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Let 𝒫(n, D) be the set of all monic polynomials in ℤ/nℤ[x] of degree D. A least common multiple for 𝒫(n, D) is a monic polynomial L ∈ ℤ/nℤ[x] of minimal degree such that f divides L for all f ∈ 𝒫(n, D). A least common multiple for 𝒫(n, D) always exists, but need not be unique; however, its degree is always unique. In this article, we establish some bounds for the degree of a least common multiple for 𝒫(n, D), present constructions for common multiples in ℤ/nℤ[x], and describe a connection to rings of integer-valued polynomials over matrix rings. [ABSTRACT FROM PUBLISHER]

The new originally capital general solution of determining the prototype filter function as the response that satisfies the specifications of all pole low-pass continual time filter functions of odd and even order is presented in this article. In this article, two new classes of filter functions are proposed using orthogonal and orthonormal Jacobi polynomials. The approximation problem of filter function was solved mathematically, most directly applying the summed Christoffel–Darboux formula for the orthogonal polynomials. The starting point in solving the approximation problem is a direct application of the Christoffel–Darboux formula for the initial set of continual Jacobi orthogonal polynomials in the finite interval in full respect to the weighting function with two free real parameters. General solution of the filter functions is obtained in a compact explicit form, which is shown to enable generation the Jacobi filter functions in a simple way by choosing the numerical values of the free real parameters. For particular specifications of free parameters, the proposed solution is used with the same criterion of approximation to generate the appropriate particular filter functions as are: the Gegenbauer, Legendre and Chebyshev filter functions of the first and second kind as well. The examples of proposed filter functions of even and odd order are illustrated and compared with classical solutions. [ABSTRACT FROM AUTHOR]

In diffusion transport, the flux at a point is typically modeled in terms of the local gradient of a potential. When heterogeneities are present, this local model can break down and it may be more appropriate to model the diffusion flux as a weighted sum of gradients present throughout the domain. Here a discrete nonlocal flux model-consistent with control-volume implementations-is developed. This scheme is referred to as the control-volume weighted flux scheme (CVWFS). The key component is the modeling of the diffusion flux at a given control-volume face in terms of a weighted sum of gradients at that face and at faces up- and downstream. Criteria for choosing the weights are proposed. This results in numerical solution schemes in which the coefficient matrix is diagonally dominant, has positive off-diagonal elements, and zero row sums. For a particular power-law weighting scheme it is shown how the CVWFS is related to the definition of the Caputo fractional derivative and the one-shift Grunwald approximation of the Riemann-Liouville fractional derivative. On developing transients and boundary condition treatments, the accuracy and suitability of the CVWFS scheme is demonstrated by solving a number of problems governed by Caputo fractional diffusion equations. [ABSTRACT FROM AUTHOR]

In this article, we give some necessary and sufficient conditions under which the Moore-Penrose inverse of a lower triangular operator has the lower and upper triangular forms, respectively. As a special case, we recover some theorems and corollaries of Chen et al. [Perturbation analysis of the generalized Bott-Duffin inverse of L-zero matrices, Linear Multilinear Algebra 51 (2003), pp. 11-20] and Deng and Chen [A note on the generalized Bott-Duffin inverse, Appl. Math. Lett., 20 (2007), pp. 746-750]. [ABSTRACT FROM AUTHOR]

Many applications in electrical engineering, signal processing and mathematical physics lead to the following approximation problem: let h be a short linear combination of nonincreasing exponentials with complex exponents. Determine all exponents, all coefficients and the number of summands from finitely many equispaced sampled data of h. This is a nonlinear inverse problem. This article is an extension of Potts and Tasche [Parameter estimation for exponential sums by approximate Prony method, Signal Process., 90 (2010), pp. 1631-1642], where only the case of sums of complex exponentials with real frequencies is considered. In the following, we present new results on an approximate Prony method (APM) which is based on Beylkin and Monzon [On approximations of functions by exponential sums, Appl. Comput. Harmon. Anal. 19 (2005), pp. 17-48]. In contrast to their results, we apply perturbation theory for a singular value decomposition of a rectangular Hankel matrix such that we can describe the properties and the numerical behaviour of APM in detail. For this inverse problem, the number of sampled data acts as regularization parameter. The first part of APM recovers the exponents. The second part computes the coefficients by an overdetermined linear Vandermonde-type system. Numerical experiments show the performance of our method. [ABSTRACT FROM AUTHOR]

Aspects of the science and the ways of doing science characteristic of the two senior scientists of Quantum Theory Project (QTP) in the period 1969-1985 are discussed from the perspective of the only QTP faculty member of that era who did not grow up in either the Lowdin or Slater schools. [ABSTRACT FROM AUTHOR]

This article deals with the uniqueness and stability of coexistence solutions of a basic N-dimensional competition model in the unstirred chemostat by Lyapunov-Schmidt procedure and perturbation technique. It turns out that if the parameter G ≠ 0, which is given in Theorem 1.1, this model has a unique coexistence solution provided that the maximal growth rates a, b of u, v, respectively, lie in a certain range. Moreover, the unique coexistence solution is globally asymptotically stable if G > 0, while it is unstable if G < 0. In the later case, the semitrivial equilibria are both stable. [ABSTRACT FROM AUTHOR]

In this paper, the condition number of singular value is discussed. The expressions of the zero-structured and patterned conditioning are also studied. The numerical tests are reported to provide interesting information about the singular value sensitivity when the perturbations of the matrix have arbitrarily assigned zero-structured and patterned case. [ABSTRACT FROM AUTHOR]

We consider a periodic pseudodifferential operator H = (- Δ)l + A (l > 0) in Rd which satisfies the following conditions: (i) the symbol of H is smooth in x, and (ii) the perturbation A has order smaller than 2l - 1. Under these assumptions, we prove that the spectrum of H contains a half-line. [ABSTRACT FROM AUTHOR]

This aritcle describes a simple alternative unified method of solving nonlinear regular perturbation problems. The procedure is based upon the manipulation of Taylor's approximation for the expansion of the nonlinear term in the perturbed equation. An essential feature of this technique is the relative simplicity used and the associated unified computational procedure that is employed. As such it should be of interest to teachers of applied mathematics courses, particularly those courses which include perturbation methods. One of the merits of this approach is that it leads on naturally to a scheme based on Taylor's expansions and, consequently, allows the regular perturbation method to be introduced into a course much earlier than is currently common. The method is illustrated by implementing it to four perturbation problems, including two algebraic equations and two initial-value problems. The new approach is compared and contrasted with the traditional perturbation scheme in order to demonstrate its relative merit. [ABSTRACT FROM AUTHOR]

In the framework of recently introduced consistent Bhatnagar-Gross-Krook (BGK) approximations of the Boltzmann equations for both reactive and nonreactive gas mixtures, the problem of an appropriate choice of relaxation times, which represent the crucial parameters of the model, is addressed. In this first approach, BGK results are tested versus the only exact analytical solutions available in the literature for mixtures, namely the Bobylev-Krook-Wu (BKW) modes for space-homogeneous inert Maxwellian molecules. Results indicate, as expected, that using the actual average collision times in the BGK equations produces a too-fast dissipation (entropy production) with respect to the Boltzmann description, and that the best fit in that trend is obtained if such times are amplified by a suitable factor, roughly of order two in the cases run. [ABSTRACT FROM AUTHOR]

This paper is concerned with asymptotic behaviour of solutions of perturbed dynamic systems on time scales. A time scale version of the Hartman-Wintner theorem is established for a class of time scales. [ABSTRACT FROM AUTHOR]

In article [Abdel-Aziz, M.R., 2004, Relative perturbation theory for matrix eigenproblems in free vibration analysis (International J. of Computer Mathematics, Vol. 81, No. 4, pp. 505–514) 1], we have introduced bounds for eigenvalues of eigenproblems in free vibration analysis, when the perturbation happens in either the mass matrix or in the stiffness matrix. This article aims to bound the additive and multiplicative relative distances and ,respectively, where ? and µ are eigenvalues and one is the perturbation of the other. These bounds are investigated when the perturbation happens in both the mass and the stiffness matrices. Numerical experiments show that the investigated bounds are satisfied and their values are depend on the element-wise perturbation of the matrix. [ABSTRACT FROM AUTHOR]

We give conditions under which a system of the formwill have constant or periodic solutions ( x ( t ), y (? t )) for and sufficiently small, such that for Here [ t ] denotes the greatest integer function. [ABSTRACT FROM AUTHOR]

A new error metric is applied for estimating accuracy of MoM solutions on purely 3-D geometries using triangle doublet basis functions. This error metric is based on checking boundary conditions performance (BCP) on scatterer surface and shown to be suited for arbitrary 3-D geometries including open ones. First, accurate expressions for the scattered field are derived to be valid at any observation points including those on the surface of triangles. Further, BCP error metric is examined for estimating accuracy of the scattering problem solution on open cube geometry, and to find the correlation of BCP error with that for near-field characteristics. Finally, BCP error metric is applied to estimate accuracy of MoM solutions on realistic car model, and to find the contributions of its elements to the total error. [ABSTRACT FROM AUTHOR]

The calculation of the self impedance in the spectral domain is rigorously investigated. This method handles the strong singularity of the dyadic Green's function in the spectral domain and can be applied to both surface and volume integrals resulting from the moment method analysis of electromagnetic problems. The singular and inefficient spatial domain integration is converted into a double spectral domain integration which can be performed efficiently. The use of a finite value for the upper limit for the spectral domain integration is the only approximation in this method. Compared with spatial domain methods, this method is simple and general, requiring no additional formulations except the spectral domain dyadic Green's function. Numerical results are provided to illustrate this method. [ABSTRACT FROM AUTHOR]

A simple, straightforward and efficient method is proposed for the calculation of the period of anharmonic oscillators as an energy series. The approach is based on perturbation theory and the method of Lindstedt— Poincaré. [ABSTRACT FROM AUTHOR]

The unwanted sensitivity of a plate pyroelectric detector to airborne acoustic noise is depends critically on the mounting conditions for the plate. We consider a plate in an isotropic acoustic pressure field oscillating at angular frequencyωd. We analyze the situation in terms of a mechanically isotropic material, and show that for one type of boundary conditions, the so-called clamped-boundary conditions, the strain-induced time-varying surface charges caused by the pressure in various regions of the plate, completely cancel out. This suggests that significantly reduced noise sensitivity of a practical free-standing pyroelectric detector can be achieved by paying careful attention to the mounting conditions. We also derive the strains for the case of the trigonal 3m class within the framework developed for the thin-plate approximation, and show that the result is unchanged. This has implications for the design of pyroelectric detectors based on a stiff material such as LiTaO3.∗Contribution of the National Institute of Standards and Technology, not subject to Copyright. [ABSTRACT FROM AUTHOR]

The ab initio calculation of molecular geometrical properties in the Hellmann-Feynman approximation is discussed in which the atomic orbitals are fixed at the positions of the nuclei at the reference geometry, thereby avoiding the calculation of derivatives of the molecular integrals with respect to the positions of the atomic orbitals. For the molecular gradient, the molecular Hessian, and the molecular dipole gradient, the convergence of the calculated properties is studied for a large number of basis sets at the Hartree-Fock level and at the CCSD(T)-R12 level. In the Hellmann-Feynman approximation, it is found to be necessary to impose explicitly rotational and translational invariance. Although small basis sets perform poorly in the Hellmann-Feynman approximation (compared with the standard approach where the atomic orbitals are moving with the displaced nuclei), satisfactory convergence is obtained for geometries and harmonic frequencies (to within 1% of the standard approximation) with the larger of the correlation-consistent core-valence cc-pCVXZ basis sets. For the infrared intensities, the agreement with the standard approach is still poor (only within 15% for the largest correlation-consistent basis). The best results are obtained with an R12 basis previously developed for the calculation of energies in the explicitly correlated R12 approximation. In this basis, the geometrical parameters and harmonic frequencies are within 0.5% of the standard approach and the infrared intensities within 5% , suggesting that the Hellmann-Feynman approximation may be useful for applications at the highly accurate MP2-R12, CCSD-R12, and CCSD(T)-R12 levels of theory. [ABSTRACT FROM AUTHOR]

The analysis of electromagnetic scattering from cavity structure is very important to many practical applications. The field iterative method (FIM) is one of the promising methods to deal with the cavity problem. In this paper, 3-dimensional (3D) FIM has been implemented using Rao-Wilton-Glisson (RWG) basis function and an accurate equivalent model of the cavity. Two testing procedures, a newly developed point matching and conventional Galerkin's methods, have been discussed for better and simpler implementation of the 3D FIM. Numerical results show that the accuracy of the 3D implementation of FIM using the newly developed point matching method is the same as that of the conventional Galerkin's method. The numerical results also show that the simpler implementation of 3D FIM using the point matching method converges very fast for all the tested cases. [ABSTRACT FROM AUTHOR]