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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]

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]

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]

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]

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]

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.

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]

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]

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]

Copyright of Revista de Ingenieria, Matematicas y Ciencias de la Informacion is the property of Corporacion Universitaria Republicana and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

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]

This article introduces a unique case study involving open curves of parabolic form situated within two-dimensional spaces. It presents a new form of a two-dimensional curve achieved by repositioning the focal point to coincide with the vertex position, resulting in what is termed a Semi-Parabolic Curve (SPC) where the focal point acts as the vertex referred to as the SPC head point. In essence, the SPC represents the path traced by a point on a plane, where its distance from a fixed point (the focus), is always greater than or equal to its distance from a fixed straight line (the directrix). Furthermore, the article provides the coordinate equations that govern the points along these curves. With the potential to pave the way for exploring additional geometric aspects relevant to this class of curves, and to enabling comparative analyses across diverse mathematical and geometric domains, particularly in three-dimensional contexts in the future.

El presente artículo se centra en el fenómeno físico sobre el movimiento de proyectiles, específicamente en el experimento desarrollado por Galileo Galilei, uno de los personajes más famosos y controvertidos de la historia. Este experimento tuvo lugar en un contexto marcado por las ideas de Aristóteles, las cuales se aceptaban como verdades absolutas. El análisis se realiza bajo la determinación de un contexto de significación como elemento fundamental de la Teoría Socioepistemológica de la Matemática Educativa (TSME). Este factor permitirá identificar elementos presentes en dicho fenómeno, los cuales serán fundamentales en la enseñanza de lo cuadrático en el futuro.

Temos como intento, neste artigo, iluminar facetas sobre como relações de sentido podem ser desencadeadas a partir do cotejamento de diferentes versões de um texto; neste caso, das Escrituras Sagradas. Para tanto, torna-se objeto de estudo uma parábola, gênero discursivo encontrado especialmente na esfera discursiva bíblica. Toma-se como referencial postulados advindos da teoria dialógica do discurso, igualmente referenciada como teoria bakhtiniana; são eles: dialogismo, entonação, palavra e estilo. Adicionalmente, apresentam-se métodos e tipos de tradução, embasamento que nos fornece elementos para conduzir este estudo. Como gesto de análise, verificou-se um índice de oscilação no que tange especialmente ao estilo, já que as traduções são pautadas por distintos métodos, e também por seu público leitor projetado. Assim, sentidos potencialmente distintos podem ser reverberados para os leitores de cada versão.

An important role in the roads design is played by plane curves, if the road lies on a plane surface and by space curves, if the road can not be embeded in a plane. This article presents some work done in the two and three dimensional roads design, by using the differential geometry methods. A study of a general case for plane roads is given.

This work, structured in two main parts, is devoted to the parabola topic. In the first, the elements of a parabola are reviewed by linking the definition of intersection of a plane and a cone with a locus of the plane. The necessary transformations for the calculation of the elements as a curve in space are pointed out as well as historical notes and properties of light are included. In the second part, the applications in teaching at a high school level or university courses are explained. A project based on the reflection of light is proposed that seeks to connect various subjects in line with the new educational paradigm of development of key competencies, joining different fields of knowledge. For university courses, applications of linear algebra are obtained in the establishment of relationships between analytical geometry in the rotation and translation of planes, and the dihedral system that is taught in technical drawing subjects. As a conclusion, it is obtained that the teaching of the parabola from different approaches allows a complete learning of diverse fields of knowledge, even of different topics of mathematics, an essential factor in the development of thought at any educational level.

There are inequable force-induced deformation, obvious cutting vibration problems during side milling of low-rigidity frame and beam parts, resulting in unsatisfactory dimensional accuracy and surface quality. Most of these parts are side-milled by end mill, which currently in use do not take into account the workpiece stiffness imparity. Therefore, the concept of gradient helix edge (GHE) is proposed in this research, and its advantages are analyzed. The general mathematical model is given. Furthermore, the parabola is taken as an example for the GHE space equation derived, of which accuracy is verified by numerical simulation. Finally, the comparison experiment with the invariant helix edge end mill was completed. The results showed that the milling force perpendicular to the milling surface of the GHE end mill was smaller and with less fluctuation, which could obtain less force-induced deformation theoretically. The milling process was smoother, and better surface quality was obtained. The excellent performance of the GHE end mill was further verified by milling different features. [ABSTRACT FROM AUTHOR]

We give a concise summary on the technical history of visualizing implicit locus equations in the widely used dynamic geometry system GeoGebra. We mainly focus on the “draggable locus” equation, introduced in GeoGebra 5 (2014), and the dynamic envelopes. We discuss how user interface improvements (the Locus Equation and Envelope tools in GeoGebra Discovery), backend updates (the Giac subsystem) and algebraic experiments (to automatically exclude degeneracy) lead to easy study of caustics of a circle or a parabola in the classroom. [ABSTRACT FROM AUTHOR]

A compact scintillator-based electron–ion Thomson Parabola Spectrometer (ei-TPS) is designed and built up, which is able to measure the spectrum of electron and ion beams simultaneously in the same angular axis and real-time mode. The energy range of electrons is around 0.27–3.8 MeV with a relative energy resolution better than 2.5%, and the energy range of proton is around 0.16–21 MeV with a relative energy resolution better than 4% at the kinetic energy of 1 MeV, which makes it suitable for laser-driven ion acceleration experiments with 100-TW level laser. Detailed mathematical modeling is performed to reveal the dependence of properties, such as energy range and resolution, response, and detection threshold, on various parameters of ei-TPS, which is useful to modify the parameters according to specific requirements of electrons and ions. [ABSTRACT FROM AUTHOR]

A mathematical model of vertical rolling process with the consideration of mechanism displacement is proposed. Using Γ function and parabola function to describe edge deformation, the corresponding 3D velocity field, strain rate field and total power functional are derived. Simultaneously, an analysis method of mechanism deformation is proposed. The deflection of vertical roller and the radial displacement of support bearings are calculated by applying superposition principle and Palmgren's modified formula respectively. The coupling calculation of slab deformation and mechanism displacement is realized through minimizing the total functional and repeated iteration. The accuracy of the presented model is verified by comparison with other models and factory measurements. Subsequently, the influences of main rolling parameters on edge deformation, rolling force and mechanism displacement are analyzed. The proposed model could provide some references for the optimization of vertical rolling process and the improvement of slab quality and yield ratio. [ABSTRACT FROM AUTHOR]

The pericope Mt 25,1-13, of the ten virgins, is one of the most commented texts in the First Gospel. In general, its textual unity is not questioned and this is one of the few consensuses among researchers regarding this pericope. The discussion about its literary genre pits those who defend it as a parable against those who consider it an allegory. This definition is not merely a formality, as it directly influences the possible hermeneutical possibilities for this text. This article therefore looks at the history of the interpretation of Mt 25,1-13. To do this, we offer a segmentation and translation of the text, notes on textual criticism, form criticism and literary genre criticism. This is followed by an exegetical commentary based on the main words in the pericope of Mt 25,1- 13. Finally, the history of the interpretation of the proposed pericope is presented. The methodology used is the Historical-Critical Method and bibliographical research. In this sense, the article demonstrates that the definition of a literary genre does not necessarily imply that there are no elements of another literary genre in the same text. With this, the pericope of Mt 25,1-13, as an open work, is contemplated in its polysemy, obviously without arbitrariness, but within the interpretative limits imposed by the text itself.. [ABSTRACT FROM AUTHOR]

In this paper, exact hydrostatic particulars equations for the centre of buoyancy curve and metacentric locus curve are given for rectangular cross section using quadratic functions. Those equations have not been given for the hyperbola range of the heel angles so far, and here it is done by using basic quadratic functions and their horizontally symmetric immersion shapes, with two new methods defined: 1. Rotation of basic cross section shapes, and 2. Hydrostatic cross section area complement method that uses homothety or scaling properties of emerged and immersed areas of the rectangular cross section. Observed metacentric curve for rectangle consists of semi-cubic parabolas and Lamé curve with 2/3 exponent and negative sign, resulting in the cusp discontinuities in the symmetry of those functions definition. In order to achieve above, two theorems are given: the theorem about scaling using hydrostatic cross section area complement and the theorem about parallelism of centre of buoyancy tangents with waterlines. After non-dimensional bounds are given for the existence of the swallowtail discontinuity of metacentric curve for rectangular cross section in the Part 1 of this paper, the proof of its position in the symmetry of rectangle vertex angle is given in this Part 2 of the paper, thus confirming its position from theory. [ABSTRACT FROM AUTHOR]

Copyright of Muiraquitã: Revista de Letras e Humanidades is the property of Muiraquita and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This work aims to present different demonstrations of the parabola, as well as possibilities of its geometric construction, using geometric design techniques and the GeoGebra dynamic geometry software. The methodology of this work is a basic theoretical research, exploratory type, in which we seek to bring a view about the parabola focused on improving its teaching as mathematical knowledge with the contribution of GeoGebra software. As a result, we bring a set of five constructions made in GeoGebra and available for use, which can be used as a methodological resource by the teacher to work in the classroom. As this work is part of an ongoing master's research, as future perspectives, we aim to develop these constructions in the classroom and collect empirical data for further analysis and discussion.

We know that circular shapes we encounter in daily life may appear to be elliptical from some viewing points. It is reasonable to expect that an ellipse may appear to be circular from certain viewpoints. In this paper, we investigate the locus of viewpoints from which an ellipse appears circular. It will be shown that the locus of viewpoints is a hyperbola passing through the two foci of the ellipse. Conversely, the locus of viewpoints from which a hyperbola looks circular is an ellipse passing through the two foci of the hyperbola. Further, the locus of viewpoints from which a parabola looks circular is itself a parabola, passing through the focus of the original parabola. There is a simple duality between the object to be observed and the observer. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Brush stroke models play an important role in robotic Chinese calligraphy as the basis of calligraphy stroke generation and are helpful for training models to write robotic Chinese calligraphy. In this study, in contrast to most current stroke models that only consider graphic generation features, we propose a novel stroke model based on composite curve and morphological dilation according to the physical characteristics and writing posture of the brush. In the proposed composite-curve-dilation brush stroke model (CCD-BSM), an oblique section of a cone and two tangent parabolas form a basic graphic, which is dilated with a fixed coefficient according to the extrusion diffusion characteristics of brush hairs. The CCD-BSM can simulate the graphics formed by various specifications of the brush touching the paper with various postures. Moreover, the parameters in CCD-BSM are measurable and controllable without any parameter estimation method or a large number of training samples. Compared with real stroke graphics written by robots, the results of several experiments prove that the proposed CCD-BSM can simulate brushstroke graphics well and show that it outperformed state-of-the-art stroke models. Compared with existing methods, the results demonstrate the advantages of our proposed model in terms of high similarity and especially show the robustness and efficacy of the method with measurable and controllable parameters. [ABSTRACT FROM AUTHOR]

"Normal" and "abnormal" are frequently used in surgical planning and to evaluate surgical results of the forefoot. However, there is no objectifiable value of metatarsophalangeal angles (MTPAs) 2–5 in the dorsoplantar (DP) view with which to objectively evaluate lesser toe alignment. We aimed to determine which angles are considered to be "normal" by orthopedic surgeons and radiologists. Thirty anonymized radiographs of feet were submitted twice in randomized order to determine the respective MTPAs 2–5. After six weeks, the anonymized radiographs and photographs of the same feet without apparent affiliation were presented again. The terms "normal," "borderline normal," and "abnormal" were assigned by the observers. Viewers considered MTP-2 alignment from 0° to −20° to be normal, and below −30° abnormal; MTP-3, 0° to −15° to be normal and below −30° abnormal; MTP-4, 0° to −10° normal and below −20° abnormal. Between 5° valgus and 15° varus was the range of MTP-5 recognized as normal. High intra-observer but low interobserver reliability with overall low correlation of clinical and radiographic aspects was observed. The assessment of the terms "normal" or "abnormal" are subject to a high degree of variation. Therefore, these terms should be used cautiously. [ABSTRACT FROM AUTHOR]

Neurons in the cerebral cortex fire coincident action potentials during ongoing activity and in response to sensory inputs. These synchronized cell assemblies are fundamental to cortex function, yet basic dynamical aspects of their size and duration are largely unknown. Using 2-photon imaging of neurons in the superficial cortex of awake mice, we show that synchronized cell assemblies organize as scale-invariant avalanches that quadratically grow with duration. The quadratic avalanche scaling was only found for correlated neurons, required temporal coarse-graining to compensate for spatial subsampling of the imaged cortex, and suggested cortical dynamics to be critical as demonstrated in simulations of balanced E/I-networks. The corresponding time course of an inverted parabola with exponent of χ = 2 described cortical avalanches of coincident firing for up to 5 s duration over an area of 1 mm2. These parabolic avalanches maximized temporal complexity in the ongoing activity of prefrontal and somatosensory cortex and in visual responses of primary visual cortex. Our results identify a scale-invariant temporal order in the synchronization of highly diverse cortical cell assemblies in the form of parabolic avalanches. The diversity of synchronized neuronal groups provides a challenge for brain theories. Here, the authors report that group size grows quadratically with duration in line with predictions for neuronal avalanches and brain dynamics being critical. [ABSTRACT FROM AUTHOR]

In complex underwater noise environment, target detection, recognition, and tracking are proceeded through the frequency spectrum analysis of the signals received by sonar systems, in which the lofargram plays a major role. Typically, if the distance of the target experiences far‐near‐far variation, there will be parabolic interference striations presented in the lofargram. However, the existing striations extraction methods sometimes lack objectivity and fail to extract the wide striations accurately. In this paper, a novel method for wide type interference striations extraction is developed based on efficient decomposition and ensemble clustering. To obtain valuable information, the lofargram is decomposed into smooth background, striations area, and noise, then a multi‐phase ensemble clustering algorithm is employed to extract parabolas from the decomposed striations area pixels. Experimental results on simulated and real‐life datasets exhibit the effectiveness of the proposed method, and verify that it has comparable performance with prevalent Hough transform while extracting striations. [ABSTRACT FROM AUTHOR]

We prove sharp bounds for the size of superlevel sets {x...ℝ²:/f(x)/>α}, where α > 0 and f: ℝ² → ... is a Schwartz function with Fourier transform supported in an R-1-neighborhood of the truncated parabola ℙ¹. These estimates imply the small cap decoupling theorem for ℙ¹ of Demeter, Guth, and Wang (2020) and the canonical decoupling theorem for ℙ¹ of Bourgain and Demeter (2015). New (lq, Lp) small cap decoupling inequalities also follow from our sharp level set estimates. [ABSTRACT FROM AUTHOR]

At the beginning of the naval architecture theory, in the 18th century, Bouguer and Euler set the foundations of naval architecture with the centre of buoyancy and metacentric curve definition. After that, in 20th century, it is determined from bifurcation and catastrophe theory developed by Thom, and its application for ships in works of Zeeman, Stewart and others, that the centre of buoyancy curve for the rectangular cross section consists of parabola and hyperbola equations, but no exact equations are given for the hyperbola segment of that curve. Therefore, the hyperbola segment of the centre of the buoyancy curve is re-examined in this paper with emphasis on belonging metacentric locus curve as the evolute of the centre of the buoyancy curve. The observed metacentric curve consists of semi-cubic parabolas and Lamé curves with 2/3 exponent and negative sign, resulting in the cusp discontinuities in the symmetry of functions definition. Belonging swallowtail discontinuity in the hyperbola range between two heel angles of the rectangular cross section deck immersion/bottom emersion angles is examined, depending on existence of extremes of belonging hyperbola curve. After that, the single condition for hyperbola extreme the existence is given with the belonging new lower and upper non-dimensional bounds of rectangle cross section dimensions. [ABSTRACT FROM AUTHOR]

Introduction: In this work, we use simulated data to quantify the different failure mechanisms of a previously presented low-cost jump height measurement system, based on widely available consumer smartphone technology. Methods: In order to assess the importance of the different preconditions of the jump height measurement algorithm, we generate a synthetic dataset of 2000 random jump parabolas for 2000 randomly generated persons without real-world artifacts. We then selectively add different perturbations to the parabolas and reconstruct the jump height using the evaluated algorithm. The degree to which the manipulations influence the reconstructed jump height gives us insights into how critical each precondition is for the method’s accuracy. Results: For a subject-to-camera distance of 2.5 meters, we found the most important influences to be tracking inaccuracies and distance changes (non-vertical jumps). These are also the most difficult factors to control. Camera angle and lens distortion are easier to handle in practice and have a very low impact on the reconstructed jump height. The intraclass correlation value ICC(3,1) between true jump height and the reconstruction from distorted data ranges between 0.999 for mild and 0.988 for more severe distortions. Conclusion: Our results support the design of future studies and tools for accurate and affordable jump height measurement, which can be used in individual fitness, sports medicine, and rehabilitation applications.

The Thomson parabola spectrometer (TPS) [1] is a well-known, universal diagnostic tool that is widely used in laser plasma experiments to measure the parameters of accelerated ions. In contrast to other popular ion diagnostics, such as semiconductor detectors or ion collectors, the TPS is not greatly affected by electromagnetic pulses generated during high-power laser interaction with matter and can be tuned to acquire data in various energy ranges of accelerated ions, depending on the goal of the experiment. Despite the many advantages of this diagnostic device, processing the collected data is a difficult task and requires a lot of caution during interpretation of gathered results. In this work, we introduce the basic principles of operation and data analysis based on the numerical tool created specifically for the TPS designed at the Institute of Plasma Physics and Laser Microfusion, present a range of data obtained during various recent experiments in which our TPS was used, and highlight the difficulties in data analysis depending on the purpose of the experiment and the experimental setup. [ABSTRACT FROM AUTHOR]

Clarifying the water invasion rule of edge and bottom water reservoirs can adjust the reservoir development mode and improve the recovery factor of edge and bottom water reservoirs in a timely manner. Influenced by the size of a reservoir water body, energy intensity and reservoir seepage capacity, the change model of reservoir water influx basically belongs to the exponential growth model of the GM (1,1) model or the self-constraint growth model of the logistic model. The above two models are used to predict and analyze the water inflow of edge and bottom water reservoirs, respectively, and it is found that the change in water inflow of the reservoir with sufficient edge and bottom water energy is more consistent with the prediction results of the GM (1,1) model, but it has a large error compared to the prediction results of the logistic model. The change in water influx in the reservoir with insufficient edge and bottom water energy is consistent with the prediction results of the logistic model and GM (1,1) model. The research shows that the strength of edge and bottom water energy of the reservoir can be determined by analyzing the error of the logistic model in predicting water influx. If we focus on the change in reservoir water influx, the improved GM (1,1) model formed by a Newton parabola interpolation polynomial is used to optimize its background value, which can further improve the prediction accuracy and reduce the prediction error of water inflow of edge and bottom water reservoirs. The method in this paper has certain reference significance for studying the water invasion rule and energy intensity of edge and bottom water reservoirs. [ABSTRACT FROM AUTHOR]

Purpose: The aim of this study was to objectively detect simulated tooth ankylosis using a novel method involving cone-beam computed tomography (CBCT). Materials and Methods: Tooth ankylosis was simulated in single-rooted human permanent teeth, and CBCT scans were acquired at different current levels (5, 6.3, and 8 mA) and voxel sizes (0.08, 0.125, and 0.2). In axial reconstructions, a line of interest was perpendicularly placed over the periodontal ligament space of 21 ankylosed and 21 non-ankylosed regions, and the CBCT grey values of all voxels along the line of interest were plotted against their corresponding X-coordinates through a line graph to generate a profile. The image contrast was increased by 30% and 60% and the profile assessment was repeated. The internal area of the resulting parabolas was obtained from all images and compared between ankylosed and non-ankylosed regions under different contrast enhancement conditions, voxel sizes, and mA levels using multi-way analysis of variance with the Tukey post hoc test (α=0.05). Results: The internal area of the parabolas of all non-ankylosed regions was significantly higher than that of the ankylosed regions (P<0.05). Contrast enhancement led to a significantly greater internal area of the parabolas of nonankylosed regions (P<0.05). Overall, voxel size and mA did not significantly influence the internal area of the parabolas (P>0.05). Conclusion: The proposed novel method revealed a relevant degree of applicability in the detection of simulated tooth ankylosis; increased image contrast led to greater detectability. [ABSTRACT FROM AUTHOR]

Rock bridges, also known as locking rock masses in landslides, influence the three-dimensional (3D) stability and deformation patterns of landslides. However, it is always difficult to simulate rock bridges with continuous grid models in three-dimensional landslides due to their discontinuous deformation. Tizicao landslide, located in Maoxian County, Southwestern China, is a typical landslide with a super-large rock mass volume of about 1,3 8 8.2 x 104 m3 and characterized by a locking segment. To explore a better rock bridge model to simulate 3D stability and deformation behaviors of the Tizicao landslide, three rock bridge models were introduced 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. Also, the two-dimensional (2D) stability of the Tizicao landslide was analyzed using the GeoStudio program. The simulation results indicate that the Tizicao landslide is stable overall under the current conditions, owing to the existence of the locking segment in the southern front, which is consistent with the field deformations and monitoring data. The overall stability and the local deformation of the landslide are found to be influenced by the locking segment when compared to 2D and 3D stability. There is a linear relationship between the locking ratio and the factor of safety (Fos) for the 2D landslide with a locking segment, while an approximate quadratic parabola for the 3D stability. Finally, the laws of the 3D Fos varying with locking ratios and strength parameters of the locking masses and the sliding surface were analyzed. The advantages and disadvantages of the three rock bridge models in simulating the 3D stability of landslides with a locking segment were also discussed. [ABSTRACT FROM AUTHOR]

We revisit in this note the Hénon's isochrone problem. By using the standard Abel inversion technique for one-dimensional motion, we recover in a simple way the Hénon's parabolae and get all isochrone central potentials under mild smoothness assumptions on the potential function. Our approach also allows us to conclude that isochronous radial periods with explicit energy dependence are necessarily Keplerian, i.e., T2 ∝ |E|−3, and that their corresponding orbits can be easily integrated by mapping them into the usual Kepler problem. It can also be employed to study some other inverse central-force problems and, in particular, it provides a proof of Bertrand's theorem. [ABSTRACT FROM AUTHOR]

Convex polygonal lines with vertices in Z + 2 and endpoints at 0 = (0 , 0) and n = (n 1 , n 2) → ∞ , such that n 2 / n 1 → c ∈ (0 , ∞) , under the scaling n 1 − 1 , have limit shape γ * with respect to the uniform distribution, identified as the parabola arc c (1 − x 1) + x 2 = c . This limit shape is universal in a large class of so-called multiplicative ensembles of random polygonal lines. The present paper concerns the inverse problem of the limit shape. In contrast to the aforementioned universality of γ * , we demonstrate that, for any strictly convex C 3 -smooth arc γ ⊂ R + 2 started at the origin and with the slope at each point not exceeding 90 ∘ , there is a sequence of multiplicative probability measures P n γ on the corresponding spaces of convex polygonal lines, under which the curve γ is the limit shape. [ABSTRACT FROM AUTHOR]

As a weak link between aggregate and mortar in concrete, interfacial transition zone (ITZ) usually plays a key role in concrete fracture. To investigate the tensile fracture property of concrete affected by the mechanical properties of ITZ numerically, the geometrical models of heterogeneous concrete were established with the parameterization modeling. They include three phases, namely, mortar, ITZ, and randomly distributed aggregates with distinct sizes and orientations. The cracking behaviors of mortar and ITZ were characterized by the bilinear cohesive zone constitutive model. Based on the experiments, the mechanical properties of ITZ were mediated by changing the water–cement ratio of mortar, the aggregate surface roughness and the content of silica fume in interfacial agent. A series of numerical simulations were conducted on the concrete models in tension after the numerical modeling method was validated. The macroscopic tensile fracture properties of concrete were quantitatively connected with some microscopic variables, including the water–cement ratio of mortar, the aggregate surface roughness and the silica fume content in interfacial agent. It was found that the tensile fracture properties of concrete have negative linear correlations with the water–cement ratio of mortar, while the effects of the aggregate surface roughness and the silica fume content in interfacial agent are very complex. The tensile fracture mechanical properties of concrete have a bilinear relationship with the aggregate surface roughness and an approximate quadratic parabola relationship with the content of silica fume in the interfacial agent. This study is beneficial to improve the fracture resistance of concrete by some interface handling measures. [ABSTRACT FROM AUTHOR]

This work is aimed at improving metrological characteristics and expanding the areas of application of optical thermometry of opaque objects, including the method of two-color compensative pyrometry with an apriori averaged adjustment previously developed by the authors. A nonlinearity equation has been obtained that connects the nonlinearity coefficient of spectral distribution of directional emissivity on an average wavelength of the registered radiation with a value of spectral directional emissivity on one of boundary wavelengths via the measured brightness temperature on three wavelengths. It has been established that the number of numerical solutions of the equation at various qualitative and quantitative characteristics of spectral distributions is from 1 to 3. To determine a correct solution, analytical and algorithmic methods of its identification have been proposed. On the basis of the equation, linear, two-range, and parabolic methods of determining spectral directional emissivity have been developed. It has been established that the methodological errors of temperature measurements by the two-range method are 5–7 times lower than the errors of the linear method. For significant nonlinearities, the parabolic method errors are 1.13–1.24 times lower than the linear method errors. In this case, for the two-range method, the combined errors for two-color compensative pyrometry are lower than the errors for spectral ratio pyrometry and for energy pyrometry 110 and 7 times respectively. [ABSTRACT FROM AUTHOR]

Circulating fluidized bed (CFB) risers using Group A particles have traditionally, mostly, been considered to operate in the fast fluidization regime, which consists of a core‐annulus flow profile with solids refluxing in the annulus layer. High gas and solids flow riser studies, however, suggest the existence of additional types of flow behaviours. Therefore, more studies are still needed to help clear uncertainties of local solids flow patterns in CFB risers of Group A particles, especially at gas and solids flow rates at or near those in commercial units. In this study, riser density and local solids flux profiles were measured in 0.3 m diameter risers of three CFB units at gas velocities of 9–16 m/s and solids fluxes of up to 700 kg/s · m2. A variety of radial solids flux profiles were obtained, including a parabolic profile with a peak flux at about the radial centre, a nearly flat profile across the riser cross‐section and an inverted parabola with peak upwards flux near the wall. At high gas velocity and solids flux, risers have no solids downflow at the wall. Multiple fluidization regimes were found to exist in the riser. The bottom dense part of the riser was in the dense suspension upflow regime, and the dilute upper part was in the dilute pneumatic transport regime. With commercial fluid catalytic cracking (FCC) risers operating at nearly similar conditions as tested here, it is likely that they also have one or both fluidization regimes and not the traditional fast fluidization regime. The data in this study fitted well on the Kim et al. fluidization regime map. [ABSTRACT FROM AUTHOR]

We applied a method of symmetry reduction to the gas dynamics equations with a special form of the equation of state. This equation of state is a pressure represented as the sum of a density and an entropy functions. The symmetry Lie algebra of the system is 12-dimensional. One, two and three-dimensional subalgebras were considered. In this article, four-dimensional subalgebras are considered for the first time. Specifically, invariants are calculated for 50 four-dimensional subalgebras. Using invariants of one of the subalgebras, a symmetry reduction of the original system is calculated. The reduced system is a partially invariant submodel because one gas-dynamic function cannot be expressed in terms of the invariants. The submodel leads to two families of exact solutions, one of which describes the isochoric motion of the media, and the other solution specifies an arbitrary blow-up surface. For the first family of solutions, the particle trajectories are parabolas or rays; for the second family of solutions, the particles move along cubic parabolas or straight lines. From each point of the blow-up surface, particles fly out at different speeds and end up on a straight line at any other fixed moment in time. A description of the motion of particles for each family of solutions is given. [Display omitted] • Method of symmetry reduction is applied to the gas dynamics equations. • Equation of state is a pressure being the sum of a density and an entropy functions. • The symmetry Lie algebra of the system is 12-dimensional. • In this article, four-dimensional subalgebras are considered for the first time. • Symmetry reduction of the original system is calculated from the subalgebra. • The reduced system leads to two families of exact solutions. • A description of the motion of particles for each family of solutions is given. [ABSTRACT FROM AUTHOR]