Block orthogonal polynomials: I. Definitions and properties

The zeros and asymptotic limits of two new classes of orthogonal polynomials, which are derived by applying two orthogonalization procedures due to Lowdin to a set of monomials, are calculated. It is established that they possess all the properties ofthe zeros of a polynomial. Their asymptotic limits are found. A Unified view of all the Lowdin orthogonal polynomials together with the standard classical orthogonal polynomials are presented in a unique graph.

The design of a gas turbine, or one of its constituentmodules, is generally approachedwith some specific operating condition in mind (its design point). Unfortunately, engine components seldom exactly meet their specifications and do not operate at just one condition, but over a range of power settings. This simplification can then lead to a product that exhibits performance worse than nominal in real-world conditions. The integration of some consideration of robustness as an active part of the design process can allow products less sensitive to the presence of the noise factors commonly found in real-world environments to be obtained. To become routinely used as a design tool, minimization of the time required for robustness analysis is paramount. In this study, a nonintrusive polynomial chaos formulation is used to evaluate the variability in the performance of a genericmodular-core compression system for a three-spool modern gas turbine engine subject to uncertain operating conditions with a defined probability density function. The standard orthogonal polynomials from the Askey scheme are replaced by a set of orthonormal polynomials calculated relative to the specific probability density function, improving the convergence of the method.

Contracted continuous distributed approximating functions (CCDAFs) have been developed. In particular, it has been shown that, continuous distributed approximating functions (CDAFs) based on standard orthogonal polynomials can be contracted to functions formed as the product of a weight function and the sinc function or a Bessel function of the first kind. The CCDAFs of Hermite type have been applied to derive new expressions for the coordinate representation of the free particle evolution operator and that of the evolution operator of harmonic oscillator. These new expressions of free particle and harmonic propagators have as compact mathematical form as Makri’s effective free propagator [N. Makri, Chem. Phys. Lett. 159, 489 (1989)] and Gaussian decay identical to that of the CDAF class free and harmonic propagators due to Kouri et al. [D. J. Kouri, W. Zhu, X. Ma, B. M. Pettitt, and D. K. Hoffman, J. Phys. Chem. 96, 9622 (1992)] and Marchioro et al. [T. L. Marchioro II, M. Arnold, D. K. Hoffman, W. Zhu, Y. Huang, and D. J. Kouri, Phys. Rev. E50, 2320 (1994)], respectively. The Gaussian decay of a CCDAF Hermite free propagator has been shown to be the result of including momentum eigenstates in the propagator which have momenta larger than the momentum of the wave packet of largest momentum that still can be well approximated by the CCDAF considered.

In the paper a comparison is made between two approximate methods of system analysis based on the use of the operational matrices of integration and differentiation. Both methods applied to model problems involving linear time-invariant differential equations yielded results of equal order of accuracy for the same number of orthogonal polynomial terms. However, the operational matrix of differentiation method is not limited to initial-value problems, but may easily be applied to boundary-value problems. In addition, due to the exact definition of this matrix, transformations from an orthogonal polynomial base to another may easily be performed with the use of the operational matrix of polynomial series transformation. The operational matrix of differentiation approach may be easily applied to the analysis of other types of systems such as time-varying and distributed parameter systems, and can be easily extended for other types of orthogonal differentiable base functions such as the Fourier trigonometric functions and the Bessel functions.

Inversion for S‐wave velocities from the amplitude variation with offset of P‐wave data is far from being a standard routine in the seismic processing sequence. However, the need for tracking the amplitude versus offset (AVO) occurs in several situations, for example in order to estimate the zero‐offset amplitude, to reveal areas with particular AVO characteristics, or to compress the AVO so that it is more easily obtainable at a later stage of the seismic processing. Furthermore, weak reflections can occasionally, due to the effect of the angle‐dependent reflectivity, have a polarity‐shift with offset, resulting in a very poor, or even vanishing, stack response. In such cases, the reflection event has to be represented by some other property than its mean amplitude or stack value.We outline how the AVO of seismic data may be extracted and classified by the use of orthogonal polynomials. The main advantage of this method compared to a general polynomial fit is that the AVO may be classified by a unique Spectrum of polynomial coefficients. This is in analogy to Fourier coefficients where the orthogonal basis is harmonic functions. The set of orthogonal polynomials is constructed entirely from the set of offset coordinates, and these polyno‐mials are defined only on the offset window considered. Compared to a Fourier transform, this is a major advantage since there is no effect of a limited spatial bandwidth.The AVO of normal‐moveout corrected data may be represented by a data gather where the orthogonal polynomial coefficients are given as time traces with each trace revealing a certain AVO characteristic. For instance, the stack is proportional to the zeroth‐order coefficient, the mean gradient is given by the firstorder coefficient, while the second‐order coefficient indicates whether the AVO increases and then decreases, or vice versa.

The theory of bi-orthogonal polynomials is exploited to investigate the location of zeros of truncated expansions in orthogonal polynomials. It turns out that, subject to additional conditions, these zeros can be confined to certain real intervals. Two general techniques are being used: the first depends on a theorem that links strict sign consistency of a generating function to loci of zeros and the second consists of re-expression of transformations from [3] in an orthogonal basis.

We extend the Mehler-Heine type formula of Jacobi polynomials to a class of multiple orthogonal polynomials of type II. The Mehler-Heine type formulas show standard orthogonal polynomials or multiple orthogonal polynomials near the endpoints of the interval of orthogonality.

Many Hankel determinant computations arising in combinatorial analysis can be done using results from the theory of standard orthogonal polynomials. Here, we will emphasize special sequences which require the inclusion of discrete Sobolov orthogonality to find their closed form.

New polynomial preserving operators on simplices: direct results

Periodic variations in patterns within a group of pixels provide important information about the surface of interest and can be used to identify objects or regions. Hence, a proper analysis can be applied to extract particular features according to some specific image properties. Recently, texture analysis using orthogonal polynomials has gained attention since polynomials characterize the pseudo-periodic behavior of textures through the projection of the pattern of interest over a group of kernel functions. However, the maximum polynomial order is often linked to the size of the texture, which implies in many cases, a complex calculation and introduces instability in higher orders leading to computational errors. In this paper, we address this issue and explore a pre-processing stage to compute the optimal size of the window of analysis called “texel.” We propose Haralick-based metrics to find the main oscillation period, such that, it represents the fundamental texture and captures the minimum information, which is sufficient for classification tasks. This procedure avoids the computation of large polynomials and reduces substantially the feature space with small classification errors. Our proposal is also compared against different fixed-size windows. We also show similarities between full-image representations and the ones based on texels in terms of visual structures and feature vectors using two different orthogonal bases: Tchebichef and Hermite polynomials. Finally, we assess the performance of the proposal using well-known texture databases found in the literature.

The activity coefficients of the binary mixtures ethanol + 1-butanol, ethanol + octane, and 1-butanol + octane were determined at temperatures of (308.15, 313.15, and 318.15) K. The determination of the vapor phase composition at equilibrium was carried out using headspace gas chromatography analysis. Multiple headspace extraction was used to calibrate the headspace gas chromatograph. Comparison of the phase diagrams produced using standard Legendre orthogonal polynomial techniques with phase diagrams from the literature showed good agreement. The composition of the azeotropes were reported, where applicable.

The known asymptotic relations interconnecting Jacobi, Laguerre, and Hermite classical orthogonal polynomials are generalized to the corresponding exceptional orthogonal polynomials of codimension m. It is proved that Xm-Laguerre exceptional orthogonal polynomials of type I, II, or III can be obtained as limits of Xm-Jacobi exceptional orthogonal polynomials of the same type. Similarly, Xm-Hermite exceptional orthogonal polynomials of type III can be derived from Xm-Jacobi or Xm-Laguerre ones. The quadratic transformations expressing Hermite classical orthogonal polynomials in terms of Laguerre ones is also extended to even X2m-Hermite exceptional orthogonal polynomials.

We derive and factorize the fourth-order q-difference equations satisfied by orthogonal polynomials obtained from some perturbations of the recurrence coefficients of q-classical orthogonal polynomials. These perturbations include the rth associated, the anti-associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified q-classical orthogonal polynomials. Moreover we find a basis of four linearly independent solutions of these fourth-order q-difference equations and express the modified families in terms of the starting ones.

This paper is concerned with a possible extension of a well-known stabilization technique for one-variable recursive digital filters to the two-dimensional case, as recently conjectured. It is shown that this problem is equivalent to considering a new class of orthogonal polynomials, the two-variable orthogonal polynomials on the hypercircle, the properties of which are investigated. As a result, the zeros of these polynomials are proved not to lie necessarily in an appropriate region compatible with the proposed conjecture, which therefore turns out to be in error.

In recent years, one of the most interesting developments in quantum mechanics has been the construction of new exactly solvable potentials connected with the appearance of families of exceptional orthogonal polynomials (EOP) in mathematical physics. In contrast with families of (Jacobi, Laguerre and Hermite) classical orthogonal polynomials, which start with a constant, the EOP families begin with some polynomial of degree greater than or equal to one, but still form complete, orthogonal sets with respect to some positive-definite measure. We show how they may appear in the bound-state wavefunctions of some rational extensions of well-known exactly solvable quantum potentials. Such rational extensions are most easily constructed in the framework of supersymmetric quantum mechanics (SUSYQM), where they give rise to a new class of translationally shape invariant potentials. We review the most recent results in this field, which use higher-order SUSYQM. We also comment on some recent re-examinations of the shape invariance condition, which are independent of the EOP construction problem.