Elliptical Region Research Articles

Riemann Problem for the Isentropic Euler Equations of Mixed Type in the Dark Energy Fluid

We are concerned with the Riemann problem for the isentropic Euler equations of mixed type in the dark energy fluid. This system is non-strictly hyperbolic on the boundary curve of elliptic and hyperbolic regions. We obtain the unique admissible shock waves by utilizing the viscosity criterion. Assuming fixed left states are in the elliptic and hyperbolic regions, respectively, we construct the unique Riemann solution for the mixed-type models with the initial right state in some feasible regions. Finally, we present numerical simulations which are consistent with our theoretical results.

Function theory in the bfd-norm on an elliptical region

Let E be the open region in the complex plane bounded by an ellipse. The B. and F. Delyon norm ‖⋅‖bfd on the space Hol(E) of holomorphic functions on E is defined by‖f‖bfd=defsupT∈Fbfd(E)⁡‖f(T)‖, where Fbfd(E) is the class of operators T such that the closure of the numerical range of T is contained in E. The name of the norm recognizes a celebrated theorem of the brothers Delyon, which implies that ‖⋅‖bfd is equivalent to the supremum norm ‖⋅‖∞ on Hol(E).The purpose of this paper is to develop the theory of holomorphic functions of bfd-norm less than or equal to one on E. To do so we shall employ a remarkable connection between the bfd norm on Hol(E) and the supremum norm ‖⋅‖∞ on the space H∞(G) of bounded holomorphic functions on the symmetrized bidisc, the domain G in C2 defined byG=def{(z+w,zw):|z|<1,|w|<1}. It transpires that there exists a holomorphic embedding τ:E→G having the property that, for any bounded holomorphic function f on E,‖f‖bfd=inf⁡{‖F‖∞:F∈H∞(G),F∘τ=f}, and moreover, the infimum is attained at some F∈H∞(G). This result allows us to derive, for holomorphic functions of bfd-norm at most one on E, analogs of the well-known model and realization formulae for Schur-class functions. We also give a second derivation of these models and realizations, which exploits the Zhukovskii mapping from an annulus onto E.

The Approximate Integrals by the Reproducing Kernel Method on the Elliptical Region Eₙ In the Space Rⁿ

In this paper, we applied the reproducing kernel method on elliptical region, we establish the formulas of the reproducing kernel from several degrees and generalized those formulas whatever the dimension of space, we obtained all surfaces, points, and their constants for compensation in the cubature formula, tables are included to show our results and examples to show the approximation solution in this method.

Spectrum design of multimode fiber Bragg gratings based on suppression of mode coupling

The direct femtosecond laser inscription method is very competitive in the fabrication of fiber Bragg gratings, but its refractive index modification region is very limited and leads to low reflectivity and wide spectra of multimode fiber gratings. In this work, the relationship between the refractive index modification region, the coupling coefficient, and the spectrum of the multimode fiber grating is investigated based on the intrinsic mechanism of mode coupling; thus the annular and eccentric elliptic refractive index modification regions are accordingly proposed to suppress mode couplings in the multimode fiber Bragg grating, so as to achieve a highly discrete spectrum and narrow spectrum of the multimode fiber Bragg grating, respectively. This work can provide a solid theoretical foundation for the spectrum design of the multimode fiber Bragg grating, which can be utilized to customize the spectrum according to application scenarios such as optical communication, fiber lasers, filtering, and sensing.

Concentric circular stress region-assisted pseudo-elliptical-ring-core polarization-maintaining few-mode fiber

In this paper, a pseudo-elliptical-ring-core polarization-maintaining few-mode fiber structure assisted by concentric circular stress-applying region is proposed. Numerical simulation results show that the structure can stably support 14 transmission modes. By introducing two elliptical stress regions with low refractive index inside the core, and a circular stress region with concentric core between the fiber core and cladding, the effective refractive index difference between higher-order modes is significantly improved. In the C + L band, the minimum refractive index difference between adjacent modes is not less than 2 × 10−4. At 1550 nm, the effective refractive index difference between the two fundamental modes reaches 4.8 × 10−4. The mode dispersions are less than |-33| ps/nm/km, and the maximum bending loss is in the order of 10−7 dB/m (bending radius ≥2.5 cm). This work may pave the way for the development of short-distance large-capacity optical fibers.

A novel data-driven method of ship collision risk evolution evaluation during real encounter situations

Aiming at realizing collision risk quantitative evaluation among encounter ships, a novel data-driven evolution model is proposed concerning encounter evolution in maritime transportation. A probabilistic velocity obstacle with an elliptic conflict region is constructed by taking into account uncertainty. The degree of and time to domain violation are introduced to quantify collision risk levels under varying velocities. Then, a ship collision risk evolution model is formulated by considering spatial attributes, macro-level and micro-level evolution based on a realistic collision avoidance decision. The model parameters and their weights are determined by statistical analysis of historical encounter scenarios and the characteristics of encounter stages. Therefore, the model encapsulates the statistical characteristics of actual data, which improves its practical values. The results of case studies indicate that the collision risk evolution model can properly reflect collision risk, so that collision evolution stages can be classified accordingly for rational anti-collision guidance. It brings new contributions to risk visualization, collision avoidance decision-making, and collision accident analysis and responsibility determination.

Closed orientable surfaces and fold Gauss maps

This paper describes how the elliptic and hyperbolic regions of a surface are related to stable Gauss maps on closed orientable surfaces immersed in three-dimensional space. We will show that for certain connected, closed, orientable surfaces containing a finite number of embedded circles that delineate two distinct types of regions, if all regions of one type are homeomorphic to a cylinder, then there exists an immersion f:M→R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f: M \\rightarrow \\mathbb {R}^3$$\\end{document} for which the Gauss map is a fold Gauss map.

Sagittal Section Hounsfield Units of the Upper Instrumented Vertebrae as a Predictor of Proximal Junctional Vertebral Fractures Following Adult Spinal Deformity Surgery.

A retrospective observational study. This study aimed to determine an accurate and convenient screening method for predicting proximal junctional fractures (PJFr) following surgery for adult spinal deformity (ASD) using computed tomography (CT)-based measurement of Hounsfield units (HUs). CT-based measurement of HUs is an alternative tool for assessing bone mineral density. However, the optimal method for predicting adjacent vertebral fractures following spinal fusion using HUs remains unclear. This retrospective observational study included 42 patients who underwent reconstructive surgery for ASD. Elliptical regions of interest (ROIs) on the axial section and rectangular ROIs on the sagittal section were placed at the upper instrumented vertebrae (UIV), UIV+1, and UIV+2. In addition, the HU value of the L2 vertebra was used as the representative. PJFr occurred in 28.6% of patients within 2 years following surgery. The HU values obtained from the axial sections of L2, UIV, UIV+1, and UIV+2 were not significantly associated with the incidence of PJFr within 2 years, except for the ROI set in the lower region of the L2 vertebra. However, the HU value of the anterior third of the UIV in the sagittal section was significantly lower in the PJFr group than in the nonPJFr group (87.0 vs. 160.3, p =0.001). A UIV HU value of <100 was associated with a higher incidence of PJFr than an HU vaue of >100 (p <0.05). Measurements of HU in the anterior one-third of the UIV in the sagittal section demonstrated predictive ability for PJFr following ASD surgery. A UIV HU value of <100 emerged as a risk factor for PJFr.

The Okubo–Weiss-type topological criteria in two-dimensional magnetohydrodynamic flows

The Okubo–Weiss (Okubo, Deep-Sea Res., vol. 17, issue 3, 1970, pp. 445–454; Weiss, Physica D, vol. 48, issue 2, 1991, pp. 273–294) criterion has been widely used as a diagnostic tool to divide a two-dimensional (2-D) hydrodynamical flow field into hyperbolic and elliptic regions. This paper considers extension of these ideas to 2-D magnetohydrodynamic (MHD) flows, and presents an Okubo–Weiss-type criterion to parameterize the magnetic field topology in 2-D MHD flows. This ensues via its topological connections with the intrinsic metric properties of the underlying magnetic flux manifold, and is illustrated by recasting the Okubo–Weiss-type criterion via the 2-D MHD stationary generalized Alfvénic state condition to approximate the slow-flow-variation ansatz imposed in its derivation. The Okubo–Weiss-type parameter then turns out to be related to the sign definiteness of the Gaussian curvature of the magnetic flux manifold. A similar formulation becomes possible for 2-D electron MHD flows, by using the generalized magnetic flux framework to incorporate the electron-inertia effects. Numerical simulations of quasi-stationary vortices in 2-D MHD flows in the decaying turbulence regime are then given to demonstrate that the Okubo–Weiss-type criterion is able to separate the MHD flow field into elliptic and hyperbolic field configurations very well.

Alternative bootstrap confidence regions for multiple correspondence analysis

A new approach for constructing elliptical bootstrap confidence regions for Multiple Correspondence Analysis is proposed, where the difference in method follows directly from the adoption of a different objective criterion to previous approaches. This includes a new correction for the well-known problems caused by the diagonal of the Burt matrix. Simulation experiments show that the method performs reasonably well in many cases, with exceptions being noted. Simulated data sets with known structure are also used to illustrate cases where the proposed method produces results in line with what would be expected, but some alternatives do not.

Characterization of Contact Pressure Distribution and Bruising Prediction of Apple under Compression Loading

The pressure distribution characteristics of an apple subjected to compressive loading were investigated using the pressure-sensitive film (PSF) technique combined with apple bruise measurements. Pressure was unevenly distributed in the elliptical contact region. The average pressure had no effect on bruising because it changed slightly in the range of 0.26–0.31 MPa with increasing load. Pressures of 0.20–0.40 MPa accounted for 72% of the total pressure area. Comparatively, the area where pressure over 0.50 MPa was distributed could be ignored and showed little contribution to the bruise area. The contact edge subjected to pressure below 0.10 MPa showed that no bruising occurred. As a result, the relationship between the ≥0.10 MPa pressure area strongly correlated with the bruise area according to a linear equation, with a correlation coefficient of ≥0.99. When this relationship was applied to determine the bruise area with FE, satisfactory predicted results were obtained with minor error rates of 0–7.89% for loads of 54–80 N. But larger prediction errors occurred when the load was above 90 N, suggesting that the linear elastic FE model may not be appropriate for accurately predicting apple bruising.

Statistical analysis of parameters and adsorption isotherm models.

The present work intends to discuss parameter estimation and statistical analysis in adsorption. The Langmuir and Tóth isotherm models are compared for a set of carbon dioxide adsorption data on 13X zeolite from literature at different temperatures: 303, 323, 373, and 423 K. Statistical analyses were performed under frequentist and Bayesian perspectives. Under the frequentist statistical view, parameters were estimated using Maximum Likelihood estimation (MLE). Statistical analyses of parameters were performed by confidence regions in terms of elliptical approximation and likelihood region, while the evaluation of models was performed by chi-square statistics. The results showed that, for these nonlinear models, the elliptical region offers a poor approximation of the parameter estimates' confidence region, especially for the most correlated parameter pairs. Additionally, the four-parameter Tóth's equation yields less correlated parameters than the three-parameter Langmuir model. From a Bayesian perspective, the Markov chain Monte Carlo (MCMC) technique facilitated the reconstruction of the probability density functions of parameters as well as enabled the propagation of parametric uncertainties in the model responses. Finally, the accurate assessment of experimental uncertainty significantly influences the evaluation of models and their respective parameters.

Probabilistic Intersection-Over-Union for Training and Evaluation of Oriented Object Detectors.

Oriented object detection is a challenging and relatively new problem. Most existing approaches are based on deep learning and explore Oriented Bounding Boxes (OBBs) to represent the objects. They are typically based on adaptations of traditional detectors that work with Horizontal Bounding Boxes (HBBs), which have been exploring IoU-like loss functions to regress the HBBs. However, extending this idea for OBBs is challenging due to complex formulations or requirement for customized backpropagation implementations. Furthermore, using OBBs presents limitations for irregular or roughly circular objects, since the definition of the ideal OBB is an ambiguous and ill-posed problem. In this work, we jointly tackle the problem of training, representing, and evaluating oriented detectors. We explore Gaussian distributions-called Gaussian Bounding Boxes (GBBs)-as fuzzy representations for oriented objects and propose using a similarity metric between two GBBs based on the Hellinger distance. We show that this metric leads to a differentiable closed-form expression that can be directly used as a localization loss term to train OBB object detectors. We also show that GBBs present a natural representation as elliptical regions (called EBBs), which inherently mitigate ambiguity representation for circular objects. Finally, we empirically show that the proposed similarity metric computed between two GBBs strongly correlates with the IoU between the corresponding EBBs, motivating the name Probabilistic Intersection-over-Union (ProbIoU). Our experiments show that results using ProbIoU as a regression loss are competitive with state-of-the-art alternatives without requiring additional hyperparameters or customized implementations, and that ProbIoU is a promising alternative to evaluate oriented object detectors. Our code is available at https://github.com/ProbIOU/.

Recategorization method based on dependence between qualitative variables using joint correspondence analysis with elliptical confidence regions

Recategorization method based on dependence between qualitative variables using joint correspondence analysis with elliptical confidence regions

The effect of myelography dye on bone density measurements utilizing Hounsfield units on CT.

Hounsfield unit (HU) measurements of bone density on CT are increasingly used for preoperative planning in spine surgery. Postmyelogram CT is another common preoperative diagnostic study. However, there is no current literature evaluating whether HU measurements on CT are affected by the presence of myelography dye. The purpose of the current study was to determine if the presence of myelography dye affects HU measurements of bone density in CT studies. Twenty-nine preoperative spine surgery patients who underwent both standard and postmyelography CT performed within 6 months of each other were identified. HU measurements were obtained from an elliptical region of interest using the available software on a standard PACS. Measurements were obtained on the axial cut at the midvertebral body on all lumbar vertebrae on three separate occasions and an average value was calculated for comparative analysis. A 6-week gap was used between measurements of the CT scans and the CT myelograms to diminish bias. The mean age of the cohort was 69 years and the average BMI was 32 kg/m2. Five patients were male and 24 were female. Six of the patients had instrumentation placed prior to the initial CT scan. The average HU measurements for CT levels L1-5 were 165, 171, 145, 154, and 225, respectively, whereas HU measurements for CT myelography of levels L1-5 were 168, 177, 148, 170, and 239, respectively. Strong correlations were noted between the HU measured on CT and CT myelography for L1 (r2 = 0.951), L2 (r2 = 0.966), L4 (r2 = 0.820), and L5 (r2 = 0.900), and moderate for L3 (r2 = 0.668). The presence of myelography dye had no clear effect on CT HU measurements of bone density. The results of this study support the use of CT myelograms for bone density assessment in the absence of standard CT images.

Stability analysis of shield tunnel face in sandy pebble strata considering the route slope angle

Unique physical characteristics of sandy pebble strata amplify disadvantages of the route slope angle for tunnel-face stability. This study uses Lanzhou Metro Line 1 as support project to investigate tunnel-face stability in sandy pebble strata. Using PFC 3D to simulate the tunnel-face failure mechanism of the horizontal tunnel and the longitudinal inclined tunnel. The results show that the slip line ahead of the tunnel face is like a logarithmic spiral, likewise, an ellipsoid failure region forms above the tunnel vault. Based on the above, the theoretical analysis model of tunnel-face stability considering the route slope angle is established. The logarithmic spiral-elliptical model comprises an upper elliptical loose region I and a logarithmic spiral slip region II. Moreover, the limit support pressure is derived by combining the upper bound theorem and associated flow rule. Finally, existing theoretical analysis models are compared with the proposed model, showing that the proposed model fits well with the tunnel-face failure mechanism in sandy pebble strata.

SIMULATION OF OSCILLATIONS OF WATER MOLECULES USING THE MODEL OF A TWO-FREQUENCY PENDULUM IN THE JAHN-TELLER POTENTIAL

The interaction potential for water molecules is considered, which corresponds to the presence of bends in hydrogen bonds in it. To explain the nature of this phenomenon, the theory of the Jahn-Teller effect (JTE) is applied. A model of the Jahn–Teller potential (JTP) was constructed, which has a minimum at a certain angle Θ. To simulate oscillations of water molecules in JTP using a pendulum, a correction of the angular potential U1 in the potential of directed forces (PDF) of intermolecular interaction is proposed using a wavelet type additive ΔU1 = c cos(mΘ)/exp(sΘ2). The parameters for the wavelet are selected based on the magnitude of the bending of hydrogen bonds in water (m=s=8, c=0…0.1). Modeling of rotational oscillations of molecules in JTP was carried out using the model of a two-frequency pendulum, which takes into account the ratio of the moments of inertia of the molecule (pendulum) along the axes k=3 and the PDF index n=8 in JTP Un =U1n. New types of pendulum oscillations in the JTP are determined in comparison with the original PDF (c=0) and their features. It is established that several types of oscillations are observed for this potential. These are new: sector, rotation of the pendulum in the potential trough – disordered or ordered, as well as types for PDF: two-frequency independent oscillations (IO) along the axes and ellipse-like oscillations (ELO) at one frequency. Oscillations in the JTP chute are observed for the main range of initial velocities inside an elliptical ring compressed along the Y axis, and only for the ELO, in an elliptical region elongated along the Y axis, as in the PDF. The methodology for calculating and analyzing the oscillation parameters of a two-frequency pendulum has been improved. The boundaries of oscillation types are determined for given parameters of the potential and a number of initial data. The difference between the patterns of oscillations is established for cases when the initial displacement of the pendulum is greater or less than the position of the minimum of the JTP. It is shown that the velocities at the boundary of the transition from the rotation of the pendulum in the potential trough to the IO correlate with the magnitude of the JTP maximum on the axis of the pendulum. The presence of stable transverse vibrations in the JTP for the case of protons of water molecules can apparently be considered as a new degree of freedom for vibrations of water molecules, which can lead to an explanation of the large contribution of rotational vibration modes to its heat capacity.

Multipaths’ statistics for scatterers with inverted elliptic–parabolic spatial density around the mobile

The joint and marginal probability densities of multipaths’ angles-of-arrival (AOA) and times-of-arrival (TOA) at the cellular base station are developed in closed form in this paper. Unlike the general simplification assumption in the open literature in which the scatterers are assumed to be located in a circular region for non-uniform spatial densities, the scatterers in this paper are assumed to be located in an elliptical region to properly model the elliptical footprint around the mobile station from the elevated base station with directional antenna. The inverted elliptic–parabolic spatial density was adopted to model the non-uniform distribution of the scatterers around the mobile. The uplink’s AOA–TOA joint distributions, AOA and TOA marginal distributions were analytically derived in closed form. How the eccentricity of the elliptical scatterer region affects the probability density functions is discussed. Furthermore, the derived AOA marginal distribution is compared to that of the elliptic conic and inverted parabolic models. The proposed model is shown to have better least-squares fit to some empirical AOA data compared to the elliptic conic and inverted parabolic models.

Event-Triggered bundled target traversing path planning using a dynamic elliptical guidance region for unmanned surface vehicles

Multiple problems such as target assignment timeliness and path energy optimization make it challenging for USVs traversing targets. This paper proposes a novel multiple targets traversing path planning algorithm for USVs, which includes Event-Triggered bundled (ET-Bundled) genetic target assignment method and dynamic elliptical guidance region (DEGR) path planning method. In terms of solving above problems, comparative simulation studies with the state-of-the-art methods demonstrate that ET-Bundled can not only ensures that the traversing time and path length do not exceed the constraint range, but also reduces the average CPU time of membrane structure by 60%, and DEGR can optimize the path energy consumption of EGR path by 34%.

Impact of moving target on underwater positioning by using state measurement

The localization of moving targets in an underwater acoustic wireless sensor network (UAWSN) is inaccurate due to the various underwater forces (viscous, hydrodynamic forces, perturbation of underwater). The false measurements in the sensor network cause position errors and velocity errors which disrupt the localization of the moving target. A randomly fluctuated spillover effect is introduced in the present paper. The absorption losses generated due to the spillover effect cause false measurements of the moving target. Theorem 1 describes the genesis of these absorption losses and their consequences in UAWSN. The measurements from each moving target in the presence of absorption losses are formulated in the elliptical region. A joint probabilistic data association (JPDA) method is proposed to quantify the false measurements in the elliptical region. A moving target state estimation (MTSE) algorithm is proposed to eliminate the false measurements from the moving targets and to measure the localization of moving targets with the help of the propagation speed of targets. The theoretical measurements of position RMSE and velocity RMSE are verified with standard methods. The proposed MTSE method improves the localization performance of the moving targets by 29.42 % and reduces 32.16 % of position errors and 36.23 % of velocity errors up to 550 ​m. The proposed algorithm will be useful for the sub-aquatic Internet of underwater things (IoUT).