Irregular Curves Research Articles

Geometric characteristics of planar quintic Pythagorean-hodograph curves

This study examines necessary and sufficient conditions for a planar quintic Bézier curve to be a Pythagorean-hodograph (PH) curve. Quintic PH curves can be categorized into two classes according to the representation of their derivatives. While the first class has been studied by Farouki (1994) to be a family of regular curves already, a more succinct proof by introducing auxiliary control points is provided in this paper. Geometric characteristics of the second class of quintic PH curves are also studied. The key technique to simplify the discussion is to represent a planar Bézier curve with a complex polynomial in Bernstein form. Benefiting from such complex expression, algebraic characteristics of quintic PH curves can be described by nonlinear complex systems with respect to control points. By treating these systems with geometric methods, conditions for a quintic planar curve to be a PH curve can be described in terms of geometric constraints on its control polygon. Furthermore, we provide methods for the construction of the second class of quintic PH curves. In particular, parameter values of cusps can be explicitly determined in advance for irregular curves.

Simulation of the discrepancy in 4DCT-based cumulative dose using digital phantoms

Objective To conduct a computer simulation to evaluate the discrepancy between the cumulative doses calculated by four-dimensional computed tomography (4DCT) images and 4DCT scans (for real-time respiratory motions) due to the patient’s irregular breathing. Methods A series of digital phantoms were generated from a patient’s 4DCT images to simulate 4DCT images and 4DCT scans (for real-time respiratory motions) resulting from various irregular breathing curves. A six-field intensity-modulated radiotherapy plan was generated. Two cumulative doses in the target were calculated. The first one, named Dall, was calculated by tracking the point displacements in the target manifested on the 4DCT images; the second one, named D4D, was calculated based on the point displacements along the whole breathing motion during 4DCT scanning. Dose discrepancy between D4D and Dall was calculated to evaluate the correlation between breathing pattern and dose discrepancy in the target. Results The dose discrepancy in the target was correlated with mean motion excursion and the standard deviation of motion excursion.ΔDmin(ΔD99) in the target increased from 2.39%(2.04%) to 11.91%(5.24%) as the mean motion excursion increased from 5 mm to 15 mm, and increased from 5.93%(2.15%) to 14.65%(5.01%) as the standard deviation of motion excursion increased from 15% to 45% of the mean motion excursion. When the mean period increased from 3 s to 5 s or the standard deviation of period increased from 10% to 40% of the mean period, ΔDmin(ΔD99) in the target was greater than 6.0%(2.0%), but less than 9.0%(3.0%). When the target diameter was 2 cm, 3 cm, and 4 cm, ΔDmin(ΔD99) in the target was 11.88%(5.50%), 6.91%(2.42%), and 7.53%(3.62%), respectively. Conclusions There is a large discrepancy between the cumulative doses calculated using 4DCT images and 4DCT scans (for real-time respiratory motions) when the patient has irregular breathing. This dose discrepancy depends on mean motion excursion and the standard deviation of motion excursion, but has little relationship with mean period, the standard deviation of period, and tumor volume. Key words: Tomography, X-ray computed; Respiratory-induced motion; Accumulative dose

An anthropomorphic breathing phantom of the thorax for testing new motion mitigation techniques for pencil beam scanning proton therapy

Motion-induced range changes and incorrectly placed dose spots strongly affect the quality of pencil-beam-scanned (PBS) proton therapy, especially in thoracic tumour sites, where density changes are large. Thus motion-mitigation techniques are necessary, which must be validated in a realistic patient-like geometry. We report on the development and characterisation of a dynamic, anthropomorphic, thorax phantom that can realistically mimic thoracic motions and anatomical features for verifications of proton and photon 4D treatments.The presented phantom is of an average thorax size, and consists of inflatable, deformable lungs surrounded by a skeleton and skin. A mobile ‘tumour’ is embedded in the lungs in which dosimetry devices (such as radiochromic films) can be inserted. Motion of the tumour and deformation of the thorax is controlled via a custom made pump system driving air into and out of the lungs. Comprehensive commissioning tests have been performed to evaluate the mechanical performance of the phantom, its visibility on CT and MR imaging and its feasibility for dosimetric validation of 4D proton treatments.The phantom performed well on both regular and irregular pre-programmed breathing curves, reaching peak-to-peak amplitudes in the tumour of <20 mm. Some hysteresis in the inflation versus deflation phases was seen. All materials were clearly visualised in CT scans, and all, except the bone and lung components, were MRI visible. Radiochromic film measurements in the phantom showed that imaging for repositioning was required (as for a patient treatment). Dosimetry was feasible with Gamma Index agreements (4%/4 mm) between film dose and planned dose >90% in the central planes of the target.The results of this study demonstrate that this anthropomorphic thorax phantom is suitable for imaging and dosimetric studies in a thoracic geometry closely-matched to lung cancer patients under realistic motion conditions.

Irregular GIS Curve Fitting based High Speed Railway Earthquake Influence Range Calculation Model

In this paper, to guarantee that the train can take measures to reduce the damage caused by the earthquake, it propose an irregular GI S curve fitting based high-speed railway earthquake influence range calculation model. Firstly, this model eliminates the abnormal points, calculates feature points and finds demarcation points of the high- speed railway GI S curve to get the processed point collection in Mercator coordinate. Secondly, though usin g the processed point collection, this model applies least square polynomial segmentation fitting method to implement complex high-speed GI S curve fitting. Thirdly, calculate the earthquake influence rang on high-seed railway line, according to the scope of the earthquake equation and the high-speed railway GI S curve fitt ed equation. Finally, the paper selects the Beijing So uth to Dezhou East high-speed railway section which is part of Beijing-Shanghai line as a case study, which proves that the model can calculate the earthquake influence scope on the railway line offering decision support for train operation to ensure safety.

AN ACTIVE, ASYNCHRONOUS COMPANION TO A REDBACK MILLISECOND PULSAR

ABSTRACT PSR J1723−2837 is a “redback” millisecond pulsar (MSP) with a low-mass companion in a 14.8 hr orbit. The system’s properties closely resemble those of “transitional” MSPs that alternate between spin-down and accretion-powered states. In this Letter, we report on long-term photometry of the 15.5 mag companion to the pulsar. We use our data to illustrate that the star experiences sporadic activity, which we attribute to starspots. We also find that the companion is not tidally locked and infer for the ratio between the rotational and orbital periods. Finally, we place constraints on various parameters, including the irradiation efficiency and pulsar mass. We discuss similarities with other redback MSPs and conclude that starspots may provide the most likely explanation for the often seen irregular and asymmetric optical light curves.

Damage to inner ear structure during cochlear implantation: Correlation between insertion force and radio-histological findings in temporal bone specimens

Cochlear implant insertion should be as least traumatic as possible in order to reduce trauma to the cochlear sensory structures. The force applied to the cochlea during array insertion should be controlled to limit insertion-related damage. The relationship between insertion force and histological traumatism remains to be demonstrated. Twelve freshly frozen cadaveric temporal bones were implanted with a long straight electrodes array through an anterior extended round window insertion using a motorized insertion tool with real-time measurement of the insertion force. Anatomical parameters, measured on a pre-implantation cone beam CT scan, position of the array and force metrics were correlated with post-implantation scanning electron microscopy images and histological damage assessment. An atraumatic insertion occurred in six cochleae, a translocation in five cochleae and a basilar membrane rupture in one cochlea. The translocation always occurred in the 150- to 180-degree region. In the case of traumatic insertion, different force profiles were observed with a more irregular curve arising from the presence of an early peak force (30 ± 18.2 mN). This corresponded approximately to the first point of contact of the array with the lateral wall of the cochlea. Atraumatic and traumatic insertions had significantly different force values at the same depth of insertion (p < 0.001, two-way ANOVA), and significantly different regression lines (y = 1.34x + 0.7 for atraumatic and y = 3.37x + 0.84 for traumatic insertion, p < 0.001, ANCOVA). In the present study, the insertion force was correlated with the intracochlear trauma. The 150- to 180-degree region represented the area at risk for scalar translocation for this straight electrodes array. Insertion force curves with different sets of values were identified for traumatic and atraumatic insertions; these values should be considered during motorized insertion of an implant so as to be able to modify the insertion parameters (e.g axis of insertion) and facilitate preservation of endocochlear structures.

Effects of building shade on photosynthesis and chlorophyll fluorescence of Euonymus fortunei

Urban shading is caused by artificial urban construction and has different effects on the photosynthesis of plant, and this shading will affect the plants in photosynthesis. The purpose of the study was to reveal the plant photosynthetic characteristics in urban shading, provide theoretical basis for improving the ecological benefits of urban vegetation and provide scientific basis for urban plant landscape configuration. We selected leaf samples of Euonymus fortunei from three typical urban light environments: full natural light, part-time shade and full urban building shade. We quantified various measures of photosynthesis and chlorophyll fluorescence using the CIRAS-2 photosynthesis and FMS-2 fluorescence systems, respectively. The results indicated that urban shading by artificial structures caused differences in both the spatial and temporal distribution of photosynthetic active radiation (PAR). Surprisingly, this was not due to differences to the air temperature (Ta), relative humidity and CO2 concentrations, which were consistent among the light conditions. Urban building shade also caused changes in leaf morphology and chloroplast pigment content of E. fortunei. Leaf area (LA) increased with part-time shade and decreased with full shade, while lamina mass per unit area (LMA) decreased significantly as the shade increased. Chlorophyll b content increased and the chlorophyll a/b ratio decreased with the decrease of PAR. Pn of E. fortunei displayed an irregular single-peak curve under full light and part-time shade, and the peak for each appeared at 10:00 and 12:00, respectively. Pn displayed a double-peak curve under full shade, with peaks appearing at 10:00 and 16:00. Tr of E. fortunei was significantly correlated with Pn. The Pn-PAR curve showed that Pmax, LSP, LCP, and Rd all decreased along with PAR, with the exception of AQY, which significantly increased. Chlorophyll fluorescence parameters also changed under the different light environments. Fo and ΦPSII both increased with the decreases in PAR, but Fv/Fm and NPQ decreased. Different levels of urban shading caused the changes in adaptive strategies of E. fortunei. When there was no direct sunlight appearing, a highest level of shading, E. fortunei presented obvious adaptive changes in its physiological photosynthetic processes, morphology, photosynthetic pigments and so on, and this type of the greatest shading caused by urban buildings or other infrastructures can obviously affect the growth of plants.

The Harer–Zagier recursion for an irregular spectral curve

We derive the Do and Norbury recursion formula for the one-loop mean of an irregular spectral curve from a variant of replica method by Brez\'in and Hikami. We express this recursion in special times in which all terms $W_1^{(g)}$ of the genus expansion of the one-loop mean are polynomials. We find a generalization of this recursion to the generalized Laguerre polynomial case.

Effect of chamber enclosure time on soil respiration flux: A comparison of linear and non-linear flux calculation methods

One of the shortcomings of closed chamber methods for soil respiration (SR) measurements is the decreased CO2 diffusion rate from soil to chamber headspace that may occur due to increased chamber CO2 concentrations. This feedback on diffusion rate may lead to underestimation of pre-deployment fluxes by linear regression techniques. Thus, usually the cumulative flux curve becomes downward concave due to the decreased gas diffusion rate. Non-linear models based on biophysical theory usually fit to such curvatures and may reduce the underestimation of fluxes. In this study, we examined the effect of increasing chamber enclosure time on SR flux rates calculated using a linear, an exponential and a revised Hutchinson and Mosier model (HMR). Soil respiration rates were measured with a closed chamber in combination with an infrared gas analyzer. During SR flux measurements the chamber was placed on fixed collars, and CO2 concentration in the chamber headspace were recorded at 1-s intervals for 45 min. Fluxes were measured in different soil types (sandy, sandy loam and organic soils), and for various manipulations (tillage, rain and drought) and soil conditions (temperature and moisture) to obtain a range of fluxes with different shapes of flux curves. The linear method provided more stable flux results during short enclosure times (few min) but underestimated initial fluxes by 15–300% after 45 min deployment time. Non-linear models reduced the underestimation as average underestimation was only about 10% after 45 min for regular flux curves. For irregular flux curves with a rapid increase in CO2 concentration immediately after chamber deployment it was shown that short enclosure times were prone to overestimation of pre-deployment fluxes, but this was mitigated by longer enclosure times (>10–15 min).

Regression models with correlated errors based on functional random design

This paper deals with the study of the estimation of the functional regression operator when the explanatory variable takes its values in some abstract space of functions. The main goal of this paper is to establish the exact rate of convergence of the mean squared error of the functional version of the Nadaraya–Watson kernel estimator when the errors come from a stationary process under long or short memory and based on random functional data. Moreover, these theoretical results are checked through some simulations with regular (smooth) and irregular curves and then with real data.

Fractal Jacobi Systems and Convergence of Fourier–Jacobi Expansions of Fractal Interpolation Functions

The fractal interpolation function (FIF) is a special type of continuous function on a compact subset of $${\mathbb{R}}$$ interpolating a given data set. They have been proved to be a very important tool in the study of irregular curves arising from financial series, electrocardiograms and bioelectric recording in general as an alternative to the classical methods. It is well known that Jacobi polynomials form an orthonormal system in $${\mathcal{L}^{2}(-1,1)}$$ with respect to the weight function $${\rho^{(r,s)}(x)=(1-x)^{r} (1+x)^{s}}$$ , $${r > -1}$$ and $${s > -1}$$ . In this paper, a fractal Jacobi system which is fractal analogous of Jacobi polynomials is defined. The Weierstrass type theorem providing an approximation for square integrable function in terms of $${\alpha}$$ -fractal Jacobi sum is derived. A fractal basis for the space of weighted square integrable functions $${\mathcal{L}_{\rho}^{2}(-1,1)}$$ is found. The Fourier–Jacobi expansion corresponding to an affine FIF (AFIF) interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is established. The closeness of the original function to the Fourier–Jacobi expansion of the AFIF is proved for certain scale vector. Finally, the Fourier–Jacobi expansion corresponding to a non-affine smooth FIF interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is investigated as well.

An investigation on modeling and compensation of synthetic geometric errors on large machine tools based on moving least squares method

This article intends to provide an efficient modeling and compensation method for the synthetic geometric errors of large machine tools. Analytical and experimental examinations were carried out on a large gantry-type machine tool to study the spatial geometric error distribution within the machine workspace. The result shows that the position accuracy of the tool-tip is affected by all the translational axes synchronously, and the position error curve shape is non-linear and irregular. Moreover, the angular error combined with Abbe’s offset during the motion of a translational axis would cause Abbe’s error and generate significant influence on the spatial positioning accuracy. In order to identify the combined effect of the individual error component on the tool-tip position accuracy, a synthetic geometric error model is established for the gantry-type machine tool. Also, an automatic modeling algorithm is proposed to approximate the geometric error parameters based on moving least squares in combination with Chebyshev polynomials, and it could approximate the irregular geometric error curves with high-order continuity and consistency with a low-order basis function. Then, to implement real-time error compensation on large machine tools, an intelligent compensation system is developed based on the fast Ethernet data interaction technique and external machine origin shift, and experiment validations on the gantry-type machine tool showed that the position accuracy could be improved by 90% and the machining precision could be improved by 85% after error compensation.

Experimental determination of nonequilibrium transport parameters reflecting the competitive sorption between Cu and Pb in slag-sand column

Competitive sorption and resulting nonequilibrium transport of Cu and Pb were investigated using slag as a primary sorbent. A series of estimation models were applied based on the equilibrium, and nonequilibrium sorption respectively, and finally calibrated by incorporating the experimentally determined batch kinetic data. When applied individually, the behavior of metals in slag-sand column were well predicted by both equilibrium and nonequilibrium models in CXTFIT code. However, coexisting Cu and Pb exhibited competition for sorption sites, generating an irregular breakthrough curves such as overshoot (higher concentration in effluent than the feed concentration) of Cu and corresponding earlier peak of Pb followed by gradual re-rising. Although two-site nonequilibrium model further considers coupled hydrochemical process, desorption of the Cu from competition made the model prediction inaccurate. However, the parameter estimation could be improved by incorporating the experimentally determined mass transfer rate, ωexp from batch kinetics. Based on the calibrated model, the fraction of instantaneous retardation, βexp of Pb decreased from 0.41 in the single system to 0.30 in the binary system, indicating the shift from equilibrium to nonequilibrium state, where which of Cu increased from 0.39 to 0.94, representing the shift towards equilibrium. The modified results were also compared with five-step sequential extraction data, confirming that the shift of particular metal fractions from the competition triggered the nonequilibrium transport.

Averaging along irregular curves and regularisation of ODEs

We consider the ordinary differential equation (ODE) dxt=b(t,xt)dt+dwt where w is a continuous driving function and b is a time-dependent vector field which possibly is only a distribution in the space variable. We quantify the regularising properties of an arbitrary continuous path w on the existence and uniqueness of solutions to this equation. In this context we introduce the notion of ρ-irregularity and show that it plays a key role in some instances of the regularisation by noise phenomenon. In the particular case of a function w sampled according to the law of the fractional Brownian motion of Hurst index H∈(0,1), we prove that almost surely the ODE admits a solution for all b in the Besov–Hölder space B∞,∞α+1 with α>−1/2H. If α>1−1/2H then the solution is unique among a natural set of continuous solutions. If H>1/3 and α>3/2−1/2H or if α>2−1/2H then the equation admits a unique Lipschitz flow. Note that when α<0 the vector field b is only a distribution, nonetheless there exists a natural notion of solution for which the above results apply.

Massive star mergers and the recent transient in NGC 4490: a more massive cousin of V838 Mon and V1309 Sco

The Galactic transient V1309 Sco was the result of a merger in a low-mass star system, while V838 Mon was thought to be a similar merger event from a more massive B-type progenitor. In this paper we study an optical/IR transient discovered in the nearby galaxy NGC4490, which appeared similar to these merger events (unobscured progenitor, irregular multi-peaked light curve, increasingly red color, similar optical spectrum, IR excess at late times), but which had a higher peak luminosity and longer duration in outburst. NGC4490-OT has less in common with the class of SN~2008S-like transients. A progenitor detected in pre-eruption HST images, combined with upper limits in the IR, requires a luminous and blue progenitor that has faded in late-time HST images. The same source was detected by Spitzer and ground-based data as a luminous IR transient, indicating a transition to a self-obscured state qualitatively similar to the evolution seen in other stellar mergers and in LBVs. The post-outburst dust-obscured source is too luminous and too warm at late times to be explained with an IR echo, suggesting that the object survived the event. The luminosity of the enshrouded IR source is similar to that of the progenitor. Compared to proposed merger events, the more massive progenitor of NGC4490-OT seems to extend a correlation between stellar mass and peak luminosity, and may suggest that both of these correlate with duration. We show that spectra of NGC4490-OT and V838 Mon also resemble light-echo spectra of eta Car, prompting us to speculate that eta Car may be an extreme extension of this phenomenon.

Stability and vibration properties of a composite laminated plate subjected to subsonic compressible airflow

The stability and nonlinear vibration of a composite laminated plate with simply supported boundary conditions in subsonic compressible airflow subjected to transverse periodic external excitation are investigated. The equation of motion of the plate is established by using the von Karman’s nonlinear plate theory. The linear potential flow theory considering the compressibility of the airflow is adopted to formulate the aerodynamic model. The nonlinear ordinary differential equation of motion of the plate is obtained by applying the assumed mode method. The critical divergence velocity and the flutter velocity of the plate are obtained by analyzing the natural frequencies of the linear system. The effects of the ply angle and the compressibility of the airflow on the critical divergence velocities and the flutter velocities of the plate are investigated. For the nonlinear system, the displacement time response of the plate subjected to accelerating airflow and transverse harmonic excitation is presented to investigate the vibration properties. The effects of the subsonic airflow on the resonance properties of the system are analyzed from the amplitude frequency curves of the plate under different flow velocities. From the study, it can be seen that with the flow velocity increasing, the natural frequencies of the plate decreases and the plate may exhibit instability of the divergence or flutter type. With the ply angle increasing from 0° to 90°, the critical divergence velocity increases and then decreases, and the flutter velocity decreases. The critical divergence velocity and the flutter velocity of the plate obtained from the compressible aerodynamic model are smaller than those obtained from the incompressible model. When the plate is in instability of divergence type, the plate vibrates with smaller amplitude around a non-zero point of equilibrium, and when the plate is in instability of flutter type, the plate vibrates with larger amplitude around a non-zero point of equilibrium. When the flow velocity exceeds the divergence velocity, the amplitude frequency curve of the plate is an irregular curve with larger vibration amplitudes and the plate may be in a chaotic motion state.

Development of cubic Bezier curve and curve-plane intersection method for parametric submarine hull form design to optimize hull resistance using CFD

Optimization analysis and computational fluid dynamics (CFDs) have been applied simultaneously, in which a parametric model plays an important role in finding the optimal solution. However, it is difficult to create a parametric model for a complex shape with irregular curves, such as a submarine hull form. In this study, the cubic Bezier curve and curve-plane intersection method are used to generate a solid model of a parametric submarine hull form taking three input parameters into account: nose radius, tail radius, and length-height hull ratio (L/H). Application program interface (API) scripting is also used to write code in the ANSYS design modeler. The results show that the submarine shape can be generated with some variation of the input parameters. An example is given that shows how the proposed method can be applied successfully to a hull resistance optimization case. The parametric design of the middle submarine type was chosen to be modified. First, the original submarine model was analyzed, in advance, using CFD. Then, using the response surface graph, some candidate optimal designs with a minimum hull resistance coefficient were obtained. Further, the optimization method in goal-driven optimization (GDO) was implemented to find the submarine hull form with the minimum hull resistance coefficient (C t ). The minimum C t was obtained. The calculated difference in C t values between the initial submarine and the optimum submarine is around 0.26%, with the C t of the initial submarine and the optimum submarine being 0.001 508 26 and 0.001 504 29, respectively. The results show that the optimum submarine hull form shows a higher nose radius (r n ) and higher L/H than those of the initial submarine shape, while the radius of the tail (r t ) is smaller than that of the initial shape.

A SmartPen for 3D interaction and sketch-based surface modeling

This paper presents an innovative, portable, and low cost system for fast and interactive design of sketched models using 3D curve networks. The system is based on a stereo camera and a wireless pen-shaped device, called the SmartPen, which can be extended or bent to facilitate concave objects’ acquisitions. Its tip position is tracked in 3D and used to trace the style lines of the object to be modeled in the virtual space. Starting from the 3D irregular curve network, progressively drawn by the user, a new Sketch-based Modeling Framework automatically creates the inferred 3D virtual model. The interactive system integrates perceptual and conceptual modeling, allowing both for the reconstruction of existing objects via user-drawn strokes on the physical object and for free-hand strokes which together contribute to realize the desired 3D curve network. Real-time reconstruction of the sketched surface interpolating the 3D curve network is obtained through an interactive curvature-based optimization approach which determines the smooth subdivision surface that best fits the traced 3D curve network.

EP-1423: Comparison of AAA and collapsed cone algorithms for planning of breast treatment with SIB technique

Results: In axis beams are well within agreement (deviation<4%) for 2 and 1 mm grid sizes. For the 4 mm grid size, at shallow depths (<< dose maximum) larger differences are observed and for narrow field sizes (widths ≤ 10 mm) those differences occur at all depths. Oblique incidence (60°) of the same beams results in an identical deviation pattern. On top of these deviations a non monotonous decay after the build-up zone is observed for the 4mm grid size for all narrow beams ≤ 14 mm. A similar behavior is also observed for offaxis fields; 1 mm and 2 mm grid sizes are equivalent for all the field sizes except at 5 mm depth but 4 mm grid sizes induces errors in output and irregular depth dose curves for field sizes 14 mm or smaller. Several Pinnacle models are evaluated for different Elekta accelerators and output factors are verified using data measured with small cylindrical and liquid ion chambers. The 8 mm field size is not included while chamber volume effects underestimate the maximal dose values. The average agreement for the 4 mm grid size between calculation values and the measured data for both detectors (SSD=90 cm depth 10 cm) for off-axis field settings is underestimation of 2.49%±3.10% while an overestimation 0.13%±1.39% is obtained using a 2 mm grid size. Those findings are confirmed by the less accurate agreement for the 4 mm calculation grid size on treatment plans for IMRT treatment verified on a Delta4 phantom for breast treatment.

Recursive Bayesian estimation of regularized and irregular quasar light curves

Recursive Bayesian estimation of regularized and irregular quasar light curves