Decomposition of Polynomials with Respect to the Cyclic Group of Orderm

Given a polynomial solution of a differential equation, its m -ary decomposition, i.e. its decomposition as a sum of m polynomials P^{[ j ]}(x)  = ∑_kα_j,kx^{λ_{j, k}}containing only exponentsλ_{j, k} with λ_{j,k  + 1} − λ_j,k = m, is considered. A general algorithm is proposed in order to build holonomic equations for the m -ary parts P^{[ j ]}(x) starting from the initial one, which, in addition, provides a factorized form of them. Moreover, these differential equations are used to compute expansions of the m -ary parts of a given polynomial in terms of classical orthogonal polynomials. As illustration, binary and ternary decomposition of these classical families are worked out in detail.     

3D radial invariant of dual Hahn moments

In this work, we propose new sets of 2D and 3D rotation invariants based on orthogonal radial dual Hahn moments, which are orthogonal on a non-uniform lattice. We also present theoretical mathematics to derive them. Thus, this paper presents in the first case new 2D radial dual Hahn moments based on polar representation of an image by one-dimensional orthogonal discrete dual Hahn polynomials and a circular function. The dual Hahn polynomials are general case of Tchebichef and Krawtchouk polynomials. In the second case, we introduce new 3D radial dual Hahn moments employing a spherical representation of volumetric image by one-dimensional orthogonal discrete dual Hahn polynomials and a spherical function, which are orthogonal on a non-uniform lattice. The 2D and 3D rotational invariants are extracts from the proposed 2D and 3D radial dual Hahn moments respectively. In order to test the proposed approach, three problems namely image reconstruction, rotational invariance and pattern recognition are attempted using the proposed moments. The result of experiments shows that the radial dual Hahn moments have performed better than the radial Tchebichef and Krawtchouk moments, with and without noise. Simultaneously, the mentioned reconstruction converges quickly to the original image using 2D and 3D radial dual Hahn moments, and the test images are clearly recognized from a set of images that are available in COIL-20 database for 2D image and PSB database for 3D image.

     

Eigenvectors of tridiagonal matrices of Sylvester type

Eigenvectors of the tridiagonal matrices of Sylvester type are explicitly determined. These are closely related to orthogonal polynomials named after Krawtchouk, (dual) Hahn and Racah as well as to q-Racah polynomials.      

The Tau Method as an analytic tool in the discussion of equivalence results across numerical methods

A Tau Method approximate solution of a given differential equation defined on a compact [a, b] is obtained by adding to the right hand side of the equation a specific minimal polynomial perturbation termH _n(x), which plays the role of a representation of zero in [a,b] by elements of a given subspace of polynomials. Neither discretization nor orthogonality are involved in this process of approximation. However, there are interesting relations between the Tau Method and approximation methods based on the former techniques. In this paper we use equivalence results for collocation and the Tau Method, contributed recently by the authors together with classical results in the literature, to identify precisely the perturbation termH(x) which would generate a Tau Method approximate solution, identical to that generated by some specific discrete methods over a given mesh Π ∈ [a, b]. Finally, we discuss a technique which solves the inverse problem, that is, to find adiscrete perturbed Runge-Kutta scheme which would simulate a prescribed Tau Method. We have chosen, as an example, a Tau Method which recovers the same approximation as an orthogonal expansion method. In this way we close the diagram defined by finite difference methods, collocation schemes, spectral techniques and the Tau Method through a systematic use of the latter as an analytical tool.     

Parameter estimation of discrete systems via Hahn polynomials

This paper presents a procedure for using Hahn polynomials for the analysis and parameter estimation of linear discrete-time single-input/single-output systems described by difference equations. An advantage is gained through transforming a linear difference equation into a set of algebraic equations of the system parameters and the Hahn coefficients of the system variables. The key to the equation conversion is the derivation of a relation between the Hahn coefficients of a function and its time-shifted ones. Examples are provided to show the utilization of the procedure.     

Partial Differential Equations and Bivariate Orthogonal Polynomials

In 1929, S. Bochner identified the families of polynomials which are eigenfunctions of a second-order linear differential operator. What is the appropriate generalization of this result to bivariate polynomials? One approach, due to Krall and Sheffer in 1967 and pursued by others, is to determine which linear partial differential operators have orthogonal polynomial solutions with all the polynomials in the family of the same degree sharing the same eigenvalue. In fact, such an operator only determines a multi-dimensional eigenspace associated with each eigenvalue; it does not determine the individual polynomials, even up to a multiplicative constant. In contrast, our approach is to seek pairs of linear differential operators which have joint eigenfunctions that comprise a family of bivariate orthogonal polynomials. This approach entails the addition of some “normalizing" or “regularity" conditions which allow determination of a unique family of orthogonal polynomials. In this article we formulate and solve such a problem and show with the help of Mathematica that the only solutions are disk polynomials. Applications are given to product formulas and hypergroup measure algebras.     

A Bessel polynomial approach for solving general linear Fredholm integro-differential–difference equations

In this paper, to find an approximate solution of general linear Fredholm integro-differential–difference equations (FIDDEs) under the initial-boundary conditions in terms of the Bessel polynomials, a practical matrix method is presented. The idea behind the method is that it converts FIDDEs to a matrix equation which corresponds to a system of linear algebraic equations and is based on the matrix forms of the Bessel polynomials and their derivatives by means of collocation points. The solutions are obtained as the truncated Bessel series in terms of the Bessel polynomials J _n (x) of the first kind defined in the interval [0, ∞). The error analysis and the numerical examples are included to demonstrate the validity and applicability of the technique.     

Higher order methods for convection-diffusion problems

This paper applies C¹ cubic Hermite polynomials embedded in an orthogonal collocation scheme to the spatial discretization of the unsteady nonlinear Burgers equation as a model of the equations of fluid mechanics. The temporal discretization is carried out by means of either a noniterative finite difference or an iterative finite difference procedure. Results of this method are compared with those of a second-order finite difference scheme and a splined-cubic Taylor's series scheme. Stability limits are derived and the matrix structure of the several schemes are compared.     

On the stability of a weighted diamond of real polynomials

Recently, a subset of the robust stability literature concentrated on the so-called diamond of polynomials. In this paper, we study the weighted diamond Q_w defined as

, where the center q^*, the weights w₀, w₁, 3dot , w_n > 0 and the radius r > 0 are given. For this family, we show that an extreme point result holds if and only if a certain ‘interlacing condition’ on the weights is satisfied.     

A further result on the oscillation of delay difference equations

In this paper, we are concerned with the delay difference equations of the form

(*)

y_n+1 − y_n + p_ny_n−k = 0, N = 0, 1, 2, …,

(*)where p_n ≥ 0 and k is a positive integer. We prove by using a new technique that

guarantees that all solutions of equation (*) oscillate, which improves many previous well-known results. In particular, our theorems also fit the case where Σⁿ⁻¹_i=n−kp_i ≤ k^k+1/(k + 1)^k+1. In addition, we present a nonoscillation sufficient condition for equation (*).     

On multidimensional analogs of Melvin’s solution for classical series of Lie algebras

A multidimensional generalization of Melvin’s solution for an arbitrary simple Lie algebra is presented. The gravitational model contains n 2-forms and l ≥ n scalar fields, where n is the rank of . The solution is governed by a set of n functions obeying n ordinary differential equations with certain boundary conditions. It was conjectured earlier that these functions should be polynomials (the so-called fluxbrane polynomials). A program (in Maple) for calculating these polynomials for classical series of Lie algebras is suggested. The polynomials corresponding to the Lie algebra D ₄ are obtained. It is conjectured that the polynomials for A _n -, B _n - and C _n -series may be obtained from polynomials for D _n+1-series by using certain reduction formulas. Talk given at the International Conference RUSGRAV-13, June 23–28, 2008, PRUR, Moscow.     

A new solution branch for the Blasius equation—A shrinking sheet problem

In this work, a similarity equation of the momentum boundary layer is studied for a moving flat plate with mass transfer in a stationary fluid. The solution is applicable to the practical problem of a shrinking sheet with a constant sheet velocity. Theoretical estimation of the solution domain is obtained. It is shown that the solution only exists with mass suction at the wall surface. The equation with the associated boundary conditions is solved using numerical techniques. Greatly different from the continuously stretching surface problem and the Blasius problem with a free stream, quite complicated behavior is observed in the results. It is seen that there are three different solution zones divided by two critical mass transfer parameters, f₀₁≈1.7028 and f₀₂≈1.7324. When f₀<f₀₁, there is no solution for this problem, multiple solutions for f₀₁<f₀≤f₀₂, and one solution when (f₀=f₀₁)(f₀>f₀₂). There is a terminating point for the solution domain and the terminating point corresponds to a special algebraically decaying solution for the current problem. The current results provide a new solution branch of the Blasius equation, which is greatly different from the previous study and provide more insight into the understanding of the Blasius equation.     

On asymptotic stability of continuous-time risk-sensitive filters with respect to initial conditions

In this paper, we consider the problem of risk-sensitive filtering for continuous-time stochastic linear Gaussian time-invariant systems. In particular, we address the problem of forgetting of initial conditions. Our results show that suboptimal risk-sensitive filters initialized with arbitrary Gaussian initial conditions asymptotically approach the optimal risk-sensitive filter for a linear Gaussian system with Gaussian but unknown initial conditions in the mean square sense at an exponential rate, provided the arbitrary initial covariance matrix results in a stabilizing solution of the (H_∞-like) Riccati equation associated with the risk-sensitive problem. More importantly, in the case of non-Gaussian initial conditions, a suboptimal risk-sensitive filter asymptotically approaches the optimal risk-sensitive filter in the mean square sense under a boundedness condition satisfied by the fourth order absolute moment of the initial non-Gaussian density and a slow growth condition satisfied by a certain Radon–Nikodym derivative.     

On the solvability of the positive real lemma equations

In this paper, we consider the classical equations of the positive real lemma under the sole assumption that the state matrix A has unmixed spectrum: σ(A)∩σ(−A)=. Without any other system-theoretic assumption (observability, reachability, stability, etc.), we derive a necessary and sufficient condition for the solvability of the positive real lemma equations.     

Radial Hahn Moment Invariants for 2D and 3D Image Recognition

Recently, orthogonal moments have become efficient tools for two-dimensional and three-dimensional (2D and 3D) image not only in pattern recognition, image vision, but also in image processing and applications engineering. Yet, there is still a major difficulty in 3D rotation invariants. In this paper, we propose new sets of invariants for 2D and 3D rotation, scaling and translation based on orthogonal radial Hahn moments. We also present theoretical mathematics to derive them. Thus, this paper introduces in the first case new 2D radial Hahn moments based on polar representation of an object by one-dimensional orthogonal discrete Hahn polynomials, and a circular function. In the second case, we present new 3D radial Hahn moments using a spherical representation of volumetric image by one-dimensional orthogonal discrete Hahn polynomials and a spherical function. Further 2D and 3D invariants are derived from the proposed 2D and 3D radial Hahn moments respectively, which appear as the third case. In order to test the proposed approach, we have resolved three issues: the image reconstruction, the invariance of rotation, scaling and translation, and the pattern recognition. The result of experiments show that the Hahn moments have done better than the Krawtchouk moments, with and without noise. Simultaneously, the mentioned reconstruction converges quickly to the original image using 2D and 3D radial Hahn moments, and the test images are clearly recognized from a set of images that are available in COIL-20 database for 2D image, and Princeton shape benchmark (PSB) database for 3D image.     

Algorithm for Computing Bernstein–Sato Ideals Associated with a Polynomial Mapping

Let f₁, . . . , f_pbe polynomials in n variables with coefficients in a fieldK. We associate with these polynomials a number of functional equations and related ideals B, B_jand BΣ ofK[ s₁, . . . , s_p] called Bernstein–Sato ideals. Using standard basis techniques, our aim is to present an algorithm for computing generators of B_jand BΣ.     

Relax, but Don’t be Too Lazy

Assume that we wish to expand the product h = fg of two formal power series f and g. Classically, there are two types of algorithms to do this: zealous algorithms first expand f and g up to order n, multiply the results and truncate at order n. Lazy algorithms on the contrary compute the coefficients of f, g and h gradually and they perform no more computations than strictly necessary at each stage. In particular, at the moment we compute the coefficient h_iof zⁱin h, only f₀, , f_iand g₀, , g_iare known.Lazy algorithms have the advantage that the coefficients of f and g may actually depend on “previous" coefficients of h, as long as they are computed before they are needed in the multiplication, i.e. the coefficients f_iand g_imay depend on h₀, , h_{i − 1}. For this reason, lazy algorithms are extremely useful when solving functional equations in rings of formal power series. However, lazy algorithms have the disadvantage that the classical asymptotically fast multiplication algorithms on polynomials—such as the divide and conquer algorithm and fast Fourier multiplication—cannot be used.In a previous paper, we therefore introduced relaxed algorithms, which share the property concerning the resolution of functional equations with lazy algorithms, but perform slightly more computations than lazy algorithms during the computation of a given coefficient of h. These extra computations anticipate the computations of the next coefficients of h and dramatically improve the asymptotic time complexities of such algorithms.In this paper, we survey several classical and new zealous algorithms for manipulating formal power series, including algorithms for multiplication, division, resolution of differential equations, composition and reversion. Next, we give various relaxed algorithms for these operations. All algorithms are specified in great detail and we prove theoretical time and space complexity bounds. Most algorithms have been experimentally implemented in C++ and we provide benchmarks. We conclude by some suggestions for future developments and a discussion of the fitness of the lazy and relaxed approaches for specific applications.This paper is intended both for those who are interested in the most recent algorithms for the manipulation of formal power series and for those who want to actually implement a power series library into a computer algebra system.     

Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials

We extend the definition of the classical Jacobi polynomials withindexes α, β>−1 to allow α and/or β to be negative integers. We show that the generalized Jacobi polynomials, with indexes corresponding to the number of boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the use of generalized Jacobi polynomials leads to much simplified analysis, more precise error estimates and well conditioned algorithms.Mathematics subject classification 1991. 65N35, 65N22, 65F05, 35J05     

Controller synthesis via the 4-polynomial approach

In some recent work it was shown that to stabilize systems with real parameter uncertainty it suffices to find a controller that simultaneously stabilizes a finite number of polynomials. These polynomials include those generated from the ‘vertex’ plants as well as some generated by some ‘fictitious’ vertex plants that involve the controller. This paper deals with the issues of existence of such a controller, controller synthesis, and conservativeness of the design. It is shown how this approach can ‘enhance’ the stability robustness of an H_∞ design.     

Sulla costruzione di un sistema di polinomi ortonormali a partire da tre suoi polinomi consecutivi

The above mentioned construction requires that the three assigned polynomials (with degreesn, n+1, n+2) verify conditions stated in Theorem II. In this case the three polynomials belong to infinite many sequences of orthogonal polynomials wich correspond to non negative measuresdϕ(x) with fixed moments μ₀, μ₁,⋯, μ_2n+4.

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Dedicato al Professor S. Faedo in occasione del suo settantesimo compleanno.