Regular Curves Research Articles

Generalized null Cartan helices and principal normal worldsheets in Minkowski 3-space

We give a method for constructing generalized null Cartan helices, which may have singularities, by using regular space-like plane curves and smooth functions in Minkowski 3-space. We also study the principle normal worldsheet, which is deeply related to generalized null Cartan helices from the viewpoint of singularity theory.

Regular and Rigid Curves on Some Calabi-Yau and General-Type Complete Intersections

Regular and Rigid Curves on Some Calabi-Yau and General-Type Complete Intersections

Nonlinear optical dynamics in Heisenberg space: Directional curves and recursive inquiry

This paper delves into the exploration of directional recursion operators within the realm of regular space curves modeled by Heisenberg systems. The central objective is to introduce a myriad of recursive flows, encompassing ferromagnetic and antiferromagnetic solutions, alongside a family of general normalization operators in the normal and binormal directions. The study employs the extended compatible and inextensible flow model of curves to examine the evolution models, providing a comprehensive understanding of their dynamics. A significant aspect of the investigation involves elucidating the evolution model in terms of anholonomy shapes and their density. The directional recursive operator, a focus of this study, demonstrates distinct results compared to traditional approaches. The reliability and applicability of the obtained results extend to the examination of various linear and nonlinear continuous dynamical systems.

Generalized Bishop frames of regular curves in $\mathbb{E}^{4}$

Generalized Bishop frames of regular curves in $\mathbb{E}^{4}$

Generalizations of quasielliptic curves

We generalize the notion of quasielliptic curves, which have infinitesimal symmetries and exist only in characteristic two and three, to a remarkable hierarchy of regular curves having infinitesimal symmetries, defined in all characteristics and having higher genera. This relies on the study of certain infinitesimal group schemes acting on the affine line and certain compactifications. The group schemes are defined in terms of invertible additive polynomials over rings with nilpotent elements, and the compactification is constructed with the theory of numerical semigroups. The existence of regular twisted forms relies on Brion's recent theory of equivariant normalization. Furthermore, extending results of Serre from the realm of group cohomology, we describe non-abelian cohomology for semidirect products, to compute in special cases the collection of all twisted forms.

Secondary electron yield (SEY) evolution of laser processed oxygen-free high conductivity copper induced by different ultrasonic cleaning durations

Laser treated material surface has highly regular curves and greatly reduces the secondary electron yield of surface. As one of the most typically used materials in the vacuum system of particle accelerators, the surface of oxygen-free high conductivity copper (OFHC) is processed by laser. Moreover, ultrasonic cleaning can effectively remove powders and impurities produced by laser ablation. The secondary electron yield (SEY), surface morphology and surface composition of laser treated oxygen-free copper samples before and after ultrasonic cleaning were compared. Experiments were conducted by using the secondary electron yield test device, X-ray diffractometer, X-ray photoelectron spectroscopy, and scanning electron microscope to characterize the SEY, crystal structures, surface composition and surface morphology of laser-treated oxygen-free copper samples, respectively. The SEY evolution of the samples was investigated and analyzed at electron doses of 7.6 × 10−6 C mm−2: after 15-min ultrasonic cleaning, the δmax of sample #1, #2 and #3 increased from 1.09, 1.23, and 1.38 to 1.19, 1.2, and 1.63, respectively, with the maximum increase of 18.12 %. Ultrasonic cleaning could greatly reduce the accumulation of spherical particles and the fibrous or flocculent structures produced by laser processing, thereby reducing the capture of secondary electrons.

JWST NIRCam Observations of SN 1987A: Spitzer Comparison and Spectral Decomposition

JWST Near Infrared Camera (NIRCam) observations at 1.5–4.5 μm have provided broadband and narrowband imaging of the evolving remnant of SN 1987A with unparalleled sensitivity and spatial resolution. Comparing with previous marginally spatially resolved Spitzer Infrared Array Camera (IRAC) observations from 2004 to 2019 confirms that the emission arises from the circumstellar equatorial ring (ER), and the current brightness at 3.6 and 4.5 μm was accurately predicted by extrapolation of the declining brightness tracked by IRAC. Despite the regular light curve, the NIRCam observations clearly reveal that much of this emission is from a newly developing outer portion of the ER. Spots in the outer ER tend to lie at position angles in between the well-known ER hotspots. We show that the bulk of the emission in the field can be represented by five standard spectral energy distributions, each with a distinct origin and spatial distribution. This spectral decomposition provides a powerful technique for distinguishing overlapping emission from the circumstellar medium and the supernova ejecta, excited by the forward and reverse shocks, respectively.

Blood cell indices as predictors of complicated appendicitis: A diagnostic study

BackgroundAppendicitis in children is associated with diagnostic complexity leading to delayed diagnosis. Such delays risk progression to complicated appendicitis. This study investigates the potential of cellular indices, including neutrophil-to-lymphocyte ratio (NLR), derived neutrophil-to-lymphocyte ratio (dNLR), platelet-to-lymphocyte ratio (PLR), and monocyte-to-lymphocyte ratio (MLR), to differentiate uncomplicated (UA) from complicated appendicitis (CA). MethodsDiagnostic study of pediatric patients with acute appendicitis admitted to our hospital from 2021 to 2022. NLR, dNLR, PLR and MLR were compared between groups. ResultsA total of 182 patients were enrolled: 116 cases with CA (63 % male, age 9.6 ± 5.5 years) and 66 patients with UA (67 % male, age 10.6 ± 2.7 years). All blood cell indices were significantly higher in the CA group than in the UA group (p= <0.0001). NLR, dNLR and PLR have a good area under the receiver operating characteristic (ROC) curve, whereas MLR has a regular curve (0.735). PLR was the most accurate predictor of CA with a sensitivity of 60 %, specificity of 85 %, positive predictive value of 88 %, area under the ROC curve of 0.802 (95 % CI 0.737–0.866) and a cut-off point for the diagnosis of CA of 213.3. The post-test probability of CA for a positive result on PLR was 80 % (95 % CI 69–88). ConclusionsPLR seems to be the most accurate cellular index to differentiate uncomplicated from complicated appendicitis in children. It may be useful for prioritizing cases for surgery as a predictor of intraoperative findings and early postoperative courses.

Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space

In this paper, we study the surfaces of osculating circles, which are the sets of all osculating circles at all points of regular curves. Since the surfaces of osculating circles may be singular, it is necessary to investigate the singular points of these surfaces. However, traditional methods and tools for analyzing singular properties have certain limitations. To solve this problem, we define the framed surfaces of osculating circles in the Euclidean 3-space. Then, we discuss the types of singular points using the theory of framed surfaces and show that generic singular points of the surfaces consist of cuspidal edges and cuspidal cross-caps.

Framed Natural Mates of Framed Curves in Euclidean 3-Space

In this study, we consider framed curves as regular or singular space curves with an adapted frame in Euclidean 3-space. We define framed natural mates of a framed curve that are tangent to the generalized principal normal of the framed curve. Subsequently, we present the relationships between a framed curve and its framed natural mates. In particular, we establish some necessary and sufficient conditions for the framed natural mates of specific framed curves, such as framed spherical curves, framed helices, framed slant helices, and framed rectifying curves. Finally, we support the concept with some examples.

Framed Curve Families Induced by Real and Complex Coupled Dispersionless-Type Equations

In this study, we investigate coupled real and complex dispersionless equations for curve families, even if they have singular points. Even though the connections with the differential equations and regular curves were considered in various ways in the past, since each curve does not need to be regular, we establish the connections for framed base curves, which generalize regular curves with linear independent conditions. Also, we give the Lax pairs of the real and complex coupled dispersionless equations from the motions of any framed curve. These give us significant conditions based on the framed curvatures and associated curvatures of the framed curves for integrability since it is well known that the Lax pair provides the integrability of differential equations.

Piecewise polynomial approximation of spatial curvilinear profiles using the Hough transform

Given a curvilinear profile P represented as a set of points in the space R3 and four families of low-degree polynomial curves that respectively depend on the parameters in the space R4, our goal is to identify the piecewise space polynomial curve best fitting the profile P. We use a parametric representation of the space curves and subdivide the profile into smaller portions that can be fitted with regular curves. We provide theoretical guarantees to the existence of such an approximation and an algorithm for the profile approximation. We take advantage of the implicit function theorem to locally project a space curve on at most two planes and to locally recognise it with a low-degree polynomial curve obtained by applying the Hough transform. Finally, we recombine the curve expressions on the two planes backwards in the space R3. The outcome of the algorithm is thus a piecewise polynomial curve approximating the profile. We validate our approach to approximate curvilinear profiles extracted from 3D point clouds representing real objects and to simplify and resample point clouds.

Selection of a Proper Maturity Index for the Mechanical Harvesting of ‘Mihong’ Peach Fruit

Peaches are one of the most popular fruits around the globe. Selecting the optimum harvesting maturity for peaches is crucial in assuring high-quality fruits. This study is a model for determining the ideal harvest time for a robot harvester. Our study was carried out over two years on ‘Mihong’ peaches during days after full bloom (DAFB) 71 to 90 in 2021 and DAFB 64 to 84 in 2022 to select the optimal maturity index through a quality survey. The fruit size, soluble solids content (SSC), titratable acidity (TA), firmness, peel color (L*, a*, b*, chroma, and hue), and ethylene production were investigated. Fruit size showed the regular double sigmoid curve, and SSC increased while firmness and TA decreased with time. The samples left in storage conditions in 2022 showed a massive change in SSC and firmness after DAFB 74, implying the optimum harvesting stage. Interestingly, color values manifest the same consequence with a*, b*, and hue by reaching a plateau with apex side color values after DAFB 74, indicating the desired maturity. Overall, the results show that color values are an outstanding non-destructive alternative to typical destructive measurements for determining the exact time to harvest ‘Mihong’ peaches.

Pattern of the heart rate performance curve in maximal graded treadmill running from 1100 healthy 18-65 Years old men and women: the 4HAIE study.

Introduction: The heart rate performance curve (HRPC) in maximal incremental cycle ergometer exercise demonstrated three different patterns such as downward, linear or inverse versions. The downward pattern was found to be the most common and therefore termed regular. These patterns were shown to differently influence exercise prescription, but no data are available for running. This study investigated the deflection of the HRPC in maximal graded treadmill tests (GXT) of the 4HAIE study. Methods: Additional to maximal values, the first and second ventilatory thresholds as well as the degree and the direction of the HRPC deflection (kHR) were determined from 1,100 individuals (489 women) GXTs. HRPC deflection was categorized as downward (kHR < -0.1), linear (-0.1 ≤ kHR ≤ 0.1) or inverse (kHR > 0.1) curves. Four (even split) age- and two (median split) performance-groups were used to investigate the effects of age and performance on the distribution of regular (= downward deflection) and non-regular (= linear or inverse course) HR curves for male and female subjects. Results: Men (age: 36.8 ± 11.9 years, BMI: 25.0 ± 3.3 kg m-2, VO2max: 46.4 ± 9.4 mL min-1. kg-1) and women (age: 36.2 ± 11.9 years, BMI: 23.3 ± 3.7 kg m-2, VO2max: 37.4 ± 7.8 mL min-1. kg-1) presented 556/449 (91/92%) downward deflecting, 10/8 (2/2%) linear and 45/32 (7/6%) inverse HRPC´s. Chi-squared analysis revealed a significantly higher number of non-regular HRPC´s in the low-performance group and with increasing age. Binary logistic regression revealed that the odds ratio (OR) to show a non-regular HRPC is significantly affected by maximum performance (OR = 0.840, 95% CI = 0.754-0.936, p = 0.002) and age (OR = 1.042, 95% CI = 1.020-1.064, p < 0.001) but not sex. Discussion: As in cycle ergometer exercise, three different patterns for the HRPC were identified from the maximal graded treadmill exercise with the highest frequency of regular downward deflecting curves. Older subjects and subjects with a lower performance level had a higher probability to show a non-regular linear or inverted curve which needs to be considered for exercise prescription.

On local properties of singular integral

UDC 517.5 Let γ be a regular curve. We study the local properties of singular integrals in the  H α α + β ( t 0 , γ ) class of functions. We obtain a strengthening of the Plemelj\–Privalov theorem for functions from the class H α α + β ( t 0 , γ ) .  It is proved that, at the point t 0 , of increased smoothness for  α + β &lt; 1 , there is only a logarithmic loss.

Ell -convex Legendre curves and geometric inequalities

In this paper we consider ell -convex Legendre curves, which are natural generalizations of strictly convex curves. We generalize various optimal geometric inequalities, isoperimetric inequality, Bonnesen’s inequality and Green–Osher inequality, for strictly convex curves to ones for ell -convex Legendre curves. Moreover we generalize the inverse curvature curve flow for this class of Legendre curves and prove that it always converges to a compact soliton after rescaling. Unlike in the class of regular curves, there are infinitely many compact solitons, which include circles and astroids.

Support Vector Machine (SVM) Application for Uniaxial Compression Strength (UCS) Prediction: A Case Study for Maragheh Limestone

The geomechanical properties of rock materials, such as uniaxial compression strength (UCS), are the main requirements for geo-engineering design and construction. A proper understanding of UCS has a significant impression on the safe design of different foundations on rocks. So, applying fast and reliable approaches to predict UCS based on limited data can be an efficient alternative to regular traditional fitting curves. In order to improve the prediction accuracy of UCS, the presented study attempted to utilize the support vector machine (SVM) algorithm. Multiple training and testing datasets were prepared for the UCS predictions based on a total of 120 samples recorded on limestone from the Maragheh region, northwest Iran, which were used to achieve a high precision rate for UCS prediction. The models were validated using a confusion matrix, loss functions, and error tables (MAE, MSE, and RMSE). In addition, 24 samples were tested (20% of the primary dataset) and used for the model justifications. Referring to the results of the study, the SVM (accuracy = 0.91/precision = 0.86) showed good agreement with the actual data, and the estimated coefficient of determination (R2) reached 0.967, showing that the model’s performance was impressively better than that of traditional fitting curves.

Sensitive Group Actions on Regular Curves of Almost ≤ n Order

Sensitive Group Actions on Regular Curves of Almost ≤ n Order

$ G^2/C^1 $ Hermite interpolation of offset curves of parametric regular curves

&lt;abstract&gt;&lt;p&gt;In this paper we presented a method of $ G^2 $ Hermite interpolation of offset curves of regular plane curves based on approximating the normal vector fields. We showed that our approximant is also $ C^1 $ Hermite interpolation of the offset curve. Our method is capable of achieving circular precision. Another advantage of our method is that if the input curve is a polynomial curve, then our method also yields a polynomial curve. Our approximation method was applied to numerical examples and its numerical results were compared to previous offset approximation methods. It was observed that our method is almost optimal with respect to the number of control points of the approximation curves for the same tolerance.&lt;/p&gt;&lt;/abstract&gt;

$ C^* $-partner curves with modified adapted frame and their applications

&lt;abstract&gt;&lt;p&gt;In this study, the curve theory, which occupies a very important and wide place in differential geometry, has been studied. One of the most important known methods used to analyze a curve in differential geometry is the Frenet frame, which is a moving frame that provides a coordinate system at each point of the curve. However, the Frenet frame of any curve cannot be constructed at some points. In such cases, it is useful to define an alternative frame. In this study, instead of the Frenet frame that characterizes a regular curve in Euclidean space $ E^3 $, we have defined a different and new frame on the curve. Since this new frame is defined with the aid of the Darboux vector, it is very compatible compared to many alternative frames in application. Therefore, we have named this new frame the "modified adapted frame" denoted by $ \{N^*, C^*, W^*\} $. Then, we have given some characterizations of this new frame. In addition to that, we have defined $ N^* $-slant helices and $ C^* $-slant helices according to $ \{N^*, C^*, W^*\} $. Moreover, we have studied $ C^* $-partner curves using this modified adapted frame. Consequently, by investigating applications, we have established the relationship between $ C^* $-partner curves and helices, slant helices.&lt;/p&gt;&lt;/abstract&gt;