Showing papers on "Basis (linear algebra) published in 1976"
TL;DR: In this paper, a computational method based on a rapidly convergent form of the unlinked cluster expansion is presented, where Ciźek's coupled-pair approximation is derived in a basis of partially non-orthogonal orbitals which transform each pair function to diagonal form; this produces a simple (non-variational) set of equations from which may be extracted the energy and coefficients of a wavefunction constructed from the Hartree-Fock function, all double excitations and all unlinked clusters of these.
88 citations
TL;DR: The use of identical basis sets for monomer and dimer, in intermolecular calculations, is discussed in this article, where evidence is presented that such a technique eliminates a basis set extension effect and gives a considerably better description of the intermolescular interaction than normal supermolecule calculations.
80 citations
TL;DR: The technique of Bakhvalov and Vasil'eva for evaluating Fourier integrals is discussed and used as the basis for providing computationally efficient and stable algorithms as discussed by the authors.
Abstract: The technique of Bakhvalov and Vasil'eva for evaluating Fourier integrals is discussed and used as the basis for providing computationally efficient and stable algorithms The method is generalized to deal with a number of weight functions including those for the infinite ranges A number of examples illustrate the methods
50 citations
01 Jan 1976
TL;DR: In this article, the abstract theorem concerning Cauchy problem solution in a Scale of Banach Spaces is given, and a strict justification of the shallow water theory in a class of analytical functions is presented.
Abstract: An exposition is given of the abstract theorem concerning Cauchy problem solution in a Scale of Banach Spaces. Its basis is the concept of quasidifferential operator. The second part is devoted to a strict justification of the shallow water theory in a class of analytical functions. This is approached by means of the abstract theorem on the example of the plane problem for unsteady periodical waves.
49 citations
TL;DR: It is shown how one can use splines, represented in the B-spline basis, to reduce the difficulties of large storage requirements in dynamic programming via approximations to the minimum-return function without the inefficiency associated with using polynomials to the same end.
35 citations
TL;DR: A procedure to construct a finite sub-set of the cycle vector space containing the elements of all minimal bases makes the generation of the required basis feasible by a finite procedure, such as Welsh's generalization of the Kruskal algorithm.
Abstract: The development of the flexibility method of analysis of skeletal structures has been hindered by the difficulty of determining a suitable statical basis on which to form the flexibility matrix. A combinatorial approach reduces the difficulty to one of selecting a minimal basis of the cycle vector space. After an introduction to flexibility analysis and a brief review of earlier work using combinatorics, the paper presents a procedure to construct a finite sub-set of the cycle vector space containing the elements of all minimal bases. This makes the generation of the required basis feasible by a finite procedure, such as Welsh's generalization of the Kruskal algorithm. It is thus possible to have an automatic method for the analysis of skeletal structures which uses an optimal combinatorial approach.
33 citations
TL;DR: In this article, the concept of local rotational invariance was introduced to increase the flexibility of floating spherical Gaussian orbital (FSGO) basis sets, and it was shown how full sets of p-and d-type atomic orbital components can be included in an FSGO basis using substantially fewer basis functions than needed in traditional lobe-function basis sets.
Abstract: An analysis is presented that shows how floating spherical Gaussian orbital (FSGO) basis sets can be related to atomic Gaussian orbital basis sets. In addition, through the development of the concept of local rotational invariance, it is shown how FSGO basis sets can, in a systematic and straightforward manner, be generalized and extended to increase the flexibility of the basis. To illustrate the concepts, it is shown how full sets of p- and d-type atomic orbital components can be included in an FSGO basis using substantially fewer basis functions than needed in traditional lobe-function basis sets.
32 citations
30 citations
Journal Article•
TL;DR: In this article, it was shown that the maximal real subfield of the n-th cyclotomic field Q(£n) is an integral basis of the principal order of the real number ζη + ε.
Abstract: Let C„ be a primitive n-th root of unity and #=(?(£,, + ζ') be the maximal real subfield of the n-th cyclotomic field Q(£n). It is proved in this paper that &»!_! {l, ς + C, . . ., (ί,, + Ο 2 } is an integral basis of*. Throughout, the following notations will be used : n a positive integer greater than 2, Q the rational number field, ζη a primitive n-th root of unity, φ (n) the Euler φ-function of «, L = (C«) the «-th cyclotomic field, Κ=ζ)(ζη + ζ~) the maximal real subfield of L, D (L), D (K) the absolute field discriminants of L and K respectively, dn = ά(ζη + C) the discriminant of the real number ζη + ζ\" over , ^L/K relative discriminant of L over K, NK/Q absolute norm taken from *over Q, A = {l, ς + C, · . ·, (Cn + C)^\"} the principal order in #generated by ζη + ζή. The following theorem is due to Lehmer [1]. Theorem 1. The discriminant dn ofthe real number ζη + ζ~ J is given by
28 citations
TL;DR: The fundamental and all encompassing nature of the generalized partitioned algorithms (hereon denoted GPA) is clearly demonstrated by showing that the GPA contain as special cases important generalizations of past well-known linear estimation algorithms, as well as whole families of such algorithms, of which all previously obtained major filtering and smoothing algorithms are special cases.
TL;DR: It is shown that the Weyl basis formed by the canonical symmetrization of an n‐dimensional, p‐rank tensor space with canonical projection operators of Sp is a Gel’fand basis of U(n).
Abstract: It is shown that the Weyl basis formed by the canonical symmetrization of an n‐dimensional, p‐rank tensor space with canonical projection operators of Sp is a Gel’fand basis of U(n). This basis may easily be generated using standard projection operator techniques.
TL;DR: In this article, it was shown that if the field directions (inclination and declination) are completely known on the surface of the earth, the geomagnetic potential can be determined uniquely except an arbitrary multiplicative constant.
Abstract: It is shown that if the field directions (inclination and declination, or other combination of two independent angles) are completely known on the surface of the earth, the geomagnetic potential can be determined uniquely except an arbitrary multiplicative constant. On the other hand, when declinations only are specified on the surface there are infinitely many potentials which satisfy exactly the same boundary conditions. Such non-uniqueness seems also to be present for the cases of other incomplete data set composed of one angle only. The uniqueness theorem serves as the basis for spherical harmonic analyses of the paleomagnetic field.
TL;DR: In this article, Mitiagin and Zobin constructed an example of nuclear Frechet space without basis, and the essential modification of their constructions gives the following results: (1) there exists such a nuclear Freche space X that for any nuclear frechet space Y the space X × Y has no basis (Sections 1 and 2).
01 Dec 1976
TL;DR: In this article, the affine tensor product is used as a basic tool to describe both homogeneous and inhomogeneous bilinear systems and the state space is constructed via Nerode equivalence.
Abstract: From the external descriptions algorithms are obtained for the construction of discrete-time, internally bilinear state-space models with unknown initial states. Both the homogeneous and the inhomogeneous bilinear systems are treated. In both cases explicit existence criteria for minimal realization as well as the state space isomorphism theorems are given. One unusual result is that for a given input/output map there exist minimal realizations which are not equivalent under basis transformations. In the special case of zero initial state the algorithm can be reduced to the one previously obtained by Isidori, and if the input-output map is linear, the algorithm reduces to that given by Ho and Kalman for linear systems. Finally, the abstract realization theory is given. We use the affine tensor product as a basic tool to describe both homogeneous and inhomogeneous bilinear systems. The state space is constructed via Nerode equivalence. This forms a theoretical basis for the realization algorithms obtained in this paper.
TL;DR: In this paper, three different series of Gaussian basis sets of different dimensions were optimized for the fluorine atom by the method of conjugate gradients, and the relation between the optima of different basis spaces and the predictability of optimum basis sets was investigated.
Abstract: Three different series of Gaussian basis sets of different dimensions were optimized for the fluorine atom. The basis sets were chosen from a series of function spaces of increasing dimension and each set was subsequently optimized by the method of conjugate gradients. The interdependence of optimum s and p type basis functions and the effects of cross representations were analyzed. The relation between the optima of different basis spaces and the predictability of optimum basis sets were investigated. The ratios of the orbital exponents showed systematic variations when the dimension of the basis space was extended.
Journal Article•
TL;DR: A simple procedure for the computation of a basis of the general solution of a finite system of linear inequalities is presented, which generalizes the well-known Burger's algorithm for homogeneous systems.
Abstract: A simple procedure for the computation of a basis of the general solution of a finite system of linear inequalities is presented. It generalizes the well-known Burger's algorithm for homogeneous systems.
TL;DR: In this paper, criteria are obtained for solvability of the multiple interpolation problem in the class, where is a proximate order (, ) for any sequence from a certain class or one of its subclasses.
Abstract: In this paper criteria are obtained for solvability of the multiple interpolation problem in the class , where is a proximate order (, ) for any sequence from a certain class or one of its subclasses. The results are applied to find criteria that the system , be a basis in its linear hull.Bibliography: 14 titles.
TL;DR: In this paper, the concept of harmonic oscillator functions is generalized to the multidimensional space of the N-body system and explicit expressions for these functions are found for them.
TL;DR: It turns out that the ordinary weight and net-weight can be easily expressed in terms of the K-versions and some auxiliary functions, moreover under some restrictions, weight andNet-weight actually coincide with the hereditary modifications ofK-weight andK-netweight, respectively.
Abstract: In this paper the notions ofK-nets andK-bases are introduced and the corresponding cardinal functions,K-netweight andK-weight, are studied. Spaces with smallK-nets orK-bases are in some sense close to compact spaces. It turns out that the ordinary weight and net-weight can be easily expressed in terms of theK-versions and some auxiliary functions, moreover under some restrictions, weight and net-weight actually coincide with the hereditary modifications ofK-weight andK-netweight, respectively.
TL;DR: A computer program POLYGRAD based on the POLYATOM/1 system is presented which evaluates analytically the energy gradient using thes-type and Cartesianp-type Gaussian basis functions.
Abstract: A computer program POLYGRAD based on the POLYATOM/1 system is presented which evaluates analytically the energy gradient using thes-type and Cartesianp-type Gaussian basis functions. Model calculations on hydrogen peroxide were made to compare the accuracy and the computer time involved in the analytical and numerical determinations of the energy gradient.
TL;DR: It will be shown that using these concepts will impact an individual's perception of information items within the space, and consideration of these factors is critical to the design of retrieval and SDI systems.
Abstract: Vector representations of information items and requests will be used as a basis for a discussion of knowledge and information spaces. The representations will allow such questions as the meaning of relationships between vectors within the spaces. It will be shown that there are a number of possibilities for establishing these relationships, even if one is restricted to a basic cosine correlation measure. Using geometric representations, it will be shown that one is able to accept a predefined space or redefine the space. This redefinition may be accomplished through projections and two types of projections will be examined. It will be shown that using these concepts will impact an individual's perception of information items within the space. Thus, consideration of these factors is critical to the design of retrieval and SDI systems.
TL;DR: In this paper, a Chernoff-Savage linear rank statistics is introduced as a basis for inference, and the principal result is an invariance principle for two-sample rank statistics, i.e., under a fixed alternative the sequence of sequential linear rank statistic converges weakly to a Wiener process.
Abstract: A sequential version of Chernoff-Savage linear rank statistics is introduced as a basis for inference. The principal result is an invariance principle for two-sample rank statistics, i.e., under a fixed alternative the sequence of sequential linear rank statistics converges weakly to a Wiener process. The domain of application of the theorem is quite broad and includes score functions which tend to infinity at the end points much more rapidly than that of the normal scores test. The method of proof involves new results in the theory of multiparameter empirical processes as well as some new probability bounds on the joint behavior of uniform order statistics. Applications of weak convergence are explored; in particular, the extension of the theory of Pitman efficiency to the sequential case.
TL;DR: In this paper, the synthesis equations are formulated on the basis of spherical relative poles, and a set of three relative-pole-equations is used for a four-precision point case.
10 Aug 1976
TL;DR: Algorithms which find a basis for a set of Gaussian rational functions and the corresponding linear equations are presented in detail and bounds on their theoretical computing times are derived.
Abstract: This paper describes a procedure for determining when a set of rational functions are pseudo-multiplicatively independent, i.e. when no non-trivial power product of the rational functions can be a constant. The method used is to derive a multiplicative basis of factors of the numerators and denominators of the rational functions; the basis is then used to derive a system of homogeneous linear equations which will have a non-trivial solution if and only if the rational functions are pseudo-multiplicatively dependent. Algorithms which find a basis for a set of Gaussian rational functions and the corresponding linear equations are presented in detail and bounds on their theoretical computing times are derived.
TL;DR: In this paper, the orthogonality conditions for the principal and supplementary series of representations of SL(2,C) in the SU(1,1) basis are discussed.
Abstract: The reduction of the principal and supplementary series of representations of SL(2,C) in the SU(1,1) basis is carried out by using a basis function which formally resembles the coupled state of two angular momenta. The spectrum of the SU(1,1) representations contained in SL(2,C) and the transformation coefficients are obtained by expanding the SU(2) in terms of the SU(1,1) bases with the help of the Sommerfeld–Watson transformation. The orthogonality conditions for the principal and supplementary series are discussed. For the principal series this follows easily from the standard Sturm–Liouville theory of the second order differential equations. For the supplementary series the orthogonality condition is obtained from the fourth order differential equation satisfied by the Fourier transform of the basis function.