Showing papers on "Basis (linear algebra) published in 1971"
TL;DR: In this article, the connections between bases and weaker structures in Banach spaces and their duals are investigated, and it is shown that every separable π-structures and finite dimensional Schauder decomposition has a basis.
Abstract: This is an investigation of the connections between bases and weaker structures in Banach spaces and their duals. It is proved, e.g., thatX has a basis ifX* does, and that ifX has a basis, thenX* has a basis provided thatX* is separable and satisfies Grothendieck’s approximation property; analogous results are obtained concerning π-structures and finite dimensional Schauder decompositions. The basic results are then applied to show that every separableℒ
p
space has a basis.
253 citations
TL;DR: A Gaussian basis set consisting of 12s-type, 6p-type and 4d-type functions has been optimized for the third row atoms, together with a 9s, 5p, 3d set for the corresponding dipositive ion as mentioned in this paper.
Abstract: A Gaussian basis set consisting of 12s-type, 6p-type and 4d-type functions has been optimized for the third row atoms, together with a 9s, 5p, 3d set for the corresponding dipositive ion. The applicability of these atomic sets for molecular calculation is discussed.
210 citations
TL;DR: In this article, it was shown that the Liouville operator of the spin hamiltonian does not reflect any symmetry inherent in the spin system, whereas the vector space (Liouville space) comprising all operators of spin system does.
Abstract: It is shown in discussing problems involving magnetic relaxation in liquids that, whilst the usual Hilbert space spanned by all the eigenkets of the spin hamiltonian does not reflect any symmetry inherent in the spin system, the vector space (Liouville space) comprising all operators of the spin system does so. The transformation properties of the Liouville operator, as reflected by those of the high resolution spin hamiltonian and relaxation operators whose effects are introduced by means of Redfield relaxation theory, with respect to arbitrary rotations of the coordinate system are investigated. The use of irreducible tensor operators as a set of basis operators spanning Liouville space is stressed, since it is shown that their super-matrix elements of the Liouville operator are given by the Wigner-Eckart theorem provided that relaxation by anisotropy of the chemical shift or anisotropic random fields is absent. These arguments are independent of the fine details of molecular reorientation in the extrem...
104 citations
TL;DR: Two efficient computer algorithms are presented for finding the dc solutions of resistive nonlinear networks containing two-terminal linear and nonlinear resistors, independent dc voltage and current sources, and linear controlled sources.
Abstract: Two efficient computer algorithms are presented for finding the dc solutions of resistive nonlinear networks containing two-terminal linear and nonlinear resistors, independent dc voltage and current sources, and linear controlled sources. The first algorithm is designed specifically for networks with multiple solutions, while the second algorithm is designed for networks with a unique solution. The first algorithm is based on the sign of the hybrid parameters associated with the linear n-port portion of the network. The second algorithm is a piecewise-linear version of the Newton-Raphson method, but it differs from the differentiable version in two important aspects. First, rather than diverging to \pm \infty , as in the usual case, the divergence phenomenon of the piecewise-linear algorithm takes the form of a cyclic repetition of two or more segment combinations. Second, the iteration formula depends not directly on the solution at the preceding iteration, but on the updated segment combination. These observations lead to an algorithm which assures that the piecewise-linear version of Newton-Raphson formula will always converge. Moreover, an important connection between the two algorithms is established on the basis that the iteration formula for the second algorithm is identical to the network equations associated with the first algorithm.
72 citations
TL;DR: In this article, a denumerable, complete and orthonormal set of functions is constructed using a denervable, complete, and orthogonal set of hyperspherical harmonics.
67 citations
TL;DR: In this article, it is shown that there exists an optimal basis which is involutory (i.e., B = B−1) for the weighted set covering problem, and a new algorithm which uses strong cutting planes.
Abstract: Some new properties associated with the special class of integer programs known as weighted set covering problems are derived. While it is well known that an optimal integer solution to the set covering problem is a basic feasible solution to the corresponding linear program, we show that there exists an optimal basis which is involutory (i.e., B = B−1). This property and others are used to develop a new algorithm which uses strong cutting planes. The cutting planes are strong in the sense that they exclude both integer and noninteger solutions. Computational experience is presented.
64 citations
TL;DR: A method for semantic description of algorithms is proposed, based on the notion of a “tail function” of an algorithm, describing all possible computations defined by this algorithm.
Abstract: A method for semantic description of algorithms is proposed. The basis of the method is the notion of a “tail function” of an algorithm, describing all possible computations defined by this algorithm. The notion of a characteristic equation for an algorithm is introduced and some relations between these equations and the corresponding tail functions are shown. An example, illustrating an application of the method, is given.
44 citations
TL;DR: The solution of the problem Ax = b; max z = ∑cj|xj| has a simplex-method solution only if cj are nonpositive (nonnegative for minimizing problems).
Abstract: Consider the problem Ax = b; max z = ∑cj|xj|. This problem cannot, in general, be solved with the simplex method. The problem has a simplex-method solution (with unrestricted basis entry) only if cj are nonpositive (nonnegative for minimizing problems).
36 citations
TL;DR: In this article, the graphical representation of angular momentum is used as the basis of a procedure for the complete evaluation of the matrix element of a Coulomb or multipole interaction operator between atomic states having any number of open shells.
Abstract: The graphical representation of angular momentum is used as the basis of a procedure for the complete evaluation of the matrix element of a Coulomb or multipole interaction operator between atomic states having any number of open shells. The method is presented in the form of a step-by-step procedure and is designed to permit straightforward extension to the evaluation of the matrix elements of other types of tensor operators and of sums of products of Coulomb matrix elements, such as occur in the perturbation theory of configuration interaction.
33 citations
TL;DR: In this paper, a complete set of angular functions for the four-body problem is given and the transformation properties of the functions are given and hence a matrix representation of the permutation group on four objects is explicitly specified.
Abstract: A complete set of angular functions for the four‐body problem is given. Such functions form the basis for irreducible representations of the orthogonal group O9, reduced according to the chain O9⊃O31×O32×O33⊃O3⊃O2. The transformation properties of the functions are given and hence a matrix representation of the permutation group on four objects is explicitly specified. The reduction of this representation yields functions suited to Bose‐Einstein or Fermi‐Dirac statistics.
TL;DR: An optimized basis set of contracted gaussian type functions (CGTF's) for ethylene is obtained by scaling "best-atom" carbon and hydrogen CGTF exponents to minimize the ethylene molecular SCF energy as discussed by the authors.
TL;DR: Approximations, accurate to ± 1 × 10−8, have been developed for the functions Fm(z) which arise in molecular calculations using Gaussian basis functions and are found to reduce integral computation time by about a factor of 5 over other methods.
Abstract: Approximations, accurate to ± 1 × 10−8, have been developed for the functions Fm(z) which arise in molecular calculations using Gaussian basis functions. These are found to reduce integral computation time by about a factor of 5 over other methods. Approximations are listed for 0 ≤ m ≤ 16.
TL;DR: In this paper, the authors considered non-linear sampled-data control systems consisting of a continuous linear part and a nonlinear pulse modulator (PM), where the PM modulates a sequence of rectangular pulses by sign, frequency and duration, as a function of a linear combination of discrete values of the system coordinates.
TL;DR: A survey of results in the theory of linear programming in reflexive linear topological spaces can be found in this article, with a brief review of the significant results for ordinary linear programs in Euclidean space.
Abstract: This paper is a survey of the principal results in the theory of linear programming in reflexive linear topological spaces. We begin with a brief review of the significant results for ordinary linear programming in Euclidean space. With this as a basis for comparison, for the general case we present a complete and self-contained account of three topics: (a) the classification scheme relating the properties of primal and dual programs, (b) the duality theory relevant to the problem of duality gaps between solutions of primal and dual programs, and (c) a “marginal cost” interpretation of solutions to the dual program.
TL;DR: A brief discussion of the use of auxiliary field techniques in a functional integral formulation of many body theory is presented in this paper. But the authors do not consider the auxiliary fields in this paper.
Abstract: A brief discussion is presented of the use of auxiliary field techniques in a functional integral formulation of many body theory. By using a Lagrangian representation for the Green functions from the outset a description is obtained in which true and auxiliary fields are treated on a more nearly equal basis than in partition function methods. The interacting fields are shown to have physical significance, their propagators leading to linear response functions of the true system. A diagrammatic representation analogous to that of a true interacting field problem is shown to follow in a straightforward manner. Transformations among fields are considered.
TL;DR: In this paper, it was shown that applying an electron-nuclear cusp constraint to a CI wavefunction is equivalent to omitting from the basis set a particular linear combination of atomic orbitals.
Abstract: It is shown that applying an electron–nuclear cusp constraint to a CI wavefunction is equivalent to omitting from the basis set a particular linear combination of atomic orbitals. This result is exploited in constructing cusp‐constrained wavefunctions for CH. Several small STO basis sets, a double‐zeta basis, Cade and Huo's basis, and a subset of their basis were used to calculate hyperfine couplings for CH. The last two sets were also used with cusp constraints. Reasonable results are obtained for the proton splitting using either the free atom 1sH orbital or the 1sH orbitals from Cade and Huo's basis. Any attempt to use optimum orbital exponents with a small basis gave unreasonable results. The cusp constraint produced little change in the results for the largest basis. Our best theoretical estimate for aH is − 18 ± 3 G. Carbon‐13 splittings are much more sensitive to the basis set. Even for the largest basis the results were still strongly affected by the cusp constraint. Our best estimate for aC is + ...
TL;DR: Using Newton-type corrections, a simultaneous improvement of a complete set of approximate factors of arbitrary degree of a polynomial is developed, which can form the basis for an iterative method of factoring the poynomial.
Abstract: Using Newton-type corrections, a simultaneous improvement of a complete set of approximate factors of arbitrary degree of a polynomial is developed, which can form the basis for an iterative method of factoring the polynomial. The simultaneous corrections for a given step are a generalization of a method introduced by Weierstrass [5] in the case of all linear factors. The present paper includes, therefore, an independent derivation of the Weierstrass formulas. From a different point of view the resulting iterative method is a generalization of a method of Samelson [3], recently analyzed by Stewart [4]. Basic properties of the iteration include the fact that the correction terms are related to a certain partial fraction decomposition. Emphasis is placed on the case of approximate quadratic factors, where formulas for the corrections are derived suitable for use in computation with the practically important case of a polynomial with real coefficients whose zeros are not necessarily all real. The formulas so...
TL;DR: The 1960 version of the Nuclear Data Sheets for A = 111 has been revised on the basis of experimental data received before February 1971, and data received during February 1971 have been added in proof.
TL;DR: A theory of conjugate approximations is developed which provides a fundamental basis for most methods of continuous piecewise approximation and it is shown that for a given finite set of base functions used in an approximation there corresponds another set of Conjugate functions which play a significant role in approximate methods of analysis.
Abstract: : In the paper a theory of conjugate approximations is developed which provides a fundamental basis for most methods of continuous piecewise approximation. It is shown that for a given finite set of base functions used in an approximation there corresponds another set of conjugate functions which play a significant role in approximate methods of analysis. In the case of finite-element approximations, it is shown that the domain of the conjugate functions includes the entire assembly of elements, and, consequently, the established method of computing stresses locally in elements based on displacement approximations is not strictly valid. Indeed, the domain of such 'local' stress fields is the entire connected system of elements. Procedures for computing derivatives and discrete analogues of linear operators which are consistent with the theory of conjugate functions are also discussed. For a given linear operator equation, the significance of the conjugate approximations in connection with the adjoint problem is also discussed. (author)
TL;DR: In this paper, the connection of group-theoretic expansions of a function defined on SO3 and SO2,1 and analytic on their complexification (except for certain singularities), is discussed by making various transformations of the Sommerfeld-Watson type.
Abstract: The connection of group-theoretic expansions (in both continuous and discrete bases) of a function defined onSO3 andSO2,1 and analytic on their complexification (except for certain singularities), is discussed. This is achieved by making various transformations of the Sommerfeld-Watson type. The analyticity assumptions needed for these transformations are considered and found to depend on the basis chosen for the representation functions. The applications of this work to multi-Reggeon expansions and, in particular, to the behaviour of Reggeon vertex functions are discussed. The connections between the various Sommerfeld-Watson and group-theoretic signatures are also given. As a preliminary, transforms are used to relate the Fourier series to the Fourier integral.
TL;DR: Results for (n − 1) -dimensional spaces over fields of n elements are obtained with the aid of combinatorial results arising from the study of hypercubes.
TL;DR: A necessary condition for equilibrium in diplo-diplo self-incompatible mating systems has been obtained and the result is completely general, applying to systems with an arbitrary number of mating types based upon an arbitrary genetic basis.
TL;DR: In this paper, the second row atoms were obtained by fitting, using a least squares criterion, to 12s-type and 9p-type Gaussian basis sets which are close to the self-consistent field atomic orbital wave functions.
Abstract: 6s-type and 4p-type gaussian basis sets are obtained for the second row atoms by fitting, using a least squares criterion, to 12s-type and 9p-type gaussian basis sets which are close to the self-consistent field atomic orbital wave functions. The small gaussian expansions are considered to be more suited for molecular calculations using double basis sets. The differences between these sets and the 10s-type, 6p-type and 9s-type, 5p-type are analysed. For molecular calculations using single gaussian basis sets the 10s-type and 6p-type would seem to be the best compromise.
TL;DR: In this article, a series of statistical tests were provided as a basis for the objective analysis of a general hydrometeorological sequence and for the purpose of constructing a best fit model.
TL;DR: In this article, the problem of the determination of the space potential is studied from some different stand points and it is conclusively described that the potential can be determined within the error of some millivolts.
Abstract: The space potential in plasma is one of the fundamental quantity which should be determined in the diagnostics. The potential of a probe immersed in the plasma is determined on the basis of the space potential. Therefore, the determination of the space potential has been discussed long time since the method of D.C. probe was established. However, it has remained with much ambiguity for a precise discussion. In this paper, the problem of the determination of the space potential is studied from some different stand points. It is conclusively described that the space potential can be determined within the error of some millivolts.