General Linear Transformation Research Articles

Methods of weyl representation of the phase space and canonical transformations. II

Thus, we have shown that the solutions (24) and (36) of the basic equation (17) for the symbol of the intertwining operator of general linear canonical transformations are convenient for constructing the Green's functions of quadratic systems and studying group representations; this makes it possible to dispense completely with the technique of coherent states. For example, for a parametrically excited oscillator we have (34), a simple expression for the symbol of the Green's function in terms of fundamental solutions of the classical equation. This fact is not fortuitous: For it is only by virtue of the characteristic property (13) of the Weyl ordering that the invariance group of the symbol, realized by unitary transformations, is identical to the dynamical symmetry groupH(2n, R) λSp(2n, R) of the classical quadratic Hamiltonians. Therefore, it is convenient to use the classification methods for diagonalizing classical systems in the symbol itself. In this way, we have been able to show that the unique δ-functional singularities of the symbol (24) determine the sliding mappings (27)–(28) on invariant subspaces, and we have also obtained the symbols (37)–(41) of the spectral projectors of arbitrary quadratic systems. Note that every irreducible subspace and the projectors acting on it are ultimately determined by the type of some Jordan cell of the matrix of the classical Hamiltonian, and therefore, the theory of canonical Jordan-Arnol'd forms [31] for many-parameter quadratic systems finds a natural quantum-mechanical realization in the formalism we have proposed.

ON THE LINEAR TRANSFORMATIONS OF GAUSSIAN TRAJEC-TORY PARAMETERS AND THEIR INFLUENCE UPON ELECTRON OPTICAL ABERRATIONS

In this paper, we describe electron trajectory by vector form and express aberra-tions in matrix form. According to the more general viewpoint, we discuss some general linear transformations of gaussian trajectory parameters. The third order aberrations, aberration coefficients and their partial differentials in an electron optical system undergone these transformations are derived. Therefore the general electron optical Fraunhofer conditions for correcting isotropic and anisotropic coma are obtained. It is shown that under this condition, astigmatism and field curvature have extreme or stationary values. According to these results, some concrete problems are discussed; for example, the effect of aperture position on aberrations in the fixed or scanned electron beam systems, and the effect of magnetic immersion on aberrations in the emission electron system (cathode lens).

Pattern recognition in molecules from a general linear transformation

The comparison of molecular structures is facilitated by a combination of graphical and numerical techniques. Conversion to a molecular reference frame makes the linear relationships between the molecules or fragments easily visualized, and can be used to produce diagrams clearly displaying the molecular similarities and differences.

Noncoherent parallel optical processor for discrete two-dimensional linear transformations

We describe a parallel optical processor, based on a lenslet array, that provides general linear two-dimensional transformations using noncoherent light. Such a processor could become useful in image- and signal-processing applications in which the throughput requirements cannot be adequately satisfied by state-of-the-art digital processors. Experimental results that illustrate the feasibility of the processor by demonstrating its use in parallel optical computation of the two-dimensional Walsh-Hadamard transformation are presented.

Simple Lagrangian for a superspace affine theory

Arguments are advanced for considering a particular Lagrangian in superspace. It is constructed solely from the connection associated with the graded group of real, general linear transformations, and is especially simple in form. The resulting equations of motion and the spontaneous-symmetry-breaking solutions, which maintain global supersymmetric invariance, are examined.

Affine and general linear equivalences of boolean functions

An algorithm is presented which determines, given two Boolean functions, the set of affine transformations and the set of general linear transformations under which the functions are equivalent. A method for detecting linear functions is developed for use in the algorithm.

A new method of matrix transformation. I. Matrix diagonalizations via involutional transformations

It is shown that two matrices A and B of order n×n which satisfy a monic quadratic equation with two roots λ1 and λ2 are connected by ATAB=TABB where TAB=A+B−(λ1+λ2) I with I being the n×n unit matrix (Theorem 1). The condition for TAB to be involutional is that the anticommutator of ?=A−(1/2)(λ1+λ2) I and ?=B−(1/2)(λ1+λ2) is a c number (Theorem 2). A 2m×2m matrix Q(2m) is introduced as a typical form of a matrix which can be diagonalized by an involutional transformation. These theorems are further extended through the matrix representation of the group of the general homogeneous linear transformations, GL(n). IUH (involutional, unitary, and Hermitian) matrices are introduced and discussed. The involutional transformations are shown to play a fundamental role in the transformations of Dirac’s Hamiltonian and of the field Hamiltonians which are quadratic in particle creation and annihilation operators in solid state physics.

Self dual l.t.a. filters

In this paper self-dual linear transformation active (l.t.a.) filter structures are introduced. The general linear transformation approach is constrained to this end to produce the necessary and sufficient conditions for self duality from which active-RC realisations of given lossless ladder filters are effected. Further degrees of freedom within the self-dual concept are then effectively utilised to produce active-RC realisations of highly modular form. These concepts are developed in this paper and are also applied to the design of two filters, the performance of which is most satisfactory.

Self dual l.t.a. filters

In this paper self-dual linear transformation active (l.t.a.) filter structures are introduced. The general linear transformation approach is constrained to this end to produce the necessary and sufficient conditions for self duality from which active-RC realisations of given lossless ladder filters are effected. Further degrees of freedom within the self-dual concept are then effectively utilised to produce active-RC realisations of highly modular form. These concepts are developed in this paper and are also applied to the design of two filters, the performance of which is most satisfactory.

Allowed supersymmetric vacuums

We construct the vacuum connection associated with a Yang-Mills theory of the graded Lie group of real, general linear transformations on superspace that is induced by supersymmetry transformations. This gives rise to a class of supersymmetric ''vacuum metrics,'' most of which are noninvertible and/or involve theories with torsion. All of these connections are nonsingular and no limiting procedures are involved, even in the case of the original Wess-Zumino-Salam-Strathdee supersymmetric line element. Many of them are a priori compatible with the internal-symmetry group SU(2) x U(1) and extensions to include color.

Yang-Mills gauge invariance on a space of Bose and Fermi coordinates

A complete formalism is developed for imposing Yang-Mills gauge invariance induced by general coordinate transformations on superspace (i.e., a space containing both commuting and anticommuting coordinates). The appropriate group is the graded pseudo-Lie group of real, general linear transformations on superspace analogous to the role played by GL(4,$R$) in general relativity. The construction of derivatives which transform covariantly under this group forces the introduction of a connection. In the usual gauge theories the connection is just the vector potential, whereas here we expect it to be a function of all the dynamical fields. In this purely affine theory, field strengths and our proposed equations of motion for them result in a self-sourced theory involving only the connection. However, we find that there exist solutions which permit us to define a metric for which an inverse does not exist. These solutions are associated with a spontaneous symmetry breakdown of the vacuum which yields only the Lorentz metric and with no restriction on the internal-symmetry group. This spontaneous symmetry breaking introduces a parameter with the dimensions of ${(\mathrm{mass})}^{2}$.

Rotational frames and Euclidean connection

The time-dependent rotational transformation, which is a special case of the time-dependent linear transformation of coordinates in Newtonian mechanics, is considered rigorously from the point of view of infinitesimal transformation. By this approach the standard techniques in differential geometry can be naturally introduced to classical dynamics. The relation between rotational reference frames and E. Cartan's Euclidean connection is obtained. It is suggested that the extension of the present theory to the (time-dependent) general linear transformation is possible by using the bundleL(M) of linear frames over a manifoldM.

Supplemented Partitioning Approach to Linear Filtering Theory

The implementation of Kalman-Bucy linear filtering theory incurs computational difficulties especially in application to large systems. These difficulties can be overcome by using the partitioning philosophy and here a new method called the Supplemented Partitioning Approach to filtering is developed. The special feature of this method is that all the subsystem filter computations are performed in parallel and the interactions between the partitioned subsystems are accounted for by supplementing the subsystem noise processes (system and measurement noise terms) by approximate terms derived using the concept of innovations processes.Various methods of partitioning are discussed and the S.P.A. filtering equations for the method of heuristic partitioning and the method of general linear transformation are derived. The former method is attractive from computational and practical aspects.The S.P.A. method is applied to a model of a chemical absorption tower and the results are compared to the optimal Kalman-Bucy filter and a suboptimal Kalman-Bucy filter in which all interactions between the subsystems are completely ignored. Finally the S.P.A. method is applied to a model of a complete nuclear power plant prototype.The application potential of the S.P.A. method is extensive and areas for future application and investigation are suggested.

Some Implications of Noncoaxiality in Finite Deformations

Two-dimensional finite deformations are analyzed by factoring and multiplying the matrices of the linear transformations representing them. A general linear transformation consists of a pure shear, a uniform dilation, and a rigid-body rotation. Coaxiality is defined for finite deformations and its effect on the resultant distortion discussed. Tests for coaxiality are devised for use on rectangular grids which are often employed in metal forming research. Formulas are derived for the initial and final directions of the resultant major principal axis in both equal and unequal noncoaxial pure shears and, in particular, conditions are found for the constancy of distortion in the second deformation.

Linear Canonical Transformations in the Theory of Vibrational-Rotational Interactions in Molecules. A Practical Closed Form N-Dimensional Procedure

A general N-dimensional linear transformation is written, wherein each original variable xi, pi (i=1, ···, N) is expressed in closed form as a linear function of the 2N transformed variables xi′, pi′ A set of 2N2-N relationships between the linear coefficients is then derived for the transformation to be canonical. This procedure provides one with the most general form of N-dimensional linear canonical quantum mechanical transformation, while giving the operators in closed form. The procedure is compared with the alternate method of using unitary generators, and its advantages over that method are discussed.

Geometrical Derivation of the Conservation Laws

Conservation laws which are customarily obtained by invoking the invariance of a Lagrangian density under the transformations of some intrinsic symmetry group may be given a completely geometrical treatment within the context of the theory by Yang and Mills. The formalism is extended from unitary transformations to general linear transformations and the concepts of parallel transfer, covariant differentiation, and intrinsic curvature tensor are discussed. Conservation laws follow from the assumed invariance of the Lagrangian under parallel transfer defined with a general affine connection which is a direct sum of intrinsic and space-time affinities. Conservation of generalized charge is a consequence of the arbitrariness of that part of the affinities which operates in the intrinsic space, and conservation of energy and momentum is related to the arbitrariness of the part of the affinities which operates on the space-time indices. Some comments on Palatini's derivation of Einstein's equation of general relativity are made.

Quantum Statistics of Coupled Oscillator Systems

The statistical properties of systems of coupled quantum-mechanical harmonic oscillators are analyzed. The Hamiltonian for the system is assumed to be an inhomogeneous quadratic form in the creation and annihilation operators, and is allowed to have an explicit time dependence. The relationship to classical theory is emphasized by expressing pure states in terms of the coherent-state vectors, and density operators by means of the $P$ representation and an analogous representation involving the Wigner function. The state which evolves from an initially coherent state of the system is found, and equations governing the time evolution of the Wigner function and the weight function for the $P$ representation are derived, in differential and integral form, for arbitrary initial states of the system. The results remain valid for couplings which do not preserve the vacuum state, and for cases in which the time dependence of the coupling parameters gives rise to large-scale amplification of the initial field intensities. The analysis is performed by first treating general linear inhomogeneous canonical transformations on the oscillator variables, and then specializing to the case in which these transformations represent the solutions for the Heisenberg operators in terms of their initial values. The results are illustrated within the context of a model of parametric amplification.

Selfconsistent calculation of nuclear equilibrium deformations

The Bogoliubov method of general linear transformations is applied to a selfconsistent calculation of deformed shapes of Rare Earth Nuclei and light nuclei around Mg24. Nucleon-nucleon forces of finite range and different spin-dependence and exchange character are used. The dependence of the deformation on the particle number and the different components of the nuclear interaction is investigated.

General linear transformations of locally convex spaces

General linear transformations of locally convex spaces

Visual pattern recognition by moment invariants

In this paper a theory of two-dimensional moment invariants for planar geometric figures is presented. A fundamental theorem is established to relate such moment invariants to the well-known algebraic invariants. Complete systems of moment invariants under translation, similitude and orthogonal transformations are derived. Some moment invariants under general two-dimensional linear transformations are also included. Both theoretical formulation and practical models of visual pattern recognition based upon these moment invariants are discussed. A simple simulation program together with its performance are also presented. It is shown that recognition of geometrical patterns and alphabetical characters independently of position, size and orientation can be accomplished. It is also indicated that generalization is possible to include invariance with parallel projection.