autocorrelation in meter induction the role of accent structure the performance of autocorrelation based meter induction was tested with two large collections of folk melodies consisting of approximately 13 000 melodies for which the correct meters were available the performance was measured by the proportion of melodies whose meter was correctly classified by a discriminant function furthermore it was examined whether including different melodic accent types would improve the classification performance by determining the components of the autocorrelation functions that were significant in the classification it was found that periodicity in note onset locations was the most important cue for the determination of meter of the melodic accents included thomassen s melodic accent was found to provide the most reliable cues for the determination of meter the discriminant function analyses suggested that periodicities longer than one measure may provide cues for meter determination that are more reliable than shorter periodicities overall the method predicted notated meter with an accuracy reaching 96 for binary classification and 75 for classification into nine categories of meter

hamiltonian formulation of systems with linear velocities within riemann liouville fractional derivatives the link between the treatments of constrained systems with fractional derivatives by using both hamiltonian and lagrangian formulations is studied it is shown that both treatments for systems with linear velocities are equivalent

calculation of a constant q spectral transform the frequencies that have been chosen to make up the scale of western music are geometrically spaced thus the discrete fourier transform dft although extremely efficient in the fast fourier transform implementation yields components which do not map efficiently to musical frequencies this is because the frequency components calculated with the dft are separated by a constant frequency difference and with a constant resolution a calculation similar to a discrete fourier transform but with a constant ratio of center frequency to resolution has been made this is a constant q transform and is equivalent to a 1 24 oct filter bank thus there are two frequency components for each musical note so that two adjacent notes in the musical scale played simultaneously can be resolved anywhere in the musical frequency range this transform against log frequency to obtain a constant pattern in the frequency domain for sounds with harmonic frequency components has been plotted this is compared to the conventio

a high resolution fundamental frequency determination based on phase changes of the fourier transform the constant q transform described recently j c brown and m s puckette an efficient algorithm for the calculation of a constant q transform j acoust soc am 92 2698 2701 1992 has been adapted so that it is suitable for tracking the fundamental frequency of extremely rapid musical passages for this purpose the calculation described previously has been modified so that it is constant frequency resolution rather than constant q for lower frequency bins this modified calculation serves as the input for a fundamental frequency tracker similar to that described by brown j c brown musical fundamental frequency tracking using a pattern recognition method j acoust soc am 92 1394 1402 1992 once the fast fourier transform fft bin corresponding to the fundamental frequency is chosen by the frequency tracker an approximation is used for the phase change in the fft for a time advance of one sample to obtain an extremely precise value for this frequency graphical examples are given for musical passages by a violin executing vibrato and glissando where the fundamental frequency changes are rapid and continuous

linear approximation of transfer function with a pole of fractional power two methods for the linear approximation of a transfer function with a pole of fractional power are presented analog circuit models are developed and their frequency response curves and step response curves are compared it was found that the pade method gives a better approximation than wang and hsia s method within the frequency limit as specified

chaos synchronization of the fractional lu system abstract chaos synchronization of the fractional lu system is theoretically and numerically studied by two methods the suitable conditions for achieving synchronization of this fractional differential system are derived by using the laplace transform theory numerical simulations coincide with the theoretical analysis

a note on optimal homotopy asymptotic method for the solutions of fractional order heat and wave like partial differential equations optimal homotopy asymptotic method oham is prolifically implemented to find the optimal solutions of fractional order heat and wave like equations we inspect the competence of the method by examining fractional order time dependent partial differential equations it is observed that oham is a prevailing and convergent method for the solutions of linear and nonlinear fractional order time dependent partial differential problems the numerical results rendering that the applied method is explicit effective and easy to use for handling more general fractional order heat and wave like models

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

the general vector solutions of such linear coupled matrix fractional differential equations by using kronecker structures in this paper we demonstrate how kronecker products and vector operator can be combined beautifully to solve and fit the non homogenous linear matrix fractional differential equations and such coupled linear matrix fractional differential equations the way exists which transform the given coupled matrix fractional differential equations by using kronecker structures into forms for which solutions may be readily computed some important and interesting special cases of the general system of non homogeneous linear matrix fractional differential equations of fractional order are also considered which includes the non homogenous linear fractional dynamical system with delays in control finally a brief comparison between the fractional and integer order vector solutions and some examples are given to illustrate our new approaches

fractional radiative transfer equation within chebyshev spectral approach in this work we report the convergence of the chebyshev polynomials combined with the s n method for the steady state transport equation using the fractional derivative the procedure is based on the expansion of the angular flux in a truncated series of orthogonal polynomials that results in the transformation of the multidimensional problem into a system of fractional differential equations the convergence of this approach is studied in the context of the multidimensional discrete ordinates equations

the perturbation theory for the drazin inverse and its applications abstract let a and e be n n matrices and b a e denote the drazin inverse of a by ad we give an upper bound for the relative error b d a d a d under certain circumstances an error bound for the solution for the singular equations ax b b r a is also considered

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

transformations between some special matrices special matrices are very useful in signal processing and control systems this paper studies the transformations and relationships between some special matrices the conditions that a matrix is similar to a companion matrix are derived it is proved that a companion matrix is similar to a diagonal matrix or jordan matrix and the transformation matrices between them are given finally we apply the similarity transformation and the companion matrix to system identification

computation of matrix exponentials of special matrices computing matrix exponentials or transition matrices are important in signal processing and control systems this paper studies the methods of computing the transition matrices related to linear dynamic systems discusses and gives the properties of the transition matrices and explores the transition matrices of some special matrices some examples are provided to illustrate the proposed methods

gradient based iterative solutions for general linear matrix equations in this paper we present a gradient based iterative algorithm for solving general linear matrix equations by extending the jacobi iteration and by applying the hierarchical identification principle convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices two numerical examples are provided to show that the proposed algorithm is effective

the short memory principle for solving abel differential equation of fractional order in this paper the short memory principle smp is applied for solving the abel differential equation with fractional order we evaluate the approximate solution at the end of required interval and construct a suitable iteration scheme employing this end point as initial value numerical experiments show that our iteration method is simple and efficient and that a proper length of memory could maintain the validity of the short memory principle

the variational iteration method an efficient scheme for handling fractional partial differential equations in fluid mechanics variational iteration method has been used to handle linear and nonlinear differential equations the main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently in this work a general framework of the variational iteration method is presented for analytical treatment of fractional partial differential equations in fluid mechanics the fractional derivatives are described in the caputo sense numerical illustrations that include the fractional wave equation fractional burgers equation fractional kdv equation fractional klein gordon equation and fractional boussinesq like equation are investigated to show the pertinent features of the technique comparison of the results obtained by the variational iteration method with those obtained by adomian decomposition method reveals that the first method is very effective and convenient the basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus

chebyshev spectral sn method for the neutron transport equation we study convergence of a combined spectral and sn discrete ordinates approximation for a multidimensional steady state linear transport problem with isotropic scattering the procedure is based on expansion of the angular flux in a truncated series of chebyshev polynomials in spatial variables that results in the transformation of the multidimensional problems into a set of one dimensional problems the convergence of this approach is studied in the context of the discrete ordinates equations based on a special quadrature rule for the scattering integral the discrete ordinates and quadrature errors are expanded in truncated series of chebyshev polynomials of degree l and the convergence is derived assuming l s t 4 p s s where s t and s s are total and scattering cross sections respectively

approximate solutions for boundary value problems of time fractional wave equation in this article we implement relatively a new analytical technique the adomian decomposition method for solving the boundary value problems of time fractional wave equation the fractional derivative is described in the caputo sense the decomposition method is used to construct analytical approximate solutions of time fractional wave equation subject to specified boundary conditions the solutions are calculated in the form of a convergent series with easily computable components some examples are given the results reveal that the adomian method is very effective and convenient

several inequalities involving khatri rao products of positive semidefinite matrices abstract the khatri rao and tracy singh products for partitioned matrices are viewed as generalized hadamard and generalized kronecker products respectively based on the connection between the khatri rao and tracy singh products we establish new inequalities involving khatri rao products of positive semidefinite matrices the results lead to inequalities involving hadamard products as a special case

variational iteration method new development and applications variational iteration method has been favourably applied to various kinds of nonlinear problems the main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently in this paper recent trends and developments in the use of the method are reviewed major applications to nonlinear wave equation nonlinear fractional differential equations nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed the confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods variational iteration method is uniquely qualified to address this challenge the flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions this paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations a new iteration formulation is suggested to overcome the shortcoming a very useful formulation for determining approximately the period of a nonlinear oscillator is suggested examples are given to illustrate the solution procedure

an efficient method for solving systems of fractional integro differential equations in this study a decomposition method for approximating the solution of systems of fractional integro differential equations is implemented the fractional derivative is considered in the caputo sense the approximate solutions are calculated in the form of a convergent series with easily computable components numerical results show that this approach is easy to implement and accurate when applied to integro differential equations

numerical steady state and hopf bifurcation analysis on the diffusive nicholson s blowflies equation1research partially supported by natural sciences and engineering research council of canada 1 for the dirichlet boundary value problem of the diffusive nicholson s blowflies equation it was shown in ref 17 that in a certain range of the parameter space there is a unique positive steady state solution in this paper we propose a scheme to compute this steady state numerically in addition we describe an iterative procedure to locate the critical values of the delay where a hopf bifurcation of time periodic solutions takes place near the steady state some numerical simulations of both schemes are given

the use of the matrix pencil method for the spectrum analysis of musical signals in the context of spectrum analysis high resolution estimation methods present interesting alternatives to classical fourier transform these methods are based on the assumption that the analyzed signal is composed of damped sinusoids and yield estimates of their parameters frequencies damping factors original amplitudes and phases this paper investigates the use of a specific high resolution estimation method the matrix pencil method for the spectrum analysis of musical signals in spite of its computational cost this technique is shown to achieve extremely good results in the case of highly damped signals short data records or close frequencies

application of the variational iteration method to the regularized long wave equation this paper applies the variational iteration method to the regularized long wave equation in this method a generalized lagrange multiplier is introduced to construct a correction functional for the discussed problem the multiplier can be identified optimally via variational theory three invariants of the motion are evaluated to determine the conservation properties of the system

instantaneous frequency and short term fourier transforms application to piano sounds for more than 30 years and until nowadays development of a system reproducing the functioning of human hearing has remained an aim difficult to reach recent methods for identification of the fundamental frequency of musical sounds obtain good results using information about the temporal evolution of the amplitude and frequency of individual sound partials piano sounds and polyphonic sounds which can have several partials that are closely spaced in the frequency domain have not been extensively tested by these procedures in this paper the instantaneous frequency if as defined by the hilbert transform is used to obtain the frequency variations of piano sounds partials the result implies that for these sounds the if may contain modulations resulting in the separation of an apparent single sinusoid signal into two or more sinusoidal components at various times in the analysis process which makes it impossible to use the temporal evolution of the frequency of partials for the procedure of note identification the separation phenomenon also appears when the short term fourier transform is used and can induce the detection of short lived parasitic spectral peaks that must be taken into account by any note identification procedure based on the use of spectral information

a quantitative version of the observation that the hadamard product is a principal submatrix of the kronecker product abstract the hadamard and kronecker products of two n m matrices a b are related by a b p t 1 a b p 2 where p 1 p 2 are partial permutation matrices after establishing several properties of the p matrices this relationship is employed to demonstrate how a simplified theory of the hadamard product can be developed during this process the well known result a b a b aa bb is extended to a b a b 1 2 aa bb ab ba aa bb showing an inherent link between the hadamard product and conventional product of two matrices this leads to a sharper bound on the spectral norm of a b a b 1 2 a 2 b 2 ab 2 1 2 a b and an improvement on the weak majorization of a b 2 a b w 1 2 2 a 2 b 2 ab w 2 a 2 b for a real non singular matrix x and invertible diagonal matrices d e the spectral condition number is shown to be if scaled bounded below as follows dxe 2 x x t 2 x x t 2 1 2 x x t 1 for a 0 we have i a 2 1 2 i a 2 a a i a 2 and a 1 2 a 1 2 2 1 2 i a a 1 a a 1 when a 0 the latter inequality is compared to styan s inequality a a 1 1 2 i a a 1 when a is a correlation matrix and is shown to possess stronger properties of ordering finally the relationship a b p t 1 a b p 2 is applied to determine conditions of singularity of certain orderings of the hadamard products of matrices

analytical solution of a time fractional navier stokes equation by adomian decomposition method abstract the aim of the present analysis is to apply adomian decomposition method for the solution of a time fractional navier stokes equation in a tube by using an initial value the explicit solution of the equation has been presented in the closed form and then its numerical solution has been represented graphically the present method performs extremely well in terms of efficiency and simplicity

new ideas for proving convergence of decomposition methods in this paper we give new formulae which calculate easily the adomian s polynomials used in decomposition methods then the proof of convergence of the adomian s technique becomes almost obvious by using a weak hypothesis on the nonlinear operator of the functional equation

kronecker products and coupled matrix riccati differential systems abstract by the application of kronecker products a riccati differential system strongly coupled in its quadratic terms is reduced to another one to which the successive approximation method is applicable an iterative algorithm for solving the problem and an error bound for the approximate solutions in terms of data are given

joint independent subspace analysis uniqueness and identifiability this paper deals with the identifiability of joint independent subspace analysis jisa jisa is a recently proposed framework that subsumes independent vector analysis iva and independent subspace analysis isa each underlying mixture can be regarded as a dataset therefore jisa can be used for data fusion in this paper we assume that each dataset is an overdetermined mixture of several multivariate gaussian processes each of which has independent and identically distributed samples this setup is not identifiable when each mixture is considered individually given these assumptions jisa can be restated as coupled block diagonalization cbd of its correlation matrices hence jisa identifiability is tantamount to cbd uniqueness in this work we provide necessary and sufficient conditions for uniqueness and identifiability of jisa and cbd our analysis is based on characterizing all the cases in which the fisher information matrix is singular we prove that non identifiability may occur only due to pairs of underlying random processes with the same dimension our results provide further evidence that irreducibility has a central role in the uniqueness analysis of block based decompositions our contribution extends previous results on the uniqueness and identifiability of isa iva coupled matrix and tensor decompositions we provide examples to illustrate our results

a new modification of the adomian decomposition method for linear and nonlinear operators in this paper we present an efficient modification of the adomian decomposition method that will facilitate the calculations we then conduct a comparative study between the new modification and the modified decomposition method the study is conducted through illustrative examples the new modification introduces a promising tool for many linear and nonlinear models

chebyshev polynomials approach for numerically solving system of two dimensional fractional pdes and convergence analysis abstract in this paper an efficient numerical technique based on the chebsyhev orthogonal polynomials is established to obtain the approximate solutions of system of two dimensional fractional order pdes with initial conditions we construct the corresponding differential operational matrix of fractional order and then transform the problem into a system of linear algebra equations compared with other analytical or semi analytical methods ours can achieve better convergence accuracy only small terms are expanded moreover the proposed algorithm is simple in theoretical derivation and numerical simulation in our study the convergence analysis of the system is emphatically investigated than other numerical approaches lastly three numerical examples are applied to test the algorithm and that the obtained numerical results show that our approach is effective and robust

nonholonomic constraints with fractional derivatives we consider the fractional generalization of nonholonomic constraints defined by equations with fractional derivatives and provide some examples the corresponding equations of motion are derived using variational principle we prove that fractional constraints can be used to describe the evolution of dynamical systems in which some coordinates and velocities are related to velocities through a power law memory function

some notes on properties of the matrix mittag leffler function abstract we have come across with publications where some properties of the matrix exponential were incorrectly extended to the matrix mittag leffler function and then used as a key tool to solve certain linear matrix fractional differential equations the main purpose of these notes is to give some clarifications on properties of the matrix mittag leffler function by explaining in detail why some identities do not hold and by providing a list of valid properties of this function a sufficient condition to identify very special cases of pairs of matrices satisfying the semigroup property is given as well as examples illustrating the results

analytical approximate solutions for nonlinear fractional differential equations we consider a class of nonlinear fractional differential equations fdes based on the caputo fractional derivative and by extending the application of the adomian decomposition method we derive an analytical solution in the form of a series with easily computable terms for linear equations the method gives exact solution and for non linear equations it provides an approximate solution with good accuracy several examples are discussed

block matrix generalizations of schur s basic theorems on hadamard products abstract in a classic 1911 paper i schur gave several useful bounds for the spectral norm and eigenvalues of the hadamard entrywise product of two matrices motivated by applications to the theory of monotone and convex matrix functions we are led to consider hadamard products in which both factors are conformally partitioned block matrices and the entries of one factor are constant within each block such products are special cases of a block kronecker tensor product and it is in this context that we present generalizations of schur s results

iterative algorithms for the minimum norm solution and the least squares solution of the linear matrix equations abstract in this paper two iterative algorithms are proposed to solve the linear matrix equations a 1 xb 1 c 1 x t d 1 m 1 a 2 xb 2 c 2 x t d 2 m 2 when the matrix equations are consistent by the first algorithm a solution x can be obtained within finite iterative steps in the absence of roundoff error for any initial value furthermore the minimum norm solution can be got by choosing a special kind of initial matrix additionally the unique optimal approximation solution to a given matrix x 0 can be derived by finding the minimum norm solution of a new matrix equations a 1 x b 1 c 1 x t d 1 m 1 a 2 x b 2 c 2 x t d 2 m 2 when the matrix equations are inconsistent we present the second algorithm to find the least squares solution with the minimum norm finally two numerical examples are tested by matlab the results show that these iterative algorithms are efficient

application of he s methods to nonlinear chemistry problems in this paper he s variational iteration method vim and homotopy perturbation method hpm are implemented for solving analytically two systems of ordinary differential equations that often appear in chemical applications in both methods the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions the solution procedure is of utter simplicity and the obtained solution is of high accuracy

non axisymmetric bending of thin annular plates due to circumferentially distributed moments abstract the non linear deformation of a thin annular plate subjected to circumferentially distributed bending moments is studied a von karman plate model is adopted to formulate the equations of motion free free boundary conditions have been applied at both inner and outer edges the matrix formulation of the differential quadrature method is used to discretise and solve the governing equations linear analysis predicts that the annular disk deforms axisymmetrically into a spherical dome however the proposed non linear analysis shows that a symmetry breaking bifurcation may occur after which the linear solution becomes unstable and the plate transitions into a non axisymmetric cylindrical deformation this is the case when at least one of two parameters reaches a critical value these parameters are the non dimensionalised ratio between applied moment and bending stiffness and the ratio between inner and outer radius furthermore it is noted that free free boundary conditions and circumferentially distributed bending moments do not break the circular symmetry of the annular disk hence the principal axes of curvature in the deformed configuration do not have a preferred orientation therefore the present work describes a shell possessing infinite identical equilibria having different yet no favoured direction and hence links to previous researches on neutrally stable structures

solving related two and three dimensional linear programming problems in logarithmic time abstract given n linear inequalities in three variables we show how to construct a corresponding spherical subdivision using great circle arcs in time o n log n and space o n this subdivision in turn allows us to compute the point in space satisfying all inequalities and maximizing any desired linear objective function in time o log n

matrix results on the khatri rao and tracy singh products abstract we establish a connection between the khatri rao and tracy singh products introduced by khatri and rao c g khatri c r rao sankhya 30 1968 167 180 and tracy and singh d s tracy r p singh statistica neerlandica 26 1972 143 157 respectively and present further results including matrix equalities and inequalities involving the two products also we give two statistical applications

collision detection algorithm of a continuous type using spherical extreme vertex diagrams abstract most collision detection algorithms are ones of a discrete type which utilize convexity and local coherence to verify the closest points however the incremental algorithms have such critical drawbacks like failure to detect high speed collisions and dependence on the time complexity on the time step size of animation in this paper in order to find a new method which can resolve the drawbacks we are concerned with a more restricted but nontrivial version of the problem given a fixed infinite plane h and a moving convex polyhedron p compute their collision time our algorithm utilizes the spherical extreme vertex diagram that is the embedding of the dual graph p onto the gauss sphere exploiting the diagram we are able to efficiently compute the sequence of the extreme vertices v i s i 1 m and find the time interval t i t i 1 during which v i is the extreme vertex of p with the sequences of the consecutive time intervals and extreme vertices we construct a distance function between p and h hence we can compute their collision time within a given tolerance by applying the interval newton method to the distance function moreover the total time complexity is independent of the time step size of animation

block kronecker products and the vecb operator this paper is concerned with two generalizations of the kronecker product and two related generalizations of the vec operator it is demonstrated that they pairwise match two different kinds of matrix partition viz the balanced and unbalanced ones relevant properties are supplied and proved a related concept the so called tilde transform of a balanced block matrix is also studied the results are illustrated with various statistical applications of the five concepts studied

analysis of a system of nonautonomous fractional differential equations involving caputo derivatives abstract we discuss existence uniqueness and stability of solutions of the system of nonlinear fractional differential equations d i y i t f i t y 1 y n y i k c k i 1 i n 1 k m i where m i i m i 1 and d i denotes caputo fractional derivative linear and nonlinear systems have been solved using banach fixed point theorem

vector least squares solutions for coupled singular matrix equations the weighted least squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition in this paper a family of algorithms are applied to solve these problems based on the kronecker structures subsequently we construct a computationally efficient solutions of coupled restricted singular matrix equations furthermore the need to compute the weighted drazin and weighted moore penrose inverses and the use of tian s work and lev ari s results are due to appearance in the solutions of these problems the several special cases of these problems are also considered which includes the well known coupled sylvester matrix equations finally we recover the iterative methods to the weighted case in order to obtain the minimum d norm g vector least squares solutions for the coupled sylvester matrix equations and the results lead to the least squares solutions and invertible solutions as a special case

an efficient algorithm for the calculation of a constant q transform an efficient method of transforming a discrete fourier transform dft into a constant q transform where q is the ratio of center frequency to bandwidth has been devised this method involves the calculation of kernels that are then applied to each subsequent dft only a few multiples are involved in the calculation of each component of the constant q transform so this transformation adds a small amount to the computation in effect this method makes it possible to take full advantage of the computational efficiency of the fast fourier transform fft graphical examples of the application of this calculation to musical signals are given for sounds produced by a clarinet and a violin

numerical approach to differential equations of fractional order in this paper the variational iteration method and the adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order the two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations in these schemes the solution takes the form of a convergent series with easily computable components this paper presents a numerical comparison between the two methods for solving systems of fractional differential equations numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order

the hamilton formalism with fractional derivatives recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives in this paper the passage from the lagrangian containing fractional derivatives to the hamiltonian is achieved the hamilton s equations of motion are obtained in a similar manner to the usual mechanics in addition the classical fields with fractional derivatives are investigated using hamiltonian formalism two discrete problems and one continuous are considered to demonstrate the application of the formalism the results are obtained to be in exact agreement with agrawal s formalism

kronecker operational matrices for fractional calculus and some applications the problems of systems identification analysis and optimal control have been recently studied using orthogonal functions the specific orthogonal functions used up to now are the walsh the block pulse the laguerre the legendre haar and many other functions in the present paper several operational matrices for integration and differentiation are studied we introduce the kronecker convolution product and expanded to the riemann liouville fractional integral of matrices for some applications it is often not necessary to compute exact solutions approximate solutions are sufficient because sometimes computational efforts rapidly increase with the size of matrix functions our method is extended to find the exact and approximate solutions of the general system matrix convolution differential equations the way exists which transform the coupled matrix differential equations into forms for which solutions may be readily computed finally several systems are solved by the new and other approaches and illustrative examples are also considered

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

an efficient algorithm for determining the extreme vertices of a moving 3d convex polyhedron with respect to a plane abstract we present an efficient algorithm for finding the sequence of extreme vertices of a moving convex polyhedron p with respect to a fixed plane h using the spherical extreme vertex diagram due to the point plane duality we are able to find such a sequence in o log n sj 1 mj time where s is the number of extreme vertices in the sequence and mj 1 j s is the number of edges of the spherical region svj corresponding to an extreme vertex vj in the sequence

spline collocation for nonlinear fractional boundary value problems abstract we consider a class of boundary value problems for nonlinear fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties and spline collocation techniques the numerical solution of boundary value problems by suitable non polynomial approximations is discussed optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given theoretical results are tested by two numerical examples

relative controllability of fractional dynamical systems with distributed delays in control this paper is concerned with the global relative controllability of fractional dynamical systems with multiple delays in control for finite dimensional spaces sufficient conditions for controllability results are obtained using schauder s fixed point theorem and the controllability grammian matrix which is defined by the mittag leffler matrix function an example is provided to illustrate the theory

on systems of linear fractional differential equations with constant coefficients abstract this paper deals with the study of linear systems of fractional differential equations such as the following system 1 y a x y b x where y is the riemann liouville or the caputo fractional derivative of order 0 2 a x a 11 x a 1 n x a n 1 x a nn x b x b 1 x b n x are matrices of known real functions in a way analogous to the usual case we show how a generalized matrix exponential function and certain fractional green function in connection with the mittag leffler type functions would allow us to obtain an explicit representation of the general solution to the system 1 when a is a constant matrix

some new connections between matrix products for partitioned and non partitioned matrices this paper is concerned with a long list of matrix products for partitioned and non partitioned matrices including kronecker khatri rao of first kind hadamard tracy singh khatri rao block kronecker block hadamard and box products sums our contribution is to gather the most useful connections of the matrix products from various sources and present them in one place several new attractive connections are also added to this collection we give as an example two applications in order to show that these connections play a central role in many applications these applications are a block diagonal least squares problem and a generalization of matrix inequalities involving khatri rao products of positive definite matrices

an iterative method for solving nonlinear functional equations an iterative method for solving nonlinear functional equations viz nonlinear volterra integral equations algebraic equations and systems of ordinary differential equation nonlinear algebraic equations and fractional differential equations has been discussed

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

weighted moore penrose drazin and group inverses of the kronecker product a b and some applications abstract the representations for the weighted moore penrose inverse the drazin inverse and the group inverse of the kronecker product a b of the two matrices a and b are given and by using these results the determinant forms for projectors aa mn a mn a a d a and a g a are deduced

musical fundamental frequency tracking using a pattern recognition method in a recent article j c brown calculation of a constant q spectral transform j acoust soc am 89 425 434 1991 the calculation of a constant q spectral transform that gives a constant pattern in the log frequency domain for sounds with harmonic frequency components has been described this property has been utilized in calculating the cross correlation function of spectra of sounds produced by musical instruments with the ideal pattern which consists of one s at the positions of harmonic frequency components therefore the position of the best approximation to the ideal pattern for the spectra produced by these instruments has been determined and in so doing the fundamental frequency for that sound has been obtained results are presented for scales produced by the piano flute and violin as well as for arpeggios played by a wide variety of instruments

approximate analytical solution for seepage flow with fractional derivatives in porous media abstract in this paper a new and more exact model for seepage flow in porous media with fractional derivatives has been proposed which has modified the well known darcy law and overcome the continuity assumption of seepage flow a new kind of analytical method of nonlinear problem called the variational iteration method is described and used to give approximate solutions of the problem the results show that the proposed iteration method requiring no linearization or small perturbation is very effective and convenient

schur complements and matrix inequalities in the lowner ordering abstract the purpose of this paper is to present a matrix inequality on the kronecker product that unifies the proofs of many existing matrix inequalities in the lowner partial ordering on the sum ordinary product and hadamard schur product schur complements serve as the basic tool

convergence of the variational iteration method for solving multi order fractional differential equations in this paper the variational iteration method vim is applied to obtain approximate solutions of multi order fractional differential equations m fdes we can easily obtain the satisfying solution just by using a few simple transformations and applying the vim a theorem for convergence and error estimates of the vim for solving m fdes is given moreover numerical results show that our theoretical analysis are accurate and the vim is a powerful method for solving m fdes

a novel method for nonlinear fractional partial differential equations combination of dtm and generalized taylor s formula in this article a novel numerical method is proposed for nonlinear partial differential equations with space and time fractional derivatives this method is based on the two dimensional differential transform method dtm and generalized taylor s formula the fractional derivatives are considered in the caputo sense several illustrative examples are given to demonstrate the effectiveness of the present method results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere results also show that the numerical scheme is very effective and convenient for solving nonlinear partial differential equations of fractional order

estimation of frequency for am fm models using the phase vocoder framework this paper proposes an extension of the applicability of phase vocoder based frequency estimators for generalized sinusoidal models which include phase and amplitude modulations a first approach called phase corrected vocoder pcv takes into account the modification of the fourier phases resulting from these modulations another approach is based on an adaptation of the principles of the time frequency reassignment and is referred to as the reassigned vocoder rv the robustness of the estimation against noise is studied both theoretically and experimentally and the performance is assessed in comparison with two state of the art algorithms an unmodified version of the reassignment method and a quadratically interpolated fast fourier transform method qifft

chaos and hyperchaos in the fractional order rossler equations abstract the dynamics of fractional order systems have attracted increasing attentions in recent years in this paper we numerically study the chaotic behaviors in the fractional order rossler equations we found that chaotic behaviors exist in the fractional order rossler equation with orders less than 3 and hyperchaos exists in the fractional order rossler hyperchaotic equation with order less than 4 the lowest orders we found for chaos and hyperchaos to exist in such systems are 2 4 and 3 8 respectively period doubling routes to chaos in the fractional order rossler equation are also found

approximate solution of nonlinear differential equations with convolution product nonlinearities in this paper a new iteration method is proposed to solve nonlinear problems special attention is paid to nonlinear differential equations with convolution product nonlinearities the results reveal the approximations obtained by the proposed method are uniformly valid for both small and large parameters in nonlinear problems furthermore the first order of approximations are more accurate than perturbation solutions at high order of approximation

on shifted jacobi spectral approximations for solving fractional differential equations in this paper a new formula of caputo fractional order derivatives of shifted jacobi polynomials of any degree in terms of shifted jacobi polynomials themselves is proved we discuss a direct solution technique for linear multi order fractional differential equations fdes subject to nonhomogeneous initial conditions using a shifted jacobi tau approximation a quadrature shifted jacobi tau q sjt approximation is introduced for the solution of linear multi order fdes with variable coefficients we also propose a shifted jacobi collocation technique for solving nonlinear multi order fractional initial value problems the advantages of using the proposed techniques are discussed and we compare them with other existing methods we investigate some illustrative examples of fdes including linear and nonlinear terms we demonstrate the high accuracy and the efficiency of the proposed techniques