For more than 20 years, variants of correspondence analysis have arisen that accommodate for the structure of ordered categorical variables using orthogonal polynomials. When the visual display from ...this analysis is the biplot, projections linking the origin to the standard coordinate of each category is a common feature. In the case when a column variable, say, consists of ordered categories, the biplot can be constructed so that their standard coordinate is determined using orthogonal polynomials which require a set of a priori scores that reflect the ordered structure of the categories. When the first two polynomials are used to construct the biplot they produce a configuration of standard coordinates that appear to be parabolic in shape. This article verifies the exact nature of this parabolic relationship and examines the various features of this configuration of points. Particular emphasis is given to the focus, vertex, intercepts and directrix of this relationship and we also briefly examine the impact of choosing different a priori scores on these features. The R function, parabola.exe(), used to perform these calculates is included as
supplementary material
to this article. Supplementary files for this article are available online.
Based on the famous Shimizu–Morioka system, this paper proposes a novel five-dimensional Shimizu–Morioka-type hyperchaotic system that has an infinite set of heteroclinic orbits. Of particular ...interest are the following observed properties of the system: (i) the existence of both ellipse-parabola-type and hyperbola-parabola-type of equilibria; (ii) the strange attractor coexisting either non-isolated equilibria or two pairs of symmetrical equilibria; (iii) the existence of the proposed strange attractors and hyperchaotic attractors bifurcated from the corresponding singularly degenerate heteroclinic cycles; (iv) the existence of an infinite set of both ellipse-parabola-type and hyperbola-parabola-type heteroclinic orbits.
In this paper, the nonlinear vibration and dynamic buckling responses of the sinusoid, parabola, and cylindrical CNT-reinforced panels with piezoelectric layer stiffened by a CNT-reinforced stiffener ...system in uniform temperature change with a piezoelectric layer are presented. An improved homogenization technique for the x- or y-direction CNT-reinforced stiffener system is utilized to determine the total stiffnesses of the considered structures. The higher-order shear deformation theory (HSDT) in conjunction with the von Kármán nonlinearities is adopted to formulate the motion equations, while the stress function for complex curvature panels is estimated using the like-Galerkin procedure. The nonlinear equation of motion is acquired by utilizing the Lagrange function and Euler-Lagrange's equations. The numerical examples use the Runge-Kutta technique to acquire the nonlinear time-amplitude curves, and the critical dynamic buckling load is determined using the Budiansky-Roth criterion. These examples evaluate the effects of stiffeners, piezoelectric layer, material, and geometrical parameters on the nonlinear vibration and dynamic buckling responses of the panels.
•Nonlinear dynamic responses of the CNT-reinforced stiffened panels are studied.•Sinusoid, parabola, and cylindrical panels with piezoelectric layer are mentioned.•The improved smeared stiffener technique for CNT-reinforced stiffeners is presented.•The like-Galerkin procedure is applied to estimate the approximate stress function.•HSDT, Euler-Lagrange's equation, and Runger-Kutta method are used.
We have constructed the empirical formula for fusion barrier height (VB), fusion barrier position (RB), and inverted parabola (ħω) of light nuclei with atomic number in the range 2≤ZC≤10. This ...formula is obtained by studying the fusion barrier characteristics of 333 projectile target combinations. The values produced by the present formula are also compared with experiments. The present semi-empirical formula produces fusion barrier characteristics of light nuclei with the simple inputs of mass number (A) and atomic number (Z) of projectile targets. This semi-empirical formula can be utilized to calculate and interpret fusion barrier characteristics in light nuclear systems.
Various 3D reconstruction methods have enabled civil engineers to detect damage on a road surface. To achieve the millimeter accuracy required for road condition assessment, a disparity map with ...subpixel resolution needs to be used. However, none of the existing stereo matching algorithms are specially suitable for the reconstruction of the road surface. Hence in this paper, we propose a novel dense subpixel disparity estimation algorithm with high computational efficiency and robustness. This is achieved by first transforming the perspective view of the target frame into the reference view, which not only increases the accuracy of the block matching for the road surface but also improves the processing speed. The disparities are then estimated iteratively using our previously published algorithm, where the search range is propagated from three estimated neighboring disparities. Since the search range is obtained from the previous iteration, errors may occur when the propagated search range is not sufficient. Therefore, a correlation maxima verification is performed to rectify this issue, and the subpixel resolution is achieved by conducting a parabola interpolation enhancement. Furthermore, a novel disparity global refinement approach developed from the Markov random fields and fast bilateral stereo is introduced to further improve the accuracy of the estimated disparity map, where disparities are updated iteratively by minimizing the energy function that is related to their interpolated correlation polynomials. The algorithm is implemented in C language with a near real-time performance. The experimental results illustrate that the absolute error of the reconstruction varies from 0.1 to 3 mm.
This study aims to compare methods for the determination of concrete properties by means of the stress diagrams present in the Brazilian standard ABNT NBR 6118: 2014. The area under the stress ...diagram, the internal reactions, and the application point of the resulting reactions for the parabola-rectangle and rectangular block diagrams are present in order to compare them. Deductions and numerical examples were used, and different results were obtained for each formulation. This is due to non-consideration of the relationship between stress and strain in the simplified rectangular block. The rectangular block is applicable only for cases in which the concrete reaches the ultimate strain. These cases are those that concrete crushing determines the section failure in compression with steel yielding in tension (domain 3) or without steel yielding (domain 4).
There are three types of affine regular polygons in AG(2, q): ellipse, hyperbola and parabola. The first two cases have been investigated in previous papers. In this note, a particular class of ...geometric one-factorizations of the complete graph Kn arising from parabolas is constructed and described in full detail. With the support of computer aided investigation, it is also conjectured that up to isomorphisms this is the only one-factorization where each one-factor is either represented by a line or a parabola.
We consider a problem in computational origami. Given a piece of paper as a convex polygon P and a point f located within, we fold every point on a boundary of P to f and compute a region that is ...safe from folding, i.e., the region with no creases. This problem is an extended version of a problem by Akitaya, Ballinger, Demaine, Hull, and Schmidt that only folds corners of the polygon. To find the region, we prove structural properties of intersections of parabola-bounded regions and use them to devise a linear-time algorithm. We also prove a structural result regarding the complexity of the safe region as a variable of the location of point f, i.e., the number of arcs of the safe region can be determined using the straight skeleton of the polygon P.
•The impact of the preceding layer stacking on rear filament winding is considered.•An improved cubic spline function approach and a novel parabola method were proposed.•The strength analysis of gas ...cylinders based on the parabola method was carried out.•The parabola method adapts well to cylinders with varied sizes and ply schemes.
Due to the characteristics of changing angle and thickness, predicting the outer contour of the vessel dome has been a challenging task in composite pressure vessel design. The impact of preceding layer fiber stacking on subsequent filament winding has never been considered in existing methods of dome thickness calculation. So we developed an improved cubic spline function approach and a novel parabola method that takes preceding layer fiber stacking into account. The example findings show that the improved cubic spline function approach has a weakness in that the selection of several critical parameters is not clearly described and heavily reliant on actual engineering knowledge. The parabola approach described in our study is not only simple in design, but also adapts well to dome contour of gas cylinders with varied sizes and ply schemes. Moreover, the strength analysis of composite gas cylinders was carried out by finite element method. The results show high consistency with hydraulic burst test results, which clearly demonstrates that the parabola method we proposed can effectively realize the precise modeling of composite pressure vessels and will be very beneficial in the design of composite pressure vessels.
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