Pointwise Variation Research Articles

Modeling longitudinal skewed functional data.

This paper introduces a model for longitudinal functional data analysis that accounts for pointwise skewness. The proposed procedure decouples the marginal pointwise variation from the complex longitudinal and functional dependence using copula methodology. Pointwise variation is described through parametric distribution functions that capture varying skewness and change smoothly both in time and over the functional argument. Joint dependence is quantified through a Gaussian copula with a low-rank approximation-based covariance. The introduced class of models provides a unifying platform for both pointwise quantile estimation and prediction of complete trajectories at new times. We investigate the methods numerically in simulations and discuss their application to a diffusion tensor imaging study of multiple sclerosis patients. This approach is implemented in the R package sLFDA that is publicly available on GitHub.

A selection theorem for essentially regulated functions of several variables

Helly's selection theorem provides a criterion for compactness of sets of single-variable functions with bounded pointwise variation. Fraňková has given a proper extension of Helly's theorem to the setting of single-variable regulated functions. We show how a similar approach yields a natural notion of regulated functions of several variables, together with an extension of the standard compactness criterion for multi-variable functions of bounded variation.

Experimental investigation of free-surface effects on flow characteristics of a torpedo-like geometry having a cambered nose

In this study, the free-surface effect in the wake region of a torpedo-like geometry having a cambered nose was investigated experimentally via Particle Image Velocimetry (PIV) and dye flow visualization methods. The Reynolds number was taken as Re = 2 × 104 and 4 × 104 while keeping the angle of attack (α =0°). The torpedo-like geometry is submerged at various positions from the free-surface in the range of 0.50 ≤ h/D ≤3.50 in which (h) is the distance from the water free-surface to the central plane of the geometry, and D is the diameter of the midsection of the torpedo-like geometry. The effect of the free-surface of water on the flow properties is also measured and presented comparatively in terms of normalized contours of instantaneous vorticity, time-averaged streamwise velocity, vorticity contours, Reynolds stress correlation, turbulent kinetic energy, velocity fluctuations in streamwise and cross-streamwise, vortex shedding frequency, spectral density distribution as well as pointwise variation. It is observed that the flow structures become asymmetric when the body is located near the free-surface for the submersion ratio of h/D ≤1.0. With the largest submersion ratio of h/D = 3.50, the flow characteristics are almost identical and symmetrical on both sides of the geometry.

The impacts of the free-surface and angle of attack on the flow structures around a torpedo-like geometry

This study presents the hydrodynamic characteristics around a torpedo-like geometry under the free-surface effect at different angles of attack using preliminary dye visualization and Particle Image Velocimetry (PIV). The optimized torpedo-like geometry is placed at submersion ratios between 0.5≤h/D≤3.50 where h is the distance to the free-surface from its centerline and D is the diameter. Throughout the experiments, angles of attack were taken as 0°≤α≤12° for two Reynolds numbers, Re = 2.0 × 104 and 4.0 × 104. The PIV method provided instantaneous vorticity and time-averaged velocity components, vorticity, streamline topology, fluctuating velocity components, Reynolds stress correlation, and the turbulent kinetic energy. This study focused on the stern section and the wake structures at h/D=1.0 . It is demonstrated that a jet-like flow region occurred between the model and the free-surface for all angles of attack at small submersion ratios of 0.5≤h/D≤1.0 while it is observed at h/D=1.5 for α=8°, a jet-like flow region occurred between the model and the free-surface. The impact of the jet-like flow was more noticeable for α=4° and 8° with velocity fluctuations in lower magnitudes. The nose section partially pierced the free-surface for h/D = 0.5, 0.75, and 1.0 in the range of 4°≤α≤12° and it prevented the wake region from connecting with the free-surface by directing the separated flow region toward the stern section. Pointwise variations of the turbulence data extracted from vertical lines within the wake region for all cases revealed that the effect of the free-surface on the turbulence statistics was negligible beyond h/D=2.0.

Functions of Bounded Variation on Complete and Connected One-Dimensional Metric Spaces

AbstractIn this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to this general setting. We show this definition of BV functions is equivalent to the BV functions introduced by Miranda [18]. Furthermore, we study the necessity of conditions on the underlying space in Federer’s characterization of sets of finite perimeter on metric measure spaces. In particular, our examples show that the doubling and Poincaré inequality conditions are essential in showing that a set has finite perimeter if the codimension one Hausdorff measure of the measure-theoretic boundary is finite.

Feature fusion network based on attention mechanism for 3D semantic segmentation of point clouds

3D scene parsing has always been a hot topic and point clouds are efficient data format to represent scenes. The semantic segmentation of point clouds is critical to the 3D scene, which is a challenging problem due to the unordered structure of point clouds. The max-pooling operation is typically used to obtain the order invariant features, while the point-wise features are destroyed after the max-pooling operation. In this paper, we propose a feature fusion network that fuses point-wise features and local features by attention mechanism to compensate for the loss caused by max-pooling operation. By incorporating point-wise features into local features, the point-wise variation is preserved to obtain a refined segmentation accuracy, and the attention mechanism is used to measure the importance of the point-wise features and local features for each 3D point. Extensive experiments show that our method achieves better performances than other prestigious methods.

Visualization of the Strain-Rate State of a Data Cloud: Analysis of the Temporal Change of an Urban Multivariate Description

One challenging problem is the representation of three-dimensional datasets that vary with time. These datasets can be thought of as a cloud of points that gradually deforms. However, point-wise variations lack information about the overall deformation pattern, and, more importantly, about the extreme deformation locations inside the cloud. This present article applies a technique in computational mechanics to derive the strain-rate state of a time-dependent and three-dimensional data distribution, by which one can characterize its main trends of shift. Indeed, the tensorial analysis methodology is able to determine the global deformation rates in the entire dataset. With the use of this technique, one can characterize the significant fluctuations in a reduced multivariate description of an urban system and identify the possible causes of those changes: calculating the strain-rate state of a PCA-based multivariate description of an urban system, we are able to describe the clustering and divergence patterns between the districts of a city and to characterize the temporal rate in which those variations happen.

Spatial variation in the occurrence of osteoarthritic lesions on the baboon distal femur is dependent on age and sex

s / Osteoarthritis and Cartilage 22 (2014) S57–S489 S59 89 SPATIAL VARIATION IN THE OCCURRENCE OF OSTEOARTHRITIC LESIONS ON THE BABOON DISTAL FEMUR IS DEPENDENT ON AGE AND SEX T.L. Bredbenner y, S.M. Levine z, J.A. Harris z, T.E. Macrini x, H.B. Coan k, L.M. Havill z, D.P. Nicolella y. y Southwest Res. Inst., San Antonio, TX, USA; z Texas BioMed. Res. Inst., San Antonio, TX, USA; x St. Mary’s Univ., San Antonio, TX, USA; kWestern Carolina Univ., Cullowhee, NC, USA Purpose: Osteoarthritis (OA) is the most common form of arthritis and the major cause of activity limitation and physical disability in older people. It is widely believed that OA results from the local mechanical environment of the joint and, particularly, the cartilage, in combination with systemic susceptibility to the disease. Preliminary study of knee OA in the baboon demonstrated that knee OA occurs commonly and naturally in male and female adults and the frequency of OA in older baboons is comparable to that in older humans. The objective of this study was to determine whether the patterns of occurrence of osteoarthritic lesions in the distal femur were related to age and sex. Methods: The right femur was collected from baboons at routine necropsy (Southwest National Primate Research Center/Texas Biomedical Research Institute; cause of death was unrelated to this study). The study sample included 123 males (16.53 5.13 years) and 300 females (18.94 5.95 years). Animals were categorized as young for age 21.7 years (w65 human years), and the remainder were categorized as middle-aged. For each animal, the distal right femur was macroscopically examined following established methods and the presence and location of any existing lesions were graded (1-4, ranging from normal to severe) and recorded on normalized maps of the articular cartilage surface. This process resulted in a total of 423 cartilage defect maps which were examined on a pointwise basis using multinomial logistic regressions to determine whether variation in cartilage grades were related to age, sex or age x sex interaction. The probability of correlation was color coded at each surface point as the statistical p-map of differences for each test variable. Correlation p-maps were corrected for multiple comparisons using the false discovery rate (FDR) method with a false discovery rate of 0.05. Additionally, a statistical trait model, based on a principal component analysis, was used to describe the spatial variation of defect grades by reducing the w490,000 highly correlated variables for each defect map to a set of 422 uncorrelated and independent principal components (e.g. composite defect traits) without loss of information. Two-way unbalanced analyses of variance were used to test the principal component weighting factors for dependence on age, sex, and age x sex. Results: Uncorrected correlation p-maps demonstrated substantial pointwise variation in the strength of correlations between cartilage grade and age, sex, and age x sex (Figure 1). FDR correction of p-values for multiple comparisons yielded a critical p-value of 0.013 for the pmap showing correlation with age, resulting in 26.1% of the tests being significant (primarily those in the medial femoral condyle). After correction, none of the pointwise tests were significant for correlation between grade and sex or age x sex. Of the 422 principal components generated, 65 were significantly dependent on age, 13 on sex, and 21 on age x sex interaction. Conclusions: Statistical p-maps demonstrated significant correlations with age, particularly in the medial femoral condyle; whereas no significant correlations were demonstrated with sex and age x sex after correction for multiple comparisons. However, it appears that statistical trait modeling may offer a promising pathway towards understanding Figure 1. Perlecan Expression as a Function of Bovine Donor Age Figure 2. Perlecan Expression as a Function of Bovine Donor Age in 2 Year

Extending vector hysteresis operators

In some recent papers we studied how to extend to BV a hysteresis operator defined on Lipschitzian inputs, preserving suitable continuity properties. More precisely we considered the so called strict metric defined by means of the essential variation. This approach may have some drawbacks from the physical point of view, therefore in the present paper we show how to extend a general hysteresis operator with respect to a notion of convergence which takes into account of the pointwise variation rather than the essential variation.

Scattering Resonances of Microstructures and Homogenization Theory

Scattering resonances are the eigenvalues and corresponding eigenmodeswhich solve the Schrödinger equation $H\psi=E\psi$ for a Hamiltonian, H, subject to the condition of outgoing radiation at infinity. We consider the scattering resonance problem for potentials which are rapidly varying in space and are not necessarily small in a pointwise sense. Such potentials are of interest in many applications in quantum, electromagnetic, and acoustic scattering, where the environment consists of microstructure, e.g., rapidly varying material properties. Of particular interest in applications are high contrast microstructures, e.g., large pointwise variations of material properties. We develop a perturbation theory for the scattering resonances and eigenvalues of such high contrast and oscillatory potentials. The expansion is proved to be convergent in a norm which encodes the degree of oscillation in the microstructure. Next, we consider the concrete example of two-dimensional microstructure potentials. These correspond, for example, to a class of photonic waveguides with transverse microstructures. The leading order behavior is given by the scattering resonances of a suitable averaged potential, as predicted by classical homogenization theory. We show that the next term in the expansion agrees with that given by a higher order homogenization multiple-scale expansion, with an error term determined by the regularity of the potential. The higher order corrections, which take into account the detailed microstructure, have been shown by the authors to be important for efficient and accurate numerical computation of radiation rates.

On a Circular Inclusion with Inhomogeneous Imperfect Interface in Plane Elasticity

In Sudak et al., a general method is developed for the rigorous solution of a circular inclusion with inhomogeneously imperfect interface in plane elasticity. In this paper, the authors extend the results presented in Sudak et al. by finding exact closed-form solutions of a more general case arising from a choice of interface parameters that leads to a coupled and seemingly intractable system of differential equations. The results again illustrate how the pointwise variation of the parameter describing the interface imperfections has a pronounced effect on the average mean stress within the inclusion.

On the analysis of cross-ply laminates with micro-cracks and inelastic deformation

For cross-ply laminates micro-matrix cracks in the 90° plies and inelastic deformation in the 0° and 90° plies are two major forms of damage which affect the long-term durability of the materials. It is crucial to develop a mechanics based approach to incorporate both micro-cracks and inelastic deformation of the composite materials based on the response of the 0° and 90° plies in order to accurately predict the response of the materials in service environments and to assist in optimizing the materials system for best performance. However, such a task leads to severe mathematical difficulties, primarily due to the zero traction conditions on the crack surfaces, the complicated constitutive relations governing the inelastic deformation and the interaction between the cracks and the inelastic deformation. In this paper, a general framework for the analysis of crossply laminates with micro-matric cracks and inelastic deformation is proposed. For this purpose admissible stress fields are constructed which satisfy equilibrium and all boundary and interface conditions. The principle of minimum complementary energy is utilized to derive a differential equation for the stress function from which the stress field of the composite can be derived. The inhomogeneous terms of the differential equation involve the inelastic strains which are loading history dependent. The Green's function of the differential equation is then obtained. Using the Green's function and a constitutive equation, two-dimensional stress and strain states in the composite at any time are represented by an integral of the Green's function and the inelastic strains accumulated up to that time. This new analysis takes into consideration the microcrack-microcrack interaction, as well as the interaction between the microcracks and the inelastic deformation, and provides a point-wise variation of the stress field instead of average stress field as most of the analytical approaches yield.The interactions of matrix cracks and creep deformation of an eight-harness satin weave (SHSW) Nextel 610/Aluminosilicate ceramic matrix composite is studied using the proposed model. The predicted creep strain of the composite shows good correlation with experimental data at different levels of temperature and stress conditions. The distribution of stresses and strains provides important information on the response of the composite.

A direct approach to solving the drift-diffusion model equations for use in certain MOSFET devices

The behavior of metal oxide semiconductor field effect transistors (MOSFETs) has frequently been modeled using the drift-diffusion partial differential equations. In this paper, we show how extensions of previously studied techniques may be applied to these equations to obtain both current-voltage relationships and pointwise variation of the potential functions arising in a number of practical cases. While our method makes use of perturbation theory, we are able to avoid the complicated asymptotic matching methods that have been widely adopted by a number of authors for studying such MOSFETs. Our numerical approach can treat the important practical case of variable doping and gives rise to accurate, and numerically stable, solutions of the model equations.

Static threshold variability in the peripheral visual field in normal subjects

We assessed the variability of point-wise static threshold values and the components of fluctuation outside the central 30 degrees field in 20 normal individuals tested on the Humphrey field analyzer. We found a mean short-term fluctuation of 2.37 dB, a long-term heterogeneous fluctuation of 5.28 dB, and a long-term homogeneous fluctuation of 1.10 dB. All components of fluctuation were greatest superiorly. Point-wise variation was highest superiorly and nasally and increased with greater eccentricity from fixation in all but the temporal quadrant. Also, point-wise variation was greater between individuals than between tests in a single individual. This study suggests that outside the central 30 degrees field, changes in individual threshold measurements in the superior and nasal quadrants should be greater than those in the temporal or inferior quadrants before they can be distinguished from normal variation.

Numerical Modeling of Diurnal Convergence Oscillations Above Sloping Terrain

Abstract We document the development and Sensitivity testing of a numerical model designed to predict diurnal cycles of boundary-layer flows of synoptic horizontal scale above sloping terrain. This application requires detailed vertical resolution of low-level jets, mixed layers and sharp inversions above terrain with large pointwise variations. The truncation errors of the pressure gradient term are of special concern. This term is computed using a thermodynamic state carried as a deviation from a standard state. Turbulent mixing is based upon a simplified diagnostic treatment. The final model produces rather stable solutions because of the rather careful pressure gradient calculation and strong physical dissipation. Sensitivity testing indicates that the model prediction of diurnal convergence cycles depends upon soil parameters. Results are also sensitive to absolute rotation and mixing parameterizations, but not equally sensitive to longwave radiative flux divergence. Applications over complex North A...

Numerical solution of the bioelectric field of the e.r.g.

A numerical solution of a model of the bioelectric field of the e.r.g. was developed utilising a finited-difference successive overrelaxation technique. The difference equations implemented by the algorithms are applicable to any spherical volume conductor problem involving axial symmetry. The nodal network utilised was characterised by a variable and nonuniform spacing between the nodes and a pointwise variation of conductivity. An excellent accuracy was demonstrated when the numerical solutions were compared with the analytical solutions for several simplified test cases. When simulating the more realistic physiological configurations the model demonstrated a reasonably good agreement with the experimental data obtained from the literature.

A probabilistic analysis of time delay extraction by the cepstrum in stationary Gaussian noise

Often, the cepstrum has been used when estimating the time delay r between elements of a composite signal embedded in noise, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n(t)</tex> . Here a probabilistic analysis is conducted dealing with the effect of stationary Gaussian noise on the characteristics of such a nonlinear processor. The input noise, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n(t)</tex> is reflected at the cepstrum's output as: 1) modulation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M_{1}(\omega)</tex> of sinusoidal information carrier with subsequent reduction in the height of echo peak at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau</tex> ; and 2) generation of background noises <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{phi}(\omega)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{psi}(\omega)</tex> with tendency to obscure that echo peak. The resulting probabilities of these terms are derived. The expected mean and standard deviation of reduction in the peak level at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau</tex> due to noise are analytically described. Similar statistical measures on the background noise are also obtained. The results point out the dependence of the-statistical measures upon the pointwise variation of input signal to noise spectra. Thus the total signal power to total noise power <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(S/N)</tex> alone is an insufficient measure for setting the cepstrum performance, and relative bandwidths of signal and noise are also needed. The detection probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{D}</tex> of a peak corresponding to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau</tex> is calculated for exponential spectra with equal bandwidths in signal and noise and with echo strength <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a = 0.4; P_{D}</tex> is found to be close to one for an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SIN</tex> greater than -4 dB; deterioration in detection is found to be rapid with thresholding near an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S/N \sim -6</tex> db where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{D}</tex> falls to 0.3.

The effect of media inhomogeneities upon intracranial electrical fields.

A three-dimensional finite difference computer simulation has been used to model entire intracranial fields from a known dipole generator in both a cat brain and an insulated cube. The model permits pointwise variation of conductivity coincident with intracranial inhomogeneities due to gray and white matter, cerebrospinal fluid, and bone. Adequacy of the model was verified by agreement between the computed results and those obtained in the cube with both homogeneous and inhomogeneous media and in the cat brain along the dipole axis. Variation of brain model parameters showed that neither increasing nor abolishing intracranial inhomogeneities would give differences great enough to be detected experimentally. On the other hand, the position of the boundary could have significant effects on the fields. The model is general enough to be applicable to a variety of multipole generators and generator locations, as well as permitting consideration of the influence of localized inhomogeneities.

Stresses Around Wellbores in Nonlinear Rock

ABSTRACT The state of stress around a vertical wellbore in rock following nonlinear stress-strain laws is examined by means of finite element analysis. The wellbore is considered an axisymmetric body with axisymmetric loading. The initial vertical and horizontal stresses are "locked" in the rock elements around the wellbore and a new state of stress is generated by the displacements which occur around the borehole. A point-wise variation of the elastic moduli is made on the basis of the new stress state and the triaxial data. The initial stresses are now reintroduced along with the changed moduli and original boundary constraints. This procedure is repeated until convergent stresses are reached. The effect of nonlinearity on stresses is examined for a 6,000-ft wellbore in a schistose gneiss and Berea sandstone using results of laboratory triaxial compression tests. The results show that the effect is restricted to one well radius from the bottom periphery of the hole. Beyond a distance of one-quarter radius, the effect of nonlinearity on stresses is almost always less than 5 percent for the cases considered. The consideration of a static pressure inside the well does not magnify the effect of nonlinearity on borehole stresses. INTRODUCTION The terms "wellbore" and "borehole" here designate cylindrical openings in the ground with vertical axis and a circular cross-section. A knowledge of the stress redistribution that occurs on excavating a wellbore is important in understanding the behavior of the lined or unlined hole, hydraulic fracture response, and the effect of stress redistribution on drillability; also it is important in predicting initial stresses in the virgin ground, and in analyzing the response of measuring instruments placed in the borehole. Our knowledge of the state of stress around a wellbore has been restricted to homogeneous, isotropic, elastic material and derives chiefly from the analysis by Miles and Topping1 and the photoelastic work of Galle and Wilhoit2 and Word and Wilhoit.3 In this investigation the state of stress is examined for a nonlinear elastic material by means of finite element analysis. Many rocks possess stress-strain curves that depart notably from straight lines in their initial or final portions. While the literature contains abundant stress-strain data from triaxial tests (axisymmetric loading) on cylindrical rock specimens, there is little information on rock deformability under nonaxisymmetric loading conditions such as occur at each point around the bottom of a wellbore. Although there is some knowledge of the effect of intermediate principal stress on rock strength, there is virtually nothing known about its effect on rock deformability; therefore, we have assumed here that the effect of intermediate principal stress can be ignored. A schistose gneiss4 and Berea sandstone5 were selected as representative rocks for this analysis. The traditional graphs of deviator stress (s1-s3) vs axial strain were reworked to give the tangent modulus as a function of the deviator stress for varying values of the minor principal stress. The result is a nesting family of skewed, bell-shaped curves for the gneiss (Fig. 1A) and the sandstone (Fig. 2A). A similar replotting of the lateral strain data defines the variation of Poisson's ratio (?) with the deviator stress and confining pressure. These curves, shown in Fig. 1B for the gneiss and in Fig. 2B for the sandstone, are not so well ordered as the tangent modulus curves. However, all of these display an increase of ? with deviator stress application, but the rate of increase diminishes with confinement. The ET and ? curves for the two rock types are tabulated in Tables 1A and 1B for use in a digital computer so that material properties corresponding to a given state of stress can be assigned by interpolation.