Showing papers on "Orthonormal basis published in 1985"
TL;DR: In this paper, the beam theory derived from variational principles is based on the sole kinematic assumption that each section is infinitely rigid in its own plane, but free to warp out of plane.
Abstract: Beam theory plays an important role in structural analysis. The basic assumption is that initially plane sections remain plane after deformation, neglecting out-of-plane warpings. Predictions based on these assumptions are accurate for slender, solid, cross-sectional beams made out of isotropic materials. The beam theory derived in this paper from variational principles is based on the sole kinematic assumption that each section is infinitely rigid in its own plane, but free to warp out of plane. After a short review of the Bernoulli and Saint-Venant approaches to beam theory, a set of orthonormal eigenwarpings is derived. Improved solutions can be obtained by expanding the axial displacements or axial stress distribution in series of eigenwarpings and using energy principles to derive the governing equations. The improved Saint-Venant approach leads to fast converging solutions and accurate results are obtained considering only a few eigenwarping terms.
156 citations
TL;DR: The new discriminant analysis with orthonormal coordinate axes of the feature space is proposed, which is more powerful than the traditional one in so far as the discriminatory power and the mean error probability for coordinate axes are concerned.
153 citations
TL;DR: In this paper, weakly convergent expansions of a plane wave are constructed in such a way that their application in Fourier integrals leads to expansions of the Fourier or inverse Fourier transform that converge with respect to the norm of either the Hilbert space L2(R3) or the Sobolev space W(1)2 (R3).
Abstract: The Fourier transform of an irreducible spherical tensor is normally computed with the help of the Rayleigh expansion of a plane wave in terms of spherical Bessel functions and spherical harmonics. The angular integrations are then trivial. However, the remaining radial integral containing a spherical Bessel function may be so complicated that the applicability of Fourier transformation is severely restricted. As an alternative, the use of weakly convergent expansions of a plane wave in terms of complete orthonormal sets of functions is suggested. The weakly convergent expansions of a plane wave are constructed in such a way that their application in Fourier integrals leads to expansions of the Fourier or inverse Fourier transform that converge with respect to the norm of either the Hilbert space L2(R3) or the Sobolev space W(1)2(R3). Accordingly, these weakly convergent expansions may be viewed as distributions that are defined on either L2(R3) or W(1)2(R3). The properties of some complete orthonormal se...
119 citations
TL;DR: In this article, the R-matrix method is used to solve the close-coupling (CC) problem of electron-atom collision theory, where wavefunctions Psi E for a system containing (N+1) electrons are expanded in terms of products of N-electron target functions multiplied by fully optimised orbital functions for a 'colliding' electron.
Abstract: In the close-coupling (CC) method of electron-atom collision theory, wavefunctions Psi E for a system containing (N+1) electrons are expanded in terms of products of N-electron 'target' functions multiplied by fully optimised orbital functions for a 'colliding' electron. The method can also be used to calculate energies En and functions Psi n for bound states of the (N+1)-electron system: this is often referred to as the 'frozen cores' (FCS) approximation. In the R-matrix method the CC problem is solved for an inner region, r or=a. The paper includes a discussion of the FCS approximation and a summary of R-matrix theory. Techniques are described for calculating orthonormal sets of outer-region solutions correct to first order in the long-range multipole potentials. Matching of inner-region to outer-region solutions gives an eigenvalue problem for the determination of the energies En which is solved using scanning techniques which do not require estimates of the En to be provided.
112 citations
TL;DR: In this article, a three-dimensional generalization of a construction originally due to Harriman is proposed, and numerical assessments are made of several new density functionals by evaluating them using accurate Hartree-Fock densities and by solving the corresponing Euler equations for electron density.
Abstract: Orthonormal orbitals systematically constructed from the electron density are employed to obtain various closed expressions for approximate atomic energy functionals. A three‐dimensional generalization of a construction originally due to Harriman is proposed. Numerical assessments are made of several new density functionals by evaluating them using accurate Hartree–Fock densities and by solving the corresponing Euler equations for electron density. The molecular virial theorem is stated and proved in a form particularly suitable for density functional theory.
97 citations
TL;DR: In this paper, an extended coherent-state theory is used to give a simple construction for the matrix elements of the proton-neutron sp(4) quasispin algebra.
34 citations
TL;DR: In this paper, the authors investigated the class of orthonormal bases in L 2 (0, 1) with respect to which all quasi-martingales become integrable in the sense of non-causal integration.
Abstract: We will investigate the problem of characterizing the class of such orthonormal bases inL
2([0, 1]) with respect to which all quasi-martingales become integrable in the sense of noncausal integration. As an application of the results, we will show that the integral of noncausal type is a natural generalization of symmetric integrals.
33 citations
TL;DR: In this article, a profile function is used to construct an appropriate orthonormal basis in a weighted L2 space in which to expand the solution and also a basis of vectors for expanding the experimental data.
Abstract: Previous work on the use of singular systems for the inversion of photon correlation functions obtained in the analysis of molecular polydispersity by laser light scattering is extended to allow the imposition of a ‘‘profile’’ function, having the measured mean and polydispersity index, to localize the recovered solution and improve its resolution. This profile function is used to construct an appropriate orthonormal basis in a weighted L2 space in which to expand the solution and also a basis of vectors for expanding the experimental data.
31 citations
TL;DR: In this article, the problem of algebraic fitting of experimental thermodynamic excess data can be solved explicitly (i) by means of Gauss method of elimination, and (ii) by employing the property of orthonormality of polynomials.
Abstract: Following the Weierstrass approximation theorem the thermodynamic excess functions are representable with arbitrary high accuracy by means of polynomials of sufficient high degrees in the mole fraction x. So, algebraic fitting of experimental thermodynamic excess data can be based upon mathematical polynomial expressions without any loss of generality. With respect to the necessary scattering of experimental results, algebraic evaluation of those data can only be solved by employing the calculus of observations. The least square method is the only principle of fitting with full justification by statistical mathematics, and which can be applied directly for algebraic fitting of experimental data by means of a computer. The general linear problem of fitting is solved explicitly (i) by means of Gauss method of elimination, and (ii) by employing the property of “orthonormality” of polynomials. In the latter case the explicit form of the “orthonormal” polynomials depends strongly on the number of experimental data which has to be fitted. A convenient procedure is presented to generate polynomials which are orthonormal with respect to an actual set of experimental data. Computer-programs in PORTRAN-language are enclosed 1) to employ Gauss method of elimination, and 2) to generate discrete orthonormal polynomials.
24 citations
TL;DR: In this article, an orthonormal system based on the classical Legendre polynomial system is incorporated in the adaptive determination of the score generating function, and the proposed sequential procedure is based on a suitably posed stopping rule.
Abstract: For the simple regression model ( containing the two—sample location model as a special case ), adaptive ( linear ) rank statistics arising in the context of ( asymtotically) efficient testing and estimation procedures are considered. An orthonormal system based on the classical Legendre polynomial system is incorporated in the adaptive determination of the score generating function, and the proposed sequential procedure is based on a suitably posed stopping rule. Various properties of this sequentailly adaptive procedure and the allied stopping rule are studied. Asymptotic linearity results ( in a shift or regression parameter ) of linear rank statistics are studied with special reference to the Legendre polynomial system and some improved rates of convergence are estabilished in this context.
24 citations
TL;DR: In this article, a canonical orthonormal basis for generic representations of the group chain SU(3) contains/implies SO(3), which is a generalization of the canonical basis for SO(2).
Abstract: A canonical orthonormal basis is given for generic representations of the group chain SU(3) contains/implies SO(3).
01 Feb 1985
TL;DR: In this paper, the existence of a uniformly bounded orthonormal basis in the space of homogeneous polynomials on the 2-dimensional complex ball B2 was shown.
Abstract: Denote by B2 the unit ball in C2. The existence is shown of a uniformly bounded orthonormal basis in H2 (B2), by constructing such systems in the spaces of homogeneous polynomials. In the second part of the paper, those spaces of homogeneous polynomials are exploited to disprove the existence of generalized analytic projections, the so-called (ip-,gp) property, for the ball algebra. Summary. In the first part of the paper we use a construction in the spaces of homogeneous polynomials on the 2-dimensional complex ball B2 to generate an orthonormal basis for the space H2(B2) which is uniformly bounded. The existence of such a system answers a question raised by W. Rudin.1 The second part is devoted to the failure of the (ip-rp) theorem, known for the disc algebra A(D), in case of the ball algebras A(Bin), m > 1. It is proved that the ideals of p-summing and p-integral operators (p # 2) in A(B2) are distinct. This fact solves negatively a problem considered in [2].
TL;DR: In this paper, the problem of the condition of matrices arising in the numerical solution of integral equations of the first kind by Galerkin and collocation schemes was investigated and a lower bound on the matrix condition number was found and its behavior as a function of the smoothness of the kernel of the original equation was studied.
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01 Jul 1985
TL;DR: In this paper, the Laplace transform transform method was used for boundary value problems with cylindrical and spherical symmetry continuous eigenvalues and Fourier integrals and generalized functions.
Abstract: Basic concepts classification of equations and boundary conditions orthonormal functions applications of Fourier's method problems involving cylindrical and spherical symmetry continuous eigenvalues and Fourier integrals the Laplace transform transform method for boundary value problems Green's functions and generalized functions the numerical approach answers to problems.
TL;DR: In this paper, a simple algorithm is given for the calculation of reduced matrix elements of the generators of the SU(3) algebra in the canonical SU (3) contains/implies SO(3).
Abstract: For pt.I see ibid., vol.18, p.1891 (1985). A simple algorithm is given for the calculation of reduced matrix elements of the generators of the SU(3) algebra in the canonical SU(3) contains/implies SO(3) basis introduced in I.
TL;DR: In this paper, a truncated model was proposed to model blocking patterns with a local approach, where stationary coherent structures were found as asymptotic solutions of the inviscid, quasi-geostrophic potential vorticity equation with a mean zonal wind with vertical and horizontal shear.
Abstract: Many recent studies have been devoted to atmospheric Patterns that persist beyond the synoptic time scale, such as those known as blocking events. In the present paper we explore the possibility that blocking patterns can be modeled with a local approach. We propose a truncated model that is a time-dependent, highly nonlinear extension of our earlier analytical theory. In this theory, stationary coherent structures were found as asymptotic solutions of the inviscid, quasi-geostrophic potential vorticity equation with a mean zonal wind with vertical and horizontal shear, in the limit of weak dispersion and weak nonlinearity. The truncated model is obtained by projecting the potential vorticity equation onto the orthonormal basis defined by the lowest order problem of the asymptotic theory and then suitably truncating the number of modes. The time-evolution of the model is investigated numerically with different truncations. The steady solutions were antisymmetric dipoles, with the anticyclone nort...
TL;DR: In this article, a table of all non-degenerate Γ1, Γ3, and Γ7 representations is given, for the axes of quantization of order 5.
Abstract: The expressions for the Γ2, Γ5, Γ6, Γ8, and Γ9 representations are given in terms of those of Γ1 representations for neighboring angular momenta. The coefficients of the Γ4 representations are expressed in terms of those of Γ7 representations. Therefore, with an arbitrary choice of orthonormal sets of Γ1, Γ3, and Γ7 representations, orthonormal sets of other kinds of representations are well defined and can be labeled with the labels of parent representations. All Clebsch–Gordan coefficients are expressed in terms of those between parent representations (and a few others). Tables of all nondegenerate Γ1, Γ3, and Γ7 representations are given, for the axes of quantization of order 5. With some degenerate Γ3 and Γ7 representations, which are also given, any representation of integer or half‐integer angular momentum up to 27 can be obtained using some usual Clebsch–Gordan coefficients of SU(2).
TL;DR: In this article, a simple approximation formula, valid to the high degree of accuracy of the Toronto approximation, is given for both the off-diagonal and diagonal matrix elements of the kappa matrices which transform the bosonic realisations of the symplectic algebra into an orthonormal basis for a unitary irreducible representation of Sp(6, R).
Abstract: A simple approximation formula, valid to the high degree of accuracy of the Toronto approximation, is given for both the off-diagonal and diagonal matrix elements of the kappa matrices which transform the bosonic realisations of the symplectic algebra into an orthonormal basis for a unitary irreducible representation of Sp(6, R).
TL;DR: In this article, the step response of a closed loop system is identified by means of a Laguerre expansion, which offers certain advantages over ARMA models, namely lack of bias in the estimates, structural flexibility and the ability to pre-compute the regressors.
19 Jun 1985
TL;DR: In this paper, a dynamic compensator for a linear time invariant multivariable system is proposed to make the eigenvector set maximally orthonormal, which can be solved by efficient and numerically stable algorithms.
Abstract: For a linear time invariant multivariable system, this paper describes a procedure for the design of a dynamic compensator that stabilizes the closed loop system and causes the closed loop system eigenstructure to be robust in the sense of making the eigenvector set maximally orthonormal The procedure consists of solution of Sylvester's equation and a least square problem Both these problems can be solved by efficient and numerically stable algorithms for which standard software is available Numerical examples will be given in later section
TL;DR: In this paper, all finite-dimensional irreducible representations of the special linear Lie superalgebra sl(1,3) are constructed and explicit formulas for the generators in an orthonormal Gel'fand-Zetlin basis of the even subalgebra gl(3) were given.
Abstract: In a series of two papers all finite‐dimensional irreducible representations of the special linear Lie superalgebra sl(1,3) are constructed. Explicit formulas for the generators in an orthonormal Gel’fand–Zetlin basis of the even subalgebra gl(3) are given. This paper develops a background for constructing the representations. Expressions for the transformation properties of the basis under the action of the generators are written down within all typical sl(1,3) modules.
TL;DR: In this article, the authors proposed a nonparametric method of estimation of density using orthonormal systems iteratively, where the asymptotic mean integrated square error of the estimate at each stage is less than or equal to that of the preceding stage.
TL;DR: In this article, the parent representations of the Γ4, Γ6, and Γ8 representations are derived in terms of Γ1 representations for neighboring angular momenta.
Abstract: Expressions for the Γ4, Γ6, and Γ8 representations are derived in terms of the Γ1 representations for neighboring angular momenta. Expressions for Γ5, Γ7, and again for Γ8 are derived in terms of the Γ2 representations. A third set of expressions for Γ8 is derived in terms of the Γ3 representations. With an arbitrary choice of orthonormal sets of the Γ1, Γ2, and Γ3 representations, orthonormal sets of all other kinds of representations are thus well defined and can be entirely labeled using the parent representations. All Clebsch–Gordan coefficients are expressed in terms of those between parent representations (and a few other ones). The Γ4 and Γ5 representations defined here are not those conventionally used, but they provide a simpler expression for the fictitious spin coefficients. Tables of the parent representations Γ1, Γ2, and Γ3 are given for quaternary and ternary axes of quantization. Using the usual Clebsch–Gordan coefficients of SU(2), these tables allow us to obtain any representation for int...
TL;DR: The purpose is to show how to get a series of lower triangular matrices by alternate orthogonal-upper triangular decompositions in different dimensions and to prove the convergence of this series.
Abstract: A generalization of the QR algorithm proposed by Francis [2] for square matrices is introduced for the singular values decomposition of arbitrary rectangular matrices. Geometrically the algorithm means the subsequent orthogonalization of the image of orthonormal bases produced in the course of the iteration. Our purpose is to show how to get a series of lower triangular matrices by alternate orthogonal-upper triangular decompositions in different dimensions and to prove the convergence of this series.
01 Jan 1985
TL;DR: A sufficient condition for the tightness of Hubert valued martingales is given in this article, where Arnold, Theodosopulu and Kotelenez give a sufficient condition.
Abstract: This paper gives a sufficient condition for the tightness of a sequence (Mn)(n∈ℕ) of Hubert valued martingales. This result applies directly to some situations of “accompanying martingales” as considered for example by L. Arnold, M. Theodosopulu and P. Kotelenez.
TL;DR: The canonical structure of the Dirac field is determined by the canonical symplectic structure of a bundle cotangent to the manifold of all the orthonormal triads on a Cauchy surface.
01 Jan 1985
TL;DR: In this paper, the authors define an orthonormal system defined on a positive measure space (X, F, μ) and a sequence of real numbers (coefficients) on the same space.
Abstract: Let (X, F, μ) be a positive measure space, {ϕi (x): i= 0, 1, …} an orthonormal system (in abbreviation: ONS) defined on X, and {ai} a sequence of real numbers (coefficients).