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

explicit inverse of a generalized vandermonde matrix in this paper the author gives an explicit closed form expression for the nxn inverse matrix v g k n 1 of the generalized nxn vandermonde matrix v g k n by using the elementary symmetric functions symbolic and numerical results are presented

the numerical rank of krylov matrices abstract in this paper we expose interesting features of krylov matrices and vandermonde matrices let s be a large symmetric matrix of order n let x be a real n vector and let k l denote the related n l krylov matrix the question considered in this paper is how the numerical rank of k l grows as l increases the key for answering this question lies in the close link between k l and the n l vandermonde matrix which is generated by the eigenvalues of s analysis of large vandermonde matrices shows that the numerical rank is expected to remain much smaller than l the proof is based on partition theorems and clustering theorems the basic tool is a new matrix equality the vandermonde pascal toeplitz equality the actual numerical rank of a vandermonde or krylov matrix depends on the distribution of the eigenvalues but often the rank is remarkable small numerical experiments illustrate these points the observation that the numerical rank of krylov matrices stays small has important practical consequences

a decomposition method for solving the nonlinear klein gordon equation in this paper adomian s decomposition scheme is presented as an alternate method for solving the nonlinear klein gordon equation the method is demonstrated by several examples comparing the scheme with existing collocation finite difference and finite element techniques shows that the present approach is highly accurate and converges rapidly

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

conditioning and accurate computations with pascal matrices a result on the ill conditioning of pascal matrices is proved however it is shown that bidiagonal factorizations of pascal matrices can be applied to perform computations with high relative accuracy

a gautschi type method for oscillatory second order differential equations we study a numerical method for second order differential equations in which high frequency oscillations are generated by a linear part for example semilinear wave equations are of this type the numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly we prove that the method admits second order error bounds which are independent of the product of the step size with the frequencies our analysis also provides new insight into the m ollified impulse method of garcia archilla sanz serna and skeel we include results of numerical experiments with the sine gordon equation

on a connection between symmetric polynomials generalized stirling numbers and the newton general divided difference interpolation polynomial this paper gives a connection between symmetric polynomials generalized stirling numbers and the newton general divided difference interpolation polynomial the generalized stirling numbers of the first and second kind denoted s n k a and s n k a respectively are given symbolically for the case n 5 as an illustrative example by using maple programming maple v r3 programming reference manual these numbers depend on n k and n distinct values a 1 a 2 a n the ordinary stirling numbers s n k and s n k may be obtained as special cases by taking a i i 1 i 1 1 n

uniform and optimal error estimates of an exponential wave integrator sine pseudospectral method for the nonlinear schrodinger equation with wave operator we propose an exponential wave integrator sine pseudospectral ewi sp method for the nonlinear schrodinger equation nls with wave operator nlsw and carry out rigorous error analysis the nlsw is nls perturbed by the wave operator with strength described by a dimensionless parameter varepsilon in 0 1 as varepsilon to0 the nlsw converges to the nls and for the small perturbation i e 0 varepsilon ll1 the solution of the nlsw differs from that of the nls with a function oscillating in time with o varepsilon 2 wavelength at o varepsilon 2 and o varepsilon 4 amplitudes for ill prepared and well prepared initial data respectively this rapid oscillation in time brings significant difficulties in designing and analyzing numerical methods with error bounds uniformly in varepsilon in this work we show that the proposed ewi sp possesses the optimal uniform error bounds at o tau 2 and o tau in tau time step for well prepared initial data and ill prepared initial

on error bounds for the gautschi type exponential integrator applied to oscillatory second order differential equations this paper studies a numerical method for second order oscillatory differential equations in which high frequency oscillations are generated by a linear time and or solution dependent part for constant linear part it is known that the method allows second order error bounds independent of the product of the step size with the frequencies and is therefore a long time step method most real world problems are not of that kind and it is important to study more general equations the analysis in this paper shows that one obtains second order error bounds even in the case of a time and or solution dependent linear part if the matrix is evaluated at averaged positions

a fast algorithm for solving linear systems of the pascal type abstract in this paper we present a fast algorithm of the complexity o n log n for solving the linear systems with coefficient matrix of the pascal type which is faster than the algorithm of the complexity o n 2 given in the paper m e a el mikkawy on solving linear systems of the pascal type applied mathematics and computation 136 2003 195 202 an application in solving non homogeneous differential equation with constant coefficients is given

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 exponential wave integrator sine pseudospectral method for the klein gordon zakharov system an exponential wave integrator sine pseudospectral method is presented and analyzed for discretizing the klein gordon zakharov kgz system with two dimensionless parameters 0 varepsilon le 1 and 0 gamma le 1 which are inversely proportional to the plasma frequency and the speed of sound respectively the main idea in the numerical method is to apply the sine pseudospectral discretization for spatial derivatives followed by using an exponential wave integrator for temporal derivatives in phase space the method is explicit symmetric in time and it is of spectral accuracy in space and second order accuracy in time for any fixed varepsilon varepsilon 0 and gamma gamma 0 in the mathcal o 1 plasma frequency and speed of sound regime i e varepsilon mathcal o 1 and gamma mathcal o 1 we establish rigorously error estimates for the numerical method in the energy space h 1 times l 2 we also study numerically the resolution of the method in the simultaneous high plasma fr

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

commentary on an open question abstract in this commentary we give the triangular recurrence relations for the elements of the matrix t t i j l 1 d 1 s 1 and its inverse t 1 t i j s 2 dl 1 1 respectively which have also answered the open question put forward by el mikkawy moreover with the recurrences a computational maple program is written to generate the elements of t and t 1

finite difference calculus invariant structure of a class of algorithms for the nonlinear klein gordon equation in a previous work the authors have presented a formalism for deriving systematically invariant symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities this formalism is herein cast in the context of exact finite difference calculus the algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus in the same sense as that of the corresponding differential equation being derivable from its associated energy function a conserved quantity a clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities which are the consistent counterpart of the invariant quantities in the continuous case results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism and that preserve energy or linear momentum are discussed in the context of finite difference calculus some

an optimal signal design for band limited asynchronous ds cdma communications an optimal signal design for band limited asynchronous direct sequence code division multiple access ds cdma communications with aperiodic random spreading sequences and a conventional matched filter receiver is considered in an additive white gaussian noise awgn channel with bandwidth defined in the strict sense two optimization problems are solved under finite bandwidth and zero interchip interference constraints first a chip waveform optimization is performed given the system bandwidth the data symbol transmission rate and the processing gain a technique to characterize a band limited chip waveform with a finite number of parameters is developed and it is used to derive optimum chip waveforms which minimize the effect of multiple access interference mai for any energy and delay profile of users next a joint optimization of the processing gain and the chip waveform is performed given the system bandwidth and the data symbol transmission rate a sufficient condition for a system to have lower average probability of bit error for any energy profile is found and it is used to derive some design strategies it is shown that the flat spectrum pulse with the processing gain leading to zero excess bandwidth results in the minimum average probability of bit error design examples and numerical results are also provided

decompositions of a matrix by means of its dual matrices with applications abstract we introduce the notion of dual matrices of an infinite matrix a which are defined by the dual sequences of the rows of a and naturally connected to the pascal matrix p i j i j 0 1 2 we present the cholesky decomposition of the symmetric pascal matrix by means of its dual matrix decompositions of a vandermonde matrix are used to obtain variants of the lagrange interpolation polynomial of degree n that passes through the n 1 points i q i for i 0 1 n

error estimates in the energy space for a gautschi type integrator spectral discretization for the coupled nonlinear klein gordon equations we propose and analyze an efficient and accurate numerical method for solving the coupled nonlinear klein gordon equations the method is based on the application of a gautschi type exponential integrator in time combined with sine spectral discretization in space the main results achieved in this paper are the rigorous error estimates in the energy space h 1 h 1 for the proposed scheme numerical tests are reported and agree with the error estimates quite well