Geometric Field Research Articles

Observation of a Photonic Orbital Gauge Field.

Gauge field is widely studied in natural and artificial materials. With an effective magnetic field for uncharged particles, many intriguing phenomena are observed in several systems like photonic Floquet topological insulator. However, previous researches about the gauge field mostly focus on limited dimensions such as the Dirac spinor in graphene materials. Here, an orbital gauge field based on photonic triangular lattices is first proposed and experimentally observed. Disclination defects with Frank angle Ω created on such lattices breaks the original lattice symmetry and generates purely geometric gauge field operating on orbital basis functions. Interestingly, it is found that bound states near zero energy with the orbital angular momentum (OAM) l = 2 are intensively confined at the disclination as gradually expanding Ω. Moreover, the introduction of a vector potential field breaks the time-reversal symmetry of the orbital gauge field, experimentally manifested by the chiral transmission of light on helical waveguides. The orbital gauge field further suggests fantastic applications of manipulating the vortex light in photonic integrateddevices.

Geometric field model of driver’s perceived risk for safe and human-like trajectory planning

Autonomous driving systems must provide safe, predictable, and consistent behaviors across diverse scenarios to enhance user experience. The geometric driver risk field (GDRF) method is proposed for human-like, interpretable, and fast risk estimations. The method models the driver’s subjectively perceived risk with fields formed by sets of geometric shapes. Unlike current safety measures adopted in trajectory planning algorithms, the proposed method aligns with driver risk perception properties. It provides risk estimations based on probable future vehicle states and the risk consequences at different locations. Compared with existing potential field methods for risk estimations, the mathematical form of the method has a high calculation efficiency while keeping the ability to model the details of the risk information. Influences of different obstacle types, multiple regions with different risk levels, and the relative motions between vehicles are modeled in a unified manner. The graph-search and optimization trajectory planning algorithms with a hierarchy framework are also designed to attain desired risk objectives in the planned trajectories. The framework mitigates the possible suboptimality problems in previous hierarchy trajectory planning algorithms. The proposed GDRF method and planning algorithms are examined in driving scenarios with different traffic states and road geometries. Results showed that the proposed risk estimation method outperforms previous risk-oriented potential field methods by more human-like and safer trajectories with evidence from existing studies and safety measures like time headway (THW), time to collision (TTC), and time to lane crossing (TLC). It also provides more consistent behaviors while maintaining a short computation time. The integration of the proposed risk estimation method and trajectory planning algorithms holds great potential for improving the safety and user experience of autonomous driving systems in real-world scenarios.

Bayesian calibration of quasi-static field distortions in HARMONI

ABSTRACT The high angular resolution monolithic optical and near-infrared Integral field spectrograph is the first light visible and near-infrared integral field spectrograph for the Extremely Large Telescope. To reach the diffraction limit of the telescope (≈ 10 mas) and maintain an optimal image quality over long exposures, an accurate measurement of geometrical distortions in the instrument’s guide star field is needed. Geometrical distortions affecting the guide stars map directly to pointing errors of the science field. The systematic contribution to the pointing error can be calibrated and removed by a corrective model. In this work, we propose a formulation of the corrective model that aims to calibrate the geometrical field distortions down to a given target residual, as well as reducing the time spent in calibrations. We also propose a calibration procedure that accounts for the uncertainties of the measurement process. We developed a tool named harmoni-pm to simulate the expected pointing error caused by geometrical distortions and validate the effectiveness of the proposed corrective model. We also relied on pseudo Zernike polynomials to formulate the model, and the Bayesian theoretical framework to study the propagation of uncertainties along the calibration process. Compared with the classical calibration algorithm, the Bayesian calibration algorithm was able to reduce the number of calibration points required to reach the same model residual. Additionally, we were able to optimize the hardware of the Geometrical Calibration Unit and reduce the time required to achieve the calibration goal.

Lie Symmetry Analysis for the Fractal Bond-Pricing Model of Mathematical Finance

The classical bond-pricing models, as important financial tools, show strong vitality in bond pricing. However, these models also expose their theoretical defects, which leads to inconsistencies with the actual observation results and usually causes the theoretical prices of bonds to be lower than the actual market prices in the financial market. In order to change this situation, considering that the price change of the underlying is regarded as a fractal transmission system, the fractal derivative is introduced into the bond-pricing equation. In order to solve the fractal bond-pricing equation, we first convert it into an equivalent equation by using a fractal two-scale transform. Only in this case can we start to study it by means of the Lie symmetry analysis method. Then the geometric vector fields, the symmetry reductions, and the exact solution to the equations are obtained. Furthermore, the dynamic behaviors of the fractal bond-pricing equation are discussed. The results show that the fractal dimension bond-pricing formula can better explain price changes in the capital market than the classical one. That is to say, the classical bond-pricing equation is only a special case of the fractal-bond pricing equation, which makes up for the defect that the theoretical bond price given by the classical bond-pricing equation is often lower than the actual market price. The results of this paper provide a basis for bond pricing in the financial market in order to seek a more appropriate and real price.

Holistically-Attracted Wireframe Parsing: From Supervised to Self-Supervised Learning.

This article presents Holistically-Attracted Wireframe Parsing (HAWP), a method for geometric analysis of 2D images containing wireframes formed by line segments and junctions. HAWP utilizes a parsimonious Holistic Attraction (HAT) field representation that encodes line segments using a closed-form 4D geometric vector field. The proposed HAWP consists of three sequential components empowered by end-to-end and HAT-driven designs: (1) generating a dense set of line segments from HAT fields and endpoint proposals from heatmaps, (2) binding the dense line segments to sparse endpoint proposals to produce initial wireframes, and (3) filtering false positive proposals through a novel endpoint-decoupled line-of-interest aligning (EPD LOIAlign) module that captures the co-occurrence between endpoint proposals and HAT fields for better verification. Thanks to our novel designs, HAWPv2 shows strong performance in fully supervised learning, while HAWPv3 excels in self-supervised learning, achieving superior repeatability scores and efficient training (24 GPU hours on a single GPU). Furthermore, HAWPv3 exhibits a promising potential for wireframe parsing in out-of-distribution images without providing ground truth labels of wireframes.

Part-through cracks computation in an Euler–Bernoulli beam model

AbstractThe reduction of the equations of a slender, brittle continuum to a one-dimensional theory and the possibility of accounting for the cracking of this body has received much attention recently. This contribution investigates the effect of cross-sectional geometry on damage localization in an Euler–Bernoulli beam. Two geometric fields characterizing the crack penetration depth on each side of the beam are introduced, from which the modulation of the bending and tensile parts of the strain energy by the crack depth is deduced. The geometry of the crack-tip envelope is obtained by energy minimization. We find that in the case of a Griffith-like dissipation potential, the characteristic size of the crack is a key element for predicting initiation at a finite external load. We can accurately predict the initiation position by regularizing the model in the spirit of Variational Brittle Fracture. Model predictions are compared against computational results obtained by the classical damage field theory.

Validation of the geometric equivalent field concept in total scatter factor calculations, for half-, quarter- and off-isocenter asymmetric square fields.

Monitor unit (MU) verification for any symmetric or asymmetric field is performed using a total scatter factor (Scp ), that is calculated based on the geometric equivalent square field (GESF) concept. In this study, we measured the Scp of various asymmetric square fields (ASFs ) and their respective GESFs. Square half-fields (SHFs ), square quarter-fields (SQFs ) and square off-isocenter fields (SOFs ), with sizes ranging from 3×3 cm2 to 20×20 cm2 were created, by varying the collimator jaws of two Varian iX Linacs (6/18 and 6/23 MV). A semi-flex ion chamber was used to measure Scp at a depth of 10cm within a water phantom, at the effective field center (EFC) of all ASFs , and at the isocenter (IC) of their respective GESFs. The later Scp values were corrected by the off-axis ratio [OAR(r)] of the 40×40 cm2 field size, where r is the distance between EFC and IC. The results show that the Scp (EFC) is independent of the type of the ASF (SHF, SQF, or SOF) and no significant difference exists between the 18 and 23 MV beams. Compared with the Scp (IC), the Scp (EFC) increased with increasing r, by up to 2% and 4% for 18/23 and 6 MV, respectively. The GESF concept provides acceptable accuracy (<2%) for the calculation of Scp of the ASFs used in most clinical situations (except from SOF with EFC at large r), and thus can be used in MU verification calculations.

A rapid performance prediction method for Two-Phase liquid metal MHD generators based on Quasi-One-Dimensional model

Liquid metal magnetohydrodynamic (LMMHD) power generation is a novel and highly efficient technology with broad application prospects. In our previous study, a novel LMMHD enhanced Closed-Brayton-Cycle power generation system for hypersonic vehicles was proposed, and its excellent power generation performance was demonstrated by thermodynamic analysis. The most important component of LMMHD system is the power generation channel, and the rapid and accurate performance prediction and optimization design methods of which are the prerequisites for further research. In this study, a quasi-one-dimensional modeling method for LMMHD power generation channels based on a two-fluid model is developed. The performance variation laws with the geometric structure, magnetic field and boundary conditions, as well as the potential loss mechanisms are investigated. Results indicate that the most optimal or appropriate design can be achieved by optimizing the channel geometry and magnetic field, such as the optimal aspect ratio of 1 and the optimal Hartmann number of 2000. But it is difficult to balance the power output and efficiency through the boundary condition optimization. The gas–liquid slip can lead to approximately 20–40% of energy losses, making the elimination and suppression of slip effects a significant research focus in the relevant field.

Early universe: Gravitational waves and tensor metric fluctuations

In this work, we develop a model for the early universe in which preinflation is followed by a warming epoch and a subsequent sub-exponential quintessential expansion. We demonstrate that gravitational waves (GW) generated at the birth of the universe leave an imprint of that event, which is significantly suppressed a few Planckian times after that moment. We calculate the strengths of both, the total transverse GW and the b-polarization, for each mode. We use a nonlinear quantum algebra to calculate the expectation value of the quantum metric tensor perturbations: δgˆαβ, which are induced by the geometric quantum field with which we extend the Riemann manifold. A measure for the temperature fluctuations at the Hubble horizon is proposed: δT/Th=−bgαβδgˆαβ(k∗,t).

A coupled approach for predicting transmission loss of silencers with consideration of three-dimensional wave effects in inlet and outlet ducts

A coupled approach combining finite element method and duct mode expansion is proposed to determine the transmission loss of large silencers with consideration of three-dimensional wave effects in inlet and outlet ducts in the presence of flow. The finite element formula is established for the linearized potential equation (LPE), and the acoustic variables in the boundary integrals on the inlet and outlet surfaces are expressed analytically in terms of mode expansion. Since the introduced modal amplitude coefficients add unknown variables, the intrinsic functions are used as weighted functions to construct integral equations on the inlet and outlet surfaces to form a closed set of equations. Expressions of incident and transmitted sound powers are derived, and the transmission loss of silencer may be determined for any-giving mode incidence. For the symmetric silencers, specific combinations of acoustic modes are presented to improve the stability and efficiency of the approach. The dissipative and reactive silencers with rectangular and circular cross-sectional inlet and outlet ducts are examined, good agreements between predictions and benchmark data verified the present approach. Effects of Mach numbers and geometric parameters, flow fields and acoustic excitations on the acoustic attenuation behavior of the silencers as well as transmission loss predictions from different strategies are discussed.

Primordial gravitational waves in generalized Palatini gravity

Extended Palatini gravity is the metric-affine gravity theory characterized by zero torsion, nonzero metricity and a quadratic of the antisymmetric Ricci curvature. It reduces dynamically to general relativity plus a geometric Proca field. In this work, we study imprints of the geometric Proca field on the gravitational waves. Our results show that the geometric Proca leaves significant signatures in the gravitational wave signal, and gravitational wave energy density could be large enough to be detectable by the next upgrade of the existing GW detectors. Our results, if confirmed observationally, will be an indication that the gravity could be non-Riemannian in nature.

Geometric magnetism and anomalous enantio-sensitive observables in photoionization of chiral molecules

Chiral molecules are instrumental for molecular recognition in living organisms. Distinguishing between two opposite enantiomers, the mirror twins of the same chiral molecule, is both vital and challenging. Geometric magnetism enables a broad class of phenomena in solids including the anomalous electron velocity, the Hall effect, and related topological observables. Here we show that ultrafast electron currents excited in chiral molecules can generate geometric fields which enable a class of anomalous enantio-sensitive observables in photoionization. Next, we introduce the first member of this class: enantio-sensitive orientation of chiral molecules via photoionization. This effect opens unexplored routes to both enantio-separation and imaging of chiral dynamics on ultrafast time scales. Our work suggests that geometric magnetism in photoionization provides the bridge between the two geometrical properties, chirality and topology.

Relativistic Mechanics in Gravitational Field within Oblate Spheroidal Coordinates Based upon Riemannian Geometry for Rotating Homogeneous Mass Distribution

The emergence of the geometrical theory of gravitation (general relativity) by Albert Einstein in his quest to unite special relativity and the Newtonian law of universal gravitation has led to several Mathematical approaches for the exact and analytical solution for all gravitational fields in nature. The first and the most famous analytical solution was the Schwarzchild’s which can be constructed by finding a mapping where the metric tensor takes a simple form i.e. the vanishing of the non-diagonal elements. In this paper, we construct exact solution of the Einstein geometrical gravitational field equation using Riemannian metric tensor called the golden metric tensor that was first developed by Howusu, in the year 2009, for the rotating homogeneous mass distribution within oblate Spheroidal Coordinates. The equations of motion for test particles in the Oblate Spheroidal Geometry were derived using the coefficient of affine connection. Then the law of conservation of momentum and energy are equivalently formulated using the generalized Lagrangian as compared to the analytical solution of the Schwarzchild’s gravitational field. We also derived the planetary equation of motion in the equatorial plane of the Oblate Spheroidal body for this gravitational field.

The design and characterization of a novel dynamic collimator system for synchrotron radiotherapy applications.

Novel synchrotron radiotherapy techniques are currently limited to using prefabricated beam-limiting blocks for field definition. For large experiments, a single square tungsten block is often used for every treatment since conformal blocks are both patient and field specific, and require long lead times for fabrication. Future synchrotron radiotherapy treatments would benefit from a dynamic collimatorsystem. We developed and tested a novel collimator design for use on the Imaging and Medical Beamline (IMBL) at the ANSTO AustralianSynchrotron. The maximum usable beam size on IMBL is 50-mm wide by 3-mm tall. Given the beam shape, targets must be vertically scanned through the synchrotron beam to cover the target volume. To shape the beam, a novel collimator design was developed, consisting of two semi-circular leaves made from 4-mm thick tungsten sheets, with each leaf capable of both vertical and horizontal movement. A software model was created to optimize motor trajectories and generate deliverable treatment fields. A series of geometric field shapes and clinical target volumes were delivered using the collimator and imaged with a digital imaging detector. Four similarity metrics (volumetric similarity, DICE, and the average and maximum Hausdorff distances) were used to measure differences between the input and planned fields, and the planned and deliveredfields. Differences between input and planned fields increased with delivery speed, and were worse for rectangular and square fields compared to circular fields. However, the differences between planned and delivered fields were small, where the maximum average deviation between the fields was 0.25mm (one pixel). Field repeatability was consistent with no difference (σ = 0 for all metrics) observed in consecutively deliveredfields. We have successfully built and demonstrated a novel collimator for synchrotron radiotherapy applications on IMBL. Several design improvements have been highlighted and will be addressed in future revisions the collimator. However, in its current state, the collimator enables dynamically delivered conformal treatment fields to be utilized on IMBL, and is ready to support the forthcoming canine treatments on IMBL.

Symmetry analysis, closed-form invariant solutions and dynamical wave structures of the Benney-Luke equation using optimal system of Lie subalgebras

Nonlinear evolution equations (NLE-Es) arise in a variety of domains, including fluid dynamics, biological sciences, the study of solid states, optical fibres, coastal engineering, ocean physics, and nonlinear complex physical systems. This work examines the Lie symmetry study of the Benney-Luke (B-L) equation relying on two nonzero real parameters. The Lie infinitesimal generators, the one-dimensional optimal system, and the geometric vector fields are all obtained using the Lie symmetry technique. To start, we find close-form invariant solutions by employing symmetry reduction of Lie subalgebras. In some reduction cases, we transform the B-L equation into a variety of non-linear ordinary differential equations (NL-ODEs), which have the benefit of providing a significant number of closed-form solitary wave solutions. The traveling wave solutions contain special functional parameter solutions, trigonometric function solutions, rational function solutions, and hyperbolic trigonometric function solutions. The dynamical profile of closed-form wave solutions exhibits periodic solitons, dark peakon solitons, bright solitons, and bright solitons (bell shape) which we reveal in our study for the first time. By using Noether’s method, we also determine the conservation laws. Finally, three-dimensional diagrams are used to reveal the dynamical investigation of some known solutions.

Optimizing efficiency in the creation of patient-specific plates through field-driven generative design in maxillofacial surgery

Field driven design is a novel approach that allows to define through equations geometrical entities known as implicit bodies. This technology does not rely upon conventional geometry subunits, such as polygons or edges, rather it represents spatial shapes through mathematical functions within a geometrical field. The advantages in terms of computational speed and automation are conspicuous, and well acknowledged in engineering, especially for lattice structures. Moreover, field-driven design amplifies the possibilities for generative design, facilitating the creation of shapes generated by the software on the basis of user-defined constraints. Given such potential, this paper suggests the possibility to use the software nTopology, which is currently the only software for field-driven generative design, in the context of patient-specific implant creation for maxillofacial surgery. Clinical scenarios of applicability, including trauma and orthognathic surgery, are discussed, as well as the integration of this new technology with current workflows of virtual surgical planning. This paper represents the first application of field-driven design in maxillofacial surgery and, although its results are very preliminary as it is limited in considering only the distance field elaborated from specific points of reconstructed anatomy, it introduces the importance of this new technology for the future of personalized implant design in surgery.

High-order field theory and a weak Euler–Lagrange–Barut equation for classical relativistic particle-field systems

In both quantum and classical field systems, conservation laws such as the conservation of energy and momentum are widely regarded as fundamental properties. A broadly accepted approach to deriving conservation laws is built using Noether’s method. However, this procedure is still unclear for relativistic particle-field systems where particles are regarded as classical world lines. In the present study, we establish a general manifestly covariant or geometric field theory for classical relativistic particle-field systems. In contrast to quantum systems, where particles are viewed as quantum fields, classical relativistic particle-field systems present specific challenges. These challenges arise from two sides. The first comes from the mass-shell constraint. To deal with the mass-shell constraint, the Euler–Lagrange–Barut (ELB) equation is used to determine the particle’s world lines in the four-dimensional (4D) Minkowski space. Besides, the infinitesimal criterion, which is a differential equation in formal field theory, is reconstructed by an integro-differential form. The other difficulty is that fields and particles depend on heterogeneous manifolds. To overcome this challenge, we propose using a weak version of the ELB equation that allows us to connect local conservation laws and continuous symmetries in classical relativistic particle-field systems. By applying a weak ELB equation to classical relativistic particle-field systems, we can systematically derive local conservation laws by examining the underlying symmetries of the system. Our proposed approach provides a new perspective on understanding conservation laws in classical relativistic particle-field systems.

Vibration Scale Model of a Converter Transformer Based on the Finite Element and Similarity Principle and Its Preparation

A similarity criterion suitable for studying the vibration characteristics of converter transformers is proposed based on comprehensive consideration of geometric dimensions, electric field, magnetic field, force field, sound field, and coupling field interface interactions. By comparing the magnetic field, stress, displacement, sound field distribution, and vibration characteristics of the scale model of the converter transformer with the initial model, the reliability of the similarity criterion was determined. Based on the vibration similarity criterion of the converter transformer, a prototype of the proportional model was designed and manufactured, and vibration signals under no-load and load conditions were tested. These signals correspond to the vibration signals of the iron core and winding in the finite element model, respectively. Through comparative analysis, the reliability of the prototype and the vibration similarity model of the converter transformer has been proven, which can provide an accurate and effective laboratory research platform for in-depth research on the vibration and noise of the converter transformer and equipment protection.

A geometrical ghost field and its cosmological implications

A geometrical ghost field and its cosmological implications

X-ray source arrays for volumetric imaging during radiotherapy treatment

This work presents a novel hardware configuration for radiotherapy systems to enable fast 3D X-ray imaging before and during treatment delivery. Standard external beam radiotherapy linear accelerators (linacs) have a single X-ray source and detector located at ± 90° from the treatment beam respectively. The entire system can be rotated around the patient acquiring multiple 2D X-ray images to create a 3D cone-beam Computed Tomography (CBCT) image before treatment delivery to ensure the tumour and surrounding organs align with the treatment plan. Scanning with a single source is slow relative to patient respiration or breath holds and cannot be performed during treatment delivery, limiting treatment delivery accuracy in the presence of patient motion and excluding some patients from concentrated treatment plans that would be otherwise expected to have improved outcomes. This simulation study investigated whether recent advances in carbon nanotube (CNT) field emission source arrays, high frame rate (60 Hz) flat panel detectors and compressed sensing reconstruction algorithms could circumvent imaging limitations of current linacs. We investigated a novel hardware configuration incorporating source arrays and high frame rate detectors into an otherwise standard linac. We investigated four potential pre-treatment scan protocols that could be achieved in a 17 s breath hold or 2–10 1 s breath holds. Finally, we demonstrated for the first time volumetric X-ray imaging during treatment delivery by using source arrays, high frame rate detectors and compressed sensing. Image quality was assessed quantitatively over the CBCT geometric field of view as well as across each axis through the tumour centroid. Our results demonstrate that source array imaging enables larger volumes to be imaged with acquisitions as short as 1 s albeit with reduced image quality arising from lower photon flux and shorter imaging arcs.