Set Of Principal Axes Research Articles

Plane-wave scattering by an ellipsoid composed of an orthorhombic dielectric-magnetic material with arbitrarily oriented constitutive principal axes

The extended boundary condition method can be formulated to study plane-wave scattering by an ellipsoid composed of an orthorhombic dielectric-magnetic material whose relative permittivity dyadic is a scalar multiple of its relative permeability dyadic, when the constitutive principal axes are arbitrarily oriented with respect to the shape principal axes. Known vector spherical wavefunctions are used to represent the fields in the surrounding matter-free space. After deriving closed-form expressions for the vector spherical wavefunctions for the scattering material, the internal fields are represented as superpositions of those vector spherical wavefunctions. The unknown scattered-field coefficients are related to the known incident-field coefficients by a transition matrix. The total scattering and absorption efficiencies are highly affected by the orientation of the constitutive principal axes relative to the shape principal axes, and the effect of the orientational mismatch between the two sets of principal axes is more pronounced as the electrical size increases. The dependence of the total scattering efficiency, but not of the absorption efficiency, on the angle of rotation about a shape principal axis can be predicted qualitatively from the variation of a scalar function with respect to the angle of rotation. The total scattering and absorption efficiencies do not depend on the polarization state of incident plane wave when the scattering material is impedance-matched to free space. The polarization state of the incident plane wave has a more discernible effect on the total scattering and absorption efficiencies for ellipsoids compared to spheres.

Using molecular principal axes for structural comparison: determining the tertiary changes of a FAB antibody domain induced by antigenic binding

BackgroundComparison of different protein x-ray structures has previously been made in a number of different ways; for example, by visual examination, by differences in the locations of secondary structures, by explicit superposition of structural elements, e.g. α-carbon atom locations, or by procedures that utilize a common symmetry element or geometrical feature of the structures to be compared.ResultsA new approach is applied to determine the structural changes that an antibody protein domain experiences upon its interaction with an antigenic target. These changes are determined with the use of two different, however comparable, sets of principal axes that are obtained by diagonalizing the second-order tensors that yield the moments-of-geometry as well as an ellipsoidal characterization of domain shape, prior to and after interaction. Determination of these sets of axes for structural comparison requires no internal symmetry features of the domains, depending solely upon their representation in three-dimensional space. This representation may involve atomic, Cα, or residue centroid coordinates. The present analysis utilizes residue centroids. When the structural changes are minimal, the principal axes of the domains, prior to and after interaction, are essentially comparable and consequently may be used for structural comparison. When the differences of the axes cannot be neglected, but are nevertheless slight, a smaller relatively invariant substructure of the domains may be utilized for comparison. The procedure yields two distance metrics for structural comparison. First, the displacements of the residue centroids due to antigenic binding, referenced to the ellipsoidal principal axes, are noted. Second, changes in the ellipsoidal distances with respect to the non-interacting structure provide a direct measure of the spatial displacements of the residue centroids, towards either the interior or exterior of the domain.ConclusionWith use of x-ray data from the protein data bank (PDB), these two metrics are shown to highlight, in a manner different from before, the structural changes that are induced in the overall domains as well as in the H3 loops of the complementarity-determining regions (CDR) upon FAB antibody binding to a truncated and to a synthetic hemagglutinin viral antigenic target.

Forecast modelling for rotations of principal axes of multidimensional data sets

A set of principal axes accounts for the largest variation directions of a data set. It is one of the important general features of the data set. The paper discusses the forecast modelling method for the rotation of principal axes in the multidimensional space. The method was used in forecasting and analysing the development of city economy in China. The movement direction of the main features of the economic development of the city group can be predicted.

Validation of 3-D curved objects: CAD model and fabricated workpiece

This paper proposes a new method to describe and identify a 3-D curved object for the purpose of validating a fabricated object to the design specification. Curved 3-D objects are, in general, difficult to represent and identify because they lack distinct properties such as edges, planes, or cylindrical surfaces which are the building blocks in representing objects. In this paper, the authors propose to use principal axes of a 3-D object to establish a reference for the representation. A method of obtaining an inertia matrix from a 3-D range image is developed. The unique set of principal axes is obtained from the inertia matrix of an object with an arbitrary 3-D position and orientation, and the object can be described uniquely on these principal axes. On the principal axes, an object is described by a set of features describing the shape of the object such as spine, section size, section orientation, and section contraction. The features are used for comparing two objects for the validation purpose. The authors also propose a direct measure of similarity between two objects as a mean-squared difference of radii. As an experiment, two 3-D object models are designed through a CAD package, and fabricated objects are compared with the designed models for validation purposes. >

Simulation of three‐dimensional strong ground motions along principal axes, San Fernando earthquake

AbstractPower spectral density which describes frequency content is considered one of the most significant properties to be taken into account when generating ground motions through the use of stochastic processes. Using a smoothed and normalized Fourier amplitude spectrum, frequency content for components of motion along a set of principal axes is estimated. Fourier amplitude spectra obtained by this moving‐window technique are presented which show the time dependency of frequency content for motions produced by the San Fernando earthquake of 9 February 1971.A mathematical model to simulate ground motion processes is proposed for which both the intensity and frequency content are non‐stationary. Using this mathematical model with parameter characteristics along principal axes similar to those of the motions recorded during the San Fernando earthquake, three‐dimensional ground motions are synthetically generated. The properties of the simulated motions show general characteristics similar to the characteristics observed in real accelerograms. The suggested model is considered adequate for engineering purposes.

三次元地震動の模擬作成に関する研究 : その 1 地震動の主軸

For certain important structural systems such as nuclear reactor facilities, dynamic response analysis would be required in the future considering multi-component effects of ground motion input. In a case of simulation of three-dimensional ground motions by use of stochastic processes, a question immediately arises "Should the components of motion be cross correlated in a statistical sense?". If correlated, in addition to properties which establish the respective components, one must specify cross-correlations for the components. An orthogonal set of principal axes is defined for three-dimensional ground motion processes. These principal axes are determined to have maximum, minimum and intermediate values of variance and have zero values of covariance. This property reveals that the corresponding components of motion along the principal axes are fully uncorrelated with one another. Since real earthquake accelerograms are well represented by Gaussian random processes, the three components of motion along a set of the principal axes are statistically independent of one another. Using the concept of principal axes and applying a moving-window technique to the accelerograms recorded during the San Fernando earthquake of February 9, 1971, time-dependent characteristics of three-dimensional ground motion along a set of principal axes are determined. Results of the analysis indicate a significant correlation between horizontal directions of the major and intermediate principal axes and directions from the recording stations to the fault slip zone. In view of the existance of principal axes, it is concluded that the three components of motion can be generated in a stochastic manner being independent of one another, provided that a set of axes along which they are directed is treated as a set of principal axes.

Electron spin resonance of Eu 2+ in KCl

Electron spin resonance of Eu 2+ in KCl has been observed at X-band using a Superheterodyne spectrometer. The spectrum was examined along [001] and [110] directions by rotating the magnetic field in (100) plane of the crystal. In a general orientation there are six non-equivalent sites with six sets of principal axes. The spectrum of each site is described by the following rhombic spin-Hamiltonian H = gβ H.S + B 2 0O 2 0 + B 4 0O 4 0 + B 6 0O 6 0 + B 2 2O 2 2 + B 4 2O 4 2 + B 6 + B 6 6O 6 6 + A h S.I where O m n are the spin operators and B m n are empirical coefficients determined by experiment. The sign of B 0 2 was determined from observations at liquid helium temperatures. The hyperfine structure arising from Eu 151 and Eu 153 isotopes have also been observed. The results can be understood on the basis of a vacancy-Eu 2+ complex in the crystal and the present results have been compared with those of Mn 2+ in KCl crystals.

Matrix Proof of a Theorem In Mechanics

An interesting application of elementary matrix theory is to prove that at any point of a rigid body there exists at least one set of principal axes of inertia.

Remarks on tucker's inter-battery method of factor analysis

Tucker's inter-battery method of factor analysis is shown to assume implicitly that, within the factor space of overlap between the vector configurations of the two test batteries involved, the locations of the two sets of principal axes and the sizes of the associated characteristic roots are identical. A suggested modification of the inter-battery method to avoid these restrictions is considered.

AN INVESTIGATION OF THE NUCLEAR RESONANCE ABSORPTION SPECTRUM OF Al27 IN A SINGLE CRYSTAL OF EUCLASE

The nuclear magnetic resonance spectrum of Al27 in a single crystal of HBeAlSiO5 (euclase) in an external magnetic field of 7000 gauss is described. The Al27 nuclei are found to occupy two kinds of lattice sites which differ from each other only in the orientation of the principal axes of the electric field gradient tensors whose eigenvalues are all distinct, but identical at both sites. The absolute value of the quadrupole coupling constant for both groups of Al27 nuclei is found to be 5173 ± 10 kc./sec. and the axial asymmetry parameter of the field gradient tensors at both the lattice positions is 0.698 ± 0.009. The two sets of principal axes are reflections of each other in the (010) plane which is consistent with the symmetry of the proposed crystal structure. The theoretical dependence of the spectrum line frequencies on the applied magnetic field is given for one group of Al27 nuclei and for one orientation of the crystal in fields up to 2000 gauss; the feasibility of checking this experimentally is discussed.