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Conic sections are commonly used in reflective X‐ray optics. Hyperbolic mirrors can focus a converging light source and are frequently paired with elliptical or parabolic mirrors in Wolter type configurations. This paper derives the closed‐form expression for a mirror‐centered hyperbolic shape, with zero‐slope at the origin. Combined with the slope and curvature, such an expression facilitates metrology, manufacturing and mirror‐bending calculations. Previous works consider ellipses, parabolas, magnifying hyperbolas or employ lengthy approximations. Here, the exact shape function is given in terms of the mirror incidence angle and the source and image distances. [ABSTRACT FROM AUTHOR]

A simple method of demonstrating apparent weightlessness in free fall is described. The Apollo 15 hammer and feather drop experiment can be recreated in the classroom using a book to shield the feather from air resistance. Plastic toys and a feather thrown together in a plastic bottle can be used to recreate an apparent weightless environment close to the surface of the Earth. Since the bottle follows a parabolic trajectory, the objects within also follow a parabolic path and are seen to float as if in a space ship in orbit around the Earth. This demonstration is useful for demonstrating to students that apparent weightlessness is not confined to outer space. The use of toy fish in the experiments makes a connection with the neutral buoyancy training of astronauts. The experiments also exemplify Newton's second law that force is mass times acceleration. [ABSTRACT FROM AUTHOR]

Laguerre spectral approximations play an important role in the development of efficient algorithms for problems in unbounded domains. In this paper, we present a comprehensive convergence rate analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove that Laguerre projection and interpolation methods of degree n converge at the root-exponential rate O(\exp (-2\rho \sqrt {n})) with \rho >0 when the underlying function is analytic inside and on a parabola with focus at the origin and vertex at z=-\rho ^2. As far as we know, this is the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions. Several important applications of our analysis are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules, the scaling factor and the Weeks method for the inversion of Laplace transform, and some sharp convergence rate estimates are derived. Numerical experiments are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Laser wakefield acceleration (LWFA) promises compact accelerators toward the high-energy frontier. However, the approach to the 100 GeV milestone faces the obstacle of the long focal length required for optimal acceleration with high-power lasers, which reaches hundreds of meters for 10–100 PW lasers. The long focal length originates from optimal laser intensity required to avoid nonlinear effects and hence large spot size and Rayleigh length. We propose a "telescope" geometry in which a micro-plasma parabola (MPP) is coupled with a short-focal-length off-axis parabola, minimizing the focal length to the meter range for LWFA under optimized conditions driven by lasers beyond 1 PW. Full-dimensional kinetic simulations demonstrate the generation of a 9 GeV electron bunch within only 1 m optical length—only one-tenth of that required with the conventional approach with the same performance. The proposed MPP provides a basis for the construction of compact LWFAs toward single-stage 100 GeV acceleration with 100 PW class lasers. [ABSTRACT FROM AUTHOR]

A set of exact solutions is suggested for the Poisson equation in a plane region of a certain form for describing a flow of Newtonian incompressible liquid in a channel at low Reynolds numbers. The motion of an ideal liquid in a region bounded by a rotating contour, torsion of a rod, deflection of a membrane at zero displacement on the boundary of its region, and heating of a rod under certain conditions are equivalent to a mathematically formulated problem of such a flow. A possibility is demonstrated to construct approximate solutions to the said problem for certain sections of a channel, having a parabolic segment of the contour, using exact solutions to this problem. Examples are given for the construction of flow regions in a prismatic channel resting on a parabolic contour. [ABSTRACT FROM AUTHOR]

We introduce a double-folded operator that, upon iterative application, generates a dynamical system with two types of trajectories: a cyclic one and, another that grows endlessly on parabolas. These trajectories produce two distinct partitions of the set of lattice points in the plane. Our object is to analyze these trajectories and to point out a few special arithmetic properties of the integers they represent. We also introduce and study the parabolic-taxicab distance, which measures the fast traveling on the steps of the stairs defined by points on the parabolic trajectories whose coordinates are based on triangular numbers. [ABSTRACT FROM AUTHOR]

Test subjects were assessed in a partial gravity environment during parabolic flight while they performed mission-critical activities that challenged their balance and locomotion. These functional activities included rising from a seated position and walking, jumping down, recovering from falls, and maintaining an upright stance. Twelve volunteers were tested during 10 parabolas that produced 0.25×g, 0.5×g, or 0.75×g, and at 1×g during level flight intervals between parabolas. Additionally, 14 other subjects were tested using identical procedures in a 1×g laboratory setting. Partial gravity altered the performance of settling after standing and navigating around obstacles. As gravity levels decreased, the time required to stand up, settle, walk, and negotiate obstacles, and the number of falls increased. Information obtained from these tests will allow space agencies to assess the vestibular, sensorimotor, and cardiovascular risks associated with different levels of partial gravity. [ABSTRACT FROM AUTHOR]

The study aims to introduce some geometric properties of dual Cartan numbers. Since a unit timelike Cartan number with lightlike vector part corresponds to a parabolic rotation around a lightlike axis, it has been shown that the unit dual Cartan number with lightlike dual vector also corresponds to a parabolic rotation and a translation transformation in the Cartan frame. Also, the angle between the directed pseudo‐null lines that tangent the same parabola is defined. Thus, the rotation and translation relationship between two directed pseudo‐null lines is obtained. Also, the E. Study transformation is given for the directed pseudo‐null lines. [ABSTRACT FROM AUTHOR]

We propose novel spectrometer designs that aim to enhance the measured spectral range of ions on a finite-sized detector. In contrast to the traditional devices that use a uniform magnetic field, in which the deflection of particles increases inversely proportional to their momentum, in a gradient magnetic field, the deflection of particles will decrease due to the reduction of the magnetic field along their propagation. In this way, low-energy ions can reach the detector because they are deflected less, compared to the uniform field case. By utilizing a gradient magnetic field, the non-linear dispersion of ions in a homogeneous magnetic field approaches nearly linear dispersion behavior. Nonetheless, the dispersion of low-energy ions, using a dipole field, remains unnecessarily high. In this article, we discuss the employed methodology and present simulation results of the spectrometer with an extended ion spectral range, focusing on the minimum detectable energy (energy dynamic range) and energy resolution. [ABSTRACT FROM AUTHOR]

We analyse 13 yr of Fermi Large Area Telescope Pass 8 events from 127 gamma-ray emitting millisecond pulsars (MSPs) in the energy range 0.1–100 GeV and significantly detect 118 MSPs. We fit the stacked emission with a log parabola (LP) spectral model that we show is preferred to two previously published models. We consider the influence of pulsar properties and observed geometric effects on spectral features by defining energy flux colours for both the individual MSPs, and our stacked model as a baseline. There is no correlation of colours with pulsar luminosity, |$\dot{E}$|⁠ , surface magnetic field, or magnetic impact angle. We also find that pulsar geometry has little effect on the observed gamma-ray spectrum, which is in tension with previous modelling of gamma-ray emission with respect to pulsar geometry. Our LP MSP model is applicable to problems where an ensemble of gamma-ray MSPs is considered, such as that of the Galactic Centre excess or in the case of emission from globular clusters. [ABSTRACT FROM AUTHOR]

The performance of diode temperature sensors depends on their base-depleted region thickness and impurity distribution profile. In this article, we study the boron diffusion process in the thin base region with a thickness of about 1 μm. We show that, for the proposed p+–n–p temperature sensor structure, the impurity distribution in the n region has the form of an asymmetric parabola, with an extreme concentration point shifted towards the "upper" p+ region. [ABSTRACT FROM AUTHOR]

Mineral exploration projects can make considerable use of a variety of geophysical techniques and datasets, including magnetic and gravity data. The interpretation of large quantities of data can be very time consuming, so semi-automatic interpretation techniques are often used to provide initial estimates of the parameters (primarily the location and depth) of the sources of anomalies. Euler deconvolution is a commonly used interpretation method for potential fields which has a number of advantages over many other techniques, such as working in the presence of remanent magnetisation, and not being restricted to a particular model such as a contact. A second-order version of Euler's equation is introduced here, which is much less affected by trends in the data than the standard method and additionally produces depth parabolas, which simplify the interpretation of results. The method was applied to aeromagnetic data from a mineral exploration project in Southern Africa and provided plausible results. [ABSTRACT FROM AUTHOR]

This work is the result of a master's investigation in Brazil, which discusses the teaching of the parabola in the initial training of mathematics teachers. Our theoretical framework addresses the relationship between intuition and the dialectics of the theory of didactic situations, which supports the analysis of the results of this study Its objective was to identify and register categories of intuitive reasoning manifested by the subjects when solving a didactic situation involving the parabola with GeoGebra software contribution. The methodology adopted was didactic engineering, experimented with eight students in initial training at a Brazilian public university, among 6th and 9th undergraduate semesters. The posterior analysis and validation allowed us to verify the need to discuss the parabola, articulating its geometric, algebraic, and analytical views, as well as to reinforce the importance of its teaching with the use of technology. [ABSTRACT FROM AUTHOR]

A passagem bíblica de Lucas 16, 19-31 é uma das mais citadas e discutidas do Novo Testamento. No entanto, quando analisada a partir das relações engendradas na sequência dos fatos narrados, uma série de programas narrativos pressupostos contribui para a elucidação do sentido desse texto. Esta pesquisa, portanto, analisará a referida passagem a partir da perspectiva da semiótica greimasiana. O método pretende desvelar o percurso gerativo de sentido e destacar os elementos intratextuais, que perfazem o trajeto de construção da significação da passagem. Ele permite analisar suas estruturas superficiais e profundas: vai do simples ao complexo, do mais abstrato ao mais concreto. Pode assim evidenciar as oposições semânticas a partir das quais se constitui o sentido do texto e a sua categoria semântica fundamental. O método tem potencial para representar a organização relacional entre as condições sociais dos sujeitos da narrativa. Além disso, fornece um suporte metodológico neutro que evita conclusões antecipadas e tendenciosidades. A análise da intertextualidade completa o processo iniciado com a Semiótica greimasiana. Isso porque uma leitura atenta de Lc16,19-31 não ignorará seu contexto imediato e mais amplo: Lucas e Atos lidos sequencialmente. Situando a parábola nessa narrativa mais ampla, os leitores de Lucas poderiam perceber elementos conectivos, ligando a história contada por Jesus ao que estava acontecendo na igreja primitiva. Assim, na boca de Jesus a história do rico e do mendigo foi contada como uma crítica à avareza, mas, no plano redacional de Lucas foi usada para desafiar os crentes a se envolverem com o cuidado dos pobres.

For any |$0 < \alpha <1$|⁠ , we construct Cantor sets on the parabola of Hausdorff dimension |$\alpha $| such that they are Salem sets and each associated measure |$\nu $| satisfies the estimate |$\left \| \widehat {f d\nu } \right \|_{L^{p}({\mathbb {R}}^{2})} \leq C_{p} \left \| f \right \|_{L^{2}(\nu)}$| for all |$p>6/\alpha $| and for some constant |$C_{p}>0$|⁠ ,which may depend on |$p$| and |$\nu $|⁠. The range |$p>6/\alpha $| is optimal except for the endpoint. The proof is based on the work of Łaba and Wang on restriction estimates for random Cantor sets and the work of Shmerkin and Suomala on Fourier decay of measures on random Cantor sets. They considered fractal subsets of |${\mathbb {R}}^{d}$|⁠ , while we consider fractal subsets of the parabola. [ABSTRACT FROM AUTHOR]

The paper presents a new technique for thermoplastic bending of glass. The technique makes it possible to produce cylindrical surfaces with a guide in the form of a parabola, ellipse, etc. for mirrors in the hard X-ray wavelength range (λ ~ 0.1 nm). Three samples of the surface of an elliptical cylinder were prepared using this technique. The production time for each sample was two days. The deviation of the guide and its local angle from the calculated values for all samples does not exceed Δy = 0.5 μm and Δα = 7 × 10–5 rad, respectively. It is shown that when such surfaces are etched for two hours, their accuracy can be improved by more than two orders of magnitude (Δy = 2 nm, Δα = 5 × 10–7 rad). [ABSTRACT FROM AUTHOR]

In the 3rd century B.C.E., Archimedes wrote his treatise on the quadrature of the parabola, in which he laid out his solution to the problem of finding the area of a parabolic segment. Archimedes attacks the problem from two sides; first, he uses classical mechanics to derive an area formula, and then he uses pure geometry to prove his formula is correct. In this paper, we will present two modern approaches to this problem using calculus. The first approach can be presented to a calculus class that has learned about optimization methods and convergent geometric sequences, and the second can be presented after they have learned the Fundamental Theorem of Calculus. [ABSTRACT FROM AUTHOR]

The problem on the optimum organization of the rectification of a multicomponent mixture in a rectifying column was considered. It is shown that, in the case where heat is supplied to the boiler of the column and is removed from its dephlegmator, the boundary of the realizable-load range of the column represents a parabola convex upwards with two characteristic parameters. The interrelation between the indicated characteristic parameters and the kinetics of the heat- and mass-transfer processes in the column, the composition of the mixture separated in it, and the mixture separation boundary selected was determined. The consequences of the possibility of parametrization of the boundary of the realizable-load range of a rectifying column are discussed. A rule for the choice of the boundary of separating a multicomponent mixture has been formulated. [ABSTRACT FROM AUTHOR]

We consider the Golden Section and Parabola Methods for solving univariate optimization problems. For multivariate problems, we use these methods as line search procedures in combination with well-known zero-order methods such as the coordinate descent method, the Hooke and Jeeves method, and the Rosenbrock method. A comprehensive numerical comparison of the obtained versions of zero-order methods is given in the present work. The set of test problems includes nonconvex functions with a large number of local and global optimum points. Zero-order methods combined with the Parabola method demonstrate high performance and quite frequently find the global optimum even for large problems (up to 100 variables). [ABSTRACT FROM AUTHOR]

We prove an (l², L6) decoupling inequality for the parabola with constant (log R)c. In the appendix, we present an application to the sixth-order correlation of the integer solutions to x² + y² = m. [ABSTRACT FROM AUTHOR]

Rock bridges, also known as locking masses in landslides, affect the three-dimensional (3D) stability and deformation patterns of landslides. However, it is always difficult to simulate rock bridges with continuous grid models in 3D landslides due to their discontinuous deformations. Tizicao landslide, located in Maoxian County, southwest China, is a typical landslide with a super-large rock mass volume of about 1388.2 × 10 4 m 3 and a locking segment. To explore a better rock bridge model used to simulate 3D stability and deformations of the Tizicao landslide, this study introduced three rock bridge models into the FLAC3D program, including the intact rock mass model (IRMM), the Jennings model (JM), and the contact surface model with high strength parameters (CSM-HSP). The CSM-HSP model was eventually used in the FLAC3D program to obtain the 3D deformation characteristics of the landslide. In addition, the two-dimensional (2D) stability of the Tizicao landslide was analyzed using the GeoStudio program. The simulation results indicate that the Tizicao landslide is generally stable under current conditions owing to the existence of the locking segment in its southern front. This inference is consistent with the field deformation and monitoring data. It was found that the general stability and local deformations of the landslide are influenced by the locking segment according to the comparison between the 2D and 3D stability. There was a linear relationship between the locking ratio and the factor of safety (Fos), which applied to the 2D stability analysis of the landslides with a locking segment each, while there existed an approximate quadratic parabola suitable for the 3D stability of the landslides. Finally, this study analyzed the laws of the 3D Fos varying with the locking ratio and strength parameters of the locking masses and the sliding surface. Furthermore, it explored the advantages and disadvantages of the three rock bridge models in the simulation of the 3D stability of landslides with a locking segment. [ABSTRACT FROM AUTHOR]

Finite element (FE) coupled thermal-mechanical analysis is widely used to predict the deformation and residual stress of wire arc additive manufacturing (WAAM) parts. In this study, an innovative single-layer multi-bead profile geometric modeling method through the isosceles trapezoid function is proposed to build the FE model of the WAAM process. Firstly, a straight-line model for overlapping beads based on the parabola function was established to calculate the optimal center distance. Then, the isosceles trapezoid-based profile was employed to replace the parabola profiles of the parabola-based overlapping model to establish an innovative isosceles trapezoid-based multi-bead overlapping geometric model. The rationality of the isosceles trapezoid-based overlapping model was confirmed by comparing the geometric deviation and the heat dissipation performance index of the two overlapping models. In addition, the FE-coupled thermal-mechanical analysis, as well as a comparative experiment of the single-layer eight-bead deposition process show that the simulation results of the above two models agree with the experimental results. At the same time, the proposed isosceles trapezoid-based overlapping models are all straight-line profiles, which can be divided into high-quality FE elements. It can improve the modeling efficiency and shorten the simulation calculation time. The innovative modeling method proposed in this study can provide an efficient and high-precision geometric modeling method for WAAM part FE coupled thermal-mechanical analysis. [ABSTRACT FROM AUTHOR]

We study the motion of a test particle in a conservative force field. In the framework of the 2D inverse problem of Newtonian dynamics, we find 2D potentials that produce a preassigned monoparametric family of regular orbits f (x , y) = c on the x y -plane (where c is the parameter of the family of orbits). This family of orbits can be represented by the "slope function" γ = f y f x uniquely. A new methodology is applied to the basic equation of the planar inverse problem in order to find potentials of a special form, i.e., V = F (x + y) + G (x − y) , V = F (x + i y) + G (x − i y) and V = P (x) + Q (y) , and polynomial ones. According to this methodology, we impose differential conditions on the family of orbits f (x , y) = c. If they are satisfied, such a potential exists and it is found analytically. For known families of curves, e.g., circles, parabolas, hyperbolas, etc., we find potentials that are compatible with them. We offer pertinent examples that cover all the cases. The case of families of straight lines is referred to. [ABSTRACT FROM AUTHOR]

The article offers the force and geometry analysis of the stone block–diamond wire saw system. The friction force of the diamond wire saw in the block is determined at the parabolic adjustment of the wire sawing trajectory. The curves of the sawing force and horizontal coordinates of the blocks are plotted at different values of the parabola focus point. The actual sawing trajectories in monoliths and blocks are described. The load increase factor at the beginning of work is obtained. Using the procedure of the actual sawing trajectory, the strength analysis of the diamond segments of the diamond wire sawing machine is performed, and the stress diagrams in the diamond segments with a sharp and rounded edge are constructed. The maximal stresses in the diamond segment with the rounded edge are plotted as function of the rounding radius. [ABSTRACT FROM AUTHOR]

Weil osteotomy is a proven procedure to restore the harmonic distal parabola of the forefoot. In addition to the proximal displacement of the head in the sagittal plane, a displacement in the transverse plane may be necessary, with the refixation of the displaced metatarsal head historically performed by screw fixation. We aimed to determine the radiological differences among 136 feet of 127 patients with 256 Weil osteotomies retrospectively enrolled and divided into groups with (n = 182) and without (n = 74) screw fixation. Demographic data, radiographic union, pre- and postoperative metatarsal angles, and differences in the dorsoplantar view were evaluated. The mean follow-up period was 3.6 months. The mean preoperative metatarsophalangeal angle was 9.24°, and the mean postoperative angle was 12.99°. The restoration of the transversal alignment plane was equally successful in both groups, with a mean extent of angle correction of 10.58°. No nonunions of the osteotomized metatarsals were observed. The radiographic comparisons revealed no significant difference between the groups (p > 0.05). However, visibility of the joint space of the metatarsophalangeal joint was achieved significantly more often in the group without screw fixation (p < 0.05). In the absence of bony malunion and the satisfactory restoration of a harmonious parabola of the forefoot, apparently there does not appear to be a necessity for regular screw fixation after Weil osteotomy based on the available data from the present study. [ABSTRACT FROM AUTHOR]

This paper presents an analytical method based on the shear flow distribution law to study the shear lag effect of thin-walled single- and double-cell box girders. The first step in this method is to determine the box girder's shear flow distribution. Subsequently, a series of novel improved longitudinal displacement functions mathematically expressed as cubic parabolas are established. The parabolic origin of these functions is located at the zero point of the shear flow corresponding to each plate; the unknown parameters used to describe the function form can be determined according to the shear flow distribution, the continuity conditions, and the axial force balance condition. Then, the variational energy method is adopted to derive the governing differential equations. The shear lag effect in thin-walled single- and double-cell box girders under several boundary conditions and load cases is studied and analytical expressions for the shear lag coefficient are derived. Finally, results obtained using the proposed method are validated via comparison with numerical results. The results show that the proposed method can provide reasonable predictions for the shear lag effect of single- and double-cell box girders, and that this method is more straightforward and practical. In addition, the shear lag coefficients at different webs are not entirely equal, which is related to the distance from the web to the zero point of the shear flow. [ABSTRACT FROM AUTHOR]

The Thomson parabola ion spectrometer is vulnerable to intense electromagnetic pulses (EMPs) generated by a high-power laser interacting with solid targets. A metal shielding cage with a circular aperture of 1 mm diameter is designed to mitigate EMPs induced by a picosecond laser irradiating a copper target in an experiment where additionally an 8-ns delayed nanosecond laser is incident into an aluminum target at the XG-III laser facility. The implementation of the shielding cage reduces the maximum EMP amplitude inside the cage to 5.2 kV/m, and the simulation results indicate that the cage effectively shields electromagnetic waves. However, the laser-accelerated relativistic electrons which escaped the target potential accumulate charge on the surface of the cage, which is responsible for the detected EMPs within the cage. To further alleviate EMPs, a lead wall and an absorbing material (ECCOSORB AN-94) were added before the cage, significantly blocking the propagation of electrons. These findings provide valuable insights into EMP generation in large-scale laser infrastructures and serve as a foundation for electromagnetic shielding design. [ABSTRACT FROM AUTHOR]

In this article we study conics inscribed in a triangle, which appear always in a group of related four. We find conditions that such a group contains simultaneously a rectangular hyperbola and a parabola. These conditions involve the de Longchamps circle of the triangle of reference, which necessarily it is obtuse. [ABSTRACT FROM AUTHOR]

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Due to carrier dynamics, the modulated light field from a directly modulated laser (DML) has an intensity envelope with a certain frequency chirp. When the chirp is linearly mapped into intensity by a frequency discriminator such as an optical filter with a linear edge, the optical field presents a new signal determined by the multiplication operation between the envelope function and the chirp function. Under a triangular drive signal, this process can contribute dark, bright and frequency-doubled bright parabolic waveforms by properly adjusting the filter window. This method is verified by both a theoretical analysis and experimental demonstrations. It not only provides a low-cost and simple scheme to generate parabola signals, but also a new method for arbitrary waveform generation. [ABSTRACT FROM AUTHOR]

We present a higher-order representation of heightfields by constructing unbounding revolved parabolas about every texel of the height texture. These surfaces of revolution do not intersect the interior of the volume defined by the height field. We present a simple generation algorithm and show that these maps can be rendered by computing intersections between lines and parabolas in the plane. We compare its quality and performance with cone step mapping. [ABSTRACT FROM AUTHOR]

In this paper, new results were presented on the symmetry reduction of gas dynamics system of partial differential equations following the general framework of Lev Ovsyannikov's article "The "podmodeli" program, Gas dynamics." We considered the gas dynamics equations with an equation of state prescribing the pressure as the sum of density function and entropy function. This system has a 12-dimensional Lie algebra and we considered its certain three-dimensional subalgebra generated by space translations, Galilean translations and pressure translation. For this subalgebra, the symmetry reduction of the original system leads to a system of ordinary differential equations. We obtained a family of exact solutions for this system, which describes the motion of particles with a linear velocity field and non-homogeneous deformation in the 3D-space. For these solutions, the trajectories of all points are either parabolas or rays. At t = 0 an instantaneous collapse occurs when all of the particles accumulate in a plane with infinitely many particles at every point of the plane. For a fixed period of time, the particles were emitted from the same point on a plane and ended up on the same line. The gas motion was vortex. A one-dimensional subalgebra embedded into three-dimensional subalgebra was considered. The invariants were written in a consistent form. It was shown that the submodel of rank one was embedded in the submodel of rank three. [ABSTRACT FROM AUTHOR]

In this paper, we present the equations of some conic sections and explore their applications in physics and astronomy. We investigate ellipses, parabolas, circles and hyperbolas that are essential in modeling a range of phenomena, from planetary orbits and satellite trajectories to projectile motion and optical systems. By discussing these geometric forms and their relationships with key concepts in physics, we examine how conic sections contribute to our understanding of celestial mechanics, gravitational forces and light reflection. This paper aims to provide a foundation for further research on the significance of conic sections in scientific exploration. [ABSTRACT FROM AUTHOR]