Dual operators and Lagrange inversion in several variables

Lagrange inversion in infinitely many variables

A comparative study of combinatorial and algebraic force methods

We present a general method of proving Lagrange inversion formulas and give new proofs of the s-variable Lagrange-Good formula [13] and the 9-Lagrange formulas of Garsia [7], Gessel [10], Gessel and Stanton [11,12] and the author [18].We also give some g-analogues of the Lagrange formula in several variables. Introduction.Let f(z) be a formal power series (fps) and g(z) a formal Laurent series (fLs) with finitely many coefficients with negative index different from zero (g(z) = J2k>i 9kZk for an integer /).Let /(0) = 0 and /'(O) ^ 0. The coefficients of the expansion g(z) -^2kezckfkiz) can be computed by the two versions of the Lagrange formula, the first of which can be written aswhere (zk) means the coefficient of zk; the second can be written as(1-2) c" = {z*)g{z)jl-forneZ.These formulas are based on the orthogonality relation (Wit) ji^ = ** for all n, k G Z (Snk is the Kronecker delta).Using Hofbauer's method [16] for an orthogonality relation (fk,fn) = Snk we can transfer certain properties of the sequence (fk)k&z to the sequence (fk)kez, where ( , ) denotes a bilinear form.Hofbauer used it to prove some one-variable Lagrange formulas.We extend this method in 4 by our Theorems 1 and 5 in order to give a unified "recipe" for proving Lagrange inversion formulas.All known finitedimensional Lagrange formulas can be treated, as we show in 5 to 8. Moreover we use this recipe to find new Lagrange inversion formulas.5 deals with the Lagrange-Good formula [13].Using our method, we give a short new proof in which the Jacobian appears in a natural way.We are also able to find multivariable generalizations even of (1.1), the "first version" of the Lagrange formula (identities (5.6) and (5.7)).

Context-free grammars, differential operators and formal power series

Une théorie combinatoire des séries formelles

In solving boundary value problems of elasticity theory in a half-strip in the form of series in Papkovich–Fadle eigenfunctions, there are always two representations for these functions. In the paper we consider Lagrange expansions in these representations. Lagrange expansions are the expansions of only one function in a series in any single system of Papkovich–Fadle eigenfunctions. This is different from the expansions that arise in solving boundary value problems where it is necessary to construct the expansions of two functions (given at the end of the half-strip) with one set of coefficients. Lagrange expansions play a fundamental role in solving boundary value problems with different boundary conditions on the long sides of the half-strip, similar to the one that trigonometric series play in periodic problems of the theory of elasticity.

T HE eigensensitivities of mechanical systems with respect to structural design parameters have become an integral part of many engineering design methodologies including optimization, structural health monitoring, structural reliability, model updating, dynamic modification, reanalysis techniques, and many other applications. Fox and Kapoor [1] computed the derivative of each eigenvector as a linear combination of all of the undamped eigenvectors. Later, Adhikari and Friswell [2] and Adhikari [3] extended the modal method to the more general asymmetric systems with viscous and nonviscous damping, respectively. Nelson [4] presented a method, which requires only the eigenvector of interest by expressing the derivative of each undamped eigenvector as a particular solution and a homogeneous solution. Later, Friswell and Adhikari [5] extended Nelson’s method to symmetric and asymmetric systems with viscous damping. Recently, Adhikari and Friswell [6] extendedNelson’s method to symmetric and asymmetric nonviscously damped systems. Fox and Kapoor [1] also suggested a direct algebraic method to calculate the eigensensitivity for symmetric undamped systems by solving a nonsingular linear system of algebraic equations. Lee et al. [7] derived an efficient algebraic method, which has a compact linear system with a symmetric coefficient matrix for symmetric systems with viscous damping. Later, Guedria et al. [8] extended the algebraic method to general asymmetric viscous damped systems. Chouchane et al. [9] reviewed the algebraic method and extended their method to the second-order and high-order derivatives of eigensolutions. Li et al. [10] extended the algebraic method to symmetric and asymmetric nonviscously damped systems. Xu andWu [11] proposed a new normalization and presented a method for the computation of eigensolution derivatives of asymmetric systemswith viscously damping. Recently,Mirzaeifar et al. [12] proposed a new method based on a combination of algebraic and modal methods for generally asymmetric viscously damped systems. More recently, Li et al. [13] proposed a method of design sensitivity analysis of asymmetric viscously damped systems with distinct and repeated eigenvalues, which can compute the left and right eigenvector derivatives separately and independently. All of the methods mentioned previously compute the eigensensitivities of asymmetric damped systems by using the left eigenvector. However, these methods have disadvantages in computational cost and storage capacity for the left eigenvector should be calculated. To avoid using the left eigenvector, an algebraic method is presented [14], which does not require the left eigenvector for asymmetric damped systems, but this method is restricted to the case of viscous damping. It should be noted that the coefficientmatrices of the algebraicmethodmay be ill conditioned due to the components of the additional constraints, and system matrices in the coefficient matrices are not all of the same order of magnitude. In addition [15], the normalization adapted in [14] and [12] will fail in some cases because it can equal zero or a very small number. This Note will present a method, which is well conditioned and can calculate the eigensensitivity of asymmetric nonviscous damped systems without using the left eigenvector. Considering an N-degree-of-freedom linear system with nonviscous (viscoelastic) damping [3,6,10]

In this article we study the fan-beam Radon transform Dm of symmet- rical solenoidal 2D tensor fields of arbitrary rank m in a unit disc D as the operator, acting from the object space L2(D ; Sm) to the data space L2((0, 2π) × (0, 2π)). The orthogonal polynomial basis s (±m) n,k of solenoidal tensor fields on the disc D was built with the help of Zernike polynomials and then a singular value decomposition (SVD) for the operator Dm was obtained. The inversion formula for the fan-beam tensor transform Dm follows from this decomposition. Thus obtained inversion formula can be used as a tomographic filter for splitting a known tensor field into potential and solenoidal parts. Numerical results are presented.

The Radon transform on the Heisenberg group and the transversal Radon transform

In this article we study the fan-beam Radon transform of symmetrical solenoidal 2D tensor fields of arbitrary rank m in a unit disc as the operator, acting from the object space L 2 (; S m ) to the data space L 2 ([0, 2π) × [0, 2π)). The orthogonal polynomial basis of solenoidal tensor fields on the disc was built with the help of Zernike polynomials and then a singular value decomposition (SVD) for the operator was obtained. The inversion formula for the fan-beam tensor transform follows from this decomposition. Thus obtained inversion formula can be used as a tomographic filter for splitting a known tensor field into potential and solenoidal parts. Numerical results are presented.

The Möbius inversion formula for a locally finite partially ordered set is realized as a Lagrange inversion formula. Schauder bases are introduced to interpret Möbius inversion.

The Lagrange inversion formula is a fundamental tool in combinatorics. In this work, we investigate an inversion formula for analytic functions, which does not require taking limits. By applying this formula to certain functions we have found an interesting arithmetic triangle for which we give a recurrence formula. We then explore the links between these numbers, Pascal's triangle, and Bernoulli's numbers, for which we obtain a new explicit formula. Furthermore, we present power series and asymptotic expansions of some elementary and special functions, and some links to the Online Encyclopedia of Integer Sequences (OEIS).

Convolution number

T. Klove (see ibid., vol.41, p.298-300, 1995) analyzed the average worst case probability of undetected error for linear [n, k; q] codes of length n and dimension k over an alphabet of size q. The following sum: S/sub n/=/spl Sigma//sub i=1//sup n/(/sub i//sup n/)(/sup i///sub n/)/sup i/((1-i)/n)/sup n-i/ arose, which also has applications in coding theory, average case analysis of algorithms, and combinatorics. Klove conjectured an asymptotic expansion of this sum, and we prove its enhanced version. Furthermore, we consider a more challenging sum arising in the upper bound of the average worst case probability of undetected error over systematic codes derived by Massey (1978). Namely S/sub n,k/=/spl Sigma//sub i=1//sup n/(/sub i//sup n-k/)(/sup i///sub n/)/sup i/((1-i)/n)/sup n-i/ for k/spl ges/0. We obtain an asymptotic expansion of S/sub n,k/, and this leads to a conclusion that Massey's bound on the average worst case probability over all systematic codes is better for every k than the corresponding Klove's bound over all codes [n, k; q]. The technique used belongs to the analytical analysis of algorithms and is based on some enumeration of trees, singularity analysis, Lagrange's inversion formula, and Ramanujan's identities. In fact, S/sub n/, turns out to be related to the so-called Ramanujan's Q-function which finds many applications (e.g. hashing with linear probing, the birthday paradox problem, random mappings, caching, memory conflicts, etc.).

Lagrange's inversion formula is usually presented in the following form. Let f be a regular (= analytic or holomorphic) complex function with the properties Then it is a standard theorem that it has a regular inverse function g , such that g ( f ( z )) = z , with similar properties. (I assume here standard results to be found in appropriate text books.