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For positive integers u and v, let Lu = [1 & 0 \\ u & 1] and Rv = [1 & v \\ 0 & 1 ]. Let Su,v be the monoid generated by Lu and Rv, and let Gu,v be the group generated by Lu and Rv. In this paper we expand on a characterization of matrices M = [a & b \\ c & d] in Sk,k and Gk,k when k ≥ 2 given by Esbelin and Gutan to Su,v when u,v ≥ 2 and Gu,v when u,v ≥ 3. We give a simple algorithmic way of determining if M is in Gu,v using a recursive function and the short continued fraction representation of b/d. [ABSTRACT FROM AUTHOR]

A recursive algorithm for the inverse factorization S-1=ZZ* of Hermitian positive definite matrices S is proposed. The inverse factorization is based on iterative refinement [A.M.N. Niklasson, Phys. Rev. B 70, 193102 (2004)] combined with a recursive decomposition of S. As the computational kernel is matrix-matrix multiplication, the algorithm can be parallelized and the computational effort increases linearly with system size for systems with sufficiently sparse matrices. Recent advances in network theory are used to find appropriate recursive decompositions. We show that optimization of the so-called network modularity results in an improved partitioning compared to other approaches. In particular, when the recursive inverse factorization is applied to overlap matrices of irregularly structured three-dimensional molecules. [ABSTRACT FROM AUTHOR]

Abstract W.O. Krawec (2012) [17] introduced a method of analyzing multi-player impartial games, and derived a recursive function capable of determining which of the n players has a winning strategy. The present paper is devoted to "Multi-player End-Nim games", abbreviated by ENim(N , n), assuming that the standard alliance matrix is adopted. The game values of ENim(N , n) are completely determined for three cases n > N + 1 , n = N + 1 and n = N. For the case n < N , by letting d = N − n ≥ 1 , the game values of ENim(n + d , n) are completely determined if n ≥ d + 4. The case 3 ≤ n ≤ d + 3 is more complicated, we present the game values for d = 1. [ABSTRACT FROM AUTHOR]

The inversion of extremely high order matrices has been a challenging task because of the limited processing and memory capacity of conventional computers. In a scenario in which the data does not fit in memory, it is worth to consider exchanging less memory usage for more processing time in order to enable the computation of the inverse which otherwise would be prohibitive. We propose a new algorithm to compute the inverse of block partitioned matrices with a reduced memory footprint. The algorithm works recursively to invert one block of a k  ×  k block matrix M , with k  ≥ 2, based on the successive splitting of M . It computes one block of the inverse at a time, in order to limit memory usage during the entire processing. Experimental results show that, despite increasing computational complexity, matrices that otherwise would exceed the memory-usage limit can be inverted using this technique. [ABSTRACT FROM AUTHOR]

This paper presents a modified Precise Integration Method (PIM) for long-time duration dynamic analysis. The fundamental solution and loading operator matrices in the first time substep are numerically computed employing a known unconditionally stable method (referred to as original method in this paper). By using efficient recursive algorithms to evaluate these matrices in the first time-step, the same numerical results as those using the original method with small time-step are obtained. The proposed method avoids the need of matrix inversion and numerical quadrature formulae, while the particular solution obtained has the same accuracy as that of the homogeneous solution. Through setting a proper value of the time substep, satisfactory accuracy and numerical dissipation can be achieved. [ABSTRACT FROM AUTHOR]

This article presents a technique for combining two doubly stochastic matrices with known spectra to create a new matrix. As application of this, we obtain a recursive method for constructing doubly stochastic matrices for the inverse eigenvalue problem and find new sufficient conditions for this problem. In addition, we improve the Soules' condition. [ABSTRACT FROM AUTHOR]

This article revisits and elaborates the significant role of positive geometry of momentum twistor Grassmannian for planar $$ \mathcal{N}=4 $$ SYM scattering amplitudes. First we establish the fundamentals of positive Grassmannian geometry for tree amplitudes, including the ubiquitous Plücker coordinates and the representation of reduced Grassmannian geometry. Then we formulate this subject, without making reference to on-shell diagrams and decorated permutations, around these four major facets: 1. Deriving the tree and 1-loop BCFW recursion relations solely from positivity, after introducing the simple building blocks called positive components for a positive matrix. 2. Applying Grassmannian geometry and Plücker coordinates to determine the signs of NMHV homological identities, which interconnect various Yangian invariants. It reveals that most of the signs are in fact the secret incarnation of the simple 6-term NMHV identity. 3. Deriving the stacking positivity relation, which is powerful for parameterizing matrix representatives in terms of positive variables in the d log form. It will be used with the reduced Grassmannian geometry representation to produce the positive matrix of a given geometric configuration, which is an independent approach besides the combinatoric way involving a sequence of BCFW bridges. 4. Introducing an elegant and highly refined formalism of BCFW recursion relation for tree amplitudes, which reveals its two-fold simplex-like structures. First, the BCFW contour in terms of (reduced) Grassmannian geometry representatives is delicately dissected into a triangle-shape sum, as only a small fraction of the sum needs to be explicitly identified. Second, this fraction can be further dissected, according to different growing modes with corresponding growing parameters. The growing modes possess the shapes of solid simplices of various dimensions, with which infinite number of BCFW cells can be entirely captured by the characteristic objects called fully-spanning cells. We find that for a given k, beyond n =4 k+1 there is no more new fully-spanning cell, which signifies the essential termination of the recursive growth of BCFW cells. As n increases beyond the termination point, the BCFW contour simply replicates itself according to the simplex-like patterns, which enables us to master all BCFW cells once for all without actually identifying most of them. [ABSTRACT FROM AUTHOR]

We develop a matrix bordering technique that can be applied to an irreducible spectrally arbitrary sign pattern to construct a higher order spectrally arbitrary sign pattern. This technique generalizes a recently developed triangle extension method. We describe recursive constructions of spectrally arbitrary patterns using our bordering technique, and show that a slight variation of this technique can be used to construct inertially arbitrary sign patterns. [ABSTRACT FROM AUTHOR]

An algorithmic approach for generating generalised Zernike polynomials by differential operators and connection matrices is proposed. This is done by introducing a new order of generalised Zernike polynomials such that it collects all the polynomials of the same total degree in a column vector. The connection matrices between these column vectors composed by the generalised Zernike polynomials and a family of polynomials generated by a Rodrigues formula are given explicitly. This yields a Rodrigues type formula for the generalised Zernike polynomials themselves with properly defined differential operators. Another consequence of our approach is the fact that the generalised Zernike polynomials obey a rather simple partial differential equation. We recall also how to define Hermite–Zernike polynomials. [ABSTRACT FROM AUTHOR]

A real square matrix A is called a sign-nonsingular (SNS) matrix if every matrix with the same sign pattern as A is not singular. An m × n matrix A with term rank m is called to have a nonzero signed row compound provided that each square submatrix of order m of A is an SNS-matrix or has an identically zero determinant. As generalizations of SNS-matrices, S ⁎ -matrices, and totally L-matrices, matrices with nonzero signed row compound have a close relationship with matrices with signed null-spaces which are applied to characterize linear systems with signed solutions. In this paper, matrices with nonzero signed row compound are characterized in terms of their signed bipartite graphs. Following these results, characterizations of matrices with signed null-spaces and full row term ranks in terms of their signed bipartite graphs are obtained too. The recursive structure of m × ( m + 2 ) row sign balanced (RSB) maximal ( 0 , 1 , − 1 ) -matrices with nonzero signed row compound (or with signed null-spaces) is also characterized. [ABSTRACT FROM AUTHOR]

We study a totally discontinuous interval map defined in [ 0 , 1 ] which is associated to a deformation of the shift map on two symbols 0 − 1 . We define a sequence of transition matrices which characterizes the effect of the interval map on a family of partitions of the interval [ 0 , 1 ] . Recursive algorithms that build the sequence of matrices and their left and right eigenvectors are deduced. Moreover, we compute the Artin zeta function for the interval map. [ABSTRACT FROM AUTHOR]

Spatially coupled low-density parity-check (LDPC) codes have attracted considerable attention due to their promising performance. Recursive encoding of the codes with low delay and low complexity has been proposed in the literature but with constraints or restrictions. In this manuscript we propose an efficient method to construct parity-check matrices for recursively encoding spatially coupled LDPC codes with arbitrarily chosen node degrees. A general principle is proposed, which provides feasible and practical guidance for the construction of parity-check matrices. According to the specific structure of the matrix, each parity bit at a coupling position is jointly determined by the information bits at the current position and the encoded bits at former positions. Performance analysis in terms of design rate and density evolution has been presented. It can be observed that, in addition to the feature of recursive encoding, selected code structures constructed by the newly proposed method may lead to better belief-propagation thresholds than the conventional structures. Finite-length simulation results are provided as well, which verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a recursive algorithm is presented to generate some exponent matrices which correspond to Tanner graphs with girth at least 6. For a J × L exponent matrix E, the lower bound Q(E) is obtained explicitly such that (J;L) QC LDPC codes with girth at least 6 exist for any circulant permutation matrix (CPM) size m ⩾ Q(E). The results show that the exponent matrices constructed with our recursive algorithm have smaller lower-bound than the ones proposed recently with girth 6. [ABSTRACT FROM AUTHOR]

A generalized Lehmer matrix with recursive entries from Kılıç et al. (2010b) is further generalized, introducing three additional parameters and taking recursive factorials instead of a term. Certain formulae are derived for the LU and Cholesky factorizations and their inverses, as well as the determinants. Then we precisely compute the elements of the inverse of the generalized Lehmer matrix. [ABSTRACT FROM AUTHOR]

In this paper we obtain a recursive formula for the shuffle product and apply it to derive two restricted decomposition formulas for multiple zeta values (MZVs). The first formula generalizes the decomposition formula of Euler and is similar to the restricted formula of Eie and Wei for MZVs with one strings of 1's. The second formula generalizes the previous results to the product of two MZVs with one and two strings of 1's respectively. [ABSTRACT FROM AUTHOR]

Abstract: In this article we present a fast recursive orthogonalization scheme for two important subspaces of the Macaulay matrix: its row space and null space. It requires a graded monomial ordering and exploits the resulting structure of the Macaulay matrix induced by this graded ordering. The resulting orthogonal basis for the row space will retain a similar structure as the Macaulay matrix and is as a consequence sparse. The computed orthogonal basis for the null space is dense but typically has smaller dimensions. Two alternative implementations for the recursive orthogonalization scheme are presented: one using the singular value decomposition and another using a sparse rank revealing multifrontal QR decomposition. Numerical experiments show the effectiveness of the proposed recursive orthogonalization scheme in both running time and required memory compared to a standard orthogonalization. The sparse multifrontal QR implementation is superior in both total run time and required memory at the cost of being slightly less reliable for determining the numerical rank. [Copyright &y& Elsevier]

Abstract: Recently, the concept of the complementary array of a Riordan array (or recursive matrix) has been introduced. Here we generalize the concept and distinguish between dual and complementary arrays. We show a number of properties of these arrays, how they are computed and their relation with inversion. Finally, we use them to find explicit formulas for the elements of many recursive matrices. [Copyright &y& Elsevier]

In this paper, a recursive sliding transform (RST) algorithm is proposed for the fast implementation of the Walsh Hadamard Transform (WHT) on sliding windows. The proposed RST algorithm accelerates the sliding WHT process by recursively reducing the order of WHT. Theoretical analysis shows that the proposed RST algorithm requires only 1.33 additions per sample for each WHT basis vector regardless of the size or dimension of the basis vector. The computational requirement of the proposed RST algorithm is the lowest among existing fast sliding WHT algorithms. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we discuss the quadratic inverse eigenvalue problem for damped gyroscopic systems, that is, we are to construct a damped gyroscopic second order system having given eigenvalues and associated eigenvectors as its full eigenstructure. Under some mild assumptions, the coefficient matrices are constructed in a recursive process. We also discuss how to obtain a damped gyroscopic system with the leading coefficient matrix being positive definite. [Copyright &y& Elsevier]

SUMMARY To improve the efficiency of the harmonic analysis, this paper presents a direct phase-domain harmonic model for long coupled transmission lines, which can take the parameter frequency dependence, the configuration asymmetry, and the untransposition of the line into consideration. The steps of the proposed modeling method are as follows. First, in the direct phase domain and on the basis of the travelling wave equation, the coupled transmission line model was derived in a nodal admittance matrix form. Then the matrix series theory was adopted to realize the numerical calculation of the nodal admittance matrix without relying on the eigenvalue and eigenvector calculations of the propagation matrix. Finally, a recursive approach was proposed to handle the nodal equation of the network containing long transmission lines. Both the accuracy and the efficiency of the proposed approach were verified by comparing with the exact method in which it is necessary to calculate the eigenvalues and eigenvectors of the propagation matrix. The computation time and memory consumption analysis indicates that the proposed approach excels the exact method in saving computation time and memory. It should be emphasized that, by utilizing the recursive approach, the proposed approach is always numerically stable and can also be applied to DC transmission lines. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We present the simplest and most straightforward derivation of the onedimensional harmonic oscillator propagator, using the Feynman path integral and recursive relations. Our calculations have pedagogical benefits for those undergraduate students beginning to learn the path integral in quantum mechanics, in that they can follow its calculations very simply with only elementarymathematical manipulation. Further, our calculations do not require cumbersome matrix algebra. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study a recursive system of integers ; which is uniquely determined by the initial values . We show under the constant initial dates for all n that the polynomial of degree is (anti) palindromic. Several explicit formulae for via Vandermonde matrix, mirrored -matrix, weighed Delannoy number, Riordan array, hypergeometric function, Jacobi polynomial, and some combinatorial identities are derived. [Copyright &y& Elsevier]

Abstract: Multivariate (n-D) polynomial matrix factorizations are basic research problems in multidimensional systems and signal processing. In this paper several extensions of the existing results on multivariate matrix factorizations are proved. Moreover, by means of the general theory of multivariate matrix factorizations recently developed, a new proof is given to the bivariate polynomial matrix factorizations. This new approach produces also a recursive algorithm for the actual computation of the general bivariate matrix factorizations, which relies on the algorithm of the Gröbner bases for modules and does not involve calculations over the function fields. [Copyright &y& Elsevier]

Abstract: Joint diagonalization of a set of matrices is an essential tool in many signal processing applications. This letter is devoted to seeking a recursive solution to nonunitary joint diagonalization. The proposed algorithm recursively minimizes an exponentially windowed least squares (LS) criterion, leading to a computationally cheaper recursive update rule for joint diagonalization. This merit enables us to develop (block) online algorithm for source separation and other applications. Simulation results on synthetic data and blind separation of real speech signals validate the effectiveness of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: We consider here the problem of tracking the dominant eigenspace of an indefinite matrix by updating recursively a rank approximation of the given matrix. The tracking uses a window of the given matrix, which increases at every step of the algorithm. Therefore, the rank of the approximation increases also, and hence a rank reduction of the approximation is needed to retrieve an approximation of rank . In order to perform the window adaptation and the rank reduction in an efficient manner, we make use of a new anti-triangular decomposition for indefinite matrices. All steps of the algorithm only make use of orthogonal transformations, which guarantees the stability of the intermediate steps. We also show some numerical experiments to illustrate the performance of the tracking algorithm. [Copyright &y& Elsevier]

A matrix decomposition algorithm was proposed for calculating exact reliability of multistate networks based on d-minimal cuts (d-MCs). This algorithm was more efficient than the inclusion-exclusion algorithm by means of decomposition principle, absorption law and definition of d-minimal cuts matrix and matrix probability. Based on recursive matrix decomposition and reduction, the algorithm obtained exact reliability by recursive matrix probability calculation. Furthermore, the algorithm reduced computational complexity by defining deletion function and dynamically selecting decomposition edge. This algorithm is characterized by clear structure, simple accomplishment and low computational complexity, which has an exponential relationship with the edge amount in the network. [ABSTRACT FROM AUTHOR]

Starting from the spectral analysis of g-circulant matrices, we study the convergence of a multigrid method for circulant and Toeplitz matrices with various size reductions. We assume that the size n of the coefficient matrix is divisible by g≥2 such that at the lower level the system is reduced to one of size n/ g, by employing g-circulant based projectors. We perform a rigorous two-grid convergence analysis in the circulant case and we extend experimentally the results to the Toeplitz setting, by employing structure preserving projectors. The optimality of the two-grid method and of the multigrid method is proved, when the number θ∈ℕ of recursive calls is such that 1< θ< g. The previous analysis is used also to overcome some pathological cases, in which the generating function has zeros located at 'mirror points' and the standard two-grid method with g=2 is not optimal. The numerical experiments show the correctness and applicability of the proposed ideas, both for circulant and Toeplitz matrices. [ABSTRACT FROM AUTHOR]

The paper continues the studies of the method of hereditary pencils for computing points of the finite spectrum of a multiparameter polynomial matrix. The method involves induction on the number of parameters and consists of two stages. At the first stage, given the coefficients of a multiparameter matrix, a sequence of (q-k)-parameter polynomial matrices (k = 1,...,q) satisfying certain recursive relations is formed. This sequence is used at the second stage. As the base case, two-parameter matrices and their spectral characteristics, which are computed by applying the method of hereditary pencils, are considered. Algorithms implementing the second stage are suggested and theoretically justified. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

The logic FBCI given by linearly ordered BCI-matrices is known not to be an axiomatic extension of the well-known BCI logic. In this paper we axiomatize FBCI by adding a recursively enumerable set of schemes of inference rules to BCI and show that there is no finite axiomatization for FBCI. [ABSTRACT FROM AUTHOR]

Abstract: Historically, there exist two versions of the Riordan array concept. The older one (better known as recursive matrix) consists of bi-infinite matrices ( implies ), 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 ( implies ), 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 . These parameters can be specialized and generate an indefinite number of other combinatorial identities which are valid in the bi-infinite realm of recursive matrices. [Copyright &y& Elsevier]

Abstract: In this paper, a new recursive symbolic computational Hessenberg matrix algorithm is presented to compute the inverse and the determinant of a Hessenberg matrix. [Copyright &y& Elsevier]

Abstract: We discuss a conjecture concerning the enumeration of nonsingular matrices over a finite field that are block companion and whose order is the maximum possible in the corresponding general linear group. A special case is proved using some recent results on the probability that a pair of polynomials with coefficients in a finite field is coprime. Connection with an older problem of Niederreiter about the number of splitting subspaces of a given dimension are outlined and an asymptotic version of the conjectural formula is established. Some applications to the enumeration of nonsingular Toeplitz matrices of a given size over a finite field are also discussed. [Copyright &y& Elsevier]

Abstract: A semi-analytical approach for a periodic planar array formed by a finite number of circular cylinders is presented using a model of layered cylindrical arrays. The method consists in extracting the reflection and transmission matrices of a cylindrically periodic array of circular cylinders and then obtaining the characteristics of layered structures by using a recursive formula. [Copyright &y& Elsevier]

For an arbitrary Steiner system S( v, k, t), we introduce the concept of a component as a subset of a system which can be transformed (changed by another subset) without losing the property for the resulting system to be a Steiner system S( v, k, t). Thus, a component allows one to build new Steiner systems with the same parameters as an initial system. For an arbitrary Steiner system S( v, k, k − 1), we provide two recursive construction methods for infinite families of components (for both a fixed and growing k). Examples of such components are considered for Steiner triple systems S( v, 3, 2) and Steiner quadruple systems S( v, 4, 3). For such systems and for a special type of so-called normal components, we find a necessary and sufficient condition for the 2-rank of a system (i.e., its rank over $\mathbb{F}_2$) to grow under switching of a component. It is proved that for k ≥ 5 arbitrary Steiner systems S( v, k, k − 1) and S( v, k, k − 2) have maximum possible 2-ranks. [ABSTRACT FROM AUTHOR]

Abstract: Combining Fourier series expansion with recursive matrix formulas, new reliable algorithms to compute the periodic, non-negative, definite stabilizing solutions of the periodic Riccati and Lyapunov matrix differential equations are proposed in this paper. First, periodic coefficients are expanded in terms of Fourier series to solve the time-varying periodic Riccati differential equation, and the state transition matrix of the associated Hamiltonian system is evaluated precisely with sine and cosine series. By introducing the Riccati transformation method, recursive matrix formulas are derived to solve the periodic Riccati differential equation, which is composed of four blocks of the state transition matrix. Second, two numerical sub-methods for solving Lyapunov differential equations with time-varying periodic coefficients are proposed, both based on Fourier series expansion and the recursive matrix formulas. The former algorithm is a dimension expanding method, and the latter one uses the solutions of the homogeneous periodic Riccati differential equations. Finally, the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples. [Copyright &y& Elsevier]

The aim of this paper is to present the structure of a class of matrices that enables explicit inverse to be obtained. Starting from an already known class of matrices, we construct a Hadamard product that derives the class under consideration. The latter are defined by 4n - 2 parameters, analytic expressions of which provide the elements of the lower Hessenberg form inverse. Recursion formulae of these expressions reduce the arithmetic operations in evaluating the inverse to O (n²). [ABSTRACT FROM AUTHOR]

Abstract: This work provides a framework for nominal and robust stability analysis for a class of discrete-time nonlinear recursive observers (DNRO). Given that the system has linear output mapping, local observability and Jacobian matrices satisfying certain conditions, the nominal and robust stability of the DNRO is defined by the property of estimation error dynamics and is analyzed using Lyapunov theory. Moreover, a simultaneous state and parameter estimation scheme is shown to be Input-to-State Stable (ISS), and adaptively reduce plant-model mismatch on-line. Three design strategies of the DNRO that satisfy the stability results are given as examples, including the widely used extended Kalman filter, extended Luenberger observer, and the fixed gain observer. [ABSTRACT FROM AUTHOR]

We investigate properties of a class of plateaued Boolean functions with support of the spectrum defined by a recursive class of matrices. For such supports of the spectrum, we find a precise number of functions with a given support. We also show that the set of these functions is the equivalence class of a function with this support of spectrum with respect to the group of shifts. [ABSTRACT FROM AUTHOR]

Abstract: Whether the determinant of the Dixon matrix equals zero or not is used for determining if a system of n +1 polynomial equations in n variables has a common root, and is a very efficient quantifier elimination approach too. But for a complicated polynomial system, it is not easy to construct the Dixon matrix. In this paper, a recursive algorithm to construct the Dixon matrix is proposed by which some problems that cannot be tackled by other methods can be solved on the same computer platform. A dynamic programming algorithm based on the recursive formula is developed and compared for speed and efficiency to the recursive algorithm. [Copyright &y& Elsevier]

Abstract: A partial difference set with parameters is said to be of Paley type. In this paper, we give a recursive theorem that for all odd constructs Paley partial difference sets in certain groups of order and . We are also able to construct Paley–Hadamard difference sets of the Stanton–Sprott family in groups of order when is a prime power and when is a prime power. Many of these are new parameters for such difference sets, and also give new Hadamard designs and matrices. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we construct the Chebyshev recursive wavelets on a unit interval of the first kind, the second kind and their corresponding weight functions. We apply wavelet collocation method to solve the natural boundary integral equation of the harmonic equation on the lower half-plane numerically. It is convenient and accurate to generate the stiffness matrix. Two numerical examples are presented. It is shown that the stiffness matrix is highly sparse when the order of the stiffness matrix becomes large. Current method allows choosing an appropriate weight function to increase the convergence rate and accuracy of the numerical results. [ABSTRACT FROM AUTHOR]

The problem of infinite eigenvalue assignment in the descriptor system [image omitted] via state feedback control u = Kx is considered. The problem is related to a group of recursive equations. By proposing a general complete parametric solution to this group of recursive equations, a general complete parametric approach is presented for the proposed infinite eigenvalue assignment problem. General parametric forms of the closed-loop eigenvectors and the feedback gain matrix are given in terms of certain parameter vectors which represent the design degrees of freedom. The approach involves mainly a singular value decomposition of the matrix E and a singular value decomposition of a lower dimension matrix, and thus is very simple and requires less computational work. Moreover, it overcomes the defects of some previous works. An example is given to illustrate the effect of the approach. [ABSTRACT FROM AUTHOR]

In this paper, we study periodicity and boundedness for the integer solutions to a minimum-delay difference equations. As an application, a recent theorem regarding absolute-difference equations is extended. [ABSTRACT FROM AUTHOR]

Abstract: The recursive method for computing the generalized LM-inverse of a constant rectangular matrix augmented by a column vector is proposed in Udwadia and Phohomsiri (2007) . The corresponding algorithm for the sequential determination of the generalized LM-inverse is established in the present paper. We prove that the introduced algorithm for computing the generalized LM-inverse and the algorithm for the computation of the weighted Moore–Penrose inverse developed by Wang and Chen (1986) in are equivalent algorithms. Both of the algorithms are implemented in the present paper using the package MATHEMATICA. Several rational test matrices and randomly generated constant matrices are tested and the CPU time is compared and discussed. [Copyright &y& Elsevier]

Abstract: Harmonic frames of prime order are investigated. The primary focus is the enumeration of inequivalent harmonic frames, with the exact number given by a recursive formula. The key to this result is a one-to-one correspondence developed between inequivalent harmonic frames and the orbits of a particular set. Secondarily, the symmetry group of prime order harmonic frames is shown to contain a subgroup consisting of a diagonal matrix as well as a permutation matrix, each of which is dependent on the particular harmonic frame in question. [Copyright &y& Elsevier]

Abstract: In this paper, by using parallel computing along with recursion, we describe a reliable symbolic computational algorithm for inverting cyclic pentadiagonal matrices. The algorithm is implemented in MAPLE. Two other symbolic algorithms are developed and the computational costs for all algorithms are given. An example is presented for the sake of illustration. [Copyright &y& Elsevier]

We consider the problem of enumerating periodic σ-juggling sequences of length n for multiplex juggling, where σ is the initial state (or landing schedule) of the balls. We first show that this problem is equivalent to choosing 1’s in a specified matrix to guarantee certain column and row sums, and then using this matrix, derive a recursion. This work is a generalization of earlier work of Chung and Graham. [ABSTRACT FROM AUTHOR]

In this paper, several recursive constructions for directed difference family and perfect directed difference family are presented by means of difference matrix and incomplete difference matrix. Finally the necessary and sufficient conditions for the existence of a ( gυ, g, 3, λ)-directed difference family in Z gυ are established. As a consequence, the necessary and sufficient conditions for the existence of a cyclic directed group divisible design with block size three and type g υ are obtained. [ABSTRACT FROM AUTHOR]