Modified Chebyshev Polynomials Research Articles

Calculation of product state distributions from resonance decay via Lanczos subspace filter diagonalization: Application to HO2

Resonance phenomena associated with the unimolecular dissociation of HO2 have been investigated quantum-mechanically by the Lanczos homogeneous filter diagonalization (LHFD) method. The calculated resonance energies, rates (widths), and product state distributions are compared to results from an autocorrelation function-based filter diagonalization (ACFFD) method. For calculating resonance wave functions via ACFFD, an analytical expression for the expansion coefficients of the modified Chebyshev polynomials is introduced. Both dissociation rates and product state distributions of O2 show strong fluctuations, indicating the dissociation of HO2 is essentially irregular.

Forbidden subsequences and Chebyshev polynomials

In (West, Discrete Math. 157 (1996) 363–374) it was shown using transfer matrices that the number | S n (123; 3214)| of permutations avoiding the patterns 123 and 3214 is the Fibonacci number F 2 n (as are also | S n (213; 1234)| and | S n (213; 4123)|). We now find the transfer matrix for | S n (123; r, r − 1,…,2, 1, r − 1)|, | S n (213; 1,2,…, r, r + 1)|, and | S n (213; r + 1, 1, 2,…, r)|, determine its characteristic polynomial in terms of the Chebyshev polynomials, and go on to determine the generating function as a quotient of modified Chebyshev polynomials. This leads to an asymptotic result for each r which collapses to the exact results 2 n when r = 2 and F 2 n when r = 3 and to the Catalan number c n as r → ∞. We observe that our generating function also enumerates certain lattice paths, plane trees, and directed animals, giving hope that these areas of combinatorics can be applied to enumerating permutations with excluded subsequences.

Linear stability analysis of plane Couette flow with viscous heating

The linear stability of a plane Couette flow of Newtonian fluids with exponential dependence of viscosity on temperature according to the Arrhenius law and Nahme law is investigated. The modified Chebyshev polynomials of the second kind are used in Galerkin’s method to solve the eigenvalue problem. The combinations of Chebyshev polynomials to form the modified ones for satisfying the required boundary conditions, and their orthogonality conditions, are derived. It is found that an analysis that does not include enough terms of the trial functions may lead to erroneous results. The results show that instability occurs for fluids with highly temperature-sensitive viscosity in a range of the Brinkman numbers; when the temperature-sensitive viscosity is small enough, instability is not observed. Furthermore, only two kinds of modes, the viscous and inviscid ones, are found for the instability of the flow system. Also, the results indicate that the fluids obeying the Arrhenius-type model are more stable than those of the Nahme-type model if both are based on the same temperature-sensitive viscosity, reference viscosity, and temperature.

Time-step sequences for parabolic differential equations

The Euler forward difference method is used for the explicit time integration of stiff systems of ordinary differential equations which originate from spatial discretization of parabolic partial differential equations. A sequence of non-equidistant time steps is derived. It primarily depends on one positive parameter, which determines an upper bound for the distance between the analytical solution and the approximation at every discrete time level in case the Jacobian matrix is constant. Use is made of modified Chebyshev polynomials. The asymptotic rate of convergence of the method is derived and its applicability in case the Jacobian matrix is non-constant is demonstrated.

A direct solver for a class of symmetric linear systems

A direct method is proposed for the numerical solution of a wide class of symmetric block-tridiagonal systems of linear equations resulting from a finite difference or finite element discretization of elliptic boundary value problems with Dirichlet boundary value conditions on rectangular domains. The method is based on a result which generalizes the classical Cramer's rule, and on an extensive use of the modified Chebyshev polynomials for the formulation of the algorithm. Numerical examples are given to illustrate the validity of the proposed approach.

Numerical experiments with a three-dimensional model of an enclosed basin

A three-dimensional numerical model, for simulating velocity and temperature variations in a basin under various forcing mechanisms, is presented. Vertical structure is analyzed using a Galerkin Method with modified Chebyshev polynomials as expansion functions. With the proper choice of these functions, the barotropic and baroclinic parts of the motion are separated naturally in the model. A time-splitting procedure is then implemented; this makes time integration very efficient.Internal seiching due to windstress is simulated in the first experiment. Results are compared with those obtained using two-layered and 3D models. The geostrophic adjustment of a river flowing into a basin is then simulated. When friction and entrainment is neglected, flow adjusts to geostrophic balance over a distance about one third of the width of the basin. Weak turbulence has a large effect on the flow. For example, with a vertical eddy coefficient of 0.001 m2 s−1, isothermals are distorted so that their angle of intersection with the basin axis is large. Geostrophic adjustment still takes place over one third of the width but, due to frictional effects the flow is now quite large all over the lake.

Model reduction by Chebyshev polynomial techniques

The problem of reduced-order modeling of high-order, linear, time-invariant, single-input, single-output systems is considered. A method is proposed, based on manipulating two Chebyshev polynomial series, one representing the frequency response characteristics of the high-order system and the other representing the approximating low-order model. The method can be viewed as generalizing the classical Padé approximation problem, with the Chebyshev polynomial series expansion being over a desired frequency interval instead of a power series about a single frequency point. Two different approaches to the problem are considered. Firstly, approximation is carried out in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> -plane by a Chebyshev polynomial series. Then, modified Chebyshev polynomials are introduced and a mapping to a new plane is defined. It turns out that in the new plane the advantages of the generalized Chebyshev-Padé approximations are retained while what is actually being solved is the classical Padé problem.

Marching Algorithms for Elliptic Boundary Value Problems. I: The Constant Coefficient Case

Several new fast direct methods for solving constant coefficient elliptic boundary value problems are presented. The methods make extensive use of the algebraic properties of the modified Chebyshev polynomials $S_n (x)$ and $C_n (x)$, which allow us to obtain operation counts of $O(n^2 )$ or $O(n^2 \log _2 ({n / k}))$ for solving problems on an $n \times n$ grid. The algorithms are shown to be numerically stable by giving a Wilkinson-style error analysis. Previously studied fast direct methods, and shooting and multiple shooting techniques are related to the algorithms.

Method of orthogonal polynomials in plane antisymmetric mixed problems of elasticity theory with two contact sections

The possibility is shown of applying the method of orthogonal polynomials to solve some-integral equations of a special kind if the eigenfunctions of the integral operator corresponding to the principal (singular) part of the kernel are unknown. Use of the classical scheme [1–3] is impossible in this case. However, by using modified Chebyshev polynomials, an integral equation of the form ∫k1 ε(ξ) ln |ξ+xξ−x| dξ = πf(x) − ∫k1 ε(ξ)G (ξ, x, λ)dξG (ξ, z, λ) = ξxG∗ (ξ, x, λ), k ⩽ x ⩽ 1, λ ∈ (0, ∞), k ∈ (0, 1) is successfully reduced to an infinite algebraic system of the first kind convenient for approximate solution. Here λ, k are dimensionless parameters, G∗ is a continuous, even, and symmetric function in ξ,x. Plane antisymmetric mixed problems of elasticity theory with two contact sections, odd in x. reduce to equations of the type (0.1). The odd function f (x) describes the shape of the boundary layer on the contact section k ⩽ | x | ⩽ 1 altered under the effect of stamps. Considered as an illustration is the problem of impressing two flat stamps into a strip.

Waveform vector analysis of orthogonal electrocardiograms: Quantification and data reduction

Quantification of vectorcardiogram (VCG) analysis by waveform vector analysis (WVA) provides a more stable data base for computer analysis of P, ST and T vectors than does polar-cardiography. Furthermore, compared with data bases composed of Cartesian or polar coordinates of ‘instantaneous’ vectors, it reduces the data and also reduces noise by smoothing. This method is applicable to electrocardiogram (ECG) analysis also. Modified Chebyshev polynomials up to the quaternary term are used as waveform functions for signal representation of scalar ECG, and corresponding waveform coefficients quantify the mean value, gradient, degree of concavity or convexity, biphasic or sigmoid shape, and triphasic or quarternary features. Applied to a vector function of time, the waveform coefficients comprise components of the five vectors—mean (MV), gradient (GV), convex (CV), sigmoid (SV), and quaternary vectors (QV). A waveform power index is derived to express the contribution of each component to overall variation. P, ST, and T vectors of 1,984 rest and exercise VCG of 992 male subjects were analyzed. Representation of the vector function was considered adequate when the approximation error was <25 μV (root mean square; RMS) for each scalar component. In 95% of the records, two waveform vectors adequately reproduced the ST vectors, four were needed for P vectors, and five for T vectors.