larger than about 4 image points as a consequence n increases with object complexity and desired scale range

first order system least squares for incompressible resistive magnetohydrodynamics magnetohydrodynamics mhd is a fluid theory that describes plasma physics by treating the plasma as a fluid of charged particles hence the equations that describe the plasma form a nonlinear system that couples navier stokes equations with maxwell s equations this paper shows that the first order system least squares fosls finite element method is a viable discretization for these large mhd systems to solve this system a nested iteration newton fosls amg approach is taken most of the work is done on the coarse grid including most of the linearizations we show that at most one newton step and a few v cycles are all that are needed on the finest grid here we describe how the fosls method can be applied to incompressible resistive mhd and how it can be used to solve these mhd problems efficiently a 3d steady state and a reduced 2d time dependent test problem are studied the latter equations can simulate a large aspect ratio tokamak the goal is to resolve as much physics from the test problems with the least amount of computational work we show that this is achieved in a few dozen work units or fine grid residual evaluations

a class of second order stationary self similar processes for 1 f phenomena we propose a class of statistically self similar processes and outline an alternative mathematical framework for the modeling and analysis of 1 f phenomena the foundation of the proposed class is based on the extensions of the basic concepts of classical time series analysis in particular on the notion of stationarity we consider a class of stochastic processes whose second order structure is invariant with respect to time scales i e e x t x spl lambda t t sup 2h spl lambda sup h r spl lambda t 0 for some x h spl infin for h 0 we refer to these processes as wide sense scale stationary we show that any self similar process can be generated from scale stationary processes we establish a relationship between linear scale invariant system theory and the proposed class that leads to a concrete analysis framework we introduce new concepts such as periodicity autocorrelation and spectral density functions by which practical signal processing schemes can be developed we give several examples of scale stationary processes including gaussian non gaussian covariance and generative models as well as fractional brownian motion as a special case in particular we introduce a class of finite parameter self similar models that are similar in spirit to the ordinary arma models by which an arbitrary self similar process can be approximated results from our study suggest that the proposed self similar processes and the mathematical formulation provide an intuitive general and mathematically simple approach to 1 f signal processing

island coalescence using parallel first order system least squares on incompressible resistive magnetohydrodynamics this paper investigates the performance of a parallel newton first order system least squares fosls finite element method with local adaptive refinement and algebraic multigrid amg applied to incompressible resistive magnetohydrodynamics in particular an island coalescence test problem is studied that models magnetic reconnection using a reduced two dimensional 2d model of a tokamak fusion reactor the results show that using an appropriate temporal and spatial resolution these methods are capable of resolving the physical instabilities accurately at small computational cost the time dependent nonlinear system of pdes is solved using work equivalent to about 50 60 simple relaxation sweeps gauss seidel iterations per time step experiments show that unless the time step is sufficiently small nonphysical numerical instabilities may occur further decreasing the time step size does not proportionally increase the cost of the computation because amg convergence is improved in addition

uniformly robust preconditioners for incompressible mhd system abstract in this paper we study efficient schemes and design robust preconditioners for solving incompressible mhd magnetohydrodynamics system we propose the first order and second order symmetric schemes with augmented terms in magnetic equations that are introduced in consideration of designing uniformly robust preconditioners we present optimal error estimates of the proposed time matching scheme further we design block diagonal preconditioners for the schemes and rigorously prove that the condition number of preconditioned system is uniformly bounded by a constant which only depends on the domain finally some numerical experiments including accuracy tests and physical benchmark problems are presented to verify the accuracy of the schemes and the robustness of the preconditioner

on the fractional linear scale invariant systems the linear scale invariant systems are introduced for both integer and fractional orders they are defined by the generalized euler cauchy differential equation the quantum fractional derivatives are suitable for dealing with this kind of systems allowing us to define impulse response and transfer function with the help of the mellin transform it is shown how to compute the impulse responses corresponding to the two half plane regions of convergence of the transfer function

unitary equivalence a new twist on signal processing unitary similarity transformations furnish a powerful vehicle for generating infinite generic classes of signal analysis and processing tools based on concepts different from time frequency and scale implementation of these new tools involves simply preprocessing the signal by a unitary transformation performing standard processing on the transformed signal and then in some cases transforming the resulting output the resulting unitarily equivalent systems can focus on the critical signal characteristics in large classes of signals and hence prove useful for representing and processing signals that are not well matched by current techniques as specific examples of this procedure we generalize linear time invariant systems orthonormal basis and frame decompositions and joint time frequency and time scale distributions these applications illustrate the utility of the unitary equivalence concept for uniting seemingly disparate approaches proposed in the literature

a least squares finite element method using elsasser variables for magnetohydrodynamic equations abstract there are various different forms of magnetohydrodynamic mhd equations and they have been studied for years due to its complicated coupling between variables this paper proposes to use an equivalently transformed mhd equations with elsasser variables and the least squares finite element method to find the approximation to them introducing new variables by combining fluid velocity and magnetic field yields a navier stokes like system then the first order system least squares method using displacement recasts the transformed mhd equations into a system of first order partial differential equations and the newton s algorithm linearizes the problem an l 2 residual functional is defined to minimize and the unique existence of corresponding weak solution is shown finally the convergence of proposed approximation is analyzed and several numerical examples are presented to verify the theory

state variable representation of a class of linear shift variant systems certain types of linear shift variant systems are rendered shift invariant by appropriate coordinate transformations changes of variable examples occur in aerial photography restoration of images degraded by coma and reconstruction of two dimensional functions from line integrals the matrices in the continuous state variable representations of such systems may be written as constant matrices multiplied by scalar functions of the independent variables while the discretized models do not appear to share this feature one system discussed in this paper does have a simple discrete time structure given a shift variant impulse response and the corresponding coordinate transformations it is shown how to construct a state variable representation experimental results from an application to image processing are presented

classical theory of runge kutta methods for volterra functional differential equations abstract for solving volterra functional differential equations vfdes a class of discrete runge kutta methods based on canonical interpolation is studied which was first presented by the same author in a previous published paper classical stability and convergence theories for this class of methods are established the methods studied and the theories established in this paper can be directly applied to non stiff non linear initial value problems in delay differential equations ddes integro differential equations ides delay integro differential equations dides and vfdes of other type which appear in practice and can be used as a necessary basis for the study of splitting methods for complex nonlinear stiff vfdes

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

multilevel first order system least squares for elliptic grid generation a new fully variational approach is studied for elliptic grid generation egg it is based on a general algorithm developed in a companion paper a l codd t a manteuffel and s f mccormick siam j numer anal 41 2003 pp 2197 2209 that involves using newton s method to linearize an appropriate equivalent first order system first order system least squares fosls to formulate and discretize the newton step and algebraic multigrid amg to solve the resulting matrix equation the approach is coupled with nested iteration to provide an accurate initial guess for finer levels using coarse level computation the present paper verifies the assumptions of the companion work and confirms the overall efficiency of the scheme with numerical experiments