General Linear Transformation Research Articles

Comment on “Comment on Dunning's correlation-consistent basis sets”

By using a general linear transformation of Dunning's contracted basis set, the contraction length can be reduced even more than noted in the Comment by Hashimoto et al. With GAUSSIAN 94, this results in a considerable savings in computer time. If one is willing to neglect small coefficients, even greater savings are possible.

The multi-configurational adiabatic electron transfer theory and its invariance under transformations of charge density basis functions

The continuum multi-configurational dynamical theory of electron transfer (ET) reactions in a chemical solute immersed in a polar solvent is developed. The solute wave function is represented as a CI expansion. The corresponding decomposition of the solute charge density generates a set of dynamical variables, the discrete medium coordinates. A new expression for the free energy surface in terms of these coordinates is derived. The stochastic equations of motion derived earlier are shown to be invariant under unitary transformations of orbitals used to build the CI expansion provided the latter is complete over the corresponding orbital subspace, and also under general linear transformations of the bases employed in expanding the charge density. The interrelation between the present general treatment and the reduced theory applied previously in terms of the two-level ET model is investigated. Finally, the explicit expression for the screening potential of medium electrons is derived in the electronic Born-Oppenheimer approximation (fast (slow) electronic timescale for solvent (solute)). The theory leads to a self-consistent scheme for practical calculations of rate constants for ET reactions involving complex solutes. Illustrative test calculations for two-level ET systems are presented, and the importance of proper boundary conditions for realistic molecular cavities is demonstrated.

An algebraic and Bayesian technology for object recognition

This paper presents a new robust, low computational cost technology for recognizing free-form objects in three-dimensional (3D) range data, or, in two dimensional (2D) curve data in the image plane. Objects are represented by implicit polynomials (i.e. 3D algebraic surfaces or 2D algebraic curves) of degree greater than two, and are recognized by computing and matching vectors of their algebraic invariants (which are functions of their coefficients that are invariant to translations, rotations and general linear transformations). Such polynomials of the fourth degree can represent objects considerably more complicated than quadrics and super-quadrics, and can realize object recognition at significantly lower computational cost. Unfortunately, the coefficients of high degree implicit polynomials are highly sensitive to small changes in the data to which the polynomials are fit, thus often making recognition based on these polynomial coefficients or their invariants unreliable. We take two approaches to the...

General linear transformations and entangled states.

In this paper we study the properties of general linear transformations with complex coefficients, and the effect of such a transformation on a multimode state which is characterized by a Gaussian-Wigner function. We present general results on the changes in the correlation characteristics and the correlation entropy. We then specialize to the two-mode (or two-particle) situation where two-mode squeezed states, subsequently squeezed separately, form a closed set under linear transformations. We derive conditions on the statistical properties of the input fields and the linear transformations such that the state of the output fields is unentangled if the input is unentangled. We present various special cases. Finally, we also discuss the production of entangled states for particles in a harmonic potential.

Quantum transformation theory in fermion Fock space

In this article a general linear quantum transformation U for the following fermion system with n modes, (b′+,b̃′)=U(b+,b̃)U−1=(b+,b̃)(B̃, A, C̃D), is studied. All above transformations in Fock space are proved to form a ray representation of a group which is isomorphic with O(2n,C). The explicit expressions of operator U in terms of the normal ordered product and non-normal ordered product are obtained. A series of auxiliary identities are given. It has been mentioned that the results are applicable to the case of fermion fields with interaction.

Requirements for correlation energy density functionals from coordinate transformations.

The behavior of the density-functional correlation energy under general linear coordinate transformations is considered. It is demonstrated that it is uniform and nonuniform scaling that yield new conditions which are fulfilled by the exact correlation energy functional, besides the well-known requirements of rotational, reflectional, and translational invariance. By considering various types of nonuniform scaling in the limit of the scale factor \ensuremath{\lambda} going to zero and infinity, a set of relations for the correlation energy is derived. These relations, which are easy to test on approximate functionals, are of the type ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}0}$(1/\ensuremath{\lambda})${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}\ensuremath{\lambda}}^{\mathit{x}\mathit{y}}$]=0 and ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}^{\mathit{x}}$]=0. In addition, a previous proof is simplified for ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$]=const, which is bounded from below, and we identify the constant as a second-order energy. Here ${\mathit{n}}_{\ensuremath{\lambda}}$(r)=${\ensuremath{\lambda}}^{3}$n(\ensuremath{\lambda}x,\ensuremath{\lambda}y,\ensuremath{\lambda}z), ${\mathit{n}}_{\ensuremath{\lambda}}^{\mathit{x}}$(r)=\ensuremath{\lambda}n(\ensuremath{\lambda}x,y,z), and ${\mathit{n}}_{\ensuremath{\lambda}\ensuremath{\lambda}}^{\mathit{x}\mathit{y}}$(r)=${\ensuremath{\lambda}}^{2}$n(\ensuremath{\lambda}x,\ensuremath{\lambda}y,z) where n is the electron density. The limiting relations are not all met by any known approximate correlation energy functional. These relations apply to a correlation energy for addition to a Hartree-Fock calculation as well as to a correlation energy for use as part of a full density-functional calculation.

Similarity transformations in one- and two-mode Fock space

The normally ordered Hilbert space image of the general (complex) linear similarity transformation in phase space is obtained in a coherent state representation. Although preserving the commutator of a and a+, these quantum mechanical images of classical transformations are generally not unitary. Remarkably, although the kets and bras produced by the non-unitary similarity transformations are not Hermitian conjugates, squeezed state analogues produced using the similarity transformation still satisfy an overcompleteness relation. The results are extended to two-mode Fock space and simple examples of the utility of the similarity transformation are presented. The evaluation of the normally ordered operators is greatly facilitated by the use of integration within ordered products.

Degeneration of point configurations in the projective plane

In this paper we construct a compactification of the moduli space Bn*=(P22⧹Δ)⧸PGL(3)of n-point configuration in the plane in general position modulo linear transformations. This compactification Bn is an analogue to the space of n-pointed trees of projective lines and a generalization of a compactification of orbit spaces of more general group actions. We relate the fibres of the universal n-point configuration Fn→Bn to schemes associated with the Bruhat-Tits Building and give explicit descriptions in the cases n=5 and n=6.

Recognition by linear combinations of models

An approach to visual object recognition in which a 3D object is represented by the linear combination of 2D images of the object is proposed. It is shown that for objects with sharp edges as well as with smooth bounding contours, the set of possible images of a given object is embedded in a linear space spanned by a small number of views. For objects with sharp edges, the linear combination representation is exact. For objects with smooth boundaries, it is an approximation that often holds over a wide range of viewing angles. Rigid transformations (with or without scaling) can be distinguished from more general linear transformations of the object by testing certain constraints placed on the coefficients of the linear combinations. Three alternative methods of determining the transformation that matches a model to a given image are proposed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The revised fundamental theorem of moment invariants

A revised fundamental theorem of moment invariants for pattern recognition which corrects the fundamental theorem proposed by M.K. Hu (1962), is presented. The correction affects neither similitude (scale) nor rotation invariants derived using the original theorem, but it does affect features invariant to general linear transformations. Four of the latter invariants were presented originally by Hu. These are revised to take the correction to the fundamental theorem into account. Furthermore, these four invariants are combined to yield three new invariants, which are additionally invariant to changes in the illumination of an image.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Feedback network with space invariant coupling

Processing images by a neural network means performing a repeated sequence of operations on the images. The sequence consists of a general linear transformation and a nonlinear mapping of pixel intensities. The general (shift variant) linear transformation is time consuming for large images if done with a serial computer. A shift invariant linear transformation can be implemented much easier by fast Fourier transform or optically, but the shift invariant transform has fewer degrees of freedom because the coupling matrix is Toeplitz. We present a neural convolution network with shift invariant coupling that nevertheless exhibits autoassociative restoration of distorted images. Besides the simple implementation, the network has one more advantage: associative recall does not depend on object position.

Video Graphics and High-Voltage Electron Microscopy For Analysis of Microtubule Distributions In Fixed Cells

Microtubules (MTs) are cytoplasmic polymers whose dynamics have an influence on cell shape and motility. MTs influence cell behavior both through their growth and disassembly and through the binding of enzymes to their surfaces. In either case, the positions of the MTs change over time as cells grow and develop. We are working on methods to determine where MTs are at different times during either the cell cycle or a morphogenetic event, using thin and thick sections for electron microscopy and computer graphics to model MT distributions.One approach is to track MTs through serial thin sections cut transverse to the MT axis. This work uses a video camera to digitize electron micrographs of cross sections through a MT system and create image files in computer memory. These are aligned and corrected for relative distortions by using the positions of 8 - 10 MTs on adjacent sections to define a general linear transformation that will align and warp adjacent images to an optimum fit. Two hundred MT images are then used to calculate an “average MT”, and this is cross-correlated with each micrograph in the serial set to locate points likely to correspond to MT centers. This set of points is refined through a discriminate analysis that explores each cross correlogram in the neighborhood of every point with a high correlation score.

On the polyatomic franck-condon factors

We derive recursion relations for the polyatomic Franck-Condon factors using a general linear transformation of boson operators.

Squeezed-state wave functions and their relation to classical phase-space maps.

Coordinate wave functions for the one-mode squeezed states produced by the quantum analog of the general linear transformation in phase space are calculated. The probability density [\ensuremath{\Vert}\ensuremath{\psi}(q)${\ensuremath{\Vert}}^{2}$] for these states is Gaussian with center predicted by the classical transformation. The quantum image (which includes the traditional two-mode squeeze operator) of a three-parameter symplectic map in two-mode phase space equally generates squeezed states having Gaussian \ensuremath{\Vert}\ensuremath{\psi}(${q}_{1}$,${q}_{1}$)${\ensuremath{\Vert}}^{2}$. The center of the two-mode Gaussian is again predicted by the classical mapping.

General two-mode squeezed states

The states\(\hat \theta \)|A1A2〉 are considered, where the operators\(\hat \theta \) are associated with a unitary representation of the groupSp(4,ℝ), and the two-mode Glauber coherent states |A1A2> are joint eigenstates of the destruction operatorsa1 anda2 for the two independent oscillator modes. We show that they are ordinary coherent states with respect to new operatorsb1 andb2, which are themselves general linear (Bogolibov) transformations of the original operatorsa1,a2 and their hermitian conjugatesaδ1,aδ2. We further show how they may be regarded as the most general two-mode squeezed states. Most previous work on two-mode squeezed states appears to be based on more restrictive definitions than our own, and thereby reduces to special cases which are unified within our treatment.

A maximum modulus theorem for linear fractional transformations

A main result in the structured singular value theory is a maximum modulus-like theorem for a special case of linear fractional transformations. We extend that result to general linear fractional transformations.

Canonical and duality transformations for the electromagnetic field

In earlier papers we have given a new hamiltonian formulation of the source-free electromagnetic field. In this paper we show that the most general linear canonical transformation with constant coefficients which leaves the form of the field equations and the hamiltonian invariant is the duality transformation. Other invariant quantities of physical interest are also discussed.

Scale dependence and the temporal persistence of spatial patterns of soil water storage

The concept of time stability as defined by Vachaud et al. (1985) is expanded to include general linear transformations in time and to account for the occurrence of spatial scale dependency. Time stability is described as the temporal persistence of a spatial pattern and is evaluated using correlation analysis of successive measurement dates. Spatial coherency analysis is suggested as a method for examining the temporal persistence of a spatial pattern as a function of spatial scale. Spatial coherency analysis was used to examine the temporal persistence of soil water storage (0–1.7 m) measured every 10 m in a 720‐m‐long transect for drying and recharge events. Soil water recharge altered the spatial pattern of water storage at small scales which was significantly related to surface (topographic) curvature. Drying did not alter the spatial pattern of soil water storage. The study supports the concept that soil water storage at a point is the product of hydrologic processes operating at different spatial scales. The analysis can be used to relate the spatial scale of processes to independent factors.

Optical associative processor for general linear transformations

A new technique for the realization of general linear transformations using associative memories is described. An optical architecture for its implementation is also presented. A low-level feature space processor using this architecture is proposed. The processor is capable of recognizing and locating objects of various shapes and uses certain linear transformations in the feature space for distortion invariance.

Model for a linear viscoelastic medium that has consistent creep and stress‐relaxation properties

The stress and strain tensors for a homogeneous isotropic viscoelastic material are related by a general linear transformation, which expresses a complete proportionality of the sum of the stress and the stress‐rate fields to the sum of the strain and the strain‐rate fields in the medium. This transformation defines the functional forms of all components in the symmetric fourth‐rank matter tensors that specify the stiffness and the compliance of the dissipative medium. Corresponding stiffness and compliance moduli (e.g., shear modulus and shear compliance) are found to have the same kind of functional dependence upon the frequency. Thus the stress‐relaxation effects exhibited by the any stiffness modulus are consistent with the creep (strain‐relaxation) effects exhibited by the corresponding compliance modulus. However, ass consequence of the tensor relation between stress and strain, stiffness (and compliance) moduli of different kinds (e.g., shear modulus and bulk modulus) vary with frequency in different ways. The matrix forms of the stiffness and compliance tensors are discussed in terms of analogous equivalent networks that can be used to represent the mechanical behavior of the viscoelastic medium. Tensor transformations and network‐analysis algebra were carried out using computer programs for symbol manipulation.