Showing papers on "Basis (linear algebra) published in 1984"

Trees and Linearly Ordered Sets

Abstract: Publisher Summary This chapter presents an introduction to several problems concerning trees and linearly ordered sets and the dualities between them. Several classes of trees and linearly ordered sets are presented and considered as set-theoretical, topological, and algebraical structures in the chapter. The chapter also discusses many fundamental problems concerning trees and linearly ordered sets, which are set-theoretical in nature. Most of the problems are undecidable on the basis of the usual axioms of set theory. Thus, a great number of results presented in the chapter are consistency results.

Locally Complemented Subspaces and ℒ︁p‐Spaces for 0 < p < 1

Abstract: We develop a theory of ℒp-spaces for 0 < p < 1, basing our definition on the concept of a locally complemented subspace of a quasi-BANACH space. Among the topics we consider are the existence of basis in ℒp-spaces, and lifting and extension properties for operators. We also give a simple construction of uncountably many separable ℒp-spaces of the form ℒp(X) where X is not a ℒp-space. We also give some applications of our theory to the spaces Hp, 0 < p < 1.

A canonical decomposition of linear periodic discrete-time systems

Abstract: In this paper a linear periodic discrete-time system is studied. On the basis of simple properties of the subspaces of controllable states and of unreconstructible ones, a canonical structure theorem is derived. This generalizes 1o such a system the classical Kalman decomposition, while preserving the constant dimensionality of the four subsystems which arise when a periodic continuous-time system is decomposed. The dynamic matrices of the non-controllable and/or non-reconstructible subsystems are shown to be non-singular at each time instant, as those for a time-invariant discrete-time system are.

Ambiguity Function and Wigner Distribution Function Applied to Partially Coherent Imagery

Abstract: The mutual intensity function is conveniently replaced by the ambiguity function or the Wigner distribution function. This allows us to treat optical systems under partially coherent illumination in a simple fashion. Many optical set-ups can be described on a geometrical basis, as is illustrated with some examples.

Effective basis sets for calculations of exchange‐repulsion energy

Abstract: Usefulness of different Gaussian basis sets for reproducing the “tail” region of the SCF wavefunctions employed in calculations of the exchange-repulsion effect is investigated for the model He-He interaction. It has been shown that extension of the monomer-centered basis set in the scheme of regularized even-tempered basis sets [M. W. Schmidt and K. Ruedenberg, J. Chem. Phys. 71, 3951 (1979)] can be more efficient than augmentation of the fully energy-optimized basis set with diffuse basis functions. It has been also found that Landshoff term vanishes and the “tail” region is well reproduced if monomer wavefunctions are calculated with the basis set of the dimer.

The Dirac equation in the algebraic approximation. II. Extended basis set calculations for hydrogenic atoms

Abstract: For pt.I see ibid., vol.17, p.L45 (1984b). The solution of the Dirac equation for hydrogen-like atoms within the algebraic approximation, that is, by using a finite basis set, is considered. It is shown that by making an appropriate choice of basis functions the problem which has been termed 'variational collapse' can be avoided. Applications using a systematic sequence of even-tempered basis sets are presented and convergence of the calculated energies within the algebraic approximation to the exact energies with increasing size of basis set is investigated.

Simultaneous Newton’s Iteration for the Eigenproblem

Abstract: For an ill-conditioned eigenproblem (close eigenvalues and/or almost parallel eigenvectors) it is advisable to group some eigenvalues and to compute a basis of the corresponding invariant subspace. We show how Newton’s method may be used for the iterative refinement of an approximate invariant subspace.

Choosing a basis for perceptual space

Abstract: If it is possible to interpret an image as a projection of rectangular forms, there is a strong tendency for people to do so. In effect, a mathematical basis for a vector space appropriate to the world, rather than to the image, is selected. A computational solution to this problem is presented. It works by backprojecting image features into 3-dimensional space, thereby generating (potentially) all possible interpretations, and by selecting those which are maximally orthogonal. In general, two solutions that correspond to perceptual reversals are found. The problem of choosing one of these is related to the knowledge of verticality. A measure of consistency of image features with a hypothetical solution is defined. In conclusion, the model supports an information-theoretic interpretation of the Gestalt view of perception.

A hierarchical basis for reordering transformations

Abstract: In this paper, we propose a new dependence baaed program representation. This representation is the union of two previously separate concepts: loop carried dependence and hierarchical abstraction. The resulting form has the property that all information necessary to reorder the set of all executions of the statements contained in a given loop exists in the representation of that loop. Thus, this representation provides an ideal basis for reordering transformations such as vectorisation and loop fusion. As evidence of this, we give efficient algorithms for these two transformations based on this representation.

Accurate Hartree–Fock wave functions without exponent optimization

Abstract: Basic functions with singularities matching those of the actual orbitals have been tested in analytical Hartree–Fock calculations. Such functions should provide the most rapidly convergent basis set expansions. Exponential singularities at r=∞, characterized by certain ‘‘asymptotic exponents,’’ have been identified by an asymptotic analysis of the Fock equation. Basis sets of Slater functions with these exponents give atomic energies and properties comparable to the most accurate existing analytical calculations, without significantly increasing the number of basis functions. No nonlinear optimizations were required. Calculations of the orbital moments 〈rn〉 show that only moments with n≤N, the number of Slater basis functions, can be evaluated with accuracy, whether or not the exponents are optimized. This effect appears to be caused by the neglect of certain irrational powers in asymptotic forms of the orbitals. The results for molecules suggest that basis functions which more adequately describe the nuc...

Sparse null bases and marriage theorems

Abstract: This dissertation considers the problem of constructing the sparsest basis for the null space of a constraint matrix. This problem arises in the design of practical algorithms for large-scale numerical optimization problems. Suprisingly, this problem can be formulated as a combinatorial optimization problem under a non-degeneracy assumption on the constraint matrix. The theory of matchings in bipartite graphs--marriage theorems--can then be used to obtain the nonzero positions in a null basis. Numerically stable matrix factorizations are used in the next stage to compute the null basis. We use conformal decompositions to characterize the columns of a sparsest null basis. Matroid theory is used to prove that a greedy algorithm constructs a sparsest null basis. We prove that finding a sparsest null basis is NP-hard by showing that associated matroidal and graph-theoretic problems are NP-complete. We propose two approximation algorithms to construct sparse null bases. Both of them make use of the Dulmage-Mendelsohn decomposition of rectangular matrices. One algorithm is a sparsity exploiting variant of the variable-reduction technique. The second is a locally greedy algorithm that constructs a null basis with an upper triangular submatrix. These results are extended to computing sparse orthogonal null bases. We show that the sparsest null basis for an n-vector computed as a product of Givens rotations has n log(,2) n nonzeros. A generalization for dense t x n matrices constructs an orthogonal null basis with nt log(,2)n/t nonzeros. We also classify all known methods for constructing null bases, and show some unexpected equivalences between some of them.

Vector processing apparatus including means for identifying the occurrence of exceptions in the processing of vector elements

Abstract: A vector processing apparatus has a number of pipeline arithmetic units operating concurrently to execute a set of vector instructions dealing with vector elements. Stack registers are provided for each arithmetic unit to hold the vector instruction address, leading vector element position and vector register internal address, so that one of the exceptions that can be detected successively by several arithmetic units during the process of the vector instructions is selected on a priority basis through the comparison of information in the stack of the currently detected exception with information of exception detected previously.

A six‐dimensional oscillator basis classified by O(6)⊇SO(2)×SU(3)⊇SO(3)

Abstract: We explicitly construct a complete set of states, useful in three‐body problems, both in a boson operator realization and in terms of coordinates which are of interest to microscopic collective models. The states carry the angular momentum quantum number L and, for the classification scheme mentioned in the title, our expressions generalize to arbitrary L the results previously available only for L=0 and 1.

A Systematic Approach to Higher-Order Necessary Conditions in Optimization Theory

Abstract: Necessary conditions for an abstract optimization problem are derived under weak assumptions. The presence of a generalized critical direction in these conditions is the basis for deriving necessary conditions of arbitrary order for various concrete problems. Two applications are considered in detail. The first concerns first- and second-order necessary conditions for a constrained optimization problem in an infinite-dimensional vector space where the cost, equality and inequality functions possess differentials of a finite-dimensional one-sided character. The second application concerns first-, second- and third-order necessary conditions for a constrained optimization problem in a Banach space with Frechet differentiability hypotheses. In both applications normality conditions are not required. Several well-known results are generalized.

Finite vibrational displacements in the Eckart—Sayvetz basis

Abstract: Curvilinear internal coordinates are considered in terms of cartesian displacements in a molecule-fixed basis determined by the Eckart-Sayvetz conditions. The latter are interpreted as a set of restrictions on the metrics of the space and define cartesian displacements of “pure” vibrational character expanded to any order in terms of internal coordinates. Explicit expressions for expansion coefficients are given as a function of contravariant components of the metric tensor taken from existing table. A compact notation is proposed for anharmonic force constants, expansion coefficients of redundancies and coupling terms of the rotation—vibration hamiltonian.

The exact order of subsets of additive bases

Abstract: Let A be a set of nonnegative integers. The h-fold sum of A, denoted hA, is the set consisting of all sums of h not necessarily distinct elements of A. The set A is an asymptotic basis of order if hA contains all sufficiently large integers. The set A is an asymptotic basis if A is an asymptotic basis of order h for some hel. If A is an asymptotic basis, the exact order of A, denoted g(A), is the smallest integer h such that A is an asymptotic basis of order h. Let kel. l'f A is an asymptotic basis, let Ik(A) denote the set of all subsets F~-A such that F has cardinality k and the set A\F is an asymptotic basis. An open problem in additive number theory is to estimate g(A\F) in terms of g(A). More precisely, define

Vector processing apparatus

Abstract: A vector processing apparatus has a number of pipeline arithmetic units (30-33) operating concurrently to execute a set of vector instructions dealing with vector elements. Stack registers (301-309) are provided for each arithmetic unit to hold the vector instruction address, leading vector element position and vector register internal address, so that one of exceptions that can be detected successively by several arithmetic units during the process of the vector instructions is selected on a priority basis through the comparison of information in the stack of the currently detected exception with information of exception detected previously.

Set operations on linear quadtrees

Abstract: Linear quadtrees provide the potential of very efficient algorithms for image processing. This paper demonstrates the correctness of and analyzes the complexity of several fundamental operations on linear quadtrees. In particular, an algorithm for computing the intersection, union, and pairwise difference of two linear quadtrees is presented and analyzed. These operations provide the basis for many more complex image processing techniques and, hence their efficiency and correctness are extremely important. It is shown that these algorithms are linear in the number of nodes in the linear quadtrees involved in the operations. The paper also provides a brief introduction to linear quadtrees and some of their properties.

Gaussian basis sets for transition metals of the second series

Abstract: A Gaussian basis set consisting of (15s, 9p, 8d) Gaussian functions has been optimized for the transition metal atoms of the second series (fourth-row atoms).

Clebsch–Gordan coefficients for E6 and SO(10) unification models

Abstract: We illustrate here a new method for computing Clebsch–Gordan coefficients (CGC) for E6 by computing CGC for the product 27⊗27 of the irreducible representation (100000) of E6 with itself. These CGC are calculated thrice: once in a weight vector basis independent of any semisimple subgroup, then in a basis which refers to SO(10)⊆E6, and finally in a basis referring to SU(5)⊆SO(10)⊆E6.