Showing papers on "Basis (linear algebra) published in 1968"
879 citations
TL;DR: In this paper, the specification of asymptotic photon states belonging to non-Fock representations is discussed, and a basis consisting of generalized coherent states is used, but in contrast to his work, these states are rigorously defined in terms of von Neumann's infinite tensor product, and the states must be given an additional label which serves to distinguish various weakly equivalent vectors, and which corresponds formally to an infinite phase factor.
Abstract: As a first step toward a treatment of soft‐photon processes which is free of infrared divergences and avoids the necessity of introducing a fictitious photon mass, the specification of asymptotic photon states belonging to non‐Fock representations is discussed. As in the work of Chung, a basis consisting of generalized coherent states is used, but in contrast to his work, these states are rigorously defined in terms of von Neumann's infinite tensor product. It is shown that the states must be given an additional label which serves to distinguish various ``weakly equivalent'' vectors, and which corresponds formally to an infinite phase factor. A nonseparable Hilbert space HIR is defined (as a subspace of the infinite tensor‐product space) which may be regarded as the space of all possible asymptotic photon states. The interaction of the electromagnetic field with a prescribed classical current distribution is discussed, and it is shown that a unitary S operator, all of whose matrix elements are finite, may...
135 citations
TL;DR: It is shown how to use knowledge of first and second order properties of the f i to obtain solutions on a digital computer using space filling curves as a basis for the concept of implicitly exhaustive search.
Abstract: The problem of finding {if314-1} in n dimensional Euclidean space such that {if314-2}, i = 1, 2, ···, N , is considered. The only assumption on the f i is that a solution exists in the quantized unit hypercube. Implicitly exhaustive solution procedures, which obtain solutions by implicitly considering every point in the quantized space without making computations at each point, are studied. The implicitly exhaustive feature is made possible by adapting “space filling curves≓ to discrete spaces of general dimensionality. Several space filling curves are surveyed, and Peano's continuous mapping from the unit interval onto the unit square is used as a basis for defining a mapping from the unit quantized interval onto the unit quantized hypercube, and inversely. Ternary arithmetic is the basis for the required functional relationships in the discrete mapping. The discrete mapping has attributes of quasi-continuity, and specific numerical bounds are derived in this respect. It is shown that these bounds are of optimal order dependence on the relevant variables. It is shown how to use knowledge of first and second order properties of the f i to obtain solutions on a digital computer using space filling curves as a basis for the concept of implicitly exhaustive search. The only global properties assumed are bounds on first, or possibly second, order variations. Concluding remarks bear on the ultimate practicality of the method, and present a limited amount of experimental data.
106 citations
TL;DR: In this article, a theoretical basis for the maximum load-carrying capacity of sharply notched aluminum columns loaded eccentrically in compression was given, and a failure criterion was developed.
85 citations
TL;DR: In this article, the relationship between the generators of R(5) expressed in the natural basis and the generators in the physical basis is discussed. But the authors focus on the traceless symmetric tensors of R (5) which are the representations appropriate to the discussion of quadrupole vibrations of the nuclear surface.
Abstract: A systematic study is made of the relationship between the generators of R(5) expressed in the ``natural basis,'' as discussed in I [J. Math. Phys. 9, 1224 (1968)], and the same generators in the ``physical basis'' in which representations of R(5) are fully reduced with respect to the physical three‐dimensional rotation group. In this paper, attention is confined to the traceless symmetric tensors of R(5) which are the representations appropriate to the discussion of quadrupole vibrations of the nuclear surface. For these representations, one quantum number in addition to the angular momentum and its projection is required to specify a state within a representation. The required extra label is found through the definition of ``intrinsic states'' in the natural basis, and a complete set of states in the physical basis is projected out of these intrinsic states by integrations over the physical rotation group manifold. Members of this set of physical states are not orthonormal; however, the overlap integrals are presented in two simple algebraic forms convenient for computer programming. The construction of the explicit representation matrices for the generators of R(5) is completed by giving the reduced matrix elements of the octopole generator between physical states in terms of the overlap integrals.
59 citations
50 citations
TL;DR: In this article, it is proved that if all bases of a Banach space are shrinking, or all bases are boundedly complete, then the space is reflexive, i.e., it is not shrinking.
Abstract: LetX be a Banach space with a basis. It is proved that if (a) all bases ofX are shrinking, or (b) all bases ofX are boundedly complete, thenX is reflexive.
50 citations
TL;DR: In this paper, it is shown that the orthogonality projection operator can be transformed in good approximation into pseudopotentials, for which explicit expressions are derived, on the basis of this transformation a new method is developed for the calculation of correlated pair functions.
Abstract: It is shown that in all previously developed methods for the calculation of correlated pair functions the mathematical difficulties arise from the fact that the pair functions must be orthogonalized to the one‐electron spin orbitals by the orthogonality projection operator (1–Ω12). Since this is an integral operator, if the pair functions are of Hylleraas type, containing r12, the orthogonalization will introduce integrals which are very time consuming to calculate. It is shown that the orthogonality projection operator can be transformed in good approximation into pseudopotentials, for which explicit expressions are derived. On the basis of this transformation a new method is developed for the calculation of correlated pair functions.
48 citations
46 citations
TL;DR: In this article, a configuration interaction calculation on the ground state of H3− has been carried out using an orbital basis set made out of 1s, 1s′, 2s, and 2p STO's.
Abstract: A configuration‐interaction calculation on the ground state of H3− has been carried out using an orbital basis set made out of 1s, 1s′, 2s, and 2p STO's. The new orbitals were included in the set one type at a time and a large number of calculations was performed to optimize the nonlinear parameters and to keep the wavefunction from becoming too complicated while including the most important configurations that can be formed from each basis set. The best wavefunction obtained from each set was analyzed in terms of natural orbitals (NO). The first two NO's were used to construct simple geminals and study H3− in terms of separated pairs. The Rank 1 functions were used to estimate the Hartree–Fock energy for the system. Electron population analysis was done to gain information about the distribution of electrons in H3−. Information on other four‐electron systems (He2 and LiH) was used in the discussion of the results obtained for H3−.
30 citations
TL;DR: In this paper, a computer program is described for obtaining such eigenfunctions by direct diagonalization of @?^2 + (1/20)@?^ 2, and examples are given.
TL;DR: In this paper, the Hartree-Fock wave function was used to describe the dipole moment and forces of the formaldehyde molecule, and the results showed that no appreciable error arises from the use of Gaussian orbitals as expansion functions for the solution of Hartree Fock equations.
Abstract: Numerous one‐electron properties of the formaldehyde molecule have been calculated using Hartree–Fock–Roothaan wavefunctions obtained from three different basis sets: an unoptimized minimum basis set of Slater orbitals, a (73 / 2) Gaussian basis set, and a (95 / 3) Gaussian basis set Comparison of the calculated and experimental values is made whenever possible, the agreement in most cases being satisfactory Use of the more flexible Gaussian wavefunctions results in significant improvements in many of the properties Consideration of the atoms shows that no appreciable error arises from the use of Gaussian orbitals as expansion functions for the solution of the Hartree–Fock equations Even though the best wavefunction is near the (sp) limit, some properties (such as the dipole moment and forces) are still inadequately described Such a basis set is just not sufficient to describe the polarizations of the atoms caused by molecular formation Inclusion of d orbitals in the basis set should substantially improve such properties
TL;DR: In this article, the NMR spin coupling constant in HD is calculated by a perturbation-variation procedure using basis sets ranging from 2 to 12 functions, and the experimental value is 42.7 ± 0.5 cps.
Abstract: The NMR spin coupling constant in HD is calculated by a perturbation‐variation procedure using basis sets ranging from 2 to 12 functions. The simplicity of the HD molecule makes it possible for the coupling constant to be calculated exactly within a given basis set. It is thus possible to determine the effect on the coupling constant not only of extension of the size of the basis set but also of the inclusion of configuration interaction. Finally, the coupling constant is calculated using very accurate ground‐state wavefunctions, and the values obtained are compared with the results of the earlier 2 to 12 basis function calculations. It is found that the main contribution to the coupling comes from the Fermi contact interaction. A value of 54.06 cps is obtained for the coupling constant. The experimental value is 42.7 ± 0.5 cps.
TL;DR: In this article, a level crossing is a degeneracy of two or more energy eigenvalues for a discrete value of H. The number of crossings is related with the degree in H of the basis vectors of the space of operators commuting with H: every crossing reduces the degree of one of these basis vectors.
01 Jan 1968
TL;DR: In this paper, the authors focus on symmetrical components and propose a method of resolving an unbalanced system of polyphase vectors into a number of balanced systems that forms the basis of the method of symmetric components.
Abstract: Publisher Summary
This chapter focuses on symmetrical components. C. L. Fortesque produced a method of resolving an unbalanced system of polyphase vectors into a number of balanced systems that forms the basis of the method of symmetrical components. In terms of a three-phase system, an unbalanced set of voltages or currents can be resolved into two symmetrical three-phase systems having opposite phase sequence , positive- and negative-phase sequence, plus a third set of equal vectors having zero-phase displacement, that is, zero-phase sequence. The method of symmetrical components may be applied to any n-phase system. The chapter discusses resolution into symmetrical components and solution of symmetrical components.
TL;DR: The experiments show that the division of polynomials is most suitable for transformation of nuclear data and spreads the cluster of descriptors in the physical field equally over the whole range of the memory.
TL;DR: In this article, the invariant subspaces of weighted shift operators are investigated, along with the problem of when a system,, forms a Riesz basis in the sequence spaces.
Abstract: In the sequence spaces the invariant subspaces of weighted shift operators are investigated, along with the problem of when a system , , forms a Riesz basis in .
TL;DR: In this article, a perturbation theory for interacting atoms or molecules in the region in which exchange becomes important is developed, using a set of linear operators which project out the components of the antisymmetrized state vector.
Abstract: A perturbation theory is developed for interacting atoms or molecules in the region in which exchange becomes important. Use of a set of linear operators which project out the components of the antisymmetrized state vector permits the derivation of a set of exact integral equations, each for a single component of the vector, which can be solved by ordinary perturbation–theoretic techniques. In this way, exchange can be treated exactly, without the necessity of obtaining an orthogonalized basis.
TL;DR: In this paper, an analytical expansion for the orbital wave functions of Ag + is presented, with short and easily manageable basis sets, and a suggestion for improving future minimum set calculations for heavy atoms.
01 Jan 1968
TL;DR: The algebraic tools are adequate as long as one deals with finite dimensional problems, even in infinite dimensional spaces as mentioned in this paper, and the abstract approach is valuable because it provides additional insight, especially if one is familiar with the classical theory of vector spaces of n -tuples, matrices and determinants.
Abstract: The ordinary vectors in ordinary space, or in “ordinary” n -dimensional space, form a vector space. Important vector spaces of functions are given by the continuous functions on an interval, the integrable functions, and the n times continuously differentiable functions. This chapter describes the algebraic machinery that is required to deal with linear problems. Coordinate-free definitions have been provided. This is essential in the case of the usual function spaces, because in these infinite dimensional spaces, there are no natural or simple coordinate systems. In the finite dimensional case the abstract approach is valuable because it provides additional insight, especially if one is familiar with the classical theory of vector spaces of n -tuples, matrices, and determinants. Matrices are defined as representations of linear transformations; the determinant of a matrix is defined in terms of the determinant of a linear transformation. The algebraic tools are adequate as long as one deals with finite dimensional problems, even in infinite dimensional spaces.
01 Jan 1968
TL;DR: In this article, the authors present an account of recent developments in mathematical system theory related to the questions: What is a system? How can it be effectively described in mathematical terms? Is there a deductive way of passing from experiments to mathematical models? How much can be said about the internal structure of a system on the basis of experimental data? What is the minimal set of components from which a system with given characteristics can be built?
Abstract: The purpose of this talk is to present, in nontechnical language, an account of recent developments in mathematical system theory. They are related to the questions: What is a system? How can it be effectively described in mathematical terms? Is there a deductive way of passing from experiments to mathematical models? How much can be said about the internal structure of a system on the basis of experimental data? What is the minimal set of components from which a system with given characteristics can be built?
01 Mar 1968
TL;DR: In this article, it was shown that the space (d) constructed by Davis and Dean [1, p. 214] has a semishrinking basis which is weakly convergent to 0.
Abstract: A Schauder basis (xi, fi) for a Banach space X is (1) shrinking provided (fi) is a basis for X*; and (2) semishrinking provided 0
TL;DR: In this article, an analytical expansion for the orbital wave functions of I -, with short and easily manageable basis sets, is presented, and the basis sets are used for the analysis of the wave functions.
TL;DR: In this article, an optimal basis set for such a problem can be determined for the particular case studies by Endres, and their prediction compares favorably with Endres' result for the general case studies.
Abstract: Endres has studied expansions of eigenfunctions of a Morse oscillator upon basis sets derived from different harmonic‐oscillator Hamiltonians. We have shown how an optimal basis set for such a problem can be determined. For the particular case studies by Endres, our prediction compares favorably with his result.
TL;DR: In this paper, the Coulombic contribution to the dynamical matrix for non-cubic lattices with a basis was analyzed and a correction was made and an expression derived for the Coulombo contribution was derived.
TL;DR: In this paper, a preliminary examination of the statistical theory of electronic energies from the view point of the 2 matrix is made, and a good representation of the Fermi hole is found for the triplet state.
Abstract: A preliminary examination of the problem of the statistical theory of electronic energies from the view‐point of the 2 matrix is made. For the two‐particle, one‐dimensional harmonic oscillator, a good representation of the Fermi hole is found for the triplet state. The employment of a discrete basis for heliumlike systems leads to problems with the virial theorem.
TL;DR: In this article, the authors evaluated screw dislocations with the kinematical theory of electron diffraction using numerical techniques and found that screw dislocations for which n = g · b = 1 will be nearly invisible for certain foil thicknesses, deviations from the Bragg angle, and dislocation positions in the foil.
Abstract: The images of screw dislocations have been evaluated with the kinematical theory of electron diffraction using numerical techniques. The advantage of this method is that the periodic term in the diffracted intensity can be treated. It is found that because of this periodic term a screw dislocation for which n = g · b = 1 will be nearly invisible for certain foil thicknesses, deviations from the Bragg angle, and dislocation positions in the foil. On the basis of these findings, it is proposed that errors in Burgers vector determinations are possible.
[Russian Text Ignored].
TL;DR: In this paper, an optimum eigenfunction-expansion basis has been found that leads to accurate solutions with only a few terms in the expansion, and the advantage of the method is that separable problems in more than one dimension can be solved almost as easily as one-dimensional problems.
Abstract: Some nonlinear boundary‐value problems in one‐ and two‐dimensional composite domains have been solved by a general eigenfunction‐expansion method. The advantage of the method is that separable problems in more than one dimension can be solved almost as easily as one‐dimensional problems. An optimum eigenfunction‐expansion basis has been found that leads to accurate solutions with only a few terms in the expansion.