Ellipsoidal Research Articles

Boundary Value Problems in Ellipsoidal Geometry and Applications

Boundary Value Problems in Ellipsoidal Geometry and Applications

New algorithm of measuring gravitational wave radiation from rotating binary system

In order to investigate the gravitational wave (GW) radiation, without resorting to the traceless transverse gauge approach to the GW formalism of the linearized general relativity, we formulate the so-called modified linearized general relativity (MLGR). As an application of the MLGR, we construct a novel paradigm of measuring the GW radiation from a binary system of compact objects, to theoretically interpret its phenomenology. To accomplish this, we formulate the mass scalar and mass vector potentials for the merging binary compact objects, from which we construct the mass magnetic field in addition to the mass electric one which also includes the mass vector potential effect. Next, defining the mass Poyinting vector in terms of the mass electric and mass magnetic fields in the MLGR, we find the GW radiation intensity profile possessing a prolate ellipsoid geometry due to the merging binary compact objects source. At a given radial distance from the binary compact objects, the GW radiation intensity on the revolution axis of the binary compact objects is shown to be twice that on the equatorial plane. Moreover, we explicitly obtain the total radiation power of the GW, which has the same characteristic as that of the electromagnetic wave in the rotating charge electric dipole moment. We also find that, in no distorting limit of the merging binary compact objects, the compact objects in the MLGR do not yield the total GW radiation power, consistent with the result of the traceless transverse gauge algorithm in the linearized general relativity.

Analytical Computation of Hyper-Ellipsoidal Harmonics

The four-dimensional ellipsoid of an anisotropic hyper-structure corresponds to the four-dimensional sphere of an isotropic hyper-structure. In three dimensions, both theories for spherical and ellipsoidal harmonics have been developed by Laplace and Lamé, respectively. Nevertheless, in four dimensions, only the theory of hyper-spherical harmonics is hitherto known. This void in the literature is expected to be filled up by the present work. In fact, it is well known that the spectral decomposition of the Laplace equation in three-dimensional ellipsoidal geometry leads to the Lamé equation. This Lamé equation governs each one of the spectral functions corresponding to the three ellipsoidal coordinates, which, however, live in non-overlapping intervals. The analysis of the Lamé equation leads to four classes of Lamé functions, giving a total of 2n + 1 functions of degree n. In four dimensions, a much more elaborate procedure leads to similar results for the hyper-ellipsoidal structure. Actually, we demonstrate here that there are eight classes of the spectral hyper-Lamé equation and we provide a complete analysis for each one of them. The number of hyper-Lamé functions of degree n is (n + 1)2; that is, n2 more functions than the three-dimensional case. However, the main difficulty in the four-dimensional analysis concerns the evaluation of the three separation constants appearing during the separation process. One of them can be extracted from the corresponding theory of the hyper-sphero-conal system, but the other two constants are obtained via a much more complicated procedure than the three-dimensional case. In fact, the solution process exhibits specific nonlinearities of polynomial type, itemized for every class and every degree. An example of this procedure is demonstrated in detail in order to make the process clear. Finally, the hyper-ellipsoidal harmonics are given as the product of four identical hyper-Lamé functions, each one defined in its own domain, which are explicitly calculated and tabulated for every degree less than five.

Equilibrium Figures of Two Liquid Masses with Synchronous Rotation. Dynamics of Double Asteroid (190166) 2005 UP156

The problem of equilibrium figures of two liquid masses in a state of tidal mutual capture is posed and solved. The condition of complete synchronous (orbital plus spin) rotation is satisfied in the system, and both bodies have the same masses and congruent ellipsoidal surfaces. For each figure, besides its own gravity and centrifugal forces, the attraction from the second body is taken into account in the tidal approximation. The spatial form of equilibrium figures as triaxial ellipsoids is found by an analytical and numerical method. The spin rotation of ellipsoidal equilibrium figures is established to occur not around small axes, as is usually assumed, but around the middle axes of the ellipsoids. This method is used to study the binary asteroid (190166) 2005 UP156, which approximately satisfies the initial conditions of the problem. The study showed that with the parameters known today, the system of two asteroids (190166) 2005 UP156 is nonequilibrium.

Effect of cement volume on biomechanical response of a spine segment treated with a PEEK polymer implant: a finite element comparative study with vertebroplasty.

In the current study, a 3D finite element study was performed to investigate the biomechanical response of an osteoporotic spine segment treated with a novel transpedicular implant (V-STRUT©, Hyprevention, France) made of PEEK (polyetheretherketone) material combined with either injections of 2, 3, 4, 5 and 6 cc of cement. The objective was to assess numerically the biomechanical performance of the implant in combination with different doses of the injected bone cement and to compare its performance with the gold standard vertebroplasty (VP) technique. A female (69yo) was selected and a 3D finite element model of an osteoporotic spine segment was built based on a Computed Tomography (CT) scan performed from T12 to L2 with corresponding intervertebral discs and ligaments. A heterogeneous distribution of bone material properties was assigned to the bone using grey scale levels. Bilateral ellipsoid geometries of the inserted cement were retained for the V-STRUT and VP models based on experimental observation performed on different patients treated with the V-STRUT device. The current study demonstrated an optimal dose of 4 cc of bilaterally injected cement for the V-STRUT and VP techniques to restore the treated segment and confirmed that the V-STRUT device in combination with bone cement is superior to VP alone in establishing the normal stiffness and in reducing the applied stress to the immediately adjacent vertebral levels.

Errors in the Introduction of Isometric Coordinates and Violation of the Property of Conformality of the Triaxial Ellipsoid Projections

The purpose of this study is to show that some coordinate systems previously described as isometric are in fact not isometric, and as a consequence, map projections derived from them do not show the expected characteristics, especially conformality. Definitions and methods of the mathematical theory of surfaces are used. The article discusses two methods of specifying coordinate systems on the surface of the triaxial ellipsoid, erroneously called isometric. A consistent mathematical study of each of them based on the initial definition of isometric coordinates showed that in the first method, the corresponding coordinate system is not orthogonal, much less isometric. In the second method, the coordinates are determined using integration, and the result of the integration depends on the path of this integration. This ambiguity shows that this method is also incorrect. The considered coordinate systems allow the creation of new projections, but these projections are not conformal. As a result of the study, errors in determining isometric coordinates were shown and proven, which in earlier studies led to a violation of the properties of projections. This analysis should lead to the development of truly conformal projections of triaxial ellipsoids. For clarity maps and cartographic grids are presented.

Constraints on Bianchi-I type universe with SH0ES anchored Pantheon+ SNIa data

We study the Bianchi-I cosmological model motivated by signals of statistical isotropy violation seen in cosmic microwave background (CMB) observations and others. To that end, we consider various kinds of anisotropic matter that source anisotropy in our model, specifically Cosmic strings, Magnetic fields, Domain walls and Lorentz violation generated magnetic fields. These anisotropic matter sources, taking one at a time, are studied for their co-evolution with standard model (isotropic) sources viz., dust-like (dark/normal) matter, and dark energy modelled as cosmological constant. We constrain the Hubble parameter, density fractions of anisotropic matter, cold dark matter (CDM), and dark energy (Λ) in a Bianchi-I universe with planar symmetry i.e., which has a global ellipsoidal geometry, and try to find signatures of a cosmic preferred axis if any.The latest compilation of Type Ia Supernova (SNIa) data from Pantheon+SH0ES collaboration is used in our analysis to obtain constraints on cosmological parameters and any preferred axis for our universe.In our analysis, we found mild evidence for a cosmic preferred axis. It is interesting to note that this preferred axis lies broadly in the vicinity of other prominent cosmic anisotropy axes reported in the literature from diverse data sets.Also we find some evidence for non-zero (negative) cosmic shear and eccentricity that characterize different expansion rates in different directions and deviation from an isotropic scale factor respectively.The energy density fractions of two of the sources considered are found to be non-zero at a2σ confidence level.To be more conclusive, we require more SNIa host galaxy data for tighter constraints on distance and absolute magnitude calibration which are expected to be available from the future JWST observations and others.

Design and Fabrication of Fibrous Spindle‐Like Constructs Using a Melt Electrohydrodynamic Writing Process

AbstractAdvanced manufacturing of 3D‐structured materials enables the production of biomimetic muscle tissues. While models of muscle tissue exist, current approaches possess a limited ability to capture essential elements of the muscle tissue microarchitecture. Therefore, this paper aims to engineer the intrinsically complex muscle spindle‐like ellipsoid geometry using a polymer melt‐based electrohydrodynamic (EHD) printing system. EHD systems have conventionally reported fiber deposition in a layerwise fashion. However, without mitigation, the observed fiber sagging and residual charge phenomena for the melt electrowriting (MEW) process limit the ability to produce layered fibrous 3D constructs with in‐plane fiber alignment. However, in this work, fiber sagging and residual charge phenomena are leveraged as part of the design intent to deposit nonoverlapping suspended fibers between two stationary walls toward spindle‐like construct fabrication. Specifically, herein the structural and mechanical properties of the MEW‐enabled spindle‐like constructs are analyzed as a function of the process and design parameters that govern control over fiber sagging and residual charge. The results indicate that the collector speed and wall‐to‐wall distance are the key parameters for tuning the spindle morphology. Moreover, cycle number and fiber diameter are identified as effective parameters for tuning the spindle mechanical properties.

Effect of Elevated Acetabular Cup on Contact and Failure Analysis in Hip Implants for Different Microseparations and Cup Inclinations Under Routine Gait Activities Using In Silico Approach.

The acetabular cup design plays a critical role in reducing contact stress between femur head acetabular cup. Many studies used ellipsoidal and spheroidal geometry in acetabular cup design to effectively reduce contact stress. The present study focuses on elevated acetabular cup rim with round corner design to reduce contact stress with round corner geometry. The cobalt chromium femur head and cup are considered for finite element (FE) model of hip resurfacing. The gait loads of routine activities of humans like normal walking, stair ascending and descending and sitting down and getting up gait activities are applied to the developed 3D FE model. Five microseparations of 0.5, 1, 1.5, 2 and 2.5mm are considered in the present study. The acetabular cup inclination angle considered for this study are 35°, 45°, 55°, 65° and 75°. The contact stress and von Mises stress plot for each gait activities under these microseparations are analyzed for betterment of longevity of implants. Overall elevated cup rim design helped in reducing contact stress to a greater extent than conventional cup with different geometries. Also, the predicted von Mises stress for all the parameters considered in the current study are well within the yield strength of CoCr material. Therefore, elevated cup rim could be used as a better alternative to spline and, ellipsoidal and circular geometries of cup.

Unipolar optical transitions in nanoclusters of ellipsoidal geometry

The quantum states of an ellipsoidal nanocluster of a heterophase system InAs/GaAs are studied using an exact analytical approach, in contrast to the generally accepted theoretical model based on the adiabatic approximation. It is shown that the spectrum of a nanoobject is formed from local groups, consisting of discrete levels, separated by terahertz frequency intervals. A double random degeneration of certain spectrum states is revealed. Allowed mid-infrared (IR) optical transitions between different spectral states are analyzed. The role of dimensional parameters and features of the shape of a nanoobject in the characteristics of unipolar transitions is assessed. The absorption spectrum for the transition in the lower part of the substructure spectrum is calculated.

Mapping Hyperion in Projections of the Triaxial Ellipsoid Based on a New Reference Network and a Digital Terrain Model

Mapping Hyperion in Projections of the Triaxial Ellipsoid Based on a New Reference Network and a Digital Terrain Model

Least Squares Comparison of Two Rigorous Math Models Used to Fit a Triaxial Ellipsoid to a Cluster of Points

Least Squares Comparison of Two Rigorous Math Models Used to Fit a Triaxial Ellipsoid to a Cluster of Points

Sensitivity of Polarization to Grain Shape. I. Convex Shapes

Aligned interstellar grains produce polarized extinction (observed at wavelengths from the far-ultraviolet to the mid-infrared) and polarized thermal emission (observed at far-infrared and submm wavelengths). The grains must be quite nonspherical, but the actual shapes are unknown. The relative efficacy for aligned grains to produce polarization at optical versus infrared wavelengths depends on particle shape. The discrete dipole approximation is used to calculate polarization cross sections for 20 different convex shapes, for wavelengths from 0.1 to 100 μm, and grain sizes a eff from 0.05 to 0.3 μm. Spheroids, cylinders, square prisms, and triaxial ellipsoids are considered. Minimum aspect ratios required by the observed starlight polarization are determined. Some shapes can also be ruled out because they provide too little or too much polarization at far-infrared and submm wavelengths. The ratio of 10 μm polarization to integrated optical polarization is almost independent of grain shape, varying by only ±8% among the viable convex shapes; thus, at least for convex grains, uncertainties in grain shape cannot account for the discrepancy between predicted and observed 10 μm polarization toward Cyg OB2-12.

Tailoring second harmonic emission by ZnO nanostructures: Enhancement of directionality

Tailoring nonlinear optical properties at the nanoscale is a hot topic in nowadays nanophotonics, promising for applications spanning from sensing to ultrafast optical communications. Here we present a numerical approach of designing a simple semiconductor nanostructure able to tailor second harmonic emission in the near- and far-field. We start from linear simulations of ZnO nanospheres, which reveal multipolar nature of the scattering. Next, we show how the same nanospheres, with radii in 30–130 nm range, excited at 800 nm, manipulate the directivity of the emitted second harmonic. We observe that the nanospheres which exhibit Kerker condition at 400 nm, emit the second harmonic field in the forward direction. We further investigate how the asymmetry (ellipsoid geometry) tailors the second harmonic directivity. We finally introduce geometry with low chiro-optical response, and observe that the second harmonic far-field depends on the handedness of the light exciting the nanostructure at 800 nm.

Mobility capillary electrophoresis-native mass spectrometry reveals the dynamic conformational equilibrium of calmodulin and its complexes.

Benefitting from the rapid evolution of artificial intelligence and structural biology, an expanding collection of high-resolution protein structures has greatly improved our understanding of protein functions. Yet, proteins are inherently flexible, and these static structures can only offer limited snapshots of their true dynamic nature. The conformational and functional changes of calmodulin (CaM) induced by Ca2+ binding have always been a focus of research. In this study, the conformational dynamics of CaM and its complexes were investigated using a mobility capillary electrophoresis (MCE) and native mass spectrometry (native MS) based method. By analyzing the ellipsoidal geometries of CaM in the solution phase at different Ca2+ concentrations, it is interesting to discover that CaM molecules, whether bound to Ca2+ or not, possess both closed and open conformations. Moreover, each individual CaM molecule actively "jumps" (equilibrium exchange) between these two distinct conformations on a timescale ranging from milli- to micro-seconds. The binding of Ca2+ ions did not affect the structural dynamics of CaM, while the binding of a peptide ligand would stabilize CaM, leading to the observation of a single, compact conformation of the resulting protein complex. A target recognition mechanism was also proposed based on these new findings, suggesting that CaM's interaction with targets may favor a conformational selection model. This enriches our understanding of the binding principles between CaM and its numerous targets.

Three-degrees-of-freedom orientation manipulation of small untethered robots with a single anisotropic soft magnet

Magnetic actuation has been well exploited for untethered manipulation and locomotion of small-scale robots in complex environments such as intracorporeal lumens. Most existing magnetic actuation systems employ a permanent magnet onboard the robot. However, only 2-DoF orientation of the permanent-magnet robot can be controlled since no torque can be generated about its axis of magnetic moment, which limits the dexterity of manipulation. Here, we propose a new magnetic actuation method using a single soft magnet with an anisotropic geometry (e.g., triaxial ellipsoids) for full 3-DoF orientation manipulation. The fundamental actuation principle of anisotropic magnetization and 3-DoF torque generation are analytically modeled and experimentally validated. The hierarchical orientation stability about three principal axes is investigated, based on which we propose and validate a multi-step open-loop control strategy to alternatingly manipulate the direction of the longest axis of the soft magnet and the rotation about it for dexterous 3-DoF orientation manipulation.

Spectropolarimetry of the Type IIP supernova 2021yja: an unusually high continuum polarization during the photospheric phase

ABSTRACT We present six epochs of optical spectropolarimetry of the Type IIP supernova (SN) 2021yja ranging from ∼25 to 95 d after the explosion. An unusually high continuum linear polarization of $p \approx 0.9~{{\ \rm per\ cent}}$ is measured during the early photospheric phase, followed by a steady decrease well before the onset of the nebular phase. This behaviour has not been observed before in Type IIP supernovae (SNe IIP). The observed continuum polarization angle does not change significantly during the photospheric phase. We find a pronounced axis of symmetry in the global ejecta that is shared in common with the Hα and Ca ii near-infrared triplet lines. These observations are consistent with an ellipsoidal geometry. The temporal evolution of the continuum polarization is also compatible with the SN ejecta interacting with aspherical circumstellar matter (CSM), although no spectroscopic features that may be associated with strong interaction can be identified. Alternatively, we consider the source of the high polarization to be an extended hydrogen envelope that is indistinguishable from low-density CSM.

Optimal shape design of spherical squeeze film bearings using genetic algorithms

The optimal shape design of hydrodynamic axisymmetric spherical bearings under conditions of transient axisymmetric pure squeeze is investigated using genetic algorithms (GA). The optimization problem is to determine the radial clearance shape which maximizes the load impulse resulting from transient ball motion to a target minimum film thickness specification. For fully-hemispherical cups, non-dimensional analysis shows that GA-based optimal designs yield significantly larger non-dimensional impulse values when compared with those obtained either randomly or from optimal ellipsoidal geometry. A sample design application with specified load history shows that an optimal GA-inspired composite clearance specification amenable to fabrication can be constructed from preferred features obtained from the GA-based optimal shapes. The optimal composite shape design is shown to provide significantly better bearing performance, as measured by minimum film thickness, after a specified time interval than an optimal ellipsoidal design. The optimal composite shape design is also found to be less sensitive to realistic deviations in clearance shape, load impulse, and viscosity variation when compared with similar variations to an optimal ellipsoidal cup design.

A magnetic fabric study of the origin of englacial debris bands at Fláajökull, Southeast Iceland

Two debris bands associated with an overdeepened basin in the terminus region of the Fláajökull glacier, Iceland, have been sampled for their ice and debris. Here the kinematics of deformation are investigated through anisotropy of magnetic susceptibly (AMS), microstructural analyses, and c-axis preferred orientations. Comparison of the crystallographic orientations of 180 grains and the AMS shape orientations of mineral inclusions in polycrystalline ice reveals a multi-maximum girdle oblique to the planar stratification and a record of near vertical triaxial ellipsoids. These two distinct fabrics can be attributed to both pure and simple shear suggesting a complex deformational history. In addition, AMS from an ice-free sediment ridge on the glacier surface indicates pure shear locally dominated strain. The physical properties of the debris bands, sedimentology and field observations suggest that they were sourced from the glacial bed and emplaced by highly pressurized water into localized fractures.

Bridging attraction of condensed bovine serum albumin solution in the presence of trivalent ions: A SANS study

The bridging attraction of condensed bovine serum albumin (BSA) solution (D2O) in the presence of yttrium chloride (YCl3) was studied by small angle neutron scattering (SANS). With increasing the concentration of YCl3 (cY) from 3 to 15 mM and from 15 to 100 mM, the intensity in low-q region increases and then decreases. Combining the tri-axial ellipsoid (TaE) geometry and the multi-component sticky hard sphere (SHS) potential, a SHS-TaE model was established to quantitatively determine the size and distribution of particles. In this way, the structural mechanism of the aggregation-redissolution process in protein solution was demonstrated and discussed. As cY increases from 3 to 100 mM, the SHS radius rL decreases from ca. 2.97 to 2.50 nm, suggesting that the relatively well dispersed BSAs may form aggregates with various polydispersities. The axis a increases from 1.88 to 2.30 nm, while b and c decrease from 3.53 to 3.23 nm and from 4.12 to 3.55 nm, respectively. (RgTaE decreases from ca. 2.57 to 2.38 nm). Moreover, the scattering length density (SLD) of BSA decreases from 3.67 to 1.56 × 10−6 Å−2. All these results consistently indicate a strengthened attraction and the BSA molecules might shrink and tune out to be more like of oblate ellipsoid with increasing the amount of YCl3.