Showing papers on "Basis (linear algebra) published in 1982"

Conditions for the definition of a strictly diabatic electronic basis for molecular systems

Abstract: A strictly diabatic electronic basis is defined as one for which all components of the nuclear momentum coupling vanish. We examine the possibility that such a basis may exist, and we find that, in general, it does not. The only important exception is for diatomic states of the same symmetry. We also consider some conditions for the definition of an approximately diabatic electronic basis. For molecular systems with three or more nuclei, one can obtain useful approximate diabatic basis sets if the transverse (solenoidal) part of the coupling is negligible; this may occur, for example, if the part of the coupling due to the internuclear‐distance dependence of the configurational wave functions is negligible as compared to that due to the internuclear‐distance dependence of the configurational coefficients. We derive a criterion showing that such approximations may be useful and accurate if the role of the coupling is important over regions of sufficiently small linear dimensions.

Robust multivariable PI-controller for infinite dimensional systems

Abstract: A robust multivariable controller is introduced for a class of distributed parameter systems. The system to be controlled is given as \dot{x} = Ax + Bu, y = Cx in a Banach space. The purpose of the control, which is based on the measurement y , is to stabilize and regulate the system so that y(t) \rightarrow y_{r}, as t \rightarrow \infty , where y r is a constant reference vector. Under the assumptions that operator A generates a holomorphic stable semigroup, B is linear and bounded, C is linear and A -bounded, and the input and output spaces are of the same dimension; a necessary and sufficient condition is found for the existence of a robust multivariable controller. This controller appears to be a multivariable PI-controller. Also, a simple necessary criterion for the existence of a decentralized controller is derived. The tuning of the controller is discussed and it is shown that the I-part of the controller can be tuned on the basis of step responses, without exact knowledge of the system's parameters. The presented theory is then used as an example to control the temperature profile of a bar, with the Dirichlet boundary conditions.

The finite element method for parabolic equations

Abstract: In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.

Reconstruction from Partial Information with Applications to Tomography

Abstract: The problem of reconstruction from projections in Hilbert space is treated. An axiomatic basis is considered which leads to techniques which provide improved reconstructions by incorporating prior knowledge to tailor the Hilbert space to the problem at hand. When applied to reconstruction of the Fourier transform of a function sampled at finitely many discrete points, the procedures lead to previously derived optimal estimation techniques. When applied to x-ray tomography, the procedures lead to new reconstruction techniques which are shown to include as a special case the minimum energy reconstruction of Logan and Shepp [Duke Math. J., 42 (1975), pp. 645–659].

Universal systematic sequence of even‐tempered exponential‐type functions in electronic structure studies

Abstract: Self‐consistent field calculations are reported for some first‐row atoms and their ions using a universal systematic sequence of basis sets consisting of exponential‐type primitive functions. Extrapolation procedures are employed to obtain estimates of the basis set limit. The convergence properties of the calculations with respect to the size of the basis set are examined and compared with previously reported calculations which employed basis sets of Gaussian primitive functions.

Design of stochastic discrete time linear optimal regulators Part I. Relationship between control laws based on a time series approach and a state space approach

Abstract: Distinct approaches to sampled data control system design use either a state space model or a ‘ controlled autoregressive moving average ’ (CARMA) model, sometimes known as Astrom's representation, One reason for the current interest in the CARMA model is that it is a useful basis for self-tuning controllers as its parameters can be readily estimated on-line. Moreover, simple transfer function controllers can be derived using κ-step-ahead prediction theory. On the other hand, these controllers can be interpreted as minimizing a single stage cost function in state space terms, and the corresponding performance can sometimes be poor. This paper explores the relationship between the κ-stop-ahead prediction approach and the state space approach, and is a generalization of the earlier work of Caines to include control weighting and time delay on the control. Two forms of state space model are used (‘ explicit ’ and ‘ implicit’ time delay models) and a new representation of the steady state Kalman filter is sho...

Theoretical basis for Coulomb matrix elements in the oscillator representation

Abstract: Hydrogenic wave functions in the spherical and parabolic bases are shown to correspond, respectively, to a restricted set of wave functions of a four‐dimensional harmonic oscillator and its coupled pair of two‐dimensional oscillators. This correspondence provides the theoretical basis for algebraic calculations of Coulomb matrix elements in the oscillator representation.

Ab initio calculation of atomic spin-orbit coupling constants using a universal systematic sequence of even-tempered exponential-type basis sets

Abstract: Ab initio calculations of spin-orbit coupling constants are reported for a number of first-row atoms and ions using a universal systematic sequence of even-tempered basis sets of exponential-type functions. The convergence of the calculations with respect to size of basis set is examined and extrapolation to the basis set limited made. The importance of the present atomic calculations to the determination of the fine structure in molecules is discussed.

Nonlinear σ Models on Symmetric Spaces and Large N Limit

Abstract: shown that there exists a third order phase transition in the large N limit of two dimensional lattice U(N) gauge theory which becomes equivalent to one dimensional chiral nonlinear (J model of N = =. In this paper, we investigate various lattice nonlinear (J models defined on symmetric spaces, which are coset spaces C/H where C is the Lie group and H is its maximum compact subgroup. We consider explicitly relevant invariant measure of our symmetric spaces. We consider the system in zero and one space dimension and calculate the energy and correlation function. It will be shown that the large N limit of these models has third order phase transition for the compact case. It will be also shown that for anisotropic Grassmannian model like Cp N 1 model, there exists a phase transition which has a discontinuity of specific heat in the large N limit. These zero and one dimensional studies may give basis of further investigations in higher dimensions. This article will be divided as follows: In § 2, we express the action by the angle variables and determine the Haar measure for various symmetric spaces. In § 3, we calculate energy of the one link and we discuss the large N behavior. In § 4, we consider CPN-l and RpN-l model as anisotropic large N cases. In § 5, two point correlation function in one dimension is considered. In § 6, S-func­ tion is derived. Section 7 is devoted to discussion. In the Appendix, we present the large N calculation for d-dimensional lattice RpN-l model.

Transfinite duals of quasireflexive Banach spaces

Abstract: The transfinite duals of a space with a neighborly basis are constructed until they become nonseparable. Let s(X) be the first ordinal a so that XG is nonseparable. It is shown that if X is nonreflexive, s(X) < w2 + I (this is best possible) and that {s(X): X separable quasireflexive of order one) = { ? 1, W + 2,2co + 1, 2wo + 2, c2 + 1). A quasireflexive space X is constructed so that X' is isomorphic to X Ef c0 and no basic sequence in X is equivalent to a neighborly basis. It is shown that the w2th dual of James space and James function space are isomorphic to subspaces of one another. Also, perhaps of interest on its own, a reflexive space with a subsymmetric basis is constructed whose inversion spans a nonreflexive space. The notions of transfinite duals and quasireflexive spaces are more intimately related than one might expect. Indeed, Perrott in [14] has shown that for many "natural" sequences {x,,) in the transfinite dual X of a nonreflexive space X, the space [xn] has a structure which often makes it quasireflexive. The author in [2] turned this around and used results about quasireflexive spaces (actually of the {xj}) to obtain results about X' and hence X. Thus the goal of this paper is to describe the transfinite duals Qf quasireflexive spaces and to see what these results imply about the transfinite duals of an arbitrary nonreflexive space. Our vehicle for this goal is the notion of a neighborly basis, defined by James in [8]. (The basic properties of neighborly bases are in ?1.) There are three reasons for restricting attention to neighborly bases. First, the result of Perrott mentioned above implies that if X is quasireflexive of order one, then for some neighborly basic sequence {en}, X' is isomorphic to X ED [en]. The second is that (to the author's knowledge) all the concrete examples of quasireflexive spaces of order one in the literature can be renormed so to have a neighborly basis. (However, the examples in ?4 will make this statement no longer true.) The final justification is that we can isometrically describe all the transfinite duals of a space with a neighborly basis until they become nonseparable. (These results are in ?2.) Indeed, if both the neighborly basis {xj} and its inversion span quasireflexive spaces, we give three different isomorphic representations of the space [x1],2. Define s( X), for a separable nonreflexive space X, to be the first ordinal a so that Xa is nonseparable. As applications, it follows that s(X) < 2 + 1 (which improves Received by the editors November 19, 1980 and, in revised form, May 11, 1981. Presented at Mathematiches Forschungsinstitut, Oberwolfach on August 7, 1981. 1980 Mathematics Subject Classification. Primary 46B10, 46B15.

The most efficacious one‐electron bases for determining and representing correlated molecular electronic wave functions. Unity in seemingly disparate electron correlation methods

Abstract: Coefficient matrices and associated operator matrices are being used increasingly in various large‐scale correlation methods. These matrices are used to find and represent the wave function directly in terms of one‐electron basis functions. They eliminate serious redundancies in computation and provide for the use of different sets of nonorthogonal external orbitals to improve convergence. These features are shown to be independent of the choice of a one‐electron basis, and illustrative calculations are presented for N2H2, HCN, and HNC.

On the vector space of 0-configurations

Abstract: Let α be a rational-valued set-function on then-element sexX i.e. α(B) eQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2 x if for everyA eA we have $$\mathop \Sigma \limits_{A \subseteqq B \subseteqq X} $$ α(B)=0. The 0-configurations form a vector space of dimension 2 n − |A| (Theorem 1). Let 0 ≦t k form a vector space of dimension $$\mathop \Sigma \limits_{t< i \leqq k} \left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right)$$ , we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).

Bases in banach spaces

Abstract: The publication of Banach's book [2] in 1932 might be regarded as marking the beginning of the systematic study of Banach spaces. Research activity in this area has expanded dramatically during the past two decades. Interesting new directions have developed and interplays between Banach space theory and other mathematics have proved to be very valuable. Most well-known classical problems have been solved, but some remain and important new problems have arisen. Our purpose is very limited. It is to discuss some of the most important and fundamental facts about bases in Banach spaces, particularly those that may be interesting and useful for mathematicians in other fields. Often, only sketches of proofs will be given, while others are given in more detail because they are relatively easy and may contribute to developing intuitive feeling for the concepts involved. It will be seen that many very important and beautiful theorems have very easy and natural proofs. A basis (or Schauder basis) for a Banach space X is a sequence (en: n > 1) of members of X which has the property that, for each x in X, there is exactly one sequence of scalars (xi} for which x = xiei in the sense that lim.nX lix - E ixieijj = 0. Some important Banach spaces have very natural bases. If (en} is a complete orthonormal sequence in Hilbert space H and x is any member of H, then there is exactly one sequence (xn} of scalars such that lim x yn- lxieill = 0. For this sequence of scalars, each xi is (x, ei) and n 00 ~~~~~1/2

Concatenation of characteristic functions in Hamiltonian optics

Abstract: A procedure is developed for the determination of the Taylor coefficients of the characteristic function of an optical system given the corresponding coefficients of its component parts. No symmetry properties are assumed, and, in principle, there is no limitation to the order to which the calculations can be carried out. Whereas earlier investigations have assumed that the elementary characteristic functions to be concatenated are those of a region bounded by two planes and containing one refracting surface, it is suggested that the homogeneous regions between two such refracting surfaces (i.e., the optical elements or the air spaces between them) are more appropriate. Since the point characteristic of such regions can be readily calculated, this concatenation procedure provides a basis for the analysis of most optical systems.