Jacobian Of Transformation Research Articles

Approximate Distributions for the Various Serial Correlograms

Saddlepoint methods are used to approximate the joint density of the serial correlogram up to lag m. Jacobian transformations also lead to approximations for the related partial correlogram and inverse correlogram. The approximations consider non-circularly and circularly defined models in both the null and the non-null settings. The distribution theory encompasses the standard non-circularly defined correlogram computed from least-squares residuals removing arbitrary fixed regressors. Connections of the general theory to the approximations given by Daniels and by Durbin in the circular setting are indicated. The double-saddlepoint density and distribution approximations are given for the conditional distribution of the non-circular lag m serial correlation given the previous lags from order 1 to m-1. This allows for the computation of p values in conditional inference when testing that the model is AR(m-1) versus AR(m). Numerical comparisons with the tests of Daniels and of Durbin suggest that their tests based on circularity assumptions are inadequate for short non-circular series but are in close agreement with the non-circular tests for moderately long series.

Generation of base station DOA distributions by Jacobi transformation of scattering areas

The author describes a simple method for transforming angular distributions of a radio scattering area located around a mobile station (MS) into a direction-of-arrival (DOA) distribution at a base station (BS). Tractable closed form solutions for circular uniform and conical (nonuniform) scattering areas, are presented.

An Adaptive Grid Method and Its Application to Steady Euler Flow Calculations

An adaptive remeshing procedure based on a cell volume deformation method is presented. Starting with an initial grid, this method offers direct cell volume control through the specification of the transformation Jacobian. Grid points are moved with appropriate grid velocities so that the specified cell volume distribution can be achieved at the end of the grid movement without adding or removing grid points. The grid velocities are determined by solving a scalar Poisson equation. This method is applied to solving the compressible Euler equations. Computational test cases of transonic flow over an airfoil are presented and demonstrate the desired control of grid sizes across shock waves.

Proof of the conjectures of H. Uhlig on the singular multivariate beta and the Jacobian of a certain matrix transformation

Uhlig proposes two conjectures. The first concerns the Jacobian of the transformation $Y = B \times B'$ where B is the matrix $m \times m$ and m and X, Y belong to the class of positive semidefinite matrices of the order of $m \times m$ of rank $n < m, S_{m,n^{\cdot}}^+$. The second is concerned with the singular multivariate Beta distribution. This article seeks to prove the two conjectures. The latter result is then extended to the case of the singular multivariate F distribution, and the respective density functions are located for the nonzero positive eigenvalues of the singular Beta and F matrices.

Analysis of the polar form of the von Mises transformation

The implementation of the classical form of the von Mises transformation in the study of fluid flow in two-dimensional curvilinear domains gives rise to an infinite Jacobian of transformation when viscous fluid flow is considered and when the streamlines of the flow become vertical at any point in the flow domain. In this work, it will be shown that in special cases the infinite nature of the Jacobian can be eliminated when the streamlines are vertical by utilizing the polar form of the transformation.

The short-lived 8Be level at Ex=11.4 MeV

Measurements of the differential cross section for the 7Li(d, alpha alpha n) reaction have been performed at incident deuteron energy Ed=19.7 MeV. An appropriate kinematical configuration was chosen to observe the population of the second excited 8Be level. The values of (11.3+or-0.2) MeV for the excitation energy and of (3.7+or-0.2) MeV for the width of the above level were found. The influence of the transformation Jacobian on the determination of the two spectroscopic parameters is discussed.

On the relationship between intensity and resolution of pulsed-source backscattering spectrometers

In a recent paper on the high-resolution pulsed-source backscattering spectrometer LAM-80ET, Ikeda et al. noted that the observed intensity I varies linearly with energy resolution Δ ɛ, on changing order of analyser reflection. This is apparently at variance with reactor folklore, which is based solely on phase-space arguments, in which it is assumed that I∝Δ ɛ 2. We discuss this problem in terms of (a) resolution-volume arguments, (b) the energy dependence of source flux, (c) the appropriate Jacobian for transformation from time-of-flight to energy scan, (d) a simple model for resolution of the spectrometer and (e) the wavelength variation of critical angle for the neutron guide. Making suitable simplifying approximations, we arrive at the form I∝Δ ɛ 4/3.

The evolution of the zodiacal dust cloud under plasma drag and lorentz forces in the latitudinally asymmetric solar wind

Orbital evolutions of zodiacal dust particles are mainly influenced by electromagnetic and by plasma Poynting-Robertson effects. Whereas the first effect acts radially symmetric, the second one depends on heliographic latitude caused by the dynamical friction of the dust particles in the ambient asymmetric solar wind flow. The solar wind mass and momentum flows vary with heliographic latitude by about 50% from the ecliptic towards the poles. Here we take into account related asymmetries in the plasma Poynting-Robertson drag. In addition, we derive formulae for a competing effect due to systematic Lorentz forces and to Lorentz scatterings of the charged dust particles at the global interplanetary magnetic field. We describe the orbital element changes occurring under these conditions. Whereas Lorentz forces by means of induced inclination drifts only cause negligible evolutionary effects, we can show that the radial migration rates due to asymmetric plasma friction forces are greater by about 5–10% for particles close to the ecliptic compared to those at high inclinations. This causes a typical variation of the inclination distribution with decreasing semimajor axis. In addition, it reveals that the dust distribution function cannot be factorized into parts solely dependent on radial distance or on inclination. We also discuss the kinetic equation for the distribution function of dust particles, and find the distribution in orbital element space from which the spatial density of the particles can then be found as an integral over this distribution function after application of a Jacobian transformation of element space differentials into configuration space differentials. Some observational consequences are drawn from this comparison and are discussed in some more detail.

Critical look on new approach for geometrically evolving domains in geomechanics

AbstractNumerical modelling of geometrically evolving domains, such as multistep construction, accretion and excavation, has always been of great interest to geotechnical engineers. Stress, strain and pore‐pressure changes depend on hydromechanical properties and the loading process. Several techniques are proposed in the literature for numerical simulation of excavation or backfilling processes. These are mostly based on a natural methodology that consists in dividing the problem into several time steps, each coinciding with the addition or removal of one layer which is then modelled by means of several elements. Unfortunately, as in these methods the evolving parameters and their relative influence are not explicit, it is not possible to determine their accuracy. A more rigorous approach lies in the geometrical transformation of the studied domain, depending on the time variable for a reference domain in which the conservation equations are verified. The application of this methodology implies a great number of calculations. By using limited development techniques of the Jacobian of transformations it is possible to reduce the volume of computations. Such a technique results in approximations which can be evaluated.

Evolution of Dust Particle Orbits under the Influence of Solar Wind Outflow Asymmetries and the Formation of the Zodiacal Dust Cloud

The secular orbital evolution of zodiacal dust particles is mainly influenced by the electromagnetic and the plasma Poynting-Robertson effects. Whereas the first effect is radially symmetric, the second one, caused by the dynamical friction of the dust particles in the ambient solar wind flow, depends on heliographic latitude. This is because the solar wind appears to have pronounced mass and momentum flow variations with respect to heliographic latitudes, with mass flows decreasing by about 50% from the ecliptic toward the poles. We take into account related asymmetries in the plasma Poynting-Robertson effect and study orbital evolution of dust particles orbiting with different inclinations. We derive equations describing the secular changes of the orbital elements under these conditions. We show that the radial migration rates are greater by about 5 to 10% for particles close to the ecliptic as compared to those at higher inclinations. This leads to typical changes in the distribution of inclinations with decreasing semimajor axes of the particles and also shows that the dust distribution function cannot be factored into parts solely dependent on radial distance or inclination alone. Furthermore, the kinetic equation for the distribution function of dust particles is solved numerically, yielding the distribution in orbital element space. The spatial density of the particles can then be found as an integral over this distribution function after application of a Jacobian transformation of element space differentials into configuration space differentials and can be compared with earlier results. Some observational consequences are drawn from this comparison and are discussed in some more detail. It is also pointed out that the plasma Poynting-Robertson effect in a two-temperature solar wind produces normal components of perturbation forces and thus tends to change the inclination of the dust particle orbits. This is of special importance in the solar wind regime inside the critical point where low-Mach-number flows prevail.

A direct approach for the development of plane elements

The derivation of an element is based on the differential equations of the problem. The second-order derivatives of equations are approximated by first-order interpolation function derivatives. The equations derived in this way, multiplied by the determinant of the Jacobian transformation matrix, represent a basis for the derivation of the element matrix. The determinant of the Jacobian is attached to the geometrical derivatives at nodes, and as a result of this, the determinant does not appear in the denominator. Consequently, explicit integration is possible. By defining the derivatives at the nodes, and at certain points inside the element, the element matrix is derived directly. Several quadrilateral plane stress elements developed by the proposed approach are explicitly defined; they give converging results of improved accuracy. This method of development of finite elements can be applied to the development of finite elements for the solution of any problem governed by similar or higher-order differential equations.

Garbage collection data structure and its application to incomplete jacobi transformations

An algorithm is developed using Jacobi transformations and implemented through the garbage collection data structure. The beauty of this algorithm is that the element having the largest magnitude is made zero after the transformation applied remains zero in the subsequent transformations. Should this element become nonzero it is stored in the garbage collection data structure with row and column pointers. This algorithm produced the required number of eigenvalues. The accuracy of the results produced is also comparable with other methods.

A field-consistent and variationally correct representation of transverse shear strains in the nine-noded plate element

We present a biquadratic Lagrangian plate bending element with consistent fields for the constrained transverse shear strain functions. A technique involving expansion of the strain interpolations in terms of Legendre polynomials is used to redistribute the kinematically derived shear strain fields so that the field-consistent forms (i.e. avoiding locking) are also variationally correct (i.e. do not violate the variational norms). Also, a rational method of isoparametric Jacobian transformation is incorporated so that the constrained covariant shear strain fields are always consistent in whatever general quadrilateral form the element may take. Finally the element is compared with another formulation which was recently published. The element is subjected to several robust bench mark tests and is found to pass all the tests efficiently.

Localization of wavefunctions from extended systems using orbital occupation numbers

An algorithm is presented which produces localized cluster region orbitals which may be useful in embedded cluster calculations. The orbital occupation number matrix, n, for a trial set of functions is evaluated using the density matrix computed from an extended system. Surprisingly, when using Hartree–Fock (HF) orbitals from finite cluster calculations, several partially occupied orbitals having nii different from 2 and 0 are found. The localization is accomplished by applying Jacobi transformation to n, mixing surrounding functions with the cluster region, so that the final set of localized functions have nii with either 2 or 0. Numerical examples of the algorithm are given for Li clusters simulating the Li(100) surface. © 1992 John Wiley & Sons, Inc.

Use of symbolic computation in robotics education

An application of symbolic computation in robotics education is described. A software package is presented which combines generality, user interaction, and user-friendliness with the systematic usage of symbolic computation and artificial intelligence techniques. The software utilizes MACSYMA, a LISP-based symbolic algebra language, to automatically generate closed-form expressions representing forward and inverse kinematics solutions, the Jacobian transformation matrices, robot pose error-compensation models equations, and Lagrange dynamics formulation for N degree-of-freedom, open chain robotic manipulators. The goal of such a package is to aid faculty and students in the robotics course by removing burdensome tasks of mathematical manipulations. The software package has been successfully tested for its accuracy using commercially available robots.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Binary trees, symplectic matrices and the Jacobi coordinates of celestial mechanics

A graph-theoretic algorithm for constructing the Jacobi coordinates in celestial mechanics is given. To every full binary tree with N leaves, there corresponds a 6N×6N symplectic matrix, which defines a Jacobi transformation. This correspondence yields a direct proof of the symplectic property for all the Jacobi coordinates; hitherto, only special examples of these transformations have been shown to be canonical.

Unique ion beam scattering technique on depth profile determination

Abstract By combination of energy and time of flight detection in ERDA or NRA measurement, the particles' masses can be determined, additionally to the depth information. This leads to unique depth profile determination even for complex targets. Combination with th concept of Jacobi transformations results in extremely fast measurements. In this paper, a way is shown how this concept can be realized even for accelerators ofenergies as low as about 1 MeV.

An exact element method for plane problem

In this paper, based on the step reduction method[1]and exact analytic method[2]a new method—exact element method for constructing finite element, is presented. Since the new method doesn't need the variational principle, it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficient. By this method, a quadrilateral noncompatible element with 8 degrees of freedom is derived for the solution of plane problem. Since Jacobi's transformation is not applied, the present element may degenerate into a triangle element. It is convenient to use the element in engineering. In this paper, the convergence is proved. Numerical examples are given at the end of this paper, which indicate satisfactory results of stress and displacements can be obtained and have higher numerical precision in nodes.

On the mixed formulation of A 9-node Lagrange shell element

A 9-node Lagrange shell element is examined using a strain-based mixed method. Starting from a modified Hellinger-Reissner principle, finite element equations are derived by assuming both the displacement and strain fields independently. The strain functions are carefully chosen in conjunction with several considerations discussed in the paper. The resulting element is free from shear and membrane locking, hence it can be used for modeling of either thin or moderately thick shells. All kinematic deformation modes have been systematically suppressed. Further, a Jacobian transformation for the strain functions is employed between the natural and lamina coordinates to reduce the element sensitivity to geometric distortions. Six examples are given to illustrate the numerical performance of the proposed element.

Central extensions of area preserving membrane algebras

Understanding the symmetries of membrane and p-brane theories is important in order to grain some insight into their quantum structure. In the light-cone gauge there is a residual symmetry of the classical theories that corresponds to the subgroup of the diffcomorphisms of the p-brane for which the transformation jacobian is equal to unity. We investigate the possible appearance of anomalies in this algebra, by looking for the most general possible central extension compatible with the Jacobi identity. We show that for membranes the most general central extension is characterized by the space of harmonic one-forms on the membrane, whose dimension is twice the genus of the surface. We generalize this algebra to a hypothetical supersymmetric theory with super-reparametrizations of the membrane surface, and show that it cannot admit any central extension.