Representation Of Curves Research Articles

A Novel Approximation Method for Solving Ordinary Differential Equations Using the Representation of Ball Curves

Numerical methods are frequently developed to investigate concepts for approximately solving ordinary differential equations (ODEs). To achieve minimal error and higher accuracy in approximate solutions, researchers have focused on developing algorithms using various numerical techniques. This study proposes the application of Ball curves, specifically the Said–Ball curve, for estimating solutions to higher-order ODEs. To obtain the best control points of the Said–Ball curve, the least squares method is used. These control points are calculated by minimizing the residual error through the sum of the squares of the residual functions. To demonstrate the proposed method, several boundary value problems are presented, and their performance is compared with existing methods in terms of error accuracy. The numerical results indicate that the proposed method improves error accuracy compared to existing studies, including those employing Bézier curves and the steepest descent method.

Measles Among the Foreign-Born Population Residing in Spain, 2014-2022: Missed Opportunities for Vaccination.

Spain has been in a measles elimination phase since 2014. No evidence exists about the distribution of measles cases among the population born outside Spain. The aim of this study was thus to describe the epidemiological situation of measles, stratified by place of birth, during the post-elimination period in Spain. This is a retrospective study of confirmed measles cases reported to RENAVE between 2014 and 2022. A descriptive analysis of case characteristics (sex, age group, vaccination status, imported case) was performed, was well as an analysis of temporal trends and geographic distribution in measles incidence rate (IR; cases/million inhabitants). All analyses were stratified by place of origin (Spain born vs. born outside Spain). We then performed a sensitivity analysis of those born outside Spain, with the representation of Kaplan-Meier curves taking into account the year of arrival in the country until the onset of measles. Between 2014 and 2022, 951 measles cases were reported in Spain (overall IR: 2.3). Among these, 18.6% (177 cases, IR: 3.0) were born outside Spain. The IRs show differences (p < 0.001) in terms of distribution by age group and origin. By age group, children under 5 years had the highest IR, but adults aged 30 years and older reported the highest proportion of cases. The incidence rate ratio (IRR) was 5-fold higher among foreign-born children under 5 years than among native-born children. The measles time trend shows the highest peak in 2019 for foreign-born and native-born (IR: 8.6 and 5.4, respectively), consistent with the European-wide scenario, while only one case of measles was reported in 2022. Geographical variability in incidence rates by region was observed: Catalonia and the Valencian Community accumulated the highest proportion of cases throughout the study period. Among those born outside Spain, the median time from arrival to onset of rash was 6 years. The incidence of measles is 40% higher in Spain's foreign-born population than in its native-born population. Taking into account the increasing migrant population in Spain, we consider that public health efforts need to be directed towards susceptible groups of people. In this context of advanced elimination, specific interventions for identifying and attending the most vulnerable populations should be designed and implemented.

Topological Properties of the Intersection Curves Between a Torus and Families of Parabolic or Elliptical Cylinders

This paper reports the research work carried out with the goal of geometrically and algebraically describing, as well as topologically classifying, the curves resulting from the intersection of a torus with families of parabolic and elliptical cylinders within a purely Euclidean framework. The parabolic cylinders under analysis have generatrices parallel to the axis of the torus, whereas the elliptical cylinders, centered at the same point as the torus, have axes either aligned with or orthogonal to the torus’s axis. For the topological classification of these intersection curves, we consider their number of connected components and self-intersection points. GeoGebra, which was used to create the 3D visual geometric representations of the intersection curves, and Maple, which was used to perform the essential symbolic algebraic calculations, were critical computational tools in the development of this work. Theoretical and computational approaches are interwoven throughout this study, with the computational work serving as the foundation for exploration and providing insights that contributed to the theoretical validation of the results revealed through GeoGebra simulations.

Enhanced curve representations using piecewise non-linear binary subdivision scheme

This study presents a piecewise non-linear binary subdivision scheme based on a four-point linear binary subdivision scheme. This scheme is first transformed into a piecewise linear subdivision scheme and then into a piecewise non-linear subdivision scheme. The non-linear subdivision approach removes artifacts and distortions from limit curves, producing better continuous curves. The study demonstrates that this technique is a workable technique for producing smooth curves, especially when avoiding bends, distortion, and oscillation.

AC Electric Conductivity of High Pressure and High Temperature Formed NaFePO4 Glassy Nanocomposite.

Olivine-like NaFePO4 glasses and nanocomposites are promising materials for cathodes in sodium batteries. Our previous studies focused on the preparation of NaFePO4 glass, transforming it into a nanocomposite using high-pressure-high-temperature treatment, and comparing both materials' structural, thermal, and DC electric conductivity. This work focuses on specific features of AC electric conductivity, containing messages on the dynamics of translational processes. Conductivity spectra measured at various temperatures are scaled by apparent DC conductivity and plotted against frequency scaled by DC conductivity and temperature in a so-called master curve representation. Both glass and nanocomposite conductivity spectra are used to test the (effective) exponent using Jonscher's scaling law. In both materials, the values of exponent range from 0.3 to 0.9, with different relation to temperature. It corresponds to the electronic conduction mechanism change from low-temperature Mott's variable range hopping (between Fe2+/Fe3+ centers) to phonon-assisted hopping, which was suggested by previous DC measurements. Following the pressure treatment, AC conductivity activation energies were reduced from EAC≈0.40 eV for glass to EAC≈0.18 eV for nanocomposite and are lower than their DC counterpart, following a typical empirical relation with the value of the exponent. While pressure treatment leads to a 2-3-orders-of-magnitude rise in the AC and apparent DC conductivity due to the reduced distance between the hopping centers, a nonmonotonic relation of AC power exponent and temperature is observed. It occurs due to the disturbance of polaron interactions with Na+ mobile ions.

The canonical representation of the Drinfeld curve

AbstractIf is a smooth projective curve over an algebraically closed field and is a group of automorphisms of , the canonical representation of is given by the induced ‐linear action of on the vector space of holomorphic differentials on . Computing it is still an open problem in general when the cover is wildly ramified. In this paper, we fix a prime power , we consider the Drinfeld curve, that is, the curve given by the equation over together with its standard action by , and decompose as a direct sum of indecomposable representations of , thus solving the aforementioned problem in this case.

Relation between the minimal Lorentz surfaces in ℝ24 and ℝ13

In this paper, we give Weierstrass-type representation formulas for the null curves and for the minimal Lorentz surfaces in the Minkowski 3-space [Formula: see text] using real-valued functions. Applying the Weierstrass-type representations for the null curves, we find a correspondence between the null curves in [Formula: see text] and the pairs of null curves in [Formula: see text]. Based on this correspondence, we obtain a relation between the minimal Lorentz surfaces in [Formula: see text] and the pairs of minimal Lorentz surfaces in [Formula: see text]. As an application, we give examples showing how we can obtain a minimal Lorentz surface in [Formula: see text] using a pair of minimal Lorentz surfaces in [Formula: see text].

Conversion from NURBS to Bézier representation

With the help of the Cox-de Boor recursion formula and the recurrence relation of the Bernstein polynomials, two categories of recursive algorithms for calculating the conversion matrix from an arbitrary non-uniform B-spline basis to a Bernstein polynomial basis of the same degree are presented. One is to calculate the elements of the matrix one by one, and the other is to calculate the elements of the matrix in two blocks. Interestingly, the weights in the two most basic recursion formulas are directly related to the weights in the recursion definition of the B-spline basis functions. The conversion matrix is exactly the Bézier extraction operator in isogeometric analysis, and we obtain the local extraction operator directly. With the aid of the conversion matrix, it is very convenient to determine the Bézier representation of NURBS curves and surfaces on any specified domain, that is, the isogeometric Bézier elements of these curves and surfaces.

A novel prediction score determining individual clinical outcome 3 months after juvenile stroke (PREDICT-score)

BackgroundJuvenile strokes (< 55 years) account for about 15% of all ischemic strokes. Structured data on clinical outcome in those patients are sparse. Here, we aimed to fill this gap by systematically collecting relevant data and modeling a juvenile stroke prediction score for the 3-month functional outcome.MethodsWe retrospectively integrated and analyzed clinical and outcome data of juvenile stroke and TIA patients treated at the LMU University Hospital, LMU Munich, Munich. Good outcome was defined as a modified Rankin Scale of 0–2 or return to baseline of function. We analyzed candidate predictors and developed a predictive model. Predictive abilities were inspected using Area Under the ROC curve (AUROC) and visual representation of the calibration. The model was validated internally.Results346 patients were included in the analysis. We observed a good outcome in n = 293 patients (84.7%). The prediction model for an unfavourable outcome had an AUROC of 89.1% (95% CI 83.3–93.1%). The model includes age NIHSS, ASPECTS, blood glucose and type of vessel occlusion as predictors for the individual patient outcome.ConclusionsHere, we introduce the highly accurate PREDICT-score for the 3-month outcome after juvenile stroke derived from clinical routine data. The PREDICT-score might be helpful in guiding individual patient decisions and designing future studies but needs further prospective validation which is already planned.Trial registration The study has been registered at https://drks.de (DRKS00024407) on March 31, 2022.

Electro-mechanical coupling isogeometric analysis of static characteristics in piezoelectric composite materials based on asymptotic homogenization method

Accurate mechanical analysis is essential for reliable utilization of piezoelectric composite materials (PCMs). Isogeometric analysis (IGA) of PCMs (termed PCMIGA) based on the asymptotic homogenization method (AHM) is presented in this study and employed to investigate the static mechanical characteristics of PCMs. PCMIGA provides accurate curve representation and shorter preprocessing time, and thus demonstrates both precision and efficiency. First, AHM is utilized to calculate the effective parameters of PCMs at different volume fractions. Next, these effective parameters are combined with the basic equations and boundary conditions of PCMs to derive equations of PCMIGA based on AHM. Finally, the results from several numerical examples are compared with the reference solution to validate the convergence and precision. PCMIGA is proven to be a reliable and accurate method for analyzing the mechanical properties of PCMs.

Dense Continuous-Time Optical Flow From Event Cameras.

We present a method for estimating dense continuous-time optical flow from event data. Traditional dense optical flow methods compute the pixel displacement between two images. Due to missing information, these approaches cannot recover the pixel trajectories in the blind time between two images. In this work, we show that it is possible to compute per-pixel, continuous-time optical flow using events from an event camera. Events provide temporally fine-grained information about movement in pixel space due to their asynchronous nature and microsecond response time. We leverage these benefits to predict pixel trajectories densely in continuous time via parameterized Bézier curves. To achieve this, we build a neural network with strong inductive biases for this task: First, we build multiple sequential correlation volumes in time using event data. Second, we use Bézier curves to index these correlation volumes at multiple timestamps along the trajectory. Third, we use the retrieved correlation to update the Bézier curve representations iteratively. Our method can optionally include image pairs to boost performance further. To the best of our knowledge, our model is the first method that can regress dense pixel trajectories from event data. To train and evaluate our model, we introduce a synthetic dataset (MultiFlow) that features moving objects and ground truth trajectories for every pixel. Our quantitative experiments not only suggest that our method successfully predicts pixel trajectories in continuous time but also that it is competitive in the traditional two-view pixel displacement metric on MultiFlow and DSEC-Flow. Open source code and datasets are released to the public.

InceptCurves: curve reconstruction using an inception network

Curve reconstruction is a fundamental task in many visual computing applications. In this paper, a data-driven approach for curve reconstruction is proposed. We present an inception layered deep neural network structure, capable of learning simultaneously the number of control points and their positions in order to reconstruct the curve. To train the network, a large set of general synthetic data is generated. The reconstructed uniform B-spline closely approximates any arbitrary input curve, with or without intersections. Because the network predicts the number of control points required for the B-spline reconstruction, redundancy is reduced in the curve representation. We demonstrate our approach on various examples.

3D Tortuosity computation as a shape descriptor and its application to brain structure analysis

In this study, we propose a novel method for quantifying tortuosity in 3D voxelized objects. As a shape characteristic, tortuosity has been widely recognized as a valuable feature in image analysis, particularly in the field of medical imaging. Our proposed method extends the two-dimensional approach of the Slope Chain Code (SCC) which creates a one-dimensional representation of curves. The utility of 3D tortuosity (τ3D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _{3D}$$\\end{document}) as a shape descriptor was investigated by characterizing brain structures. The results of the τ3D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _{3D}$$\\end{document} computation on the central sulcus and the main lobes revealed significant differences between Alzheimer’s disease (AD) patients and control subjects, suggesting its potential as a biomarker for AD. We found a p<0.05\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p<0.05$$\\end{document} for the left central sulcus and the four brain lobes.

A comparative study of analytical representations of potential energy curves for CO in its ground electronic state

AbstractIn this work, a comparative study of four functional forms used to represent potential energy curves (PECs) is presented. The starting point is the Hulburt‐Hirschfelder, followed by the Extended Rydberg potential function, ending with two of the most recent potentials presented in the literature for diatomic systems: the Araújo‐Ballester potential and the Improved Extended Lennard‐Jones potential. The chosen potentials have in common the fact that all their parameters are algebraically calculated, without any fitting procedure, and all of them have direct dependence on Dunham's parameters. The mathematical behavior of these functions for the short‐ and long‐range regions is discussed. As study case, the diatomic system in its ground electronic state was selected. To quantify the accuracy of the potential energy functions, the least‐squares Z‐test method, proposed by Murrell and Sorbie, is used. Furthermore, the main spectroscopic constants and vibrational energy levels are calculated and compared for all potentials.

Atmospheric Retrievals of the Phase-resolved Spectra of Irradiated Brown Dwarfs WD-0137B and EPIC-2122B

We present an atmospheric retrieval analysis of Hubble Space Telescope/Wide Field Camera 3/G141 spectroscopic phase curve observations of two brown dwarfs, WD-0137B and EPIC-2122B, in ultrashort period orbits around white dwarf hosts. These systems are analogous to hot and ultra-hot Jupiter systems, enabling a unique and high-precision comparison to exoplanet systems. We use the PHOENIX Exoplanet Retrieval Algorithm retrieval suite to test various analysis setups, including joint-phase retrievals, multiple temperature structures, and nonuniform abundances. We find that WD-0137B has a dayside that closely resembles that of other ultra-hot Jupiters with inverted temperature structures and H− opacity, but quickly transitions to a mostly noninverted temperature structure on the nightside. Meanwhile, EPIC-2122B’s atmosphere remains inverted at all constrained longitudes, with dominant H− opacity. Retrievals with multiple temperature profiles and nonuniform vertical abundances were generally not statistically justified for this data set, but retrievals with dayside-dilution factors were found to be justified. Retrieving all phases simultaneously with a linear combination of a dayside and nightside atmosphere was found to be an adequate representation of the entire phase curve once a longitudinal temperature gradient free parameter was included in the retrieval. Comparing to global circulation models, we attribute behavior in the 1D retrievals to the inclined viewing geometry of the systems, which results in always-visible irradiated and inverted portions of the atmosphere contaminating spectra measured from the nightside hemisphere. This study sheds light on the similarities between these irradiated brown dwarf systems and hot and ultra-hot Jupiters, but also their unique differences, including the influence of the inclined viewing geometry.

On Intersections of B-Spline Curves

Bézier and B-spline curves are foundational tools for curve representation in computer graphics and computer-aided geometric design, with their intersection computation presenting a fundamental challenge in geometric modeling. This study introduces an innovative algorithm that quickly and effectively resolves intersections between Bézier and B-spline curves. The number of intersections between the two input curves within a specified region is initially determined by applying the resultant of a polynomial system and Sturm’s theorem. Subsequently, the potential region of the intersection is established through the utilization of the pseudo-curvature-based subdivision scheme and the bounding box detection technique. The projected Gauss-Newton method is ultimately employed to efficiently converge to the intersection. The robustness and efficiency of the proposed algorithm are demonstrated through numerical experiments, demonstrating a speedup of 3 to 150 times over traditional methods.

Insights on Stability Constants and Structures of Complexes between Coumarin Derivatives and Pb(II) in Aqueous Media.

In the frame of a systematic study on the sequestering ability of natural antioxidants towards metal cations, here the complexation of coumarin-3-carboxilic acid (HCCA) with Pb(II) and the overall stability constants of the resulting complexes, at 37 °C and in 0.16 M NaClO4, are discussed. Reaction of Pb(ClO4)2 with HCCA in an aqueous medium at a pH range from 2 to 6 and various ratios (1:1-1:10) yielded the Pb-CCA complexes, which were characterized spectrometrically by laser desorption ionization mass spectrometry (LD-MS). LD-MS has provided the composition and structure of Pb-CCA species according to the speciation model proposed on the basis of the potentiometric data. The graphic representation of the complex's concentration curves is given by the distribution diagram, which provides a whole depiction of the species present in the solution at the selected pH ranges.

A New Representation for Slant Curves in Sasakian 3-Manifolds

In this paper, we define a new kind of curve called $N$-slant curve whose principal normal vector field makes a constant angle with the Reeb vector field $\xi$ in Sasakian $3$-manifolds. Then, we give some characterizations of $N$-slant curves in Sasakian $3$-manifolds and we obtain some properties of the curves in $\mathbb{R}^{3}(-3)$. Moreover, we investigate the conditions of $C$-parallel and $C$-proper mean curvature vector fields along $N$-slant curves in Sasakian $3$-manifolds. Finally, we study $N$-slant curves of type $AW(k)$ where k=1,2 or 3.

Proposed Improvements to AISC 360-22 Chapter F for Built-Up I-Section Members

Recent analytical and experimental data has shown that the AISC 360–22 Ch. F lateral-torsional buckling (LTB) resistance equations significantly overpredict the strengths of a wide range of built-up I-girders. These overpredictions are mainly attributed to unaccounted-for yielding effects and an insufficient characterization of the plateau strengths by the AISC 360–22 compact- and noncompact-web limits. The objective of the current study is to present and justify the implementation of a new set of recommended AISC 360 Ch. F provisions (AISC 360 Rec.) for built-up I-girders. These recommended provisions were developed by making six targeted improvements to AISC 360–22, which consist of changes to the web classification limits, the inelastic/elastic LTB strength threshold, the handling of moment gradient, and the representation of the LTB strength curve itself. Two independent studies evaluate the performance of the AISC 360 Rec. equations. In the first study, professional factors (Mtest/Mn) were calculated and statistically evaluated using a newly updated LTB experimental database. The second study compares the LTB strength curves for targeted cross sections to the results from nonlinear shell finite element analysis (FEA) test simulations. The findings demonstrate that the recommended approach provides accurate-to-conservative strength predictions that meet or exceed a targeted minimum reliability index of 2.6, which AISC 360–22 does not achieve for a range of I-section member characteristics. The development of the AISC 360 Rec. provisions is significant as it fixes the strength overprediction problems recently observed for built-up I-girders while maintaining good accuracy for members where the AISC 360–22 equations perform well.

A Systematic Approach of Global Sensitivity Analysis and Its Application to a Model for the Quantification of Resilience of Interconnected Critical Infrastructures

We consider a model for the resilience analysis of interconnected critical infrastructures (ICIs) that describes the dependencies among the subsystems within the ICIs and their time-varying behavior. The model response is a function of uncertain inputs comprising ICIs design parameters and failure magnitudes of vulnerable elements in the system, etc. In this methodological paper, we present a systematic approach based on an innovative blend of methods to perform a sensitivity analysis for identifying the most relevant variables affecting the system resilience at different stages, during a disruptive event. The methods considered include the following: the use of the graphical representation of Cusunoro curves for a visualization of the impact of an input on the resilience metric and an understanding of whether the associated dependence is monotonic, increasing, or decreasing; the introduction of an ensemble of indicators related to different properties of the resilience metric to allow the prioritization of variable importance and avoid false negatives, meaning to regard a variable as non-influential when, instead, it plays a relevant role in the determination of the model response; the calculation of first-order variance-based sensitivity indices to have an appreciation on the relevance of interactions when inputs are independent; and a data approach to visually identify relevant second-order interactions. All the sensitivity methods considered are performed on a provided sample, and do not require additional model evaluations. They allow the analyst to post-process the data to extract, simultaneously, several desirable insights. The systematic approach proposed to apply these methods allows us to identify the model input variables and parameters that are not very relevant, while it enables the identification of the relevant ones which allows prioritizing interventions on the vulnerable elements of the system for its resilience at different stages during a disruptive event. Given the methodological nature of the work, a simplified infrastructure model describing an interconnected gas network and electric power grid is taken as case study: this allows us to show that the approach is straightforward to understand and implement, and the results obtained show the usefulness of the approach in providing meaningful insights that can be used by stakeholders and decision makers to inform strategies for the improvement of system resilience. By the application of the simplified ICIs model to the case study, it is shown that the approach can be straightforwardly implemented to identify the most relevant variables on system resilience and obtain the most important subsystems. The key factors which affect system resilience in multiple initial failures scenarios are found; this allows us to identify the key resilience improvement measurements, and their priorities.