Ellipsoidal Surface Research Articles

A phase separation problem and geodesic disks on Cassinian oval surfaces

We conduct numerical investigations into the shape of equilibrium patches on Cassinian oval surfaces. Such patches arise as minimizers of a Landau-type free energy subject to a conservation constraint. The equilibrium patch shape and location depends on a number of factors, including the grid size, a diffusion coefficient, a conservation parameter, and the initial phase distribution. We develop a scheme to distinguish between those patches that closely approximate geodesic disks and those patches that are poor approximations to geodesic disks. We find that the poor approximations form when the patch is relatively large, in contrast to what other researchers found for ellipsoid surfaces.

On uniform ultimate boundedness of linear systems with time-varying delays and peak-bounded disturbances

This paper is devoted to a problem of uniform ultimate boundedness (UUB) for a class of linear systems with interval time-varying delays and peak-bounded disturbances. By using the Lyapunov–Krasovskii functional method, a general UUB condition and an estimation of the ultimate bound on system trajectories are developed. On the basis of the above results, linear matrix inequality (LMI) conditions on UUB of the perturbed systems under consideration are derived by constructing an appropriate Lyapunov–Krasovskii functional. Meanwhile, the ultimate bound for a prescribed disturbance attenuation level is given in the form of an ellipsoidal surface. The feasibility and effectiveness of the derived results are illustrated by two numerical examples.

Surface tractions on an ellipsoid in Stokes flow: Quadratic ambient fields

The surface tractions on an ellipsoid undergoing rigid body motion in an ambient Stokes flow field are known to be surprisingly simple—essentially of the same form as the boundary condition for the disturbance velocity multiplied by the dot product of the position vector at the ellipsoid surface and the outward surface normal at that point. The direct derivation of these remarkable classical results via the Newtonian constitutive equation is straightforward but requires many pages of algebra. More recently, a more satisfying explanation of this result was derived by proving that the Stokes flow double layer operator acting on the surface of an ellipsoid is self-adjoint for a particular weighted metric space (discussed in this work). In the present paper, this result, which the authors call the weighted ellipsoidal metric space theorem, is exploited to derive simple forms for the surface tractions on a force-free ellipsoid in a quadratic ambient field. After even more pages of algebra than encountered for the classical flows, the tractions reduce to a simple form: the weighting function times a quadratic in the position vector (without the guidance of the general theory, the initial expressions for the traction would have taken several columns or pages of this journal). The nature of the derivation along with the corresponding analysis of the eigensystem of the double layer operator suggests that for the general polynomial ambient field, the surface traction is a polynomial in the position vector of the same degree, multiplied by the weighting function.

Magnetic Anomaly Signal Space Analysis and Its Application in Noise Suppression

The decomposition of magnetic dipole signal in orthonormal basis functions (OBFs) is widely used in underwater target detection. Traditionally, only the squared sum of three coefficients is constructed as a magnetic anomaly detector, but the space distribution characteristic of coefficients is not fully utilized. In this letter, the constraint of coefficients is analyzed, and their space distribution constraint that located on an ellipsoid surface is obtained. Moreover, when the magnetization direction of target is nearly horizontal position, the constraint relationship of coefficients in the signal space can be compressed into an elliptical plane. Then, we project the decomposed coefficients of the measured signal onto the elliptical plane to further reduce the noise by about 1/3, and the squared sum of renewed coefficients is constructed as a detector for anomaly judgment. The simulation is conducted, compared with the traditional OBF method, and it can achieve an incremental detection probability of about 6% in the case of low signal-to-noise ratio.

Взаимодействие электромагнитной H-волны с тонкой металлической пленкой на диэлектрической подложке в случае анизотропной поверхности Ферми металла

The interaction of electromagnetic H-waves with the thin metal film subject to the shape of the ellipsoidal Fermi surface and a constant mean free path of electrons for various angles of incidence of the electromagnetic waves of theta and different from each other of the coefficients of specularity of q1 and q2 in the reflection of electrons from surfaces of the film is calculated. The metal film is enclosed between two media with permittivity ε1 and ε2. The behavior of reflection coefficients R, passage T and absorption A from the effective mass of conduction electrons is analyzed.

A Configuration Space Model for Intermediate‐Scale Ionospheric Structure

AbstractStochastic ionospheric structure is characterized by spectral density functions (SDFs), which are formally the average intensity of Fourier transformations of the electron density structure. Structure elongation along magnetic field lines is typically accommodated by constraining contours of constant spatial correlation to field‐aligned ellipsoidal surfaces. Structure realizations are generated by imposing the square root of the SDF onto uncorrelated random Fourier modes. The approach has been used successfully for interpreting both in situ and remote propagation diagnostics. However, the only connection to the field‐aligned structures generated by the underlying instability mechanism is the correlation scale. This paper introduces a configuration space model that constructs realizations as summations of field‐aligned elemental striations with a prescribed scale and peak electron density. We show that by choosing the contributing scales according to a bifurcation rule and imposing a power law intensity scaling, the corresponding SDF closely approximates an inverse power law. Thus, configuration space realizations can be structured to reproduce prescribed or measured SDFs from one‐dimensional scans. Stochastic variation comes from an imposed random distribution of striation locations, which are defined by their intercept in a reference slice plane. To conform more directly to physics‐based simulations, the striations can be interpreted as voids in the background electron density. The model is designed to support propagation simulations with arbitrary propagation angles relative to the local magnetic field direction. Relations between in situ structure and diagnostic measurements as well as phase screen equivalence can be explored.

Three-Dimensional Cloud Volume Reconstruction from the Multi-angle Imaging SpectroRadiometer

Characterizing 3-D structure of clouds is needed for a more complete understanding of the Earth’s radiative and latent heat fluxes. Here we develop and explore a ray casting algorithm applied to data from the Multi-angle Imaging SpectroRadiometer (MISR) onboard the Terra satellite, in order to reconstruct 3-D cloud volumes of observed clouds. The ray casting algorithm is first applied to geometrically simple synthetic clouds to show that, under the assumption of perfect, clear-conservative cloud masks, the reconstruction method yields overestimation in the volume whose magnitude depends on the cloud geometry and the resolution of the reconstruction grid relative to the image pixel resolution. The method is then applied to two hand-picked MISR scenes, fully accounting for MISR’s viewing geometry for reconstructions over the Earth’s ellipsoidal surface. The MISR Radiometric Camera-by-camera Cloud Mask (RCCM) at 1.1-km resolution and the custom cloud mask at 275-m resolution independently derived from MISR’s red, green, and blue channels are used as input cloud masks. A wind correction method, termed cloud spreading, is applied to the cloud masks to offset potential cloud movements over short time intervals between the camera views of a scene. The MISR cloud-top height product is used as a constraint to reduce the overestimation at the cloud top. The results for the two selected scenes show that the wind correction using the cloud spreading method increases the reconstructed volume up to 4.7 times greater than without the wind correction, and that the reconstructed volume generated from the RCCM is up to 3.5 times greater than that from the higher-resolution custom cloud mask. Recommendations for improving the presented cloud volume reconstructions, as well as possible future passive remote sensing satellite missions, are discussed.

О преимуществе системы нормальных высот

The author analyzes the arguments in the report by Robert Kingdon, Petr Vanicek and Marcelo Santos “The shape of the quasigeoid” (IX Hotin-Marussi Symposium on Theoretical Geodesy, Italy, Rome, June 18 June 22, 2018), which presents the criticisms for the basic concepts of Molodensky’s theory, the normal height and height anomaly of the point on the earth’s surface, plotted on the reference ellipsoid surface and forming the surface of a quasigeoid. The main advantages of the system of normal heights, closely related to the theory of determining the external gravitational field and the Earth’s surface, are presented. Despite the fact that the main advantage of Molodensky’s theory is the rigorous determining the anomalous potential on the Earth’s surface, the use of the system of normal heights can be shown and proved separately. To do this, a simple example is given, where the change of marks along the floor of a strictly horizontal tunnel in the mountain massif is a criterion for the convenience of the system. In this example, the orthometric heights show a change of 3 cm per 1.5 km, which will require corrections to the measured elevations due the transition to a system of orthometric heights. The knowledge of the inner structure of the rock mass is also necessary. It should be noted that the normal heights are constant along the tunnel and behave as dynamic ones and there is no need to introduce corrections. Neither the ellipsoid nor the quasi-geoid is a reference for normal heights, because so far the heights are referenced to initial tide gauge. The points of the earth’s surface are assigned a height value; this is similar to the ideas of prof. L. V. Ogorodova about the excessive emphasis on the concept of quasigeoid. A more general term is the height anomaly that exists both for points on the Earth’s surface and at a distance from it and decreases together with an attenuation of the anomalous field.

Coupled axisymmetric pulsar magnetospheres

We present solutions of force-free pulsar magnetospheres coupled with a uniform external magnetic field aligned with the dipole magnetic moment of the pulsar. The inclusion of the uniform magnetic field has the following consequences: The equatorial current sheet is truncated to a finite length, the fraction of field lines that are open increases, and the open field lines are confined within a cylindrical surface instead of becoming radial. A strong external magnetic field antiparallel to the dipole allows for solutions where the pulsar magnetic field is fully enclosed within an ellipsoidal surface. Configurations of fully enclosed or confined magnetospheres may appear in a double neutron star (DNS) where one of the components is a slowly rotating, strongly magnetised pulsar and the other a weakly magnetised millisecond pulsar. Depending on the geometry, twisted field lines from the millisecond pulsar could form an asymmetric wind. More dramatic consequences could appear closer to merger: there, the neutron star with the weaker magnetic field may undergo a stage where it alternates between an open and a fully enclosed magnetosphere releasing up $10^{38}$erg. Under favourable combinations of magnetic fields DNSs can spend up to 10$^3$ yr in the coupled phase, implying a Galactic population of $0.02$ systems. Potential observational signatures of this interaction are discussed including the possibility of powering recurring Fast Radio Bursts (FRBs). We also note that the magnetic interaction cannot have any significant effect on the spin evolution of the two pulsars.

A helical interpolation precision truing and error compensation for arc-shaped diamond grinding wheel

In the grinding of aspherical optical surface, an arc-shaped diamond grinding wheel is generally employed, whereas its profile errors greatly influence the form accuracy of ground workpiece. A high efficiency helical interpolation precision truing method for the arc-shaped diamond grinding wheel is proposed in this paper. Helical interpolation trajectory and decreasing interpolation radius with the wear of a rotary truing wheel ensure a constant contact length and a uniform wear of the diamond grinding wheel and truer. The mathematical models of tool setting errors and measurement error of truer diameter in the truing process were established firstly. The prediction models considering the wear of truer are coincided well with the truing experiment results. Based on the mathematical error models, an error compensation truing process was proposed. The profile accuracy of an arc-shaped diamond grinding wheel was improved to 5 μm (PV) effectively by the error compensation truing process. An ellipsoidal surface of a fused silica workpiece was ground by the well-trued diamond grinding wheel. A form error (in PV) below 4.5 μm was obtained after precision grinding.

Non-phononic mechanism of superconductivity in pure semi-metallic Bismuth single crystal at sub-milli Kelvin

Very recently, the elusive phenomenon of superconductivity in pure semi-metallic Bismuth single crystal has been experimentally observed at extremely low transition temperature of 0.53 mK. Electrons, charged fermions, in this crystal form a very rare gas with interparticle separation of nearly 185A˚. But it is a highly quantum mechanical degenerate gas at such a low temperature with thermal de Broglie wavelength equal to a fraction of a million Angstrom. In such a dilute system the peculiar oscillatory behavior of the generalized electronic dielectric function at large distances can give rise to an attractive interaction between two electrons. The interaction of electrons therefore, can be described by a pseudo-potential with a negative scattering length that gives rise to the formation of Cooper pairs and then bulk superconductivity in the framework of BCS theory. It turns out that the value of scattering length of 3.265A˚ can explain the observed superconducting transition temperature. The complexity of ellipsoidal Fermi surfaces of Bismuth and different values of electronic mass along its three crystal axes, consistent with the symmetry of its crystal structure, have been incorporated in the analysis. It follows that there will not be any Isotope effect. The suggested theoretical argument is akin to what has been proposed recently in neutral atomic rare gas of fermions in magnetic traps to study the presence of superfluidity in these systems.

Linear discriminants described by disjoint tangent configurations

Abstract In this paper, a new interpretation of parametric linear discriminants for binary classification problems is presented. Linear discriminants are described in terms of Disjoint Tangent Configurations (DTC) established between the ellipsoidal level surfaces resulting from the means and covariance matrices of the distributions. This is a new framework that allows, first, a new interpretation and analysis of several well-known linear discriminants and, second, the design of new discriminants with very interesting properties. In particular, it is shown that the analytical expression of the Bayes Linear Discriminant —whose explicit expression is still unknown— can be derived from a particular DTC. Besides the Bayes discriminant, other classical linear discriminants are also described according to the DTC analysis, in particular, the Fisher and the Scatter-based Linear Discriminants. On the other hand, two new linear discriminants for the minimax and the Bayesian solutions are obtained from the DTC analysis. Both have a direct analytical expression in contrast to the existing iterative solutions, with which they are compared. The first DTC discriminant, which is called MPDH-DTC, is the solution of the Minimax Probabilistic Decision Hyperplane (MPDH) problem, the same solution that the Minimax Probability Machine (MPM) method approximates by an iterative convex optimization. The second discriminant, called Quasi-Bayes-DTC Linear Discriminant, is designed to be an approximation to the Bayes Linear Discriminant, which requires a search procedure to find the solution. Considering both the accuracy over several synthetic and real problems and the computational cost, the Quasi-Bayes-DTC is the preferred discriminant due to its high performance and low computational cost, unless a minimax solution is required, in that case the MPDH-DTC is preferred.

Simulation Study of an Impulse Radiation Antenna Array

A picosecond-pulse electric field, matching ultra-wideband antenna, can be applied to biomedical noninvasive therapy. Its effect depends largely on the electric field intensity on the target. Because the amplitude enhancement of a picosecond-pulse source is restricted by many factors, ultra-wideband time-domain antenna research has become important. Here, the intensity of the electromagnetic radiation propagated by the antenna was greatly improved by arraying the antenna feed structure. For targeted focusing, an ellipsoidal reflective surface is commonly utilized, but this structure limits the formation feasibility of the feed structure. Here, the feed structure and the reflector were arrayed at the same time, while satisfying the time and spatial consistency of the electromagnetic waves at the target, the intensity of the electric field at the target was greatly improved, and the size of the focal spot was reduced simultaneously. A Gaussian pulse with a 1-V amplitude and a 200-ps width was used as the excitation source. Using this antenna array, a focal spot less than $1\times 1\times 1$ cm was formed at the target. Also, with the increased number of elements in the antenna array, the electric field intensity of the target increased. The relationship between the field intensity at the target and the number of combined antennas is discussed. The increased field strength eventually levels off, which is also consistent with the theory.

Ptychographic X-ray CT characterization of the osteocyte lacuno-canalicular network in a male rat's glucocorticoid induced osteoporosis model

Ptychographic X-ray computed tomography (PXCT) is a quantitative imaging modality that non-destructively maps the 3D electron density inside an object with tens of nanometers spatial resolution. This method provides unique access to the morphology and structure of the osteocyte lacuno-canalicular network (LCN) and nanoscale density of the tissue in the vicinity of an osteocyte lacuna. Herein, we applied PXCT to characterize the lacunae and LCN in a male Wistar rat model of glucocorticoid-induced osteoporosis (GIO). The ptychographic images revealed significant (p < 0.05) differences in the number of canaliculi originating from the lacuna per ellipsoidal surface unit, Ca.Nb (p = 0.0106), and the 3D morphology of the lacuna (p = 0.0064), between GIO and SHAM groups. Moreover, the mean canalicular diameter, Ca.Dm, was slightly statistically un-significantly smaller in GIO (152 ± 6.5) nm than in SHAM group (165 ± 8) nm (p = 0.053). Our findings indicate that PXCT can non-destructively provide detailed, nanoscale information on the 3D organization of the LCN in correlative studies of pathologies, such as osteoporosis, leading to improved diagnosis and therapy.

Problems of Hydrodynamics for a Triaxial Ellipsoid

Problems of motion of a triaxial ellipsoid in an ideal liquid and in a viscous liquid in the Stokes approximation and also equilibrium shapes of the rotating gravitating liquid mass are considered. Solutions of these problems expressed via four quadratures depending on four parameters are significantly simplified because they are expressed via the only function of two arguments. The efficiency of the proposed approach is demonstrated by means of analyzing the velocity and pressure fields in an ideal liquid, calculating the added mass of the ellipsoid, determining the viscous friction, and studying the equilibrium shapes and stability of the rotating gravitating capillary liquid. The pressure on the triaxial ellipsoid surface is expressed via the projection of the normal to the impinging flow velocity. The shape of an ellipsoid that ensures the minimum viscous drag at a constant volume is determined analytically. A simple equation in elementary functions is derived for determining the boundary of the domains of the secular stability of the Maclaurin ellipsoids. An approximate solution of the problem of equilibrium and stability of a rotating droplet is presented in elementary functions. A bifurcation point with non-axisymmetric equilibrium shapes branching from this point is found.

Design of a See-through Off-Axis Head-Mounted-Display Optical System with an Ellipsoidal Surface

A new method to design a see-through off-axis head-mounted-display (OA-HMD) optical system with an ellipsoidal surface is proposed, in which a tilted ellipsoidal surface is used as the combiner, which yields the benefits of easier fabrication and testing compared to a freeform surface. Moreover, we realize a coaxial structure in the relay lens group, which is simple and has looser tolerance requirements, thus making assembly easier. The OA-HMD optical system we realize has a simple structure and consists of a combiner and 7 pieces of coaxial relay lenses. It has a 48° × 36° field of view (FOV) and 12-mm exit pupil diameter.

A 17.5 kWel high flux solar simulator with controllable flux-spot capabilities: Design and validation study

A new High Flux Solar Simulator (HFSS) has been designed and built for the research of medium/ high-temperature solar material testing and solar thermochemical processes in Mexico. The HFSS was designed using seven 2.5 kWel Xenon short arc lamps, each close-coupled to a 2 m focal length truncated ellipsoidal specular reflector made of polished aluminum. A Monte Carlo (MC) ray tracing technique was employed to design the shape of the ellipsoidal reflectors. The estimated radiative flux obtained with the MC ray-tracing was 1700 kW m−2 peak flux for all the seven lamps in a flux spot of 100 mm in diameter. A peak flux of 267 kW m−2 was obtained for a single lamp-reflector unit. The design, dimensions and material specifications are presented. The control system is composed of an attenuator curtain and a servo-controlled dynamic test bench that allows variation of the spot size and irradiance incident in the focal plane. A study to compare the theoretical design and the real ellipsoidal reflector surface was conducted with the use of the photogrammetry technique. This was done in order to know if differences in the theoretical flux compared to that of the measured flux come from defects in manufacturing of the reflector. A data acquisition module and an artificial vision system provide information and allow for a monitoring flux spot. Obtaining an experimental peak flux of 194 kW m−2 and a flux spot diameter of 120 mm per lamp, the results were consistent with the desired design. In addition, in order to quantify the risk of our own installation and implement a safety protocol, an experimental campaign of ultraviolet radiation monitoring was conducted; the assessment of ultraviolet radiation shows that the recommended dose per 8 h is exceeded in a few seconds when close to the focal plane (∼0.2 m). In this paper, due to extension, preliminary design, construction, and validation of the HFSS are presented.

The effect of cosmic ray acceleration on supernova blast wave dynamics

Non-relativistic shocks accelerate ions to highly relativistic energies provided that the orientation of the magnetic field is closely aligned with the shock normal (quasi-parallel shock configuration). In contrast, quasi-perpendicular shocks do not efficiently accelerate ions. We model this obliquity-dependent acceleration process in a spherically expanding blast wave setup with the moving-mesh code {\sc arepo} for different magnetic field morphologies, ranging from homogeneous to turbulent configurations. A Sedov-Taylor explosion in a homogeneous magnetic field generates an oblate ellipsoidal shock surface due to the slower propagating blast wave in the direction of the magnetic field. This is because of the efficient cosmic ray (CR) production in the quasi-parallel polar cap regions, which softens the equation of state and increases the compressibility of the post-shock gas. We find that the solution remains self-similar because the ellipticity of the propagating blast wave stays constant in time. This enables us to derive an effective ratio of specific heats for a composite of thermal gas and CRs as a function of the maximum acceleration efficiency. We finally discuss the behavior of supernova remnants expanding into a turbulent magnetic field with varying coherence lengths. For a maximum CR acceleration efficiency of about 15 per cent at quasi-parallel shocks (as suggested by kinetic plasma simulations), we find an average efficiency of about 5 per cent, independent of the assumed magnetic coherence length.

Calculating the High-Frequency Electrical Conductivity of a Thin Metallic Layer for an Ellipsoidal Fermi Surface

The conductivity of a thin metallic layer in an ac electric field is calculated with respect to different specular reflection coefficients of the layer surfaces for an ellipsoidal Fermi surface. The ratio between the free path of electron conduction and layer thickness is not limited. The dependences of the absolute value and the argument of dimensionless conductivity on the dimensionless layer’s thickness, dimensionless external electric field frequency, and specular reflection coefficient of one of the layer surfaces at different ellipticity parameters of the Fermi surface are analyzed.

Geomorphometry on the surface of a triaxial ellipsoid: towards the solution of the problem

ABSTRACTExisting algorithms of geomorphometry can be applied to digital elevation models (DEMs) given with plane square grids or spheroidal equal angular grids on the surface of an ellipsoid of revolution or a sphere. Computations on spheroidal equal angular grids are trivial for modelling of the Earth, Mars, the Moon, Venus, and Mercury. This is because: (a) forms of these celestial bodies can be described by an ellipsoid of revolution or a sphere and (b) for these surfaces, there are well-developed theory and algorithms to solve the inverse geodetic problem as well as to determine spheroidal trapezoidal areas. It is advisable to apply a triaxial ellipsoid for describing the forms of small moons and asteroids. However, there are no geomorphometric algorithms intended for such a surface. In this article, first, we formulate the problem of geomorphometric modelling on a triaxial ellipsoid surface. Then, we recall definitions and formulae for coordinate systems of a triaxial ellipsoid and their transformation. Next, we present analytical and computational solutions, which provide the basis for geomorphometric modelling on the surface of a triaxial ellipsoid. The Jacobi solution for the inverse geodetic problem has a fundamental mathematical character. The Bespalov solutions for determination of the length of meridian/parallel arcs and the spheroidal trapezoidal areas are computationally efficient. Finally, we describe easy-to-code algorithms for derivation of local and non-local morphometric variables from DEMs based on a spheroidal equal angular grid of a triaxial ellipsoid.