Singular Regions Research Articles

Chaos in the vicinity of a singularity in the three-body problem: The equilateral triangle experiment in the zero angular momentum limit

We present numerical simulations of the gravitational three-body problem, in which three particles lie at rest close to the vertices of an equilateral triangle. In the unperturbed problem, the three particles fall towards the center of mass of the system to form a three-body collision, or singularity, where the particles overlap in space and time. By perturbing the initial positions of the particles, we are able to study chaos in the vicinity of the singularity. Here we cover both the singular region close to the unperturbed configuration and the binary-single scattering regime where one side of the triangle is very short compared to the other two. We make phase space plots to study the regular and ergodic subsets of our simulations and compare them with the outcomes expected from the statistical escape theory of the three-body problem. We further provide fits to the ergodic subset to characterize the properties of the left-over binaries. We identify the discrepancy between the statistical theory and the simulations in the regular subset of interactions, which only exhibits weak chaos. As we decrease the scale of the perturbations in the initial positions, the phase space becomes entirely dominated by regular interactions, according to our metric for chaos. Finally, we show the effect of general relativity corrections by simulating the same scenario with the inclusion of post-Newtonian corrections to the equations of motion.

A factorisation-aware matrix element emulator

In this contribution I present a neural network based model to emulate matrix elements. This model improves on existing methods by taking advantage of the known factorisation properties of matrix elements to drastically improve per-point accuracy. By building in the factorisation properties into the model we can control the behaviour of simulated matrix elements when extrapolating into more singular regions than the ones used for training the neural network. I apply this model to the case of leading-order jet production in e + e −collisions with up to five jets where this model can reproduce the matrix elements with errors below the one-percent level on the phase-space covered during fitting and testing, with a robust extrapolation to the parts of the phase-space where the matrix elements are more singular than seen at the fitting stage. Finally I discuss the usage and performance of the model on a GPU system.

Possible scenarios transgressing the nondegeneracy theorem

In contrast to the nondegeneracy theorem, we present various scenarios in one-dimensional quantum mechanics that demonstrate how the Wronskian of two bound-state eigenfunctions with the same energy eigenvalue can be zero without implying that the eigenfunctions are linearly dependent. It is shown that the nondegeneracy theorem fails only when the potential makes different bound-state solutions corresponding to the same energy vanish at the singular point or region of singularity.

Boundary Layer Lines of Solutions to Singularly Perturbate Equations With a Saddle Point

In this paper, we study the asymptotic behavior of solutions of singularly perturbed equations in complex domains. The considered equations have a saddle point. The main goal is to prove the existence of boundary-layer lines, boundary-layer regions and regular, singular regions, identifying their features in comparison with previous studies. Harmonic functions and their level lines are involved in solving the set problem. Using the level line, geometric constructions are carried out. The area under consideration is divided into parts and integration paths are chosen that ensure the convergence of some functions with respect to a small parameter. Using the method of successive approximations, the existence and boundedness of the solution of the equation is proved. The features of the boundary lines are revealed.

Asymptotic Behavior of a Function of a Complex Variable With a Parameter and Zeros

The study of the asymptotic behavior of solutions of singularly perturbed equations in complex domains reduces to the study of integrals of exponential functions of a complex variable with a parameter. The problem is to study the asymptotic behavior of such functions. The functions in the exponent have zeros. The study of such functions is hampered by the fact that it is necessary to select a certain part from a given region and choose integration paths that ensure the boundedness of the considered functions with respect to a small parameter. The saddle point method is not applicable to such integrals. To solve the problem, the level lines of harmonic functions generated by analytic functions are applied. The level lines divide the area in the complex plane into parts. Integration paths are chosen that ensure boundedness of integrals with respect to a small parameter. Boundary-layer lines, areas where the integrals have no limit with respect to a small parameter, but are limited with respect to the modulus, are revealed; regular domains (integrals have a limit); singular regions (integrals are not limited). All constructions are accompanied by corresponding drawings. In the future, the results of this paper can be used for the theory of singularly perturbed equations in a complex domain.

Evolutionary Invariant of the Structure of DNA Double Helix in RNAP II Core Promoters.

Eukaryotic and archaeal RNA polymerase II (POL II) machinery is highly conserved, regardless of the extreme changes in promoter sequences in different organisms. The goal of our work is to find the cause of this conservatism. The representative sets of aligned promoter sequences of fifteen organisms belonging to different evolutional stages were studied. Their textual profiles, as well as profiles of the indexes that characterize the secondary structure and the mechanical and physicochemical properties, were analyzed. The evolutionarily stable, extremely heterogeneous special secondary structure of POL II core promoters was revealed, which includes two singular regions—hexanucleotide “INR” around TSS and octanucleotide “TATA element” of about −28 bp upstream. Such structures may have developed at some stage of evolution. It turned out to be so well matched for the pre-initiation complex formation and the subsequent initiation of transcription for POL II machinery that in the course of evolution there were selected only those nucleotide sequences that were able to reproduce these structural properties. The individual features of specific sequences representing the singular region of the promoter of each gene can affect the kinetics of DNA-protein complex formation and facilitate strand separation in double-stranded DNA at the TSS position.

Accurate fracture analysis of multi-material V-notched Reissner plates under bending or twisting

An accurate fracture analysis of multi-material V-notched Reissner plates under bending or twisting is performed by a new coupling approach of isogeometric analysis (IGA) and eigenfunction expansion method. In this approach, the overall plate is divided into two regions: a singular region near the notch tip (near field) and a regular region far away from the notch tip (far field). The overall model is discretized by a T-spline-based IGA. Asymptotic solutions of singular stress fields are applied to the near field, and therefore, the large number of fundamental unknowns of control points is reduced into a small set of undetermined coefficients. The unknowns in the far field remain unchanged. Consequently, the computational cost is significantly reduced, and explicit expressions of singular stress components in the vicinity of notch tip are obtained without post processing. Comparisons are presented to demonstrate the accuracy and convergence of the present approach. Effects of key influencing factors on the singularity order and singular items of stresses are investigated also.

A new algebraic solution for transforming Cartesian to geodetic coordinates

An exact and stable algebraic solution based on solving a quartic equation with respect to the cosine function of the reduced latitude is proposed to transform Cartesian into geodetic coordinates. The unique proper root of the equation appropriate to the transformation is chosen from all possible roots by rigorous analyses and the singular region of the transformation that in which there at least one component of the geodetic coordinates is indeterminate or non-single-valued characteristics are determined strictly. The new algorithm does not need any approximation and the instability problems incurred in other algebraic solutions are overcome. For practical applications, the algorithm performs comparably to that of [Vermeille, H., 2011. An analytical method to transform geocentric into geodetic coordinates. Journal of geodesy, 85 (2), 105–117.] and shows a certain superiority in the singular disc over Vermeille's algorithm.

Spherically symmetric vacuum solutions and horizons in covariant f(T) gravity theory

In this paper we study properties that the vacuum must possess in the minimal extension to the teleparallel equivalent of general relativity (TEGR) where the action is supplemented with a quadratic torsion term. No assumption is made about the weakness of the quadratic term although in the weak-field regime the validity of our previously derived perturbative solution is confirmed. Regarding the exact nature of the vacuum, it is found that if the center of symmetry is to be regular, the mathematical conditions on the tetrad at the isotropy point mimic those of general relativity. With respect to horizons it is found that, under very mild assumptions, a smooth horizon cannot exist unless the quadratic torsion coupling, $\alpha$, vanishes, which is the TEGR limit (with the Schwarzschild tetrad as its solution). This analysis is then supplemented with computational work utilizing asymptotically Schwarzschild boundary data. It is verified that in no case studied does a smooth horizon form. For $\alpha > 0$ naked singularities occur which break down the equations of motion before a horizon can form. For $\alpha < 0$ there is a limited range of $\alpha$ where a vacuum horizon might exist but, if present, the horizon is singular. Therefore physically acceptable black hole horizons are problematic in the studied theory at least within the realm of vacuum static spherical symmetry. These results also imply that static spherical matter distributions generally must have extra restrictions on their spatial extent and stress-energy bounds so as to render the vacuum solution invalid in the singular region and make the solutions finite.

Cosmologies, singularities and quantum extremal surfaces

Following [1], we study quantum extremal surfaces in various families of cosmologies with Big-Crunch singularities, by extremizing the generalized entropy in 2-dimensional backgrounds which can be thought of as arising from dimensional reduction. Focussing first on the isotropic AdS Kasner case, introducing a spatial regulator enables relating the locations in time of the quantum extremal surface and the observer. This shows that the quantum extremal surface lags behind the observer location. A potential island-like region, upon analysing more closely near the island boundary, turns out to be inconsistent. Similar results arise for other holographic cosmologies. We then study certain families of null Kasner singularities where we find that the quantum extremal surface can reach the near singularity region although the on-shell generalized entropy is generically singular. We also study other cosmologies including de Sitter (Poincare slicing) and FRW cosmologies under certain conditions.

An introduction to classical monodromy: Applications to molecules in external fields

An integrable Hamiltonian system presents monodromy if the action-angle variables cannot be defined globally. As a prototype of classical monodromy with azimuthal symmetry, we consider a linear molecule interacting with external fields and explore the topology structure of its phase space. Based on the behavior of closed orbits around singular points or regions of the energy–momentum plane, a semi-theoretical method is derived to detect classical monodromy. The validity of the monodromy test is numerically illustrated for several systems with azimuthal symmetry.

Асимптотика решений однородного бисингулярно возмущенного дифференциального уравнения в теории обобщенных функций

In the space of generalized functions, a homogeneous system of singularly perturbed differential equations in the case of stability change is considered. A theorem on generalized solutions of the corresponding degenerate system of the equation is proved. At special points, the asymptotic closeness of the solutions of the perturbed and unperturbed problems in the singular domain is established. The novelty of the work lies in the fact that, for the first time, an estimate for the singular region was obtained. A degenerate system has a special point. At this point, we solve the equation in generalized functions. In turn, this is also a novelty, because previously performed works only considered the classical solution. The following novelty of the work lies in the fact that we take the starting point in an unstable interval and also head towards the unstable interval. This property is not characteristic of previously published works.

Single-impurity Anderson model out of equilibrium: A two-particle semianalytic approach

We apply a two-particle semianalytic approach to a single Anderson impurity attached to two biased metallic leads. The theory is based on reduced parquet equations justified in critical regions of singularities in the Bethe-Salpeter equations. It offers a way to treat one-particle and two-particle thermodynamic and spectral quantities on the same footing. The two-particle vertices are appropriately renormalized so that spurious transitions into the magnetic state of the weak-coupling approximations are suppressed. The unphysical hysteresis loop in the current-voltage characteristics is thereby eliminated. Furthermore, in the linear response regime, we qualitatively reproduce the three transport regimes with the increasing temperature: from the Kondo resonant tunneling through the Coulomb-blockade regime up to a sequential tunneling regime. Far from equilibrium, we find that the bias plays a similar role as the temperature in destroying the Kondo resonant peak when the corresponding energy scale is comparable with the Kondo temperature. Aside from that, the applied voltage in low bias is shown to develop spectral peaks around the lead chemical potentials as observed in previous theoretical and experimental studies.

Inverse geodetic problem for long distance based on improved Vincenty’s formula

Abstract Solutions of geodetic problem is an important technical problem in geodesy. Based on the comprehensive analysis of the characteristics of Bessel’s formula and Vincenty’s formula, an improved algorithm is proposed. Firstly, arctangent formula is replaced by arccosine formula to calculate spherical surface angular distance. Secondly, Bessel’s formula is used to replace Vincenty’s formula in the singular region. Finally, the test formulas of geodetic line and azimuth are given. According to the test formula, four points are selected for the test, which are located in the equator, low latitude, middle latitude and high latitude. The geodesic length range of the test is 1000–18000 km and the azimuth range is 0 ∘ 0{\hspace{0.1667em}^{\circ }} – 360 ∘ 360{\hspace{0.1667em}^{\circ }} . The experimental results show that the improved algorithm can solve the problem of singularity and quadrant judgment well. The solution precision of geodetic line is better than 1 mm, and the precision of azimuth is less than 0.0001 second.

On the reducibility of the ellipsoidal system

Lamé introduced a triaxial ellipsoidal system, and via some ingenious arguments, he managed to spectrally decompose the Laplace operator and define ellipsoidal harmonic functions. Since then, many authors modified the Lamé system and proposed some related ellipsoidal coordinate variables. Perhaps the most important of them is the system introduced by Jacobi, in connection with the geodesic curves on an ellipsoid. All other attempts to introduce ellipsoidal coordinates are basically modifications of the Jacobi or the Lamé system. However, an important question in the theory of ellipsoidal harmonics is the way in which this theory reduces to the spheroidal and spherical systems. This is by no means a straightforward procedure, since the relative limiting cases lead to generic underdetermined forms. The basic difficulty is due to the different dimensionality of the singularity regions corresponding to each system. Note that this region has zero dimensions in the case of the sphere, one dimension in the case of the prolate spheroid, and two dimensions in the case of the oblate spheroid and the ellipsoid. The present work provides a systematic way to obtain these reductions and to establish a correspondence between ellipsoidal, spheroidal, and spherical harmonics. We have also demonstrated how our approach applies to the proposed modified Jacobi ellipsoidal system when we connect the Lamé with the Jacobi system through certain non‐degenerable relations. The importance of these geometrical degeneracies lies in the fact that the solution of any boundary value problem in ellipsoidal geometry provides immediately the solutions in all special geometries of prolate and oblate spheroids, discs, needles, or spheres.

Nonlinear error control of five-axis machining singular region

Aiming at the singular problem of large nonlinear error caused by the extreme change of rotation in five axis NC machining of complex surface. An optimization algorithm of tool path in singular region is proposed. Taking a-c double turntable five axis linkage NC machine tool as an example, firstly, the mathematical models of tool local milling ability, machining bandwidth and tool axis inclination are established. Based on the two constraint models, the mechanism of singularity and the size of singular region are analyzed, and it is proposed to adjust the side angle with the tool contact as the rotation center to avoid the singular region. The modified tool axis vector that does not meet the accuracy is interpolated recursively according to the interpolation principle. This method solves the problem of large nonlinear error caused by the extreme rotation of the rotating axis in the singular region on the basis of meeting the requirement of no curvature interference. At the same time, the processing bandwidth before and after modification is guaranteed to remain unchanged. Simulation results show that the algorithm can effectively detect the singular region and improve the machining quality in the singular region.

The role of polarseaworthiness in shipping planning for infrastructure projects in the Arctic: The case of Yamal LNG plant

The numerous Artic port industrial complex projects are currently marked by a lack of both railroad and road infrastructures and require a heavy cargo shipping service. Yet, navigation in the Arctic implies facing harsh sailing conditions. To prevent risk arising from this singular region, the IMO Polar Code imposes mandatory tools as the Polar Ship Certificate and suggests others such as POLARIS. At the crossroads of container and tramp market, scholars have paid little attention to the transportation of heavy cargo. The objective of this research is to assess the impact of the Polar Code policy framework and its tools on the management of a highly strategic project. To do so, we shall rely on the case of the transport and assembly of the Yamal LNG plant modules. Yamal LNG started its production in late 2019 and has already produced 19 million tons per year. The paper contemplates a ship routing and scheduling optimization model that considers different fleet types and ice conditions and applies the POLARIS Risk Index Outcome (RIO) in Arctic waters. Even though the modules are required on site in summer, due to the extreme weather conditions and limited accessibility in winter, our results highlight that the use of Polar Class vessels allowing year-round navigation in Arctic waters is critical to ensure the success of such projects. Indeed, Polar Class 3 vessels, as ships with the greatest possible “polarseaworthiness”, are capable of significantly reducing delays in project’s completion. It also emphasizes the paramount role of the Polar Code and related tools in the shipping risk management of Arctic infrastructure projects.

Neurophysiological considerations for visual implants.

Neural implants have the potential to restore visual capabilities in blind individuals by electrically stimulating the neurons of the visual system. This stimulation can produce visual percepts known as phosphenes. The ideal location of electrical stimulation for achieving vision restoration is widely debated and dependent on the physiological properties of the targeted tissue. Here, the neurophysiology of several potential target structures within the visual system will be explored regarding their benefits and downfalls in producing phosphenes. These regions will include the lateral geniculate nucleus, primary visual cortex, visual area 2, visual area 3, visual area 4 and the middle temporal area. Based on the existing engineering limitations of neural prostheses, we anticipate that electrical stimulation of any singular brain region will be incapable of achieving high-resolution naturalistic perception including color, texture, shape and motion. As improvements in visual acuity facilitate improvements in quality of life, emulating naturalistic vision should be one of the ultimate goals of visual prostheses. To achieve this goal, we propose that multiple brain areas will need to be targeted in unison enabling different aspects of vision to be recreated.

An Improved Weighted Gradient Projection Method for Inverse Kinematics of Redundant Surgical Manipulators.

Different from traditional redundant manipulators, the redundant manipulators used in the surgical environment require the end effector (EE) to have high pose (position and orientation) accuracy to ensure the smooth progress of the operation. When analyzing the inverse kinematics (IK) of traditional redundant manipulators, gradient-projection method (GPM) and weighted least-norm (WLN) method are commonly used methods to avoid joint position limits. However, for the traditional GPM and WLN method, when joints are close to their limits, they stop moving, which greatly reduces the accuracy of the IK solution. When robotic manipulators enter a singular region, although traditional damped least-squares (DLS) algorithms are used to handle singularities effectively, motion errors of the EE will be introduced. Furthermore, selecting singular region through trial and error may cause some joint velocities exceed their corresponding limits. More importantly, traditional DLS algorithms cannot guide robotic manipulators away from singular regions. Inspired by the merits of GPM, WLN, and DLS methods, an improved weighted gradient projection method (IWGPM) is proposed to solve the IK problem of redundant manipulators used in the surgical environment with avoiding joint position limits and singularities. The weighted matrix of the WLN method and the damping factor of the DLS algorithm have been improved, and a joint limit repulsive potential field function and singular repulsive potential field function belong to the null space are introduced to completely keep joints away from the damping interval and redundant manipulators away from the unsafe region. To verify the validity of the proposed IWGPM, simulations on a 7 degree of freedom (DOF) redundant manipulator used in laparoscopic surgery indicate that the proposed method can not only achieve higher accuracy IK solution but also avoid joint position limits and singularities effectively by comparing them with the results of the traditional GPM and WLN method, respectively. Furthermore, based on the proposed IWGPM, simulation tests in two cases show that joint position limits have a great impact on the orientation accuracy, and singular potential energy function has a great impact on the position accuracy.

A factorisation-aware Matrix element emulator

In this article we present a neural network based model to emulate matrix elements. This model improves on existing methods by taking advantage of the known factorisation properties of matrix elements. In doing so we can control the behaviour of simulated matrix elements when extrapolating into more singular regions than the ones used for training the neural network. We apply our model to the case of leading-order jet production in e+e− collisions with up to five jets. Our results show that this model can reproduce the matrix elements with errors below the one-percent level on the phase-space covered during fitting and testing, and a robust extrapolation to the parts of the phase-space where the matrix elements are more singular than seen at the fitting stage.