Scalar Magnitude Research Articles

An Asymmetric Micropolar Moment Tensor Derived from a Discrete-Block Model for a Rotating Granular Substructure

Abstract I define an asymmetric moment tensor for individual slip events and for an average over multiple slip events using a discrete rigid-block model to account for the brittle deformation of a granular material. For the brittle crust, the grains are taken to be fault-bounded blocks. Permanent deformation accumulates by slip events on the boundaries of the blocks. The deformation is described by two independent motions: the local relative motion of the block centroids and the local rigid rotation of the blocks about their centroids. Averaging each of these local motions over multiple slip events in a volume defines both the macrodeformation, which consists of the macrostrain and macrorotation, and the microrotation. An asymmetric local micropolar moment tensor and an asymmetric micropolar moment-density tensor are defined from the local and the averaged motions, respectively. The model shows (1) the symmetric part of the micropolar moment tensors depends on the constant-volume local shear strain of the block centroids or its averaged equivalent, the macrostrain; (2) the antisymmetric part depends on an objective quantity defined as the difference between the rotational component associated with the centroid deformation and the local block rotation, or their averaged equivalents the macrorotation and the microrotation; and (3) the symmetric and antisymmetric parts of the micropolar moment-density tensor can be inferred up to a scalar magnitude by a micropolar inversion of standard seismic focal mechanisms. Three field tests show consistency with the theory, but definitive tests are thwarted by insufficient quantitative information or insufficient resolution of the available data.

Seismic hazard scenario and attenuation model of the Garhwal Himalaya using near-field synthesis from weak motion seismometry

In this paper, we present a seismic hazard scenario for the Garhwal region of the north-western Himalayan range, in terms of the horizontal Peak Ground Acceleration. The scenario earthquake of moment magnitude Mw 8.5 has a 10% exceedance probability over the next 50 years. These estimates, the first for the region, were calculated through a stepwise process based on: An estimation of the Maximum Credible Earthquake from the seismicity of the region and Global Seismic Hazard Assessment Program considerations, and four seismotectonic parameters abstracted from near field weak-motion data recorded at five stations installed in Chamoli District of the Garhwal region in the aftermath of the 1999 Chamoli earthquake. The latter include The frequency dependent power law for the shear wave quality factor, QS the site amplification at each station using horizontal-to-vertical-spectral ratio and generalized inversion technique source parameters of various events recorded by the array and application of the resulting relations between the scalar seismic moment M0 (dyne-cm) and moment magnitude Mw and the corner frequency, ƒc (Hz) and moment magnitude Mw to simulate spectral acceleration due to higher magnitude events corresponding to the estimated Maximum Credible Earthquake, and regional and site specific local spectral attenuation relations at different geometrically central frequencies in the low, moderate and high frequency bands.

The ellipsoidal area ratio: an alternative anisotropy index for diffusion tensor imaging.

In the processing and analysis of diffusion tensor imaging (DTI) data, certain predefined morphological features of diffusion tensors are often represented as simplified scalar indices, termed diffusion anisotropy indices (DAIs). When comparing tensor morphologies across differing voxels of an image, or across corresponding voxels in different images, DAIs are mathematically and statistically more tractable than are the full tensors, which are probabilistic ellipsoids consisting of three orthogonal vectors that each has a direction and an associated scalar magnitude. We have developed a new DAI, the "ellipsoidal area ratio" (EAR), to represent the degree of anisotropy in the morphological features of a diffusion tensor. The EAR is a normalized geometrical measure of surface curvature in the 3D diffusion ellipsoid. Monte Carlo simulations and applications to the study of in vivo human data demonstrate that, at low noise levels, EAR provides a similar contrast-to-noise ratio (CNR) but a higher signal-to-noise ratio (SNR) than does fractional anisotropy (FA), which is currently the most popular anisotropy index in active use. Moreover, at the high noise levels encountered most commonly in real-world DTI datasets, EAR compared with FA is consistently much more robust to perturbations from noise and it provides a higher CNR, features useful for the analysis of DTI data that are inherently noise sensitive.

Voxel-wise comparisons of the morphology of diffusion tensors across groups of experimental subjects

Water molecules in the brain diffuse preferentially along the fiber tracts within white matter that form the anatomical connections across spatially distant brain regions. A diffusion tensor (DT) is a probabilistic ellipsoid composed of three orthogonal vectors, each having a direction and an associated scalar magnitude, that represent the probability of water molecules diffusing in each of those directions. The 3D morphologies of DTs can be compared across groups of subjects to reveal disruptions in structural organization and neuroanatomical connectivity of the brains of persons with various neuropsychiatric illnesses. Comparisons of tensor morphology across groups have typically been performed on scalar measures of diffusivity, such as Fractional Anisotropy (FA) rather than directly on the complex 3D morphologies of DTs. Scalar measures, however, are related in nonlinear ways to the eigenvalues and eigenvectors that create the 3D morphologies of DTs. We present a mathematical framework that permits the direct comparison across groups of mean eigenvalues and eigenvectors of individual DTs. We show that group-mean eigenvalues and eigenvectors are multivariate Gaussian distributed, and we use the Delta method to compute their approximate covariance matrices. Our results show that the theoretically computed mean tensor (MT) eigenvectors and eigenvalues match well with their respective true values. Furthermore, a comparison of synthetically generated groups of DTs highlights the limitations of using FA to detect group differences. Finally, analyses of in vivo DT data using our method reveal significant between-group differences in diffusivity along fiber tracts within white matter, whereas analyses based on FA values failed to detect some of these differences.

Turbulent liquid spray mixing and combustion–fundamental simulations

Fundamental simulations are used to investigate the ignition process of turbulent nheptane liquid fuel spray jets. A DNS quality Eulerian method is used to solve the carrier gas flow field, while a Lagrangian method is used to track the liquid fuel droplets. Two-way coupling between the phases is included through the exchange of mass, momentum and energy. A detailed mechanism with 33 species and 64 reactions is used to describe the chemical reactions. The simulation approach allows studies of larger scale interaction of sprays and turbulence, including evaporation, mixing, and detailed chemical reaction. Both time developing and spatially developing liquid spray jets are studied. The initial carrier gas temperature was 1500 K. Several cases were simulated with different droplet radii (from 10 microns to 30 microns) and two initial velocities (100 m/s and 150 m/s). In the time developing case it was found that evaporative cooling and turbulence mixing play important roles in the ignition process of liquid fuel spray jets. Ignition first occurs at the edges of the jets where the fuel mixture is lean, and the scalar dissipation rate and vorticity magnitude are low. For smaller droplets, ignition occurs later than larger droplets due to increased evaporative cooling. Higher initial droplet velocity enhances turbulence mixing and evaporative cooling. For smaller droplets, higher initial droplet velocity causes the ignition to occur earlier, whereas for larger droplets, higher initial droplet velocity delays the ignition time. In the spatially developing liquid jets, ignition and flame lift-off characteristics similar to diesel sprays are observed. Near the injector, combustion development progresses very rapidly along the stoichiometric surface. In the downstream region of the spray, combustion develops with steep temperature fronts in a flamelet mode.

Statistics and geometry of passive scalars in turbulence

We present direct numerical simulations of the mixing of the passive scalar at modest Taylor microscale (10⩽Rλ⩽42) and Schmidt numbers larger than unity (2⩽Sc⩽32). The simulations resolve below the Batchelor scale up to a factor of 4. The advecting turbulence is homogeneous and isotropic, and is maintained stationary by stochastic forcing at low wave numbers. The passive scalar is rendered stationary by a mean scalar gradient in one direction. The relation between geometrical and statistical properties of scalar field and its gradients is examined. The Reynolds numbers and Schmidt numbers are not large enough for either the Kolmogorov scaling or the Batchelor scaling to develop and, not surprisingly, we find no fractal scaling of scalar level sets, or isosurfaces, in the intermediate viscous range. The area-to-volume ratio of isosurfaces reflects the nearly Gaussian statistics of the scalar fluctuations. The scalar flux across the isosurfaces, which is determined by the conditional probability density function (PDF) of the scalar gradient magnitude, has a stretched exponential distribution towards the tails. The PDF of the scalar dissipation departs distinctly, for both small and large amplitudes, from the log-normal distribution for all cases considered. The joint statistics of the scalar and its dissipation rate, and the mean conditional moment of the scalar dissipation, are studied as well. We examine the effects of coarse-graining on the probability density to simulate the effects of poor probe-resolution in measurements.

Revised world seismicity catalog (1950-1997) for strong (Ms≧ 6) shallow (h≦ 70 km) earthquakes

AbstractUsing the standard earthquake catalogs provided by the International Seismological Centre (ISC), its predecessor the International Seismological Summary (ISS), and the National Earthquake Information Center (NEIC), I analyze the worldwide consistency of teleseismic reporting, completeness of the seismicity record, and homogeneity of magnitude determination, for strong shallow earthquakes (surface-wave magnitude, Ms ≥ 6; depth, h ≤ 70 km) for the period 1950 to 1997. Under the postulates that the rate of earthquake occurrence for the entire world is constant on a time scale of decades and that well after the installation of the World-Wide Standardized Seismograph Network in the middle 1960s the earthquake catalog for strong (Ms ≥ 6) shocks is complete, and the seismicity rates are typical of all periods in the century, we find that, due to the use of different formulations and criteria to calculate the parameter magnitude, the Ms of moderate (6 ≤ Ms < 7) events during the period 1950 to 1963 was systematically overestimated by as much as 0.5 magnitude unit, relative to the Ms assigned to shocks occurring after 1963. When this correction is taken into account, the new catalog of events with Ms (corrected) ≥ 6 in the period 1950 to 1997 becomes largely homogeneous in Ms. Under the foregoing postulates, this new catalog is shown to list nearly all and only the strong shocks [Ms (corrected) ≥ 6; h ≤ 70 km] that occurred in the Earth during the period, a notable exception being the time span from 1964 to 1968. The revised catalog, including the scalar moment and moment magnitude for each event, is listed on the worldwide web page http://www.ldc.usb.ve/~ojperez/catalog.

Parallel manipulator kinematics learning using holographic neural network models

The forward kinematic problem of parallel manipulators is resolved using a holographic neural paradigm. In a holographic neural model, stimulus–response (input–output) associations are transformed from the domain of real numbers to the domain of complex vectors. An element of information within the holographic neural paradigm has a semantic content represented by phase information and a confidence level assigned in the magnitude of the complex scalar. Networks are trained on a database generated from the closed-form inverse kinematic solutions. After the learning phase, the networks are tested on trajectories which were not part of the training data. The simulation results, given for a planar three-degree-of-freedom parallel manipulator with revolute joints and for a spherical three-degree-of-freedom parallel manipulator, show that holographic neural network models are feasible to solve the forward kinematic problem of parallel manipulators.

Non-spherical source-surface model of the heliosphere: a scalar formulation

Abstract. The source-surface method offers an alternative to full MHD simulation of the heliosphere. It entails specification of a surface from which the solar wind flows normally outward along straight lines. Compatibility with MHD results requires this (source) surface to be non-spherical in general and prolate (aligned with the solar dipole axis) in prototypical axisymmetric cases. Mid-latitude features on the source surface thus map to significantly lower latitudes in the heliosphere. The model is usually implemented by deriving the B field (in the region surrounded by the source surface) from a scalar potential formally expanded in spherical harmonics, with coefficients chosen so as to minimize the mean-square tangential component of B over this surface. In the simplified (scalar) version the quantity minimized is instead the variance of the scalar potential over the source surface. The scalar formulation greatly reduces the time required to compute required matrix elements, while imposing essentially the same physical boundary condition as the vector formulation (viz., that the coronal magnetic field be, as nearly as possible, normal to the source surface for continuity with the heliosphere). The source surface proposed for actual application is a surface of constant F̃ ≡ r-kB̃, where r is the heliocentric distance and B̃ is the scalar magnitude of the B field produced by currents inside the Sun. Comparison with MHD simulations suggests that k ≈ 1.4 is a good choice for the adjustable exponent. This value has been shown to map the neutral line on the source surface during Carrington Rotation 1869 (May–June 1993) to a range of latitudes that would have just grazed the position of Ulysses during that month in which sector structure disappeared from Ulysses' magnetometer observations.

On Robustness in Statistical FDI Methods

Abstract We describe a statistical approach to the robustness issue in model-based FDI methods. The main idea consists in using confidence ellipsoids for plugging the uncertainty in the shape of the (no-fault and fault) hypotheses to be tested, instead of using a confidence interval around the decision function itself. This can be viewed as a multi-parameter extension of a scalar minimum magnitude of fault to be detected, which in turn has proven an efficient alternative to the more intuitive idea of using multiple thresholds. This simple idea turns out to be useful for the robustness against both changes in the functioning modes and under-modelling of the system to be monitored

The Mid-German Conductivity Anomaly and its Contribution to the Determination of the Variscan Deep Structure

It is possible to construct a precise isoline map of the scalar magnitude of the induction arrows for Germany in the period range 10 to 20 min based on the results of different study groups. In the central part of Germany, a minimum zone indicates high crustal conductivity to the North of the renewed southward increase of these scalar magnitudes and it corresponds to the location of the Mid-German Crystalline Zone in the border area between Saxothuringicum and Rhenoherzynicum. Using magneto telluric, refraction seismic, reflection seismic, geothermal, magnetic and gravimetric results and the results of geological and geochemical studies, the author tries to chose geotectonic models of the Variscan which can be harmonised with the present knowledge about the causes of high electric conductivity in the central part of the Earth’s crust.

Cold, warm, and composite (cool) cosmic string models

The dynamical behaviour of a cosmic string is strongly affected by any reduction of the effective string tension T below the constant value, T = m 2 say, that typifies a simple, longitudinally Lorentz invariant Goto-Nambu type string model, where m is a fixed mass scale determined by the internal structure of an underlying Nielsen-Olesen type vacuum vortex. Such a reduction of tension occurs in the standard “warm” cosmic string model in which the effect of thermal perturbations of a simple Goto-Nambu model is represented by an effective tension T given in terms of the corresponding effective temperature, Θ say, by T 2 = m 2(m 2 − 1 3 πΘ 2) . A qualitatively similar though analytically more complicated tension reduction phenomenon occurs in “cold” conducting cosmic string models of the kind whose existence was first proposed by Witten, where the role of the temperature is played by an effective mass or chemical potential μ that is constructed as the scalar magnitude of the energy momentum covector obtained as the gradient of the phase ϕ of a bosonic condensate in the core of the vacuum vortex. The present article describes the construction and essential mechanical properties of a new category of composite “cool” cosmic string models that are intermediate between these “warm” and “cold” limit cases. These composite models are the string analogues of the standard Landau model for a two-constituent finite temperature superfluid, and as such involve two independent currents interpretable as that of the entropy on the one hand and that of the bosonic condensate on the other. It is surmised that the stationary (in particular ring) equilibrium states of such “cool” cosmic strings may be of cosmologicl significance.

Characterising the performance of a confocal acoustic microscope

The transfer functions of confocal optical imaging systems are applied to equivalent ultrasonic imaging systems. The ideal complex amplitude line-scan signals across a step edge with various degrees of defocus, excluding attenuation and aberrations, are generated. The scalar magnitudes of the resulting ideal complex image profiles are differentiated to obtain the unique theoretical impulse response functions (IRFs) and a Fourier transform is applied to the data to obtain the corresponding transfer functions (TFs). The experimental single-frequency and broadband image profiles for a step edge are differentiated and a Fourier transform is applied to obtain TFs. The effects of attenuation and aberrations are investigated and the experimental data are compared with the theoretical data. The relationship between broadband and single-frequency acoustic images is described. A practical model for the detection limits for 2D cracks obtained from the response for a pair of step edges is proposed. The numerical calculations are compared with experimental data. The strong contrast and small limit of detection usually seen in the imaging of surface cracks are associated with the excitation of surface acoustic waves (SAWs). In this study, a similar sensitivity to detect cracks is achieved by the defocus effect of pure longitudinal waves, which can be clearly explained by theory.

Characterizing the performance of acoustic microscopes

The transfer functions for focused and defocused, coherent and confocal imaging systems developed in optics are applied to the equivalent ultrasonic imaging systems. Assuming that both of the two systems are free of attenuation and aberrations except defocus, the ideal complex amplitude line-scan signals across a step edge, with various degrees of defocus, are generated. The scalar magnitude of ideal complex image profiles are differentiated to get the unique theoretical IRF’s and Fourier transformed to obtain the corresponding TF’s. The acoustic system has been operated with a 50-MHz lens-rod design focused transducer. The experimental single frequency and broadband image profiles of a step edge, with varying degree of defocus, are differentiated and Fourier transformed. The experimental IRF’s and TF’s are compared with the corresponding theoretical data. As the initial theoretical analysis is for the wave propagating in an ideal fluid, the effects of attenuation and aberrations are investigated to enable the experimental data to be compared with the theoretical data. The relation between broadband and single frequency acoustic images is described. A practical model for the detection limits for 2-D cracks using the response for a pair of step edges is proposed. The numerical calculations are compared with experimental data. Usually the strong contrast seen in the imaging of cracks is associated with the excitation of Rayleigh waves in surface. In this study, a similar sensitivity to detect cracks is achieved by the defocus effect of pure longitudinal waves, which can be clearly explained by theory.

Pulsation periods at mid-latitudes—A seven-station study

A pulsation array was set up in 1984 of seven stations in Central and Northern Europe. High resolution dynamic spectra of a dozen events were analyzed. It was found that on several occasions latitude dependent and independent periods occur in the events simultaneously so that the changing period branch starts from the constant period one around L ∼ 2. The constant period corresponds to that expected on the basis of the actual scalar magnitude of the interplanetary magnetic field, the changing period increases sometimes to the highest latitude station in the array and may indicate the width of the primary spectrum. In 1984 the change of the periods with latitude is large, and that is perhaps why less latitude dependent events are found.

Darlington's theorem and complex normalization

AbstractTwo novel proofs are presented to establish the equivalence between complex normalization and Darlington representation of the loads. Schwarz reflection of analytic functions and Hermitian metrics serve in these proofs as natural concepts for the analysis of lossless multiports. By use of these tools, the algebraic structure that underlies the physical notion of losslessness is analysed, and a class of matrix representations of lossless 2n‐ports is derived. the fundamental transformations between these matrices are established to facilitate easy conversion of results related to different choices of co‐ordinates (voltages/currents, waves, impedances, reflectances). By virtue of this theory, complex normalization is tacitly extended to include not only impedances but also reflectances as references. Explicit expressions for a 2n‐port normalized to complex loads are derived. Invariance of the complex normalized scattering matrix in a lossless cascade circuit is shown to be the most comprehensive generalization of the well known invariance of scalar reflection magnitudes. the relation between this viewpoint and the general theory of invariants of linear fractional maps in terms of cross‐ratios is pointed out explicitly. the factorization problem, common to complex normalization as well as Darlington representations, is also reviewed, and it is shown that the concept of minimal factorization (well known in linear prediction theory) leads to more satisfactory results than those of established factorization recipes in complex normalization.

Derivation of motor line-current waveforms from the DC-link current of an inverter

A technique for reconstructing motor line currents using information from a single current transducer in the DC-link of an inverter is presented. All the instantaneous motor line currents can be reconstructed with a bandwidth limited to half the inverter switching frequency. These waveforms are used to produce a highbandwidth signal, proportional to the scalar magnitude of the line current, and which is suitable for motor control purposes. A simple, low-cost implementation of the scheme demonstrates accurate results in a practical motor drive.

Scalar control: an alternative AC drive philosophy

A new control philosophy for inverter driven induction motors causes the airgap flux to vary over a wide range in proportion to load-induced changes in the scalar magnitude of the machine current. A controller based on this principle outputs a constant slip frequency at high efficiency under all useful operating conditions. It is stable with good damping factors and good response times, yet it requires no external transducers. It allows for accurate torque estimates, but it can be implemented with low cost microprocessors that can be interfaced to computers for interactive plant control and data analysis. Measured results from a practical inverter are shown to be in close agreement with both the theoretical analysis presented and with computer simulations.

Interrelationship of some electrocardiogram amplitudes, time intervals and respiration in the horse

The relationship of the scalar and vector magnitudes of the QRS waveform to RR, PR and respiration was examined in 10 horses in regular rhythm using an orthogonal lead system. There were no significant changes in the durations of RR and PR with respiration. The amplitudes of the scalar variables XR and YR were directly related and that of XQ (or XS) and ZR were inversely related to PR in this group of horses. The peak spatial vector magnitude and its H-plane angle were directly related to PR and inversely related to RR. The effect of inspiration on the QRS was a decrease in YR and ZR, an increase in YS and peak spatial vector magnitude and a reduction in its angle of elevation. Examined over the 10 horses these changes were only small (1.8 to 3.8 per cent) and considered to be not significant clinically. The possible mechanisms of the relationship to PR and RR and the effect of respiration on the QRS waveform are discussed. The results suggest the possibility of a vagal effect on intraventricular conduction. The effect of respiration was shown to be independent of this effect and it might be due to changes in lung electrical conductivity during respiration.

Direct conversion of los acceleration data to vertical gravity anomalies: A new approach

Most of the existing lunar and planetary gravity data are in the form of LOS (line-of-sight) components which cannot be used for conventional geophysical modelling. Current methods to invert LOS data yield non-unique or poorly constrained results or results of low spatial resolution. An alternate method presented here promises to produce unique, detailed and more reliable results. It utilizes the fact that three non-coplanar LOS acceleration vectors determined at different times at some point of observation uniquely define the total acceleration vector at that point. Vector analysis shows that the local cartesian componentsgi of the total gravity anomaly vector may be obtained by inversion of the system $$a_{ij} g_j = A_i^2 $$ where theaij andAi's are, respectively, the local cartesian components and scalar magnitudes of the three required LOS acceleration vectors.