Projector Matrix Research Articles

Generalized formulation of quantum and classical crystallography using Green's functions

AbstractQuantum crystallography is a methodology by which structural information about a crystalline material obtained from X‐ray crystallography is combined with quantum mechanical methods. The objective is to enhance the data obtained from the X‐ray diffraction experiment, which are related to the atomic structure of the crystal, and to predict the properties and efficacy of those chemical compounds from which the crystals are derived. One approach in quantum crystallography is to use a projector matrix with a normalized trace. In this approach, quantum mechanical parameters in the projector matrix are fit into crystallographic data. During this fitting, the properties of the projector matrix called idempotency and normalization are used. Throughout this implementation procedure, Clinton's iteration scheme has been used in addition to the least‐squares technique. The purpose of the present study is to generalize Clinton's iterative equations in quantum crystallography by means of single‐particle Green's functions with the aid of the equal atoms model in the theory of direct methods. Convergency characters of the novel iterative equations are discussed by the steepest descent procedure. Furthermore, whether the calculations are valid in nonorthogonal bases was also examined. The iteration schemes widely used in quantum crystallography have been generalized but, in addition, the generalized expressions relating to the phase determination procedure and the probabilities of the sign relations between the structure factors are obtained and discussed comprehensively. The phrase order of crystallography has been put forward as a new concept. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

On compressions of normal matrices

Let A be a normal matrix and consider the polygon NR[A]={x *Ax: ∥x∥=1}. If υ *Aυ∈ int NR[A] , a projector matrix P υ is defined such that NR[ P * υ AP υ ] is supported by all or some edges of a polygon.

Quantum crystallography, a developing area of computational chemistry extending to macromolecules

We describe the concept of quantum crystallography (QCr) and present examples of its potential as a technique for facilitating computational chemistry, particularly, applications of quantum mechanics. Structural information has been used to facilitate quantum-mechanical calculations for several decades. Recent advances in theory and computational facilities have led to research opportunities that could be considered only in the past several years. We focus on the feasibility of applications of quantum mechanics to macromolecules. The approach used involves the concept of calculations based on fragments of molecules. The method for constructing fragments, their composition, and how they are assembled to form a projector matrix are discussed without the introduction of mathematical detail. Papers that provide the theoretical basis for QCr and our method for making fragment calculations are referenced, and some initial calculations are described here.

Quantum crystallography

Quantum Crystallography (QCr) concerns the combining of crystallographic data with quantum-mechanical techniques in such a way that it should be possible to obtain information of enhanced value. The enhancement could be increased accuracy or information more readily obtained, or both. Our approach to quantum crystallography makes use of molecular orbitals and a single density matrix (projector matrix) formalism to form a quantum-mechanical model that includes information from a crystal structure analysis. Some experiences involving data for crystalline maleic anhydride and also various substituted cubanes are described. A method is also discussed for performing ab initio quantum mechanics on large structures for which the calculation time is essentially a linear function of the complexity. This method is based on a property of overlap integrals that permits the ab initio calculations to be performed on fragments of the structure. The fragment calculations are combined to form an accurate density matrix for the entire structure.

Co-rotational flat facet triangular elements for shell instability analyses

This paper investigates the formulation of co-rotational flat facet triangular elements for the numerical analysis of instability phenomena in shell structures. The elements have three nodes with six degrees of freedom at each node. The term ‘co-rotational’ relates here to the provision of a local system that continuously rotates and translates with the element. Following mainly Nour-Omid and Rankin [B. Nour-Omid and C.C. Rankin, Finite rotation analysis and consistent linearization using projectors, Comput. Methods Appl. Mech. Engrg. 93 (1991) 353–384], the definition of an element resorts to a change of variables from the local frame to the global one. This is done through the use of a projector matrix which relates the variations of the local displacements to the variations of the global ones, by extracting the rigid body modes from the latter. The main difference from the original formulation lies in the parameterization of 3D finite rotations. In contrast to the paper by Nour-Omid and Rankin, a parameterization based on the rotational vector is here adopted and thus, an additional change of variables has to be performed. As a result, the rotational variables become additive and the necessity of a special updating procedure is avoided. The main feature of the adopted formulation is its independence of the local assumptions used to derive the internal forces and tangent stiffness in local coordinates. For a certain class of elements (i.e. elements with the same number of nodes and degrees of freedom) the main co-rotational framework is the same. Using this property, three types of local formulations are considered. A set of carefully chosen test problems is used in order to assess the performances of the three element types.

A unifying representation of some case-deletion influence measures in univariate and multivariate linear regression

In recent years numerous measures have been proposed for assessing the influence of observations on least-squares regression results. Various influence measures were introduced based on different motivational arguments and each was designed to measure the influence of observations on different aspects of various regression results. The chief motivation and result of this paper is the determination of the main mathematical components common to many multiple-case regression diagnostics, and to make use of these components in formulating an algebraic representation which unifies the existing measures. The representation, which we refer to as the J I -class, is based on a few common interpretable components. These components are functions of the elements of two orthogonal projector matrices; the projector matrix for the column space of the explanatory variables, and the projector matrix for the residuals. The proposed representation enables one to perceive such existing measures not only in their original diversified formulations, but also in a manner that allows associations or similarities to be recognized among them in a clear, concise fashion. The class possesses several properties that can be used to study the relationships among the existing influence measures, especially those with seemingly different motivations and initial characterizations. The J I -class, which we use for a unifying representation of some existing influence measures, can also be used to generate new influence measures.

Quantum crystallography and the use of kernel projector matrices

Quantum crystallography is a developing technique for extracting quantum mechanically valid properties from X-ray diffraction experiments. Quantum mechanics and crystallography are joined through the fact that the electron distributions around atoms are the source of X-ray diffraction and electron density distributions are observables that lend themselves readily to quantum mechanical description. Direct contact with the X-ray diffraction data is made by equating the structure factor magnitudes, which are readily obtained from the measured X-ray diffraction intensities, with the magnitudes of certain Fourier transforms of the quantum mechanical description of the electron distribution. The quantum mechanical description of the electron density involves molecular orbitals and an associated matrix. By requiring that the associated matrix be a projector with a normalized trace, while optimizing the fit to the experimental structure factor magnitudes, strong constraints are imposed on the relationship between the X-ray data and the quantum mechanical description of the electron density distribution. The final result should be a wave function that is in good agreement with the X-ray diffraction information and from which a variety of properties, e.g., electron densities and electrostatic potentials, could be extracted. The method for making the fit to the X-ray data involves the use of least-squares calculations in which the defining equations are the structure factor equations and equations that arise from the conditions defining a projector matrix with a normalized trace. The variables are the elements of the projector. It is also possible to refine such parameters as atomic coordinates. The calculations are facilitated by use of good initial projector matrices. For this purpose, a method has been developed for generating projector matrices for large molecules from the sum of kernel matrices. Kernel matrices are obtained from the use of fragments of the known atomic coordinates of the substance of interest and the application of molecular orbital methods in quantum mechanics. The use of fragments is justified by the fact that overlap integrals rapidly approach zero as the distances between atoms increase. © 1995 John Wiley & Sons, Inc.

Unlearning dynamics in Hopfield neural network

The dynamics of a convergent iterative unlearning algorithm proposed earlier [7, 8] is examined. A self-consistent system of equations of the spectral dynamics of a synaptic matrix is obtained at the thermodynamic limit. The unlearning intensity (which varies during the iteration process) that optimizes the algorithm's rate of convergence on the projector matrix is found. The synaptic-matrix spectrum dynamics for optimal unlearning is determined.

Finite rotation analysis and consistent linearization using projectors

A systematic procedure is presented that may be used to extend the capabilities of an existing linear finite element to accomodate finite rotation analysis. This procedure is a generalization of the work presented in Computers and Structures, Volume 30, pp. 257–267. The basis of our approach is the element-independent co-rotational algorithm, where the element rigid body motion (translations and rotations) is separated from the deformational part of its total motion. The variation of this co-rotational relation results in a projector matrix, with the property that a consistent internal force vector is invariant under its action. The consistent tangent stiffness matrix is shown to depend on this invariance condition through the derivative of the projector. This results in an unsymmetric tangent matrix whose anti-symmetric part depends on the out-of-balance force vector. In this paper we prove that using the symmetric part of the tangent matrix, the Newton iteration retains its quadratic rate of convergence. This approach has been used to solve a number of large rotation test example problems. The results demonstrate that it is possible to analyze structures undergoing large rotations within a general co-rotational framework, using simple and economical finite elements. The resulting improvements in the performance of these simple elements are brought about through the use of convenient software utilities as pre- and post-processors to the element routines.

Layered neural networks

Exact solutions for the dynamics of layered feedforward neural networks are presented. These networks are expected to respond to an input by going through a sequence of preassigned states on the various layers. The family of networks considered has a variety of interlayer couplings: linear and non-linear Hebbian, Hebbian with Gaussian synaptic noise and with various kinds of dilution. In addition, the authors also solve the problem of layered networks with the pseudoinverse (projector) matrix of couplings. In all cases the solutions take the form of layer-to-layer recursions for the mean overlap with a (random) key pattern and for the width of the embedding field distribution. The dynamics is governed by the fixed points of these recursions. For all cases, non-trivial domains of attraction of the memory states are found and graphically displayed.

Moshinsky brackets for light nuclei

A new formula for calculation of the Moshinsky brackets is derived. As its consequence, simple formulas for the angle-averaged Pauli projector matrix elements and for the transformed two-particle oscillator wave functions are given.NUCLEAR STRUCTURE New formulas for Moshinsky brackets, Pauli projector and Moshinsky transformed states.