on a connection between the pascal vandermonde and stirling matrices ii this article is an extension of the work presented in on a connection between the pascal vandermonde and stirling matrices i applied mathematics and computation 2003 to appear it gives a full matrix t t for which t 1 is exactly the lower triangular stochastic matrix t given in on a connection between the pascal vandermonde and stirling matrices i applied mathematics and computation 2003 to appear a new strategy to generate the elements of the matrix t t 1 is investigated a new conjecture and three combinatorial identities are also given

a note on the stirling matrix of the second kind in this article a new algorithm for generating the elements of the n 1 by n 1 stirling matrix of the second kind is presented based on this algorithm a computational maple procedure is written and a sample output of this procedure is also given for some values of n

infinite triangular matrices q pascal matrices and determinantal representations we use basic properties of infinite lower triangular matrices and the connections of toeplitz matrices with generating functions to obtain inversion formulas for several types of q pascal matrices determinantal representations for polynomial sequences and identities involving the q gaussian coefficients we also obtain a fast inversion algorithm for general infinite lower triangular matrices

application of the θ method to a telegraphic model of fluid flow in a dual porosity medium abstract this work focuses mainly on the study of numerical solutions which are obtained using the method of a generalized warren and root model that includes a second order wave like equation in its formulation the solutions approximately describe the single phase hydraulic head in fractures by considering the finite velocity of propagation by means of a cattaneo like equation the corresponding discretized model is obtained by utilizing a non uniform grid and a non uniform time step a simple relationship is proposed to give the time step distribution convergence is analyzed by comparing results from explicit fully implicit and crank nicolson schemes with exact solutions a telegraphic model of fluid flow in a single porosity reservoir with relaxation dynamics the warren and root model and our studied model which is solved with the inverse laplace transform we find that the flux and the hydraulic head have spurious oscillations that most often appear in small time solutions but are attenuated as the solution time progresses furthermore we show that the finite difference method is unable to reproduce the exact flux at time zero obtaining results for oilfield production times which are in the order of months in real units is only feasible using parallel implicit schemes in addition we propose simple parallel algorithms for the memory flux and for the explicit scheme

the linear algebra of the generalized pascal matrix abstract this paper discusses three kinds of generalized pascal matrix and generalizes the results of r brawer and m pirovino

high accuracy wave simulation revised derivation numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation the nearly analytic integration discrete naid method for solving the two dimensional acoustic wave equation has been fully mathematically revised analyzed and tested the naid method is an alternative numerical modeling method for generating synthetic seismograms the acoustic wave equation is first transformed into a system of first order ordinary differential equations odes with respect to time variable t and then directly integrated at a small time interval of tn tn 1 to obtain semi discrete ordinary differential equations the integral kernel is expanded into a truncated taylor series to which the integration operator is explicitly applied the high order temporal derivatives involved in the integral kernel are replaced by high order spatial derivatives which then are approximately calculated as a weighted linear combination of the displacement the particle velocity and their spatial gradients in this article we investigate the theoretical properties of the revised naid method including the discrete error and the stability criteria numerical results for constant and layered velocity models show that comparing to the lax wendroff correction lwc scheme and the staggered grid finite difference method the naid method can effectively suppress the numerical dispersion and source noises caused by the discretization of the acoustic wave equation with too coarse spatial grids or when models have strong velocity contrasts between adjacent layers the proposed naid method has been applied in computing the acoustic wavefields for two heterogeneous models the corner edge model and the marmousi model promising numerical results illustrate that the naid method provides an encouraging tool for large scale and complex wave simulation and inversion problems based on the acoustic equation

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

generalized pascal matrix and recurrence sequences this paper gives a product formula of the generalized pascal matrix n x y from this getting the striking simplicity of the powers of n x y furthermore this paper defines the iegp matrix n x y it is shown that not only can n x y be factorized by special summation matrices but also has the closely relation with generalized fibonacci sequences wn x y an explicit diagonalization of n x y is given

equivalent accuracy at a fraction of the cost overcoming temporal dispersion abstractnumerical dispersion in finite difference fd modeling produces coherent artifacts severely constraining the resolution of advanced imaging and inversion techniques conventionally numerical dispersion is reduced by increasing the order of accuracy of the fd operators and we resign ourselves to paying the high computational cost that is incurred assuming no spatial dispersion we have found that fd time dispersion is independent of the medium velocity and the spatial grid for propagation and only depends on the time stepping scheme and the propagation time based on this observation we have devised postpropagation filters to collapse the time dispersion effect of fd modeling our dispersion correction filters are designed by comparing the input waveform with dispersive waveforms obtained by 1d forward modeling these filters are then applied on multidimensional shot records to eliminate the time dispersion by two schemes 1 stationary filtering plus interpolation and 2 nonstationary fil

matrices related to the bell polynomials abstract in this paper we study the matrices related to the partial exponential bell polynomials and those related to the bell polynomials with respect to as a result the factorizations of these matrices are obtained which give unified approaches to the factorizations of many lower triangular matrices moreover some combinatorial identities are also derived from the corresponding matrix representations

catalan matrix and related combinatorial identities we introduce the notion of the catalan matrix c n x whose non zero elements are expressions which contain the catalan numbers arranged into a lower triangular toeplitz matrix inverse of the catalan matrix is derived correlations between the matrix c n x and the generalized pascal matrix are considered some combinatorial identities involving catalan numbers binomial coefficients and the generalized hypergeometric function are derived using these correlations moreover an additional explicit representation of the catalan number as well as an explicit representation of the sum of the first m catalan numbers are given

on several q special matrices including the q bernoulli and q euler matrices abstract in the spirit of our earlier articles 12 14 11 and our work 13 we define two dual q bernoulli polynomials with corresponding vector and matrix forms following aceto trigiante 1 the q l matrix the indefinite q integral of the q pascal matrix is the link between the q cauchy and the q bernoulli matrix the q analogue of the bernoulli complementary argument theorem can be expressed in matrix form through the diagonal a n matrix for the q euler polynomials corresponding results are obtained the umbral calculus for generating functions of q appell polynomials is shown to be equivalent to a transform method which maps polynomials to matrices a true q analogue of arponen 6 this is manifested by the vein 21 matrix which occurs as the transform of the q difference operator the aceto trigiante shifted q bernoulli matrix has a simple connection to the q bernoulli arponen matrix through the q pascal matrix we reintroduce certain q stirling numbers z q from 12 which will be needed for the polynomial matrix definitions

the pseudospectral method comparisons with finite differences for the elastic wave equation the pseudospectral or fourier method has been used recently by several investigators for forward seismic modeling the method is introduced here in two different ways as a limit of finite differences of increasing orders and by trigonometric interpolation an argument based on spectral analysis of a model equation shows that the pseudospectral method for the accuracies and integration times typical of forward elastic seismic modeling may require in each space dimension as little as a quarter the number of grid points compared to a fourth order finite difference scheme and one sixteenth the number of points as a second order finite difference scheme for the total number of points in two dimensions these factors become 1 16 and 1 256 respectively in three dimensions they become 1 64 and 1 4 096 respectively in a series of test calculations on the two dimensional elastic wave equation only minor degradations are found in cases with variable coefficients and discontinuous interfaces

notes on particular symmetric polynomials with applications this paper presents some applications using several properties of three important symmetric polynomials elementary symmetric polynomials complete symmetric polynomials and the power sum symmetric polynomials the applications includes a simple proof of el mikkawy conjecture in m e a el mikkawy appl math comput 146 2003 759 769 and a very easy proof of the newton girard formula in addition a generalization of stirling numbers is obtained

matrices determined by a linear recurrence relation among entries abstract a matrix a a ij is called a 7 matrix if its entries satisfy the recurrence relation a i 1 j 1 a i 1 j a ij where are fixed numbers a 7 matrix is completely determined by its first row and first column in this paper we determine the structure of 7 matrices and investigate the sequences represented by columns of infinite 7 matrices

generalized leibniz functional matrices and factorizations of some well known matrices in this paper we introduce the generalized leibniz functional matrices and study some algebraic properties of such matrices to demonstrate applications of these properties we derive several novel factorization forms of some well known matrices such as the complete symmetric polynomial matrix and the elementary symmetric polynomial matrix in addition by applying factorizations of the generalized leibniz functional matrices we redevelop the known results of factorizations of stirling matrices of the first and second kind and the generalized pascal matrix

removing false images in reverse time migration the concept of de primary abstractprimary reflections like multiples can generate false images when reverse time migration rtm algorithms are used the false images are formed by the zero lag correlation of the source wavefields and primary reflections which are propagated by the migration algorithm along nonphysical paths these paths are generated by strong velocity gradients or reflection interfaces when the two way wave equation is used conceptually this type of artifact can be removed by separating up and downgoing waves but such separation may be impractical because it often requires storing the entire wavefields at all time steps we have developed a de primary rtm method in which such separation can be accomplished without saving the wavefields the computational cost of the proposed method was only approximately 33 higher than that of conventional rtm algorithms using field and synthetic data sets we have demonstrated the existence of this endemic rtm problem and verified the effectiveness of the de primary rt

an optimized wave equation for seismic modeling and reverse time migration the acoustic wave equation has been widely used for the modeling and reverse time migration of seismic data numerical implementation of this equation via finite difference techniques has established itself as a valuable approach and has long been a favored choice in the industry to ensure quality results accurate approximations are required for spatial and time derivatives traditionally they are achieved numerically by using either relatively very fine computation grids or very long finite difference operators otherwise the numerical error known as numerical dispersion is present in the data and contaminates the signals however either approach will result in a considerable increase in the computational cost a simple and computationally low cost modification to the standard acoustic wave equation is presented to suppress numerical dispersion this dispersion attenuator is one analogy of the antialiasing operator widely applied in kirchhoff migration when the new wave equation is solved numerically using finite difference schemes numerical dispersion in the original wave equation is attenuated significantly leading to a much more accurate finite difference scheme with little additional computational cost numerical tests on both synthetic and field data sets in both two and three dimensions demonstrate that the optimized wave equation dramatically improves the image quality by successfully attenuating dispersive noise the adaptive application of this new wave equation only increases the computational cost slightly

stirling matrix via pascal matrix abstract the pascal type matrices obtained from the stirling numbers of the first kind s n k and of the second kind s n k are studied respectively it is shown that these matrices can be factorized by the pascal matrices also the ldu factorization of a vandermonde matrix of the form v n x x 1 x n 1 for any real number x is obtained furthermore some well known combinatorial identities are obtained from the matrix representation of the stirling numbers and these matrices are generalized in one or two variables

the algebraic properties of the generalized pascal functional matrices associated with the exponential families abstract in this paper pascal matrices are generalized to functional matrices by using the exponential families and the restricted sequences of polynomials of binomial type some algebraic properties of the pascal functional matrices are presented and proved in addition several important combinatorial identities are obtained

q double riordan matrices abstract in this paper the concept of q double riordan matrices is given by using binary operations q 2 and 1 q 2 we give a q analogue of the fundamental theorem of double riordan matrices and study the structure of the entries of the multiplications of q double riordan matrices as an application q analogue of the pascal fibonacci array can be obtained by a q double riordan representation

symmetric bernoulli matrix and its cholesky decomposition abstract this study associates the symmetric bernoulli matrix with various combinatorial matrices such as pascal vandermonde and stirling matrices of the first and second kind including catalan numbers by way of dual matrices of the matrix a recurrence relation related to the entries of the cholesky factor of a symmetric positive definite matrix is presented which sheds new light on examining positive definiteness and cholesky s method of a symmetric matrix this relation is applied to give a necessary and sufficient condition for the positive definiteness of a symmetric matrix which leads to a slightly modified form of the outer product formulation for cholesky s method of a symmetric matrix

divided differences and a general explicit formula for sequences of mansour mulay shattuck divided differences is a very important tool often used both in theory and in practice for the approximation of functions in this paper we employ divided differences with repetitions to find a general explicit formula for the sequences satisfying mansour mulay shattuck s two term recurrence with arbitrary b i i 0

a matrix approach to some identities involving sheffer polynomial sequences a simple but elegant method was adopted in youn and yang 2011 in order to derive a differential equation and recursive formulas for sheffer polynomials namely they used the so called the generalized pascal functional matrix of an analytic function and the wronskian matrix of several analytic functions in this paper we will use their method to find some identities satisfied by sheffer polynomials

high order finite differences and the pseudospectral method on staggered grids finite difference approximations for the first derivative valid halfway between equidistant gridpoints are in general much more accurate than the corresponding approximations which are valid at gridpoints the pseudospectral fourier method can be viewed as the limit of finite difference approximations when the order of accuracy tends to infinity a fundamentally different and more accurate pseudospectral method is obtained from the halfway approximations this study derives these and related methods and discusses their accuracies

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

a matrix approach to sheffer polynomials abstract this paper deals with a unified matrix representation for the sheffer polynomials the core of the proposed approach is the so called creation matrix a special subdiagonal matrix having as nonzero entries positive integer numbers whose exponential coincides with the well known pascal matrix in fact sheffer polynomials may be expressed in terms of two matrices both connected to it as we will show one of them is strictly related to appell polynomials while the other is linked to a binomial type sequence consequently different types of sheffer polynomials correspond to different choices of these two matrices

kernel optimum nearly analytical discretization kond algorithm applied to parabolic and hyperbolic equations abstract two applications of the kernel optimum nearly analytical discretization kond algorithm to the parabolic and the hyperbolic type equations are presented in detail to lead to novel numerical schemes with very high numerical accuracy it is demonstrated numerically that the two dimensional kond p scheme for the parabolic type yields much less numerical error by over 2 3 orders measured quantitatively by the root mean square deviation from analytical solutions and reduces the cpu time to about 1 5 for a common numerical accuracy compared with the conventional explicit scheme of reference it is also demonstrated numerically that the kond h scheme for the hyperbolic type yields much less diffusive error and has fairly high stability for both of the linear and the nonlinear wave propagations compared with other conventional schemes origins of numerical errors in data processing in general numerical schemes are discussed

some identities on catalan numbers and hypergeometric functions via catalan matrix power abstract in this paper we use the catalan matrix power as a tool for deriving identities involving catalan numbers and hypergeometric functions for that purpose we extend earlier investigated relations between the catalan matrix and the pascal matrix by inserting the catalan matrix power and particulary the squared catalan matrix in those relations we also pay attention to some relations between catalan matrix powers of different degrees which allows us to derive the simplification formula for hypergeometric function 3 f 2 as well as the simplification formula for the product of the catalan number and the hypergeometric function 3 f 2 some identities involving catalan numbers proved by the non matrix approach are also given

combinatorial and hypergeometric identities via the legendre polynomials a computational approach in this paper old and new combinatorial and hypergeometric identities are obtained via the legendre polynomials as one of the most commonly used orthogonal polynomials the results are tested using maple computer algebra system cas

functions and jordan canonical forms of riordan matrices abstract this paper collects results about riordan arrays in the framework of matrix functions actually the following methodology applies to any square matrix m m with exactly one eigenvalue of algebraic multiplicity m n generalized lagrange bases are used to construct hermite polynomials that interpolate a family of functions moreover we show a parallel application of such functions via jordan canonical forms and case studies are given

matrix factorization of the n n shift bell matrix abstract let b n b n k n k 0 be the bell matrix define the n n shift bell matrix p n k by p n k i j b k i 1 k j 1 for i j 1 2 n and k 0 1 2 in this paper matrix factorizations of the n n shift bell matrix and the n n generalized riordan matrix are studied as a result many lower triangular matrices related to bell polynomials can be factorized by the corresponding matrices and some identities are derived from the matrix representations in addition some harmonic number identities are obtained from the riordan array method

the linear algebra of the pascal matrix abstract pascal s triangle can be represented as a square matrix in two basically different ways as a lower triangular matrix pn or as a full symmetric matrix qn it has been found that the pn pnt is the cholesky factorization of qn pn can be factorized by special summation matrices it can be shown that the inverses of these matrices are the operators which perform the gaussian elimination steps for calculating cholesky s factorization by applying linear algebra we produce combinatorial identities and an existence theorem for diophantine equation systems finally an explicit formula for the sum of the kth powers is given

symmetric functions and the vandermonde matrix this work deduces the lower and the upper triangular factors of the inverse of the vandermonde matrix using symmetric functions and combinatorial identities the l and u matrices are in turn factored as bidiagonal matrices the elements of the upper triangular matrices in both the vandermonde matrix and its inverse are obtained recursively the particular value x i 1 q qi 1 in the indeterminates of the vandermonde matrix is investigated and it leads to q binomial and q stirling matrices it is also shown that q stirling matrices may be obtained from the pascal matrix

generalized leibniz functional matrices and divided difference form of the lagrange burmann formula abstract in this paper we derive two versions of divided difference form of the lagrange burmann expansion formula the first version expresses all coefficients by a form of a determinant by means of the factorizations of the generalized leibniz functional matrices with respect to a sequence we obtain the second version with coefficients represented by a finite sum some new matrix inversions are also investigated

the g spectral estimator abstract in this article a modified definition of the g spectral estimator is given it is shown that the resulting estimator is a method of moments autoregressive moving average arma spectral estimator that does not require an estimate of the moving average parameters as a result a new formula for the power spectrum of an arma process is given that does not explicitly involve the moving average ma parameters this formula then leads to a closed form expression for the ma parameters and their corresponding moment estimators

forward modeling by a fourier method a fourier or pseudospectral forward modeling algorithm for solving the two dimensional acoustic wave equation is presented the method utilizes a spatial numerical grid to calculate spatial derivatives by the fast fourier transform time derivatives which appear in the wave equation are calculated by second order differcncing the scheme requires fewer grid points than finite diffcrcnce methods to achieve the same accuracy it is therefore believed that the fourier method will prove more efficient than finitedifference methods especially when dealing with threedimensional models the fourier forward modeling method was tested against two problems a single layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling the numerical results agreed with both the analytic and physical model results furthermore the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account

other manifestations of the schur complement abstract many series and sequence transformations used in numerical analysis are defined as ratios of determinants they are implemented by recursive algorithms based on determinantal identities the aim of this paper is to study the connections of these transformations and algorithms with the schur complement of a matrix the schur s formula is extended to the vector case thus providing the same treatment for vector sequence transformations and the corresponding recursive algorithms thanks to these connections particular rules for avoiding division by zero or numerical instability are obtained for these algorithms some fixed point methods and continued fractions also fit in this framework

an extension of the levin sidi class of nonlinear transformations for accelerating convergence of infinite integrals and series in this paper we extend levin and sidi s nonlinear d and d transformations for accelerating convergence of infinite integrals and series the simple extension is made by viewing all these transformations as generalized jackknifes the new transformations appear to be more accurate in general and are as easily implemented as d and d transformations the new transformations have also some computational advantages as a result more accurate approximations can be obtained in many cases as is shown in the examples which are included

groups of generalized pascal matrices abstract we present a new approach to the study of generalized pascal matrices that yields general results about group structures and explicit formulas for powers and inverses of the pascal matrices shows how the groups of pascal matrices are related to groups of 2 2 matrices and clarifies the way in which symmetries and duality relate to the algebraic aspects of the theory we consider bilaterally infinite generalized pascal matrices obtained as the matrix representations of certain linear operators on spaces of formal laurent series their column generating functions are simple rational functions closely related to the linear fractional maps of the complex plane we obtain explicit expressions for the inverses and the powers of the pascal matrices that generalize most of the analogous results in the literature

a novel extension of the lagrange burmann expansion formula abstract aking use of factorization and inverse of the leibniz functional matrix in yang generalized leibniz functional matrices and factorizations of some well known matrices linear algebra appl 430 2009 511 531 we establish a formal extension of the classical lagrange burmann expansion formula in connection with this result some new matrix inversions are also investigated

identities induced by riordan arrays historically there exist two versions of the riordan array concept the older one better known as recursive matrix consists of bi infinite matrices d n k n k is an element of z k n implies d n k 0 deals with formal laurent series and has been mainly used to study algebraic properties of such matrices the more recent version consists of infinite lower triangular arrays d n k n k is an element of n k n implies d n k 0 deals with formal power series and has been used to study combinatorial problems here we show that every riordan array induces two characteristic combinatorial sums in three parameters n k m is an element of z these parameters can he specialized and generate an indefinite number of other combinatorial identities which are valid in the hi infinite realm of recursive matrices

the linear algebra of the generalized pascal functional matrix abstract in this paper we generalize pascal s matrix by defining the polynomials factorial binomial then using this generalization we introduce a two variable pascal s matrix and state its related theorems and prove them finally we introduce pascal s functional matrix associated with a sequence a a n n 0 and obtain several important combinatorial identities

remarks on two symmetric polynomials and some matrices by means of complete symmetric polynomials this paper gives a new proof for the vandermonde determinant formula another alternative proof for this formula is obtained via the collocation matrices it also gives a generalized relationship between the vandermonde the pascal and the stirling matrices a new approach to obtain the explicit inverse of the vandermonde matrix is investigated closed form expressions for some vandermonde related determinants are obtained

extended symmetric pascal matrices via hypergeometric functions in this paper we give a matrix representation of the hypergeometric functions of the type 2f 1 a b c x as a result we obtain a connection between the hypergeometric functions the legendre polynomials and the delannoy numbers moreover it is shown that each entry of p n x y p n x y t can be represented by the hypergeometric functions where p n x y i j x i jy i j 2i 1j 1 is the extended generalized pascal matrix which is defined by zhang and liu linear algebra appl 271 1998 169

an extension of the generalized pascal matrix and its algebraic properties abstract the extended generalized pascal matrix can be represented in two different ways as a lower triangular matrix n x y or as a symmetric n x y these matrices generalize p n x q n x and r n x which are defined by zhang and by call and velleman a product formula for n x y has been found which generalizes the result of call and velleman it is shown that not only can n x y be factorized by special summation but also n x y as q n xy s t y 1 x or n x y p n t y x finally the inverse of n x y and the values of det n x y det n 1 x y det n x y and det n 1 x y are given

general identities on bell polynomials the exponential partial bell polynomials are polynomials in an infinite number of variables x 1 x 2 and it is well known that some special combinatorial sequences e g stirling numbers of both kinds lah numbers and idempotent numbers can be obtained from the bell polynomials in this paper we study these polynomials by making appropriate choices of the variables x 1 x 2 which are related to associated sequences binomial sequences and sheffer sequences as a consequence many general identities on bell polynomials are proposed from these general identities we can obtain series of identities on bell polynomials it can also be found that many results presented before are special cases of the general identities of this paper

dispersion relation preserving stereo modeling method beyond nyquist frequency for acoustic wave equation abstractaccording to nyquist shannon sampling theory two sampling points of the displacements per minimal wavelength are required to reconstruct a wavefield however high frequency components above the nyquist frequency can be theoretically recovered by introducing spatial derivatives of the displacements we have developed a new finite difference scheme for solving the acoustic equation named the dispersion relation preserving stereo modeling method which is practical for seismic forward modeling beyond the nyquist frequency the key idea in this model is to approximate the high order spatial derivatives with wavenumber domain optimization in wavefields represented by the wave displacements and their gradients we investigated the theoretical properties of this method including the absolute error of spatial operators numerical error stability criterion and computational efficiency our results indicated that the method can effectively suppress the numerical dispersion on extremely coarse grids wi

implementing the jackknife the jackknife is a well known statistical procedure for bias reduction it is given in terms of a ratio of two determinants this ratio is exactly the same as in some sequence transformations used in numerical analysis for accelerating the convergence of a sequence by extrapolation for such a transformation a recursive algorithm for avoiding the computation of the determinants is known the so called e algorithm we show how properties of the jackknife follow from similar properties for the e algorithm and how the e algorithm can be used for implementing the jackknife

generalized pascal matrices generate classes closed under multiplication abstract classes of matrices which are the hadamard product of a fixed lower triangular generating matrix p and any toeplitz matrix are studied these classes are generalizations of the special case when a toeplitz matrix is generated by the vector of powers and p is either also toeplitz or a pascal triangle the matrix p which lead to the class being closed under matrix multiplication is fully characterized explicit formulae for inverses are derived and commutativity of products within each class proven examples using lower triangular matrices with binomial coefficients are given

generalized pascal functional matrix and its applications in this paper we introduce the generalized pascal functional matrix and show that the existing variations of pascal matrices are special cases of this generalization we study some algebraic properties of such generalized pascal functional matrices in addition we demonstrate a direct application of these properties by deriving several novel combinatorial identities and a nontraditional approach for lu decompositions of some well known matrices such as symmetric pascal matrices

simulation of acoustic wavefields in heterogeneous media a robust method for automatic suppression of numerical dispersion numerical dispersion limits the application of numerical simulation methods for solving the acoustic wave equation in largescalecomputation thenearlyanalyticdiscretemethodnadm and its improved version for suppressing numerical dispersion weredevelopedrecently thisnewmethodisarefinementoftwo previousmethodsandfurtherincreasestheabilityofsuppressing numerical dispersion for modeling acoustic wave propagation in 2d heterogeneous media which uses acoustic wave displacement particle velocity and their gradients to restructure the acoustic wavefield via the truncated taylor expansion and the high order interpolation approximate method for the method proposed weinvestigateitsimplementationandcompareitwith the higher order lax wendroff correction lwc scheme the original nearly analytic discrete method nadm and its improved version with regard to numerical dispersion computational costs and computer storage requirements the numerical dispersionrelationsprovidedbytherefinedalgorithmfor1dand 2d cases are analyzed as well as the numerical results obtained bythismethodagainsttheexactsolutionforthe2dacousticcase numerical results show that the refined method gives no visible numericaldispersionforverylargespatialgridincrements itcan simulatehigh frequencywavepropagationforagivengridinterval and automatically suppress the numerical dispersion when the acoustic wave equation is discretized when too few samples per wavelength are used or when models have large velocity contrasts unlike the high order lwc methods our present methodcansavesubstantialcomputationalcostsandmemoryrequirements because very large grid increments can be used the refined method can be used for the simulation of large scale wavefields

extended local rytov fourier migration method we develop a novel depth migration method termed the extended local rytov fourier elrf migration method it is based on the scalar wave equation and a local application of the rytov approximation within each extrapolation interval wavefields are fourier transformed back and forth between the frequency space and frequency wavenumber domains during wavefield extrapolation the lateral slowness variations are taken into account in the frequency space domain the method is efficient due to the use of a fast fourier transform algorithm under the small angle approximation the elrf method leads to the split step fourier ssf method that is unconditionally stable the elrf method and the extended local born fourier elbf method that we previously developed can handle wider propagation angles than the ssf method and account for the phase and amplitude changes due to the lateral variations of slowness whereas the ssf method only accounts for the phase changes the stability of the elrf method is controlled

extended bernoulli and stirling matrices and related combinatorial identities abstract in this paper we establish plenty of number theoretic and combinatoric identities involving generalized bernoulli polynomials and stirling numbers of both kinds which generalize various known identities these formulas are deduced from pascal type matrix representations of bernoulli and stirling numbers for this we define and factorize a modified pascal matrix corresponding to bernoulli and stirling cases

matrix approach to polynomials 2 abstract we continue our study of the structure initiated in t arponen a matrix approach to polynomials linear algebra appl 359 2003 181 196 our main emphasis is exploring further structure into the formalism introduced in t arponen a matrix approach to polynomials linear algebra appl 359 2003 181 196 this formalism reveals beautiful interplay between certain elementary operators and provides tools for example for checking handling and generalizing combinatorial identities as we show in examples in addition to that we discover a group structure among vandermonde matrices a fascinating diophantine equation and new proofs and generalizations of recently found related results

elimination of numerical dispersion in finite difference modeling and migration by flux corrected transport finite difference acoustic wave modeling and reverse time depth migration based on the full wave equation are general approaches that can take into account arbitrary variations in velocity and density and can handle turning waves as well however conventional finite difference methods for solving the acoustic or elastic wave equation suffer from numerical dispersion when too few samples per wavelength are used the flux corrected transport fct algorithm adapted from hydrodynamics reduces the numerical dispersion in finite difference wavefield continuation the flux correction procedure endeavors to incorporate diffusion into the wavefield continuation process only where needed to suppress the numerical dispersion incorporating the flux correction procedure in conventional finite difference modeling or reverse time migration can provide finite difference solutions with no numerical dispersion even for impulsive sources the fct correction which can be applied to finite difference approximations of any order in space and time is an efficient alternative to use for finite difference approximations of increasing order through demonstrations of modeling and migration on both synthetic and field data we show the benefits of the fct algorithm as well as its inability to fully recover resolution lost when the spatial sampling becomes too coarse

generalized higher order bernoulli number pairs and generalized stirling number pairs abstract from a delta series f t and its compositional inverse g t hsu defined the generalized stirling number pair s n k s n k in this paper we further define from f t and g t the generalized higher order bernoulli number pair b n z b n z making use of the bell polynomials the potential polynomials as well as the lagrange inversion formula we give some explicit expressions and recurrences of the generalized higher order bernoulli numbers present the relations between the generalized higher order bernoulli numbers of both kinds and the corresponding generalized stirling numbers of both kinds and study the relations between any two generalized higher order bernoulli numbers moreover we apply the general results to some special number pairs and obtain series of combinatorial identities it can be found that the introduction of generalized bernoulli number pair and generalized stirling number pair provides a unified approach to lots of sequences in mathematics and as a consequence many known results are special cases of ours

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

on solving linear systems of the pascal type in the current paper we study the pascal matrix of order n an algorithm is developed to find its inverse in an explicit form another algorithm for solving any linear system with coefficient matrix of this type is also developed an illustrative example for solving a linear system of the pascal type is given for the case n 6 the implementation of the two algorithms in maple maple v programming reference manual is presented

3d weak dispersion reverse time migration using a stereo modeling operator reliable 3d imaging is a required tool for developing models of complex geologic structures reverse time migration rtm as the most powerful depth imaging method has become the preferred imaging tool because of its ability to handle complex velocity models including steeply dipping interfaces and large velocity contrasts finite difference methods are among the most popular numerical approaches used for rtm however these methods often encounter a serious issue of numerical dispersion which is typically suppressed by reducing the grid interval of the propagation model resulting in large computation and memory requirements in addition even with small grid spacing numerical anisotropy may degrade images or worse provide images that appear to be focused but position events incorrectly recently stereo operators have been developed to approximate the partial differential operator in space these operators have been used to develop several weak dispersion and efficient stereo modeling methods that have been found to be superior to conventional algorithms in suppressing numerical dispersion and numerical anisotropy we generalized one stereo modeling method fourth order nearly analytic central difference nacd from 2d to 3d and applied it to 3d rtm the rtm results for the 3d seg eage phase a classic data set 1 and the seg advanced modeling project model demonstrated that even when using a large grid size the nacd method can handle very complex velocity models and produced better images than can be obtained using the fourth order and eighth order lax wendroff correction lwc schemes we also applied 3d nacd and fourth order lwc to a field data set and illustrated significant improvements in terms of structure imaging horizon layer continuity and positioning we also investigated numerical dispersion and found that not only does the nacd method have superior dispersion characteristics but also that the angular variation of dispersion is significantly less than for lwc

the factorization of block matrices with generalized geometric progression rows abstract generalized geometric progression gp block matrices are introduced and it is shown that such matrices can be factored as the product of one lower triangular matrix and several upper triangular reduced pascal matrices p k x which were introduced by cheon and kim the determinant formula for any gp block matrix follows readily from this factorization this lu factorization and determinant formula are a generalization of results presented by yang and leida as direct applications of the new results we rederive factorizations of the extended generalized symmetric pascal matrix introduced by zhang and liu and the vandermonde matrix in addition determinants of three types of generalized vandermonde matrices are immediate consequences of our main theorem

theoretical analysis of the stability for extended trapezoidal rules abstract in this paper a rigorous analysis of the linear stability for two classes of methods both generalizing the trapezoidal rule is done the information on the coefficients of the methods useful for the analysis are obtained by an extensive use of properties of the pascal matrix some of which are derived in this work

symmetric polynomials pascal matrices and stirling matrices we consider lower triangular matrices consisting of symmetric polynomials and we show how to factorize and invert them since binomial coefficients and stirling numbers can be represented in terms of symmetric polynomials these results contain factorizations and inverses of pascal and stirling matrices as special cases this work generalizes that of several other authors on pascal and stirling matrices

factorial stirling matrix and related combinatorial sequences abstract in this paper some relationships between the stirling matrix the vandermonde matrix the benoulli numbers and the eulerian numbers are studied from a matrix representation of k s n k which will be called the factorial stirling matrix where s n k are the stirling numbers of the second kind as a consequence a number of interesting and useful identities are obtained

anisotropic wave propagation through finite difference grids an algorithm is presented to solve the elastic wave equation by replacing the partial differentials with finite differences it enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models the space derivatives are calculated using discrete convolution sums while the time derivatives are replaced by a truncated taylor expansion a centered finite difference scheme in cartesian coordinates is used for the space derivatives leading to staggered grids the use of finite difference approximations to the partial derivatives results in a frequency dependent error in the group and phase velocities of waves for anisotropic media the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the hook sum this interpolation induces an additional error in the wave properties the overall error depends on the precision of the derivative and interpolation operators the anisotropic symmetry system its orientation and the degree of anisotropy the dispersion relation for the homogeneous case was derived for the proposed scheme since we use a general description of convolution sums to describe the finite difference operators the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model in particular phase and group velocities of the three wave types can be determined in any direction we demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed

third order symplectic integration method with inverse time dispersion transform for long term simulation the symplectic integration method is popular in high accuracy numerical simulations when discretizing temporal derivatives however it still suffers from time dispersion error when the temporal interval is coarse especially for long term simulations and large scale models we employ the inverse time dispersion transform itdt to the third order symplectic integration method to reduce the time dispersion error first we adopt the pseudospectral algorithm for the spatial discretization and the third order symplectic integration method for the temporal discretization then we apply the itdt to eliminate time dispersion error from the synthetic data as a post processing method the itdt can be easily cascaded in traditional numerical simulations we implement the itdt in one typical exiting third order symplectic scheme and compare its performances with the performances of the conventional second order scheme and the rapid expansion method theoretical analyses and numerical experiments show that the itdt can significantly reduce the time dispersion error especially for long travel times the implementation of the itdt requires some additional computations on correcting the time dispersion error but it allows us to use the maximum temporal interval under stability conditions thus its final computational efficiency would be higher than that of the traditional symplectic integration method for long term simulations with the aid of the itdt we can obtain much more accurate simulation results but with a lower computational cost we use the inverse time dispersion transform to the 3rd order symplectic integration it can greatly reduce the time dispersion error especially for long travel times it is superior to the conventional 2nd and high order finite difference methods it allows the maximum temporal interval under stability conditions thus is efficient we can obtain more accurate simulation results but with a lower computational cost

characterization and construction of classical orthogonal polynomials using a matrix approach abstract we introduce a new algebraic approach for the study of orthogonal polynomial sequences we use simple properties of infinite matrices to obtain some basic results in the theory of orthogonal polynomial sequences including several characterizations of the classical sequences some of which are equivalent to the ones obtained by bochner maroni and al salam and chihara we also obtain explicit formulas for the coefficients of the recurrence relations of all the classical orthogonal polynomials in terms of four parameters the polynomial coefficients and the eigenvalues that appear in bochner s differential equation are also found explicitly

viscoelastic finite difference modeling real earth media disperse and attenuate propagating mechanical waves this anelastic behavior can be described well by a viscoelastic model we have developed a finite difference simulator to model wave propagation in viscoelastic media the finite difference method was chosen in favor of other methods for several reasons finite difference codes are more portable than for example pseudospectral codes moreover finite difference schemes provide a convenient environment in which to define complicated boundaries a staggered scheme of second order accuracy in time and fourth order accuracy in space appears to be optimally efficient because of intrinsic dispersion no fixed grid points per wavelength rule can be given instead we present tables which enable a choice of grid parameters for a given level of accuracy since the scheme models energy absorption natural and efficient absorbing boundaries may be implemented merely by changing the parameters near the grid boundary the viscoelastic scheme is

letter to the editor a new algorithm for linear systems of the pascal type in this paper we give an algorithm for solving linear systems of the pascal matrices the method is based on the explicit factorization of the pascal matrices the algorithm costs no multiplications and o n 2 additions the linear systems of the generalized pascal matrices are also considered some examples are given

pascal k eliminated functional matrix and its property abstract in this paper we shall first define the pascal k eliminated functional matrix for two variables over an arbitrary field f 1 8 then using the previous results we obtain several interesting combinatorial identities we also investigate the relationship between these matrices and cesaro matrices 9 finally we give an affirmative answer to an open problem raised in 1 for minimal polynomial of pascal k eliminated functional matrix over the field z p

an optimal nearly analytic discrete weighted runge kutta discontinuous galerkin hybrid method for acoustic wavefield modeling abstractthe newly developed optimal nearly analytic discrete onad and the weighted runge kutta discontinuous galerkin wrkdg methods can effectively suppress the numerical dispersion caused by discretizing wave equations but it is difficult for onad to implement on flexible meshes whereas the wrkdg has high computational cost for wavefield simulations we have developed a new hybrid algorithm by combining the onad method with the wrkdg method in this hybrid algorithm the computational domain was split into several subdomains in which the subdomain for the onad method used regular cartesian grids whereas the subdomain for the wrkdg method used triangular grids the hybrid method was at least third order spatially accurate we have applied the proposed method to simulate the scalar wavefields for different models including a homogeneous model a rough topography model a fracture model and a cave model the numerical results found that the hybrid method can deal with complicated geometrical struc

generation of finite difference formulas on arbitrarily spaced grids on etablit des recurrences simples pour le calcul des poids dans les formules aux differences finies compactes pour les derivees de tous ordres avec une precision d ordre arbitraire sur des grilles a une dimension d espacement arbitraire

the cyclic groups via the pascal matrices and the generalized pascal matrices abstract in this paper given a positive integer m we consider the multiplicative order of upper and lower triangular matrices and symmetric matrices derived from pascal s triangle when read modulo m we also consider the corresponding problems when the binomial coefficients i j are replaced by x i j i j and x i j i j where x is an integer satisfying certain divisibility hypotheses we also offer some open conjectures on these orders

dual operators and lagrange inversion in several variables the original lagrange inversion formula which gives explicitly the inverse under composition of a formal series was obtained by lagrange in 1770 through formal computations involving logarithms of inlmite products see lagrange 121 there exists an extensive literature on various versions of the inversion formula obtained by algebraic analytic and combinatorial methods comtet 3 gives a survey and an ample bibliography during the last ten years there have appeared several papers on generalizations and new proofs of the inversion formula in 1974 abhyankar obtained an inversion formula using algebraic methods see bass connell and wright 11 several authors have related lagrange inversion with rota s finite operator calculus 151 for example joni 8 garsia and joni 4 hofhauer 7 and roman and rota 14 joyal and labelle used a combinatorial theory of formal series to obtain inversion formulas see 9 l i and viskov 161 used ideas from lie algebra theory to prove formulas similar to abhyankar s in the present paper we use operator methods to obtain inversion formulas in several variables that generalize the formulas of abhyankar joni and viskov we get our results studying an algebra with involution of linear operators on the algebra of formal laurent series in several indeterminates we also show that by means of an anti isomorphism of operator algebras which we call the operator bore1 transform the study of finite operator calculus can be reduced to the study of certain groups of operators on the algebra of formal laurent series our approach may be considered as an algebraic analogue of the complex variable methods used by good 5 to obtain the lagrange inversion formula as a consequence of the change of variables theorem for the