Showing papers on "Basis (linear algebra) published in 1983"

Efficient diffuse function‐augmented basis sets for anion calculations. III. The 3‐21+G basis set for first‐row elements, Li–F

Abstract: The relatively small diffuse function-augmented basis set, 3-21+G, is shown to describe anion geometries and proton affinities adequately. The diffuse sp orbital exponents are recommended for general use to augment larger basis sets.

MINOS 5.0 User's Guide.

Abstract: : MINOS is a large-scale optimization system, for the solution of sparse linear and nonlinear programs. The objective function and constraists may be linear or nonlinear, or a mixture of both. The nonlinear functions must be smooth. Stable numerical methods are employed throughout. Features include a new basis package(for maintaining sparse LU factors of the basis matrix), automatic scaling of linear constraints, and automatic estimation of some or all gradients. Upper and lower bounds on the variables are handled efficiently. File formats for constraint and basis data are compatible with the industry MPS format. The source code is suitable for machines with a Fortran 66 or 77 compiler and at least 500K bytes of storage. (Author)

State-space and singular-value decomposition-based approximation methods for the harmonic retrieval problem

Abstract: We present new high-resolution methods for the problem of retrieving sinusoidal processes from noisy measurements. The approach taken is by use of the so-called principal-components method, which is a singular-value-decomposition-based approximate modeling method. The low-rank property and the algebraic structure of both the data matrix and the covariance matrix (under noise-free conditions) form the basis of exact modeling methods. In a noisy environment, however, the rank property is often perturbed, and singular-value decomposition is used to obtain a low-rank approximant in factored form. The underlying algebraic structure of these factors leads naturally to least-squares estimates of the state-space parameters of the sinusoidal process. This forms the basis of the Toeplitz approximation method, which offers a robust Pisarenko-like spectral estimate from the covariance sequence. Furthermore, the principle of Pisarenko’s method is extended to harmonic retrieval directly from time-series data, which leads to a direct-data approximation method. Our simulation results indicate that favorable resolution capability (compared with existing methods) can be achieved by the above methods. The application of these principles to two-dimensional signals is also discussed.

Unitary representations of non-compact supergroups

Abstract: We give a general theory for the construction of oscillator-like unitary irreducible representations (UIRs) of non-compact supergroups in a super Fock space. This construction applies to all non-compact supergroupsG whose coset spaceG/K with respect to their maximal compact subsupergroupK is “Hermitean supersymmetric”. We illustrate our method with the example of SU(m, p/n+q) by giving its oscillator-like UIRs in a “particle state” basis as well as “supercoherent state basis”. The same class of UIRs can also be realized over the “super Hilbert spaces” of holomorphic functions of aZ variable labelling the coherent states.

Interframe coding apparatus

Abstract: An interframe coding apparatus performs motion detection and vector quantization on the basis of a block of interframe differential signals and includes a vector quantizer. The data of the interframe differential signal or the block scanning predictive error signal is compressed using a vector quantizer. The vector quantizer receives a plurality of suppressed predictive error signals (624) as input signals in block form and operates to form input vectors. The data is analysed by a motion detector (652) and where motion is detected, the relevant block is coded as a significant block. A coder (638-642) operates to determine an output vector having the least distortion with respect to an input vector by various techniques. A coded output or output vector codetable address (625) is output from the coder of the device for receipt by a decoder (639,642,654,655) which constructs a proper output vector signal.

A Fast Beamforming Algorithm for Large Arrays

Abstract: This beamforming algorithm is written specifically for array radars in which the number of array elements K is very large compared with the number of jammers L the radar is designed to suppress. It uses a set of M noise vectors to construct a basis for the jammer component of the antenna output vectors. The component of the quiescent weight vector orthogonal to each basis vector is calculated, renormalized to unit length, and identified as the adapted weight vector. This algorithm is effective in the suppression of many types of jammers. The number of noise samples M required in the construction of the adapted weight vector is approximately equal to L. In the special case of L narrowband noise jammers, for example, a choice of M = L usually reduces the receiver output jammer power to a few dBs above the white noise background. It is permissible to have M

The finite dimensional basis problem with an appendix on nets of Grassmann manifolds

Abstract: It is well-known that if B is a normed space, then for every n-dimensional (or ncodimensional) subspace F there is a projection from B onto F of norm not exceeding ~/-n. But it has so far been an open question whether for every B of dimension greater than n there is some n-dimensional subspace F onto which there is a projection of norm o(Vr-n) or even K, with K an absolute constant (2). In this paper we will construct a 2ndimensional space in which every projection of rank n in B has norm greater than Cx/-nwhere C is an absolute positive constant. This example, which is in an obvious sense the be s t -o r the worst, depending on the point of view--possible , also settles in the negative \" the finite-dimensional basis p rob lem\" , i.e. the question whether there is an absolute constant K such that every finite-dimensional space has a basis with basis constant ~

Linear Ordering and Application to Placement

Abstract: Given a set of interconnected elements, linear ordering generates a linear sequence of elements of the set, which is the basis for most constructive initial-placement methods. This paper presents a new strategy for linear ordering. The important difference of the new technique from the previous ones is that it starts the ordering process from the most lightly connected seed. It was applied to various placement problems including standard cell and gate array and produced very good results.

Systems of numeration

Abstract: A numeration system is a set of integers (basis elements) such that every integer can be represented uniquely over the set using integer digits of bounded size. Such systems are scattered in many fields in mathematics and computer science. Many of the known ones and new ones are unified and derived from a basic result on recursively defined basis elements. Applications are indicated.

Solving low density subset sum problems

Abstract: The subset sum problem is to decide whether or not the 0-1 integer programming problem Σi=1n aixi = M; all xi = 0 or 1; has a solution, where the ai and M are given positive integers. This problem is NP-complete, and the difficulty of solving it is the basis of public key cryptosystems of knapsack type. We propose an algorithm which when given an instance of the subset sum problem searches for a solution. This algorithm always halts in polynomial time, but does not always find a solution when one exists. It converts the problem to one of finding a particular short vector v in a lattice, and then uses a lattice basis reduction algorithm due to A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovasz to attempt to find v. We analyze the performance of the proposed algorithm. Let the density d of a subset sum problem be defined by d=n/log2(maxi ai). Then for "almost all" problems of density d ≪ .645 the vector v we are searching for is the shortest nonzero vector in the lattice. We prove that for "almost all" problems of density d ≪ 1/n the lattice basis reduction algorithm locates v. Extensive computational tests of the algorithm suggest that it works for densities d ≪ dc (n), where dc (n) is a cutoff value that is substantially larger than 1/n. This method gives a polynomial time attack on knapsack public key cryptosystems that can be expected to break them if they transmit information at rates below dc (n), as n → ∞.

The immersion under feedback of a multidimensional discrete-time non-linear system into a linear system

Abstract: This paper deals with the problem of finding conditions which ensure the reproducibility of the input-output behaviour of a linear analytic discrete-time system by means of a linear discrete-time system. On this basis the problem of modifying the system by feedback in order for it to enjoy such a property is posed and solved,

On the basis set superposition error in potential surface investigations. I. Hydrogen‐bonded complexes with standard basis set functions

Abstract: The basis set superposition errors on the energy of the complexes FH ⋅⋅⋅ OH2 and FH ⋅⋅⋅ NCH are investigated at the SCF level. The two‐dimension energy‐potential surfaces, corresponding to the slow and fast frequencies of the hydrogen bridge, are calculated without and with the counterpoise correction. The corresponding quadratic, cubic, and quartic force constants are computed. Similar calculations are reported for the electric dipole moment and its derivatives. These investigations show that the basis set superposition error: (i) does not decrease, in the vicinity of the equilibrium configuration of the complex, when standard high‐quality basis sets are used; (ii) shifts the minimum of the potential surface, distorts this surface, and changes the force constants in a significant but not drastic manner.

Gaussian basis sets for the fourth‐row main group elements, In–Xe

Abstract: Gaussian basis sets, consisting of 15 s-type, 11 p-type, and 6 d-type functions, for the fourth-row main group elements, In-Xe, are presented. In order to compare these basis sets with larger ones, calculations have been performed in I2 and TeO2.

Basis set quality. II. Information theoretic appraisal of various s- orbitals†

Abstract: The expectation values 〈rk〉 (−2 ⩽ k ⩽ 4, k = 10), values of the charge density ρ(r) at selected points, and coefficients in the MacLaurin expansion of ρ(r) are used to test the quality of 71 orbital basis sets used for the atomic helium Hartree–Fock problem. These tests in position space are complementary to the momentum space tests previously carried out [Int. J. Quantum Chem. 21, 419 (1982)]. Information theoretic measures with respect to either or both position and momentum space properties are subsequently defined and the orbitals are ranked accordingly. These measures indicate that, for a given orbital, momentum space properties are more poorly predicted than position space ones. Moreover an improvement in the virial ratio does not necessarily lead to a more balanced orbital with respect to position and momentum space properties. Basis sets containing Slater-type orbitals are again found to be rather accurate. The exponentially damped rational function is confirmed to be the outstanding two-parameter unconventional orbital.

Finite spectrum assignment problem for systems with delay in state variables

Abstract: This correspondence is concerned with the finite spectrum assignment problem for linear systems with delay in state variables. A systematic procedure for constructing the feedback law is presented on the basis of advanced elementary row operations on matrices with two variables s and z = e^{-sh} .

Large-scale linear programming: Geometry, working bases and factorizations

Abstract: This paper is concerned with linear programming problems in which many of the constraints are handled implicitly by requiring that the vector of decision variables lie in a polyhedronX. It is shown that the simplex method can be implemented using a working basis whose size is the number of explicit constraints as long as the local structure ofX around the current point is known. Various ways of describing this local structure lead to known implementations whenX is defined by generalized or variable upper bounds or flow conservation constraints. In the general case a decomposition principle can be used to generate this local structure. We also show how to update factorizations of the working basis.

Molecular basis set generation: Accurate slater basis sets for LiH− ground and excited state and Li2− ground state

Abstract: A procedure for generating basis sets for diatomic molecule electronic structure calculations is described. In essence, this procedure maps the results of nearly exact numerical Hartree–Fock calculations into basis set form. Two applications of the procedure are proposed: (a) generation of very high accuracy basis sets, and (b) investigation of basis sets for unusual systems. The latter application is illustrated by some results for diatomic anions.

Scattering in View of the Titchmarsh-Weyl Theory

Abstract: The Titchmarsh-Weyl theory for a singular second-order differential equation is presented. The equations, however, are arranged such that the theory is immediately applicable to the multichannel formulation. The mathematical basis of the Titchmarsh-Weyl theory (TWT) is outlined and the Titchmarsh-Weyl m function is presented. Relations to the associated Green’s function and the spectral function are given. It is shown how some quantities common in scattering theory can be expressed in terms of the TWT. The method of complex rotation is then applied to the TWT to create a complex rotated analog of the Titchmarsh-Weyl theory. This is then extended as exterior complex scaling is introduced. An outline of a proof for the method of exterior complex scaling is presented and some numerical results of the complete theory are given.

Three measures for synchronic dependence in Petri nets

Abstract: Three notions of distance for measuring the synchronic dependence of transition firings in Petri nets are studied. For the first two notions we present algorithms for obtaining a basis of the linear space of weight vectors for which the distance is finite, and methods for computing the distance for any given weight vector by examining a finite set of vectors obtained a priori. For the third notion of distance, the problem of deciding whether a given weight vector yields a finite distance is shown to be equivalent to the reachability problem. Finally it is shown that a basis of the linear space of weight vectors for which the weighted sum of token counts is bounded over all reachable markings can be obtained effectively. Also some complexity results are given.

An algorithmic approach to the linear stability of the Ekman layer

Abstract: The linear stability of the stationary Ekman-layer flow near a plane boundary is considered. Analytical formulas for the eigenfunctions are derived by a spectral analysis. Standard optimization algorithms are used to calculate critical points, maximum growth rates and neutral-stability curves. The near approach provides a better basis for both a linear and a nonlinear stability analysis than the well-known methods have done. The method may also be applied to other boundary-layer problems.

Differential forms as a basis for vector analysis-with applications to electrodynamics

Abstract: I give a concise and self‐contained presentation of the theory of differential forms and how it subsumes all of classical vector analysis when applied to a three‐dimensional space. The differential form analog of all vector operators, identities, and theorems are given (many of which are proved), all in the context of classical electrodynamics.

On cycling in the network simplex method

Abstract: There are well-known examples of cycling in the linear programming simplex method having basis size two and requiring only six pivots. We prove that any example having basis size two for the network simplex method requires at least ten pivots. We also present an example that achieves this lower bound. In addition, we show that an attractive variant of Cunningham's noncyling method does admit cycling.

Patterson search – an alternative to direct methods

Abstract: On the basis of the solution of six unknown structures by a real-space Patterson search the merits and limitations of the method are discussed. It is shown how chemical information can be used to provide a reliable starting point for direct methods when there is no automatic solution. Two examples demonstrate the potential usefulness of forcefield calculations for generating the geometries of appropriate search fragments in the absence of related crystal structures. The further discussion deals with future applications of the vector search method to determine large structures. A Patterson search program, which incorporates all necessary features of a modern program and is compatible with SHELX84, is being developed.