Elliptic Paraboloid Research Articles

Bernstein problem of affine maximal type hypersurfaces on dimension N ≥ 3

Bernstein problem for affine maximal type equation(0.1)uijDijw=0,w≡[det⁡D2u]−θ,∀x∈Ω⊂RN has been a core problem in affine geometry. A conjecture proposed firstly by Chern (1977) [6] for entire graph and then extended by Trudinger-Wang (2000) [14] to its fully generality asserts that any Euclidean complete, affine maximal type, locally uniformly convex C4-hypersurface in RN+1 must be an elliptic paraboloid. At the same time, this conjecture was solved completely by Trudinger-Wang for dimension N=2 and θ=3/4, and later extended by Jia-Li (2009) [12] to N=2,θ∈(3/4,1] (see also Zhou (2012) [16] for a different proof). On the past twenty years, many efforts were done toward higher dimensional issues but not really successful yet, even for the case of dimension N=3. In this paper, we will construct non-quadratic affine maximal type hypersurfaces which are Euclidean compete forN≥3,θ∈(1/2,(N−1)/N).

Tips and tricks in linear imaging polarimetry of extended sources with FORS2 at the VLT

Context. Polarimetry is a very powerful tool for uncovering various properties of astronomical objects that otherwise remain hidden in standard imaging or spectroscopic observations. While common observations only measure the intensity of light, polarimetric measurements allow us to distinguish and measure the two perpendicular components of the electric field associated with the incoming light. By using polarimetry it is possible to unveil asymmetries in supernova explosions, properties of intervening dust, characteristics of atmosphere of planets, among others. However, the reliable measurement of the low polarization signal from astronomical sources requires a good control of spurious instrumental polarization induced by the various components of the optical system and the detector. Aims. We perform a detailed multi-wavelength calibration study of the FORS2 instrument at the VLT operating in imaging polarimetric mode to characterize the spatial instrumental polarization that may affect the study of extended sources. Methods. We used imaging polarimetry of high signal-to-noise ratio blank field BVRI observations during the full moon, when the polarization is expected to be constant across the field of view and deviations originate from the instrument, and a crowded star cluster in broad-band RI and narrow-band Hα filters, where the individual polarization values of each star across the field can be measured. Results. We find an instrumental polarization pattern that increases radially outwards from the optical axis of the instrument reaching up to 1.4% at the edges, depending on the filter. Our results are closely approximated by an elliptical paraboloid down to less than ∼0.05% accuracy, and ∼0.02% when using non-analytic fits. We present 2D maps to correct for this spurious instrumental polarization. We also give several tips and tricks for analyzing polarimetric measurements of extended sources. Conclusions. FORS2 is a powerful instrument that allows the linear polarimetry of extended sources to be mapped. We present and discuss a methodology that can be used to measure the polarization of such sources, and to correct for the spatial polarization induced in the optical system. This methodology could be applied to polarimetric measurements using other dual-beam polarimeters.

Machine learning based stochastic dynamic analysis of functionally graded shells

This paper presents stochastic dynamic characterization of functionally graded shells based on an efficient Support Vector Machine assisted finite element (FE) approach. Different shell geometries such as cylindrical, spherical, elliptical paraboloid and hyperbolic paraboloid are investigated for the stochastic dynamic analysis. Monte Carlo Simulation is carried out in conjunction with the machine learning based FE computational framework for obtaining the complete probabilistic description of the natural frequencies. Here the coupled machine learning based FE model is found to reduce the computational time and cost significantly without compromising the accuracy of results. In the stochastic approach, both individual and compound effect of depth-wise source-uncertainty in material properties of FGM shells are considered taking into account the influences of different critical parameters such as the power-law exponent, temperature, thickness and variation of shell geometries. A moment-independent sensitivity analysis is carried out to enumerate the relative significance of different random input parameters considering depth-wise variation and collectively. The presented numerical results clearly indicate that it is imperative to take into account the relative stochastic deviations (including their probabilistic characterization) of the global dynamic characteristics for different shell geometries to ensure adequate safety and serviceability of the system while having an economical structural design.

Analysis of advancing speed effect in gas safety extraction channels and pressure-relief gas extraction

Combined with the coal mining process of a high gas coal mine in Jinzhong, Shanxi Province, this paper discussed the evolution mechanism of the gas safety extraction channels induced by the mining of overburden fissures in the fully mechanized mining working face and carried out the physical similar simulation test. According to the theory of mining fracture elliptic paraboloid zone, the working face advanced speed as the key parameter, mathematical modelling of the gas safety extraction channel profile was performed. The effect of advancing speeds on the evolution of gas safety extraction channels in overburden fissures was comprehensively analysed. The results suggested that increasing advancing speed reduced the height of the caving, fractured and bending zone, enlarged average weighting intervals, and the range of development was small. What's more, with increasing advancing speed, gas safety extraction channels presented gradually reduced bed-separation volume and continuity, and the fractal dimensions (7.0 < 5.0 < 3.0 m/d) also decreased. By conducting on-site high-level drilling field tests, parameters of the high-level boreholes were optimized based on specific advancing speeds, which ensured production safety and efficiency of the mining working face. The present work contributes a theoretical guides for the identification of gas extraction area under different advancing speeds.

Дифференцируемое отображение, порожденное комплексами эллиптических параболоидов

In three-dimensional equiaffine space, we consider a differentiable map generated by complexes with three-parameter families of elliptic paraboloids according to the method proposed by the author in the mate­rials of the international scientific conference on geometry and applica­tions in Bulgaria in 1986, as well as in works published earlier in the sci­entific collection of Differ. Geom. Mnogoobr. Figur. The study is carried out in the canonical frame, the vertex of which coincides with the top of the generating element of the manifold, the first two coordinate vectors are conjugate and lie in the tangent plane of the elliptic paraboloid at its vertex, the third coordinate vector is directed along the main diameter of the generating element so that the ends are, respectively, the sums of the first and third, and also the sums of the second and third coordinate vec­tors lay on a paraboloid, while the indicatrixes of all three coordinate vec­tors describe lines with tangents, parallel to the first coordinate vector. The existence theorem of the mapping under study is proved, according to which it exists and is determined with the arbitrariness of one function of one argument. The systems of equations of the indicatrix and the main directions of the mapping under consideration are obtained. The indicatrix and the cone of the main directions of the indicated mapping are geomet­rically characterized.

The question of finding the shortest distance between two points of different diversity

Digital education at the university implies application of information technologies, active learning methods and comprehensive algorithmizing of the process of solving various tasks. It is also supposed to create application software packages, which can find their application at the junction of various disciplines. (1) One area of digital education is to solve the problems of motion optimization and localization of various objects. In particular, the question of finding the shortest distance between points lying on two different varieties (curves, plane, etc.) is frequently solved (more than 10) Here are examples of the solving. The first example describes an algorithm for finding the shortest distance between two parabolas lying in the same plane; the second example suggests a solution course for an elliptical paraboloid and a point on the coordinate axis. Finally, the third example deals with the distance between a cube parabola and a hemisphere with the center at the beginning of the coordinates.

Thermo-solutal growth of a dendritic crystal in the form of an elliptical paraboloid with forced convection

The steady-state analytical solutions describing the thermo-solutal growth of a dendritic crystal in the form of an elliptical paraboloid with allowance for a forced convective flow are found. The system of transcendental equations consisting of the undercooling balance and the stability criterion that describes a stable mode of growth of sections of elliptical dendrites is formulated and analyzed. It is shown that the temperature distributions in the melt, the growth and flow Peclét numbers as well as the dendrite tip diameter and tip velocity in the case of elliptical crystals essentially differ from their analogs in the case of symmetric shapes of dendrites in the form of paraboloids of revolution.

Robust automated reading of the skin prick test via 3D imaging and parametric surface fitting.

The conventional reading of the skin prick test (SPT) for diagnosing allergies is prone to inter- and intra-observer variations. Drawing the contours of the skin wheals from the SPT and scanning them for computer processing is cumbersome. However, 3D scanning technology promises the best results in terms of accuracy, fast acquisition, and processing. In this work, we present a wide-field 3D imaging system for the 3D reconstruction of the SPT, and we propose an automated method for the measurement of the skin wheals. The automated measurement is based on pyramidal decomposition and parametric 3D surface fitting for estimating the sizes of the wheals directly. We proposed two parametric models for the diameter estimation. Model 1 is based on an inverted Elliptical Paraboloid function, and model 2 on a super-Gaussian function. The accuracy of the 3D imaging system was evaluated with validation objects obtaining transversal and depth accuracies within ± 0.1 mm and ± 0.01 mm, respectively. We tested the method on 80 SPTs conducted in volunteer subjects, which resulted in 61 detected wheals. We analyzed the accuracy of the models against manual reference measurements from a physician and obtained that the parametric model 2 on average yields diameters closer to the reference measurements (model 1: -0.398 mm vs. model 2: -0.339 mm) with narrower 95% limits of agreement (model 1: [-1.58, 0.78] mm vs. model 2: [-1.39, 0.71] mm) in a Bland-Altman analysis. In one subject, we tested the reproducibility of the method by registering the forearm under five different poses obtaining a maximum coefficient of variation of 5.24% in the estimated wheal diameters. The proposed method delivers accurate and reproducible measurements of the SPT.

The Self-Calibration Method for the Vertex Distance of the Elliptical Paraboloid Array

The elliptical paraboloid array plays an important role in precision measurement, astronomical telescopes, and communication systems. The calibration of the vertex distance of elliptical paraboloids is of great significance to precise 2D displacement measurement. However, there are some difficulties in determining the vertex position with contact measurement. In this study, an elliptical paraboloid array and an optical slope sensor for displacement measurement were designed and analyzed. Meanwhile, considering the geometrical relationship and relative angle between elliptical paraboloids, a non-contact self-calibration method for the vertex distance of the elliptical paraboloid array was proposed. The proposed self-calibration method was verified by a series of experiments with a high repeatability, within 3 μ m in the X direction and within 1 μ m in the Y direction. Through calibration, the displacement measurement system error was reduced from 100 μ m to 3 μ m . The self-calibration method of the elliptical paraboloid array has great potential in the displacement measurement field, with a simple principle and high precision.

An analytical approach for the three-dimensional water entry problem of solids creating elliptical contact regions

The purpose of the present study is to approach analytically the water entry problem of solids creating elliptical contact regions with the free surface during impact. The governing boundary value problem has been linearized using asymptotic analysis and considering the early stages of the phenomenon. The fundamental assumption is made that the contact region between the solid and the free surface is always an ellipse. This allows to consider the underlying boundary value problem in elliptical coordinates exploiting the properties of elliptical harmonics. It is shown that for elliptical contact regions the solution sought can be detached from the angular dependence. This finding is extremely beneficial as it allows the derivation of a perturbation sequence of mathematical systems of mixed type, involving Fredholm integral equations, which can be treated using rigorous methods of solution for mixed boundary value problems in potential theory. The theory developed for elliptical contact regions during water entry situations, is accordingly applied for anticipating the details of the slamming phenomenon during the free-surface impact of a nonaxisymmetric prolate spheroid assuming that the contact line between the liquid and the solid is always an ellipse. The solid is assumed to descend steadily, without rotating, towards an initially undisturbed free surface. The perturbation technique requires that the slenderness parameter is small compared to unity, while slenderness affects significantly the hydrodynamic load exerted on the solid during water entry. Numerical results are presented and discussed using both the von Karman and the Wagner models of water impact. Finally, the study was extended to consider the 3D Wagner water impact problem of an elliptic paraboloid and to compare computations against available data reported in the past.

Beam-Scanning Lens Antenna Based on Elliptical Paraboloid Phase Distribution Metasurfaces

A two-dimensional (2-D) beam-scanning lens antenna is proposed for Ka-band satellite applications, which is composed of a feed horn that can move in plane and a metasurface lens with elliptical paraboloid phase distribution. Due to the paraboloid phase compensation, the lens antenna possesses wide scanning angles in the azimuth and elevation. The fabricated lens has a size of 10.5 λ 0 × 10.5 λ 0 × 0.2 λ 0 ( λ 0 is wavelength at the center frequency of 30 GHz) and the focal-to-diameter ratio of 0.2. The total system can realize the boresight gain of 18.5 dB, and the scanning coverage of ±60° with a gain tolerance of 3.7 dB. The design methodology features the advantages of low cost, compact size, and light weight. It can find potential applications in multibeam or beam-scanning antenna systems that are comparable with array antennas.

Near-Optimal MPC Algorithm for Actively Damped Grid-Connected PWM-VSCs With LCL Filters

This paper proposes and investigates a novel near-optimal finite-control-set model predictive control (NOP-MPC) algorithm to control the grid-connected, pulsewidth-modulator-driven voltage source converters with LCL filters. Exploiting the convex and elliptical paraboloid properties of the cost error, NOP-MPC adopts a systematic iterative algorithm within each control cycle to progressively synthesize finite sets of virtual voltage vectors (VVs) for the control optimization stage. The synthesis has the inherent features of respecting the converter voltage limits and converging the sets of VV candidates toward the global optimal point. The fixed-switching-frequency feature of NOP-MPC is expected to ease the LCL filter design. Effects of computational delay, pulsewidth modulation delay, and deadtime are considered and compensated successfully. A two-vector-variable cost function is used to actively damp the inherent LC resonance through an adjustable, weighting-factor-based damping level. This paper is substantiated by theoretical consideration, simulation and experimental results, parameter sensitivity study, and a comparative study with the standard finite-control-set model predictive control that uses only actual VVs.

Buckling mechanism of cable-stiffened lattice shells with bolted connections

The cable-stiffened lattice shell is a new structural system for its translucence and lighting. This article discusses the effect of the connections’ behavior and geometric imperfection on the structural stability and reveals the buckling mechanism of the cable-stiffened lattice shell. The spring stiffness for bolted connections of cable-stiffened lattice shells is deduced from the spring in series model. The buckling mechanism of cable-stiffened lattice shells with three types of joints have been studied based on the prototypical static experiments of bolted connections. The decrease of bolted connections’ stiffness would lead to the change in the displacement distribution for the lattice shell under its ultimate load. The buckling loads and initial structural stiffness of cable-stiffened lattice shells with shim-strengthened bolted joints are approximately 80% of those for cable-stiffened lattice shells with rigid joints. The result indicates that the buckling loads of cable-stiffened lattice shells with bolted connections decrease much more slowly than the decrease of bolted connections’ stiffness. The cable-stiffened lattice shell with SBP connections is more sensitive to the initial geometric imperfection. Finally, a formula has been proposed for estimating buckling loads of elliptic paraboloid cable-stiffened lattice shells with bolted connections.

A Novel 2D Micro-Displacement Measurement Method Based on the Elliptical Paraboloid

The micro-displacement measurement system with 2D/3D has become increasingly important in the field of scientific research and technology application. In order to explore the application of an optical surface in micro-displacement measurement, a novel and simple 2D micro-displacement measurement method based on the elliptical paraboloid was designed and subjected to experiment. The measurement system takes advantage of the elliptical paraboloid instead of a plane mirror in the optical structure of an autocollimator which has been ameliorated to adapt to curved surface measurement. Through the displacement of the light spot on the CCD (Charge Coupled Device) detector, the displacement of the target could be measured with a linear correlation coefficient of 0.9999. The accuracy of the system is about ± 0.3 μm in a wide range in two dimensions. The results were in good agreement with the theoretical analysis and indicated the potential applicability of the proposed system in the detection of geometric errors of CNC (Computerized Numerical Control) machine tools.

L1-NORM FITTING OF ELLIPTIC PARABOLOIDS WITH PRIOR INFORMATION FOR ENHANCED CONIFEROUS TREE LOCALIZATION IN ALS POINT CLOUDS

Abstract. Airborne laser scanning (ALS) is an established tool for deriving various tree characteristics in forests. In some applications, an accurate pointwise estimate of the tree position is required. For dense data acquired by TLS or UAV-mounted scanners, this can be achieved by locating the stem, whose center coordinates are then used for deriving the planimetric tree position. However, in case of standard ALS data this is often not an option due to the low probability of obtaining stem hits in operational scenarios of forest mapping campaigns. This paper presents an alternative, indirect approach where the tree position is approximated as the center of a quadric surface which best represents the tree crown shape. The study targets coniferous trees due to their distinct crown shape which may be approximated by an elliptic paraboloid. It is assumed that individual tree point clusters are given and the task is to find the tree center for each cluster. We first consider the general problem of fitting an elliptic paraboloid with a known axis and an L1 residual norm error criterion, which is more robust to outliers compared to least-squares fitting. We formulate this problem as a quadratically constrained quadratic program (QCQP), and show how prior knowledge on the crown shape and center position can be incorporated. Next, a computationally simpler problem is considered where the paraboloid semiaxis lengths are constrained to be equal, and a corresponding linear program is constructed. Experiments on ALS datasets of forest plots from Bavaria, Germany and Oregon, USA reveal that a reduction in median tree position error of up to 20% can be attained compared to both least-squares fitting and other baseline techniques, resulting in an absolute error of ca. 22 cm on both datasets.

Classification of the relative positions between a small ellipsoid and an elliptic paraboloid

We classify all the relative positions between an ellipsoid and an elliptic paraboloid when the ellipsoid is small in comparison with the paraboloid (small meaning that the two surfaces cannot be tangent at two points simultaneously when one is moved with respect to the other). This provides an easy way to detect contact between the two surfaces by a direct analysis of the coefficients of a fourth degree polynomial.

定义于椭圆抛物面上的多元Lagrange插值问题

定义于椭圆抛物面上的多元Lagrange插值问题

Lattice points in elliptic paraboloids

We consider the lattice point problem corresponding to a family of elliptic paraboloids in Rd with d ≥ 3 and we prove the expected to be optimal exponent, improving previous results. This is especially noticeable for d = 3 because the optimal exponent is conjectural even for the sphere. We also treat some aspects of the case d = 2, getting for a simple parabolic region an Ω-result that is unknown for the classical circle and divisor problems.

Heat Transfer in a Half-Space with Transversal Anisotropy Under the Action of a Lumped Heat Source

The authors have obtained a novel analytical solution to the problem of the theory of heat conduction in an anisotropic half-space with an anisotropy of the transversal type when the anisotropy takes place in planes formed by two coordinate axes, whereas in the direction of the third coordinate axis guided deep into the body, the material is isotropic. The solution was obtained by two applications of the variablewise Fourier transformation in the anisotropy plane and of the specially developed method of substitution in the direction of the third coordinate. The obtained results point to the drift of the heat fluxes and temperature fields toward the principal axis with a larger coefficient of the thermal-conductivity tensor, with this coefficient being the larger the higher the degree of anisotropy in the indicated direction. It has been established that isothermal surfaces have the shapes of elliptic paraboloids.

Thermoelastic free vibration response of graphene reinforced laminated composite shells

This article deals with the free vibration analyses of graphene reinforced singly and doubly curved laminated composite shell panels in thermal environment using finite element method. The curved shell panels are mathematically modeled using higher order shear deformation theory (HSDT) while the distortion of the geometry due to thermal loading is manifested through geometric stiffness considering Green Lagrange type of nonlinearity. Four different types of graphene reinforcement configurations are considered in different types of shells (cylindrical, elliptic paraboloid, hyperbolic paraboloid, conoidal and hypar). The extended Halpin–Tsai approach is used to obtain the temperature dependent material properties of the laminated composite shells. Eight noded isoparametric shell elements consisting of seven degrees of freedom per node are considered to discretize the geometry. The accuracy and convergence of the formulation are verified by solving various examples available in the literature. Finally, parametric study is presented to investigate the influence of various parameters on the free vibration response of the graphene reinforced laminated composite shells.