a signal processing approach to symmetry detection we present an algorithm that detects rotational and reflectional symmetries of two dimensional objects both symmetry types are effectively detected and analyzed using the angular correlation ac which measures the correlation between images in the angular direction the ac is accurately computed using the pseudopolar fourier transform which rapidly computes the fourier transform of an image on a near polar grid we prove that the ac of symmetric images is a periodic signal whose frequency is related to the order of the symmetry this frequency is recovered via spectrum estimation which is a proven technique in signal processing with a variety of efficient solutions we also provide a novel approach for finding the center of symmetry and demonstrate the applicability of our scheme to the analysis of real images

fuzzy fractional differential equations with the generalized atangana baleanu fractional derivative abstract in this paper we introduce a generalization of atangana baleanu type fractional calculus with respect to the generalized mittag leffler kernel which has been named as the generalized atangana baleanu gab type fractional calculus existence and uniqueness results for the initial value problems of fuzzy differential equations involving a gab fractional derivative in the caputo sense are established by employing the method of successive approximation and by means of fixed point theorems to visualize the theoretical results some examples and numerical simulations are given

calculus for interval valued functions using generalized hukuhara derivative and applications this paper is devoted to studying differential calculus for interval valued functions by using the generalized hukuhara differentiability which is the most general concept of differentiability for interval valued functions conditions examples and counterexamples for limit continuity integrability and differentiability are given special emphasis is set to the class f t c g t where c is an interval and g is a real function of a real variable here the emphasis is placed on the fact that f and g do not necessarily share their properties underlying the extra care that must be taken into account when dealing with interval valued functions two applications of the obtained results are presented the first one determines a delta method for interval valued random elements in the second application a new procedure to obtain solutions to an interval differential equation is introduced our results are relevant to fuzzy set theory because the usual fuzzy arithmetic extension functions and mathematical analysis are done on a cuts which are intervals

a new discrete impulse response gramian and its application to model reduction some fundamental properties of an impulse response gramian for linear time invariant asymptotically stable discrete single input single output siso systems are derived this gramian is system invariant and can be found by solving a lyapunov equation the connection with standard controllability observability and cross gramians is proven the significance of these results in model order reduction is highlighted with an efficient procedure

the discrete zak transform application to time frequency analysis and synthesis of nonstationary signals algorithms to compute coefficients of the finite double sum expansion of time varying nonstationary signals and to synthesize them from a finite number of expansion coefficients are presented the algorithms are based on the computation of the discrete zak transform dzt fast algorithms to compute dzt are presented modifications to the algorithms which increase robustness are given

the angular difference function and its application to image registration the estimation of large motions without prior knowledge is an important problem in image registration in this paper we present the angular difference function adf and demonstrate its applicability to rotation estimation the adf of two functions is defined as the integral of their spectral difference along the radial direction it is efficiently computed using the pseudopolar fourier transform which computes the discrete fourier transform of an image on a near spherical grid unlike other fourier based registration schemes the suggested approach does not require any interpolation thus it is more accurate and significantly faster

volume registration using the 3 d pseudopolar fourier transform this paper introduces an algorithm for the registration of rotated and translated volumes using the three dimensional 3 d pseudopolar fourier transform which accurately computes the fourier transform of the registered volumes on a near spherical 3 d domain without using interpolation we propose a three step procedure the first step estimates the rotation axis the second step computes the planar rotation relative to the rotation axis the third step recovers the translational displacement the rotation estimation is based on euler s theorem which allows one to represent a 3 d rotation as a planar rotation around a 3 d rotation axis this axis is accurately recovered by the 3 d pseudopolar fourier transform using radial integrations the residual planar rotation is computed by an extension of the angular difference function to cylindrical motion experimental results show that the algorithm is accurate and robust to noise

error control of a numerical formula for the fourier transform by ooura s continuous euler transform and fractional fft in this paper we consider a method for fast numerical computation of the fourier transform of a slowly decaying function with given accuracy in a given range of the frequency recently some useful formulas for the fourier transform have been proposed to resolve the difficulty of the computation caused by the slow decay and the oscillation of the integrand in particular ooura proposed formulas with continuous euler transformation and showed their effectiveness it has however also been reported that their errors become large outside some ranges of the frequency then for an illustrative representative of the formulas in order to compute the fourier transform with given accuracy in a given frequency range we choose the parameters in the formula based on its error analysis furthermore by combining the formula and fractional fft a generalization of the fast fourier transform fft we execute the computation in the same order of computation time as that of the fft

fractional calculus for interval valued functions abstract we use a generalization of the hukuhara difference for closed intervals on the real line to develop a theory of the fractional calculus for interval valued functions the properties of riemann liouville fractional integral riemann liouville fractional derivative and caputo fractional derivative for interval valued functions are investigated several examples are presented to illustrate the concepts and results

a survey on fuzzy fractional variational problems abstract this paper focuses on introducing and studying generalized fuzzy euler lagrange equation and fuzzy isoperimetric problem according to the concept of caputo type fuzzy fractional derivative in the sense of the generalized fuzzy differentiability we extend and establish some definitions on fuzzy fractional calculus of variation and provide some necessary conditions to obtain the fuzzy fractional euler lagrange equation for both constrained and unconstrained fuzzy fractional variational problems the fuzzy isoperimetric problem is also investigated in relation to the generalized fuzzy euler lagrange finally two examples are given to describe the proposed approach

pseudopolar based estimation of large translations rotations and scalings in images one of the major challenges related to image registration is the estimation of large motions without prior knowledge this work presents a fourier based approach that estimates large translations scalings and rotations the algorithm uses the pseudopolar pp fourier transform to achieve substantial improved approximations of the polar and log polar fourier transforms of an image thus rotations and scalings are reduced to translations which are estimated using phase correlation by utilizing the pp grid we increase the performance accuracy speed and robustness of the registration algorithms scales up to 4 and arbitrary rotation angles can be robustly recovered compared to a maximum scaling of 2 recovered by state of the art algorithms the algorithm only utilizes one dimensional fast fourier transform computations whose overall complexity is significantly lower than prior works experimental results demonstrate the applicability of the proposed algorithms

fuzzy fractional functional integral and differential equations this paper is devoted to considering the existence and uniqueness results for fuzzy fractional functional integral equations employing the contraction principle moreover the approach is followed to prove the existence and uniqueness of solutions for the fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives finally the solution to fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives by a modified fractional euler method mfem is presented the method is illustrated by solving some examples

discrete generalized fresnel functions and transforms in an arbitrary discrete basis the idea of generalized fresnel functions which traces back to expressing a discrete transform as a linear convolution is developed in this paper the generalized discrete fresnel functions and the generalized discrete fresnel transforms for an arbitrary basis are considered this problem is studied using a general algebraic approach to signal processing in an arbitrary basis the generalized fresnel functions for the discrete fourier transform dft are found and it is shown that dft of even order has two generalized fresnel functions while dft of odd order has a single generalized fresnel function the generalized fresnel functions for the conjunctive and walsh transforms and the generalized fresnel transforms induced by these functions are considered it is shown that the generalized fresnel transforms induced by the walsh basis and the corresponding generalized fresnel functions are unitary and that the generalized fresnel transforms induced by the conjunctive basis and the corresponding generalized fresnel functions consist of powers of the golden ratio it is also shown that the fresnel transforms induced by the generalized fresnel functions for the walsh and conjunctive transforms have fast algorithms

generalized differentiability of fuzzy valued functions in the present paper using novel generalizations of the hukuhara difference for fuzzy sets we introduce and study new generalized differentiability concepts for fuzzy valued functions several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them characterization and relatively simple expressions are provided for the new derivatives

h model reduction for discrete time singular systems this paper investigates the problem of h sub infinity model reduction for linear discrete time singular systems without decomposing the original system matrices necessary and sufficient conditions for the solvability of this problem are obtained in terms of linear matrix inequalities lmis and a coupling non convex rank constraint set when these conditions are feasible an explicit parametrization of the desired reduced order models is given particularly a simple lmi condition without rank constraint is derived for the zeroth order h sub infinity approximation problem finally an illustrative example is provided to demonstrate the applicability of the proposed approach

a projection approach to covariance equivalent realizations of discrete systems covariance equivalent realization theory has been used recently in continuous systems for model reduction 1 4 and controller reduction 2 5 in model reduction this technique produces a reduced order model that matches q 1 output covariances and q markov parameters of the full order model in controller reduction it produces a reduced controller that is close to matching q 1 output covariances of the full order controller and q markov parameters of the closed loop system for discrete systems a method was devised to produce a reduced order model that matches the q 1 covariances 6 but not any markov parameters this method requires a factorization to obtain the input matrix and since the dimension of this matrix factor depends upon rank properties not known a priori this method may not maintain the original dimension of the input vector hence this method 6 is obviously not suitable for controller reduction a new projection method is described that matches q covariances and q markov parameters of the original system while maintaining the correct dimension of the input vector

a new reconstruction of multivariate normal orthant probabilities a new method is introduced for geometrically reconstructing orthant probabilities for non singular multivariate normal distributions orthant probabilities are expressed in terms of those for auto regressive sequences and an efficient method is developed for numerical approximation of the latter the approach allows more efficient accurate evaluation of the multivariate normal cumulative distribution function than previously for many situations where the original distribution arises from a graphical model an implementation is available as a package for the statistical software r and an application is given to multivariate probit models

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

fuzzy fractional differential equations under caputo katugampola fractional derivative approach abstract in this work an initial value problem of caputo katugampola ck fractional differential equations in fuzzy setting is considered and an idea of successive approximations under generalized lipschitz condition is used to prove the existence and uniqueness results of the solution to the given problem in order to obtain the above results some necessary comparison theorems in real valued differential equation under ck fractional derivative are established finally a new technique to find analytical solutions of ck fuzzy fractional differential equations by using the solutions of fuzzy integer order differential equations is proposed some examples illustrating the applications of our results are presented

an accurate solver for forward and inverse transport this paper presents a robust and accurate way to solve steady state linear transport radiative transfer equations numerically our main objective is to address the inverse transport problem in which the optical parameters of a domain of interest are reconstructed from measurements performed at the domain s boundary this inverse problem has important applications in medical and geophysical imaging and more generally in any field involving high frequency waves or particles propagating in scattering environments stable solutions of the inverse transport problem require that the singularities of the measurement operator which maps the optical parameters to the available measurements be captured with sufficient accuracy this in turn requires that the free propagation of particles be calculated with care which is a difficult problem on a cartesian grid a standard discrete ordinates method is used for the direction of propagation of the particles our methodology to address spatial discretization is based on rotating the computational domain so that each direction of propagation is always aligned with one of the grid axes rotations are performed in the fourier domain to achieve spectral accuracy the numerical dispersion of the propagating particles is therefore minimal as a result the ballistic and single scattering components of the transport solution are calculated robustly and accurately physical blurring effects such as small angular diffusion are also incorporated into the numerical tool forward and inverse calculations performed in a two dimensional setting exemplify the capabilities of the method although the methodology might not be the fastest way to solve transport equations its physical accuracy provides us with a numerical tool to assess what can and cannot be reconstructed in inverse transport theory

a unified framework for the fractional fourier transform the paper investigates the possibility for giving a general definition of the fractional fourier transform frt for all signal classes one dimensional 1 d and multidimensional continuous and discrete periodic and aperiodic since the definition is based on the eigenfunctions of the ordinary fourier transform ft the preliminary conditions is that the signal domain periodicity be the same as the ft domain periodicity within these classes a general frt definition is formulated and the frt properties are established in addition the multiplicity which is intrinsic in a fractional operator is clearly developed the general definition is checked in the case in which the frt is presently available and moreover to establish the frt in new classes of signals

new results on the spectral analysis of multi h cpm signals this paper deals with the spectral analysis of multi h continuous phase modulation cpm signals a new solution to the problem is obtained by identifying an inner hidden markov model in the cpm signal which can be accomplished by expressing the modulator as the cascade of a sequential machine sm and a bank of pam modulators an in depth study of the resulting markov chain shows that it is both cyclostationary and reducible and proper tools are identified to address these peculiarities as a result the multi h cpm spectrum is expressed as a combination of fourier transforms of portions of a cpm signal thus allowing an efficient numerical evaluation through fast fourier transforms ffts

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

the structure stability of periodic solutions for first order uncertain dynamical systems abstract this paper studies the structural stability of periodic solutions for first order fuzzy differential equations fdes understood as differential inclusions i e first order uncertain dynamical systems the existence and uniqueness of periodic solutions for this first order fuzzy problems have been obtained on general fuzzy number space when the forcing function has specific perturbations the structural stability of the periodic solutions are discussed by using the support function the dini theorem and the convergence theorem in the differential inclusion theory

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

model reduction using balanced realizations with improved low frequency behaviour abstract model reduction using balanced realization techniques is known to give a better approximation of the high frequency behaviour than the low frequency one in this paper a simple modification to the technique is proposed to get a better approximation of the low frequency behaviour of the original system the proposed technique is illustrated by numerical examples

design of low order linear phase iir filters via orthogonal projection this paper presents an indirect linear phase iir filter design technique based on a reduction of linear phase fir filters the desired filter is obtained by minimizing the l sub 2 norm of the difference between the original fir filter and the lower order iir filter we first establish a relationship between the hankel singular values of the discarded part of the fir filter and the l sub 2 norm of the corresponding filter approximation error based on model truncation this result motivates us to propose a simple finite search method that will achieve better approximation results than commonly used truncation methods such as the balanced truncation bt and the impulse response gramian irg methods we then develop an iterative algorithm for finding an optimal iir filter based on a matrix projection of the original fir filter the convergence of the proposed algorithm is established filters designed using the proposed algorithm are compared with those obtained by other techniques with respect to the amplitude response and group delay characteristics in the passband numerical examples show that the proposed algorithm offers the best performance

fast computation of multidimensional fourier integrals bailey and swarztrauber siam j sci comput 15 1994 pp 1105 1110 have developed an algorithm that uses fast fourier transforms to numerically evaluate one dimensional fourier integrals whose integration ranges are finite and symmetric returning a symmetric range of values in transform variable space their algorithm is here extended to evaluate multidimensional fourier integrals possessing asymmetric integration and transform variable ranges using a second order open quadrature rule to admit integrands with endpoint singularities

the groupoid of bifractional transformations bifractional transformations which lead to quantities that interpolate between other known quantities are considered they do not form a group and groupoids are used to describe their mathematical structure bifractional coherent states and bifractional wigner functions are also defined the properties of the bifractional coherent states are studied the bifractional wigner functions are used in generalizations of the moyal star formalism a generalized berezin formalism in this context is also studied

the use of model reduction techniques for designing iir filters with linear phase in the passband a technique to design iir filters with linear phase in the passband is presented this technique is based on model reduction of an fir prototype using frequency weighting to improve the approximation in the transition region filters designed using the proposed technique are compared with linear phase fir designs and to iir phase equalized designs with respect to computational complexity and group delay it is shown that for highly selective filters the proposed technique offers a good compromise

an integrated fractional fourier transform abstract for 0 f of wiener type closely related to fractional levy brownian motion x is established by means of this transform transform the reproducing kernel hilbert space and the covariance operator of x are characterized and the connection between x and the laplacian is shown as a further application a blaschke fuglede type theorem for a fractional radon transform is derived

the fractional fourier transform and applications this paper describes the fractional fourier transform which admits computation by an algorithm that has complexity proportional to the fast fourier transform algorithm whereas the discrete fourier transform dft is based on integral roots of unity e 2 pi i n the fractional fourier transform is based on fractional roots of unity e 2 pi i alpha where alpha is arbitrary the fractional fourier transform and the corresponding fast algorithm are useful for such applications as computing dfts of sequences with prime lengths computing dfts of sparse sequences analyzing sequences with noninteger periodicities performing high resolution trigonometric interpolation detecting lines in noisy images and detecting signals with linearly drifting frequencies in many cases the resulting algorithms are faster by arbitrarily large factors than conventional techniques

combining local filtering and multiscale analysis for edge ridge and curvilinear objects detection this paper presents a general method for detecting curvilinear structures like filaments or edges in noisy images this method relies on a novel technique the feature adapted beamlet transform fabt which is the main contribution of this paper it combines the well known beamlet transform bt introduced by donoho with local filtering techniques in order to improve both detection performance and accuracy of the bt moreover as the desired feature detector is chosen to belong to the class of steerable filters our transform requires only o n log n operations where n n 2 is the number of pixels besides providing a fast implementation of the fabt on discrete grids we present a statistically controlled method for curvilinear objects detection to extract significant objects we propose an algorithm in four steps 1 compute the fabt 2 normalize beamlet coefficients 3 select meaningful beamlets thanks to a fast energy based minimization and 4 link beamlets together in order to get a list of objects we present an evaluation on both synthetic and real data and demonstrate substantial improvements of our method over classical feature detectors

model reduction of linear discrete systems via weighted impulse response gramians abstract a new method for model reduction of linear discrete systems is proposed it is based on the weighted impulse response gramians proposed here for discrete systems these gramians are extensions of the ones proposed for linear continuous systems agathoklis and sreeram 1988 a 1990 and contain information on the input output behaviour of the system the reduced order models are obtained by retaining the states corresponding to the dominant eigenvalues of these gramians and are stable if the original system is stable the method is illustrated by a numerical example and is compared with well known model reduction techniques

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

the generation of discrete time q markov covers via inverse solution of the lyapunov equation a method for generating q markov covers for siso discrete time systems is proposed the method is based on computing first the impulse response gramian from the markov parameters and covariances and then solving the lyapunov equation inversely to get the system matrices in controllability canonical form the conditions for existence of q markov covers are also derived the method is illustrated by a numerical example and is shown to be computationally simple

two and three dimensional image rotation using the fft techniques for rotating two and three dimensional 2 d and 3 d images using fast fourier transforms ffts are presented the methods are applications of the multidimensional chirp algorithm in the 2 d case one chirp transformation is sufficient requiring four 2 d ffts in the 3 d case two successive chirp transformations are required needing six 3 d ffts

a framework for discrete integral transformations i the pseudopolar fourier transform the fourier transform of a continuous function evaluated at frequencies expressed in polar coordinates is an important conceptual tool for understanding physical continuum phenomena an analogous tool suitable for computations on discrete grids could be very useful however no exact analogue exists in the discrete case in this paper we present the notion of pseudopolar grid pp grid and the pseudopolar fourier transform ppft which evaluates the discrete fourier transform at points of the pp grid the pp grid is a type of concentric squares grid in which the radial density of squares is twice as high as usual the pp grid consists of equally spaced samples along rays where different rays are equally spaced in slope rather than angle we develop a fast algorithm for the ppft with the same complexity order as the cartesian fast fourier transform the algorithm is stable invertible requires only one dimensional operations and uses no approximate interpolations we prove that the ppft is invertible and develop two algorithms for its inversion iterative and direct both with complexity o n 2 log n where n times n is the size of the reconstructed image the iterative algorithm applies conjugate gradients to the gram operator of the ppft since the transform is ill conditioned we introduce a preconditioner which significantly accelerates the convergence the direct inversion algorithm utilizes the special frequency domain structure of the transform in two steps first it resamples the pp grid to a cartesian frequency grid and then recovers the image from the cartesian frequency grid

fast transform from an adaptive multi wavelet representation to a partial fourier representation we present a fast algorithm to compute the partial transformation of a function represented in an adaptive pseudo spectral multi wavelet representation to a partial fourier representation such fast transformations are useful in many contexts in physics and engineering where changes of representation from a piece wise polynomial basis to a fourier basis the algorithm is demonstrated for a gaussian in one and in three dimensions for 2d we apply this approach to a gaussian in a periodic domain the accuracy and the performance of this method is compared with direct summation

identification and model reduction of 2 d systems via the extended impulse response gramians the horizontal and vertical extended impulse response gramians eirg of causal recursive separable denominator crsd 2 d digital systems are defined in this paper based on the 2 d eirg a new identification and model reduction algorithm is presented for 2 d crsd systems it is shown that the reduced order system has three important properties first the 2 d eirg of the reduced order 2 d system approximate the 2 d eirg of the original 2 d system second the 2 d system obtained is a suboptimal solution to the minimization of the energy of the 2 d error system between the original 2 d system and the reduced order system finally the structure of the reduced order 2 d system is convergent to a balanced realization furthermore the present algorithm can handle impulse response data of deterministic or stochastic nature for 2 d system identification application

on the structure of balanced and other principal representations of siso systems a new matrix w co which can be considered as a cross gramian matrix which contains information about both controllability and observability is defined for single input single output linear systems using this matrix the structural properties of linear systems are studied in the context of principal component analysis the matrix w co can be used in obtaining balanced and other principal representations without computation of the controllability and the observability gramians the importance of this matrix in model order reduction is highlighted

a survey on the initial value problems of fuzzy implicit fractional differential equations abstract the purpose of this paper is to present some basic theories of various classes of initial value problems of implicit fractional differential equation involving the caputo fuzzy type riemann liouville fuzzy type and hadamard fuzzy type concepts of fractional derivative in the case of the order 0 1 in particular the existence and uniqueness results of solution for the given initial value problems of implicit fractional differential equations is investigated in addition some results concerning continuous dependence of solutions on perturbations of the given parameters namely of the initial conditions of the function and parameters on the right hand side of the order of fractional derivative are analyzed finally some examples are given to illustrate our main results

a survey on fuzzy fractional differential and optimal control nonlocal evolution equations abstract we survey some representative results on fuzzy fractional differential equations controllability approximate controllability optimal control and optimal feedback control for several different kinds of fractional evolution equations optimality and relaxation of multiple control problems described by nonlinear fractional differential equations with nonlocal control conditions in banach spaces are considered

analysis of preconditioners for hyperbolic partial differential equations preconditioners for conjugate gradient cg like iterative methods are analyzed the systems of equations arise from discretizations of time dependent hyperbolic partial differential equations pdes in two space dimensions incomplete lu ilu and block ilu preconditioners are considered for a model problem subject to periodic boundary conditions the spectra of the preconditioned coefficient matrices are determined by fourier analysis it is found that the condition number is not improved by the ilu preconditioners furthermore the number of distinct eigenvalues is not decreased a semicirculant preconditioner applied to a problem subject to dirichlet boundary conditions at the inflow boundaries is also examined analytical formulas for the eigenvalues and the eigenvectors are derived for this purpose a special analysis based on spectral decomposition and residue theory is developed when the grid ratio in space is less than one the spectrum asymptotically becomes two finite curve segments which are independent of the number of gridpoints for the restarted generalized minimal residual gmres iteration a slight reduction of the grid ratio from one substantially improves the convergence rate this is also predicted by an asymptotic analysis

fractional differential systems a fuzzy solution based on operational matrix of shifted chebyshev polynomials and its applications in this paper a new formula of fuzzy caputo fractional order derivatives 0 v leq 1 in terms of shifted chebyshev polynomials is derived the proposed approach introduces a shifted chebyshev operational matrix in combination with a shifted chebyshev tau technique for the numerical solution of linear fuzzy fractional order differential equations the main advantage of the proposed approach is that it simplifies the problem alike in solving a system of fuzzy algebraic linear equations an approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors is derived furthermore we also discuss the convergence of the proposed method from the fuzzy perspective experimentally we show the strength of the proposed method in solving a variety of fractional differential equation models under uncertainty encountered in engineering and physical phenomena i e viscoelasticity oscillations and resistor capacitor rc circuits comparisons are also made with solutions obtained by the laguerre polynomials and the fractional euler method

the solvability of fuzzy fractional partial differential equations under caputo gh differentiability the foundation of the concepts of fuzzy fractional integral and caputo gh partial for fuzzy valued multivariable functions is defined as a result fuzzy fractional partial differential equations are considered and the appropriateness of local boundary value problems for hyperbolic equations is proved we present two new results on the existence of two kinds of gh weak solutions of these problems the first result is based on the banach fixed point theorem with the lipschitz condition of functions on the right hand side of equations the second result is based on the nonlinear alternative of the schauder type for fuzzy valued continuous functions without the lipschitz condition moreover we indicate the boundedness and continuous dependence of solutions on the initial data of the problems our models are embedded in the sense of caputo gh differentiability some examples are presented to illustrate the results

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

balanced gains and their significance for l 2 model reduction the singular values do not give enough information for model reduction in the l2sense other invariants are introduced which help describe the contribution of each state to the l2magnitude of the impulse response this leads to another suboptimal balanced truncation criterion

the discrete time q markov cover models with improved low frequency approximation a new method of obtaining q markov cover models for linear discrete systems based on singular perturbation technique is proposed the reduced order models obtained not only match initial markov parameters and high frequency power moments like conventional q markov covers but also match the initial time moments and low frequency power moments of the original system

an exact and fast computation of discrete fourier transform for polar and spherical grid numerous applied problems of two dimensional 2 d and 3 d imaging are formulated in continuous domain they place great emphasis on obtaining and manipulating the fourier transform in polar and spherical coordinates however the translation of continuum ideas with the discrete sampled data on a cartesian grid is problematic there exists no exact and fast solution to the problem of obtaining discrete fourier transform for polar and spherical grids in the literature in this paper we develop exact algorithms to the above problem for 2 d and 3 d which involve only 1 d equispaced fast fourier transform with no interpolation or approximation at any stage the result of the proposed approach leads to a fast solution with very high accuracy we describe the computational procedure to obtain the solution in both 2 d and 3 d which includes fast forward and inverse transforms we find the nested multilevel matrix structure of the inverse process and we propose a hybrid grid and use a preconditioned conjugate gradient method that exhibits a drastic improvement in the condition number

a note on initial value problems for fractional fuzzy differential equations abstract in this note we present some remarks on solutions of fractional fuzzy differential equation in general a fractional fuzzy differential equation and a fractional fuzzy integral equation are not equivalent in this paper we give an appropriate condition so that this equivalence is valid some examples are provided to illustrate the theory

identification reduction to a balanced realization via the extended impulse response gramian a new identification reduction algorithm for linear discrete time invariant ldti systems is proposed which is based on the extended impulse response gramian first defined here for ldti systems the reduction algorithm employs singular value decomposition to retain states corresponding to dominant singular values of these gramians the proven properties of the reduced order models include convergence to a balanced realization with conditional controllability observability and asymptotic stability a suboptimal property of the model in minimizing an impulse response error norm is also demonstrated the proposed technique can handle impulse response data of deterministic or stochastic nature for system identification application