Singular Points Research Articles

Polar Bloch points in strained ferroelectric films

Topological domain structures have drawn great attention as they have potential applications in future electronic devices. As an important concept linking the quantum and classical magnetism, a magnetic Bloch point, predicted in 1960s but not observed directly so far, is a singular point around which magnetization vectors orient to nearly all directions. Here we show polar Bloch points in tensile-strained ultrathin ferroelectric PbTiO3 films, which are alternatively visualized by phase-field simulations and aberration-corrected scanning transmission electron microscopic imaging. The phase-field simulations indicate local steady-state negative capacitance around the Bloch points. The observation of polar Bloch points and their emergent properties consequently implies novel applications in future integrated circuits and low power electronic devices.

Unified framework for the separation property in binary phase-segregation processes with singular entropy densities

Abstract This paper investigates the separation property in binary phase-segregation processes modelled by Cahn-Hilliard type equations with constant mobility, singular entropy densities and different particle interactions. Under general assumptions on the entropy potential, we prove the strict separation property in both two and three-space dimensions. Namely, in 2D, we notably extend the minimal assumptions on the potential adopted so far in the literature, by only requiring a mild growth condition of its first derivative near the singular points $\pm 1$ , without any pointwise additional assumption on its second derivative. For all cases, we provide a compact proof using De Giorgi’s iterations. In 3D, we also extend the validity of the asymptotic strict separation property to the case of fractional Cahn-Hilliard equation, as well as show the validity of the separation when the initial datum is close to an ‘energy minimizer’. Our framework offers insights into statistical factors like particle interactions, entropy choices and correlations governing separation, with broad applicability.

Solutions with moving singularities for a one-dimensional nonlinear diffusion equation

AbstractThe aim of this paper is to study singular solutions for a one-dimensional nonlinear diffusion equation. Due to slow diffusion near singular points, there exists a solution with a singularity at a prescribed position depending on time. To study properties of such singular solutions, we define a minimal singular solution as a limit of a sequence of approximate solutions with large Dirichlet data. Applying the comparison principle and the intersection number argument, we discuss the existence and uniqueness of a singular solution for an initial-value problem, the profile near singular points and large-time behavior of solutions. We also give some results concerning the appearance of a burning core, convergence to traveling waves and the existence of an entire solution.

A replaceable-component method to construct single-degree-of-freedom multi-mode planar mechanisms with up to eight links

Abstract. The multi-mode planar mechanisms (MMPMs) are excellent-performance reconfigurable mechanisms, which not only inherit structural characteristics of planar mechanisms but also have the multi-task, multi-working-condition application advantages of multi-mode mechanisms. However, lacking common bifurcation analysis and construction methods, their industrial application and development are seriously hindered. This paper presents a replaceable-component method to construct a set of single-degree-of-freedom (single-DOF) MMPMs based on the branch graphs of the corresponding planar mechanisms and the proposed multi-mode modules (MMMs). First, according to the established loop equations, all the kinematic information of the original planar mechanism is obtained by the branch graphs and singularity points using Maple. Then, compared to the relationship between the concepts of the branch and motion mode, the number and continuity of branches are taken as the index to identify the potential bifurcation and mode conversion ability for the corresponding planar mechanisms. Subsequently, the MMM is presented to help the planar mechanisms break the singularity positions to form the corresponding MMPMs, and the steps of constructing single-DOF MMPMs are summarized. Finally, a single-DOF Stephenson six-bar three-mode planar mechanism, a Watt six-bar three-mode planar mechanism, and an eight-bar four-mode planar mechanism are constructed for the first time, and the corresponding multi-mode motion analyses are made. The results can give the available configuration for the design of corresponding MMPMs. The proposed method will provide strong guidance for the configuration design of MMPMs.

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

Fringe-based depth segmentation via minimum-fringe-period-based singular points extraction

Fringe-based depth segmentation via minimum-fringe-period-based singular points extraction

Pole-skipping and chaos in hot\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{M}{\ ext{QCD}}$$\\end{document}

We address the question of whether thermal QCD at high temperature is chaotic from the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{M}$$\\end{document} theory dual of QCD-like theories at intermediate coupling as constructed in [1]. The equations of motion of the gauge-invariant combination Zs(r) of scalar metric perturbations is shown to possess an irregular singular point at the horizon radius rh. Very interestingly, at a specific value of the imaginary frequency and momentum used to read off the analogs of the “Lyapunov exponent” λL and “butterfly velocity” vb not only does rh become a regular singular point, but truncating the incoming mode solution of Zs(r) as a power series around rh, yields a “missing pole”, i.e., Cn,n+1 = 0, det M(n) = 0, n ∈ \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb{Z}}^{+}$$\\end{document} is satisfied for a single n ≥ 3 depending on the values of the string coupling gs, number of (fractional) D3 branes (M)N and flavor D7-branes Nf in the parent type IIB set [2], e.g., for the QCD(EW-scale)-inspired N = 100, M = Nf = 3, gs = 0.1, one finds a missing pole at n = 3. For integral n > 3, truncating Zs(r) at \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}\\left({\\left(r-{r}_{h}\\right)}^{n}\\right)$$\\end{document}, yields Cn,n+1 = 0 at order n, ∀n ≥ 3. Incredibly, (assuming preservation of isotropy in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb{R}}^{3}$$\\end{document} even with the inclusion of higher derivative corrections) the aforementioned gauge-invariant combination of scalar metric perturbations receives no \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}\\left({R}^{4}\\right)$$\\end{document} corrections. Hence, (the aforementioned analogs of) λL, vb are unrenormalized up to \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}\\left({R}^{4}\\right)$$\\end{document} in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{M}$$\\end{document} theory.

Fast numerical integration of highly oscillatory Bessel transforms with a Cauchy type singular point and exotic oscillators

Fast numerical integration of highly oscillatory Bessel transforms with a Cauchy type singular point and exotic oscillators

Limit Cycles and Local Bifurcation of Critical Periods in a Class of Switching Z2 Equivariant Quartic System

In this paper, the limit cycles and local bifurcation of critical periods for a class of switching Z2 equivariant quartic system with two symmetric singularities are investigated. First, through the computation of Lyapunov constants, the conditions of the two singularities to become the centers are determined. Then, we prove that there are at most 18 limit cycles with a distribution pattern of 9-9 around the two symmetric singular points of the system. Numerical simulation is conducted to validate the obtained results. Furthermore, by calculating the period constants, we determine the conditions for the critical point to be a weak center of finite order. Finally, the number of local critical periods that bifurcate from the equilibrium point under the center conditions is discussed. This study presents the first example of a quartic switching smooth system with 18 limit cycles and 4 local critical periods bifurcating from two symmetric singular points.

Deformation in the wrinkle–crease transformation

We investigate deformation in the wrinkle–crease transformation to reveal the missing link between the wrinkling and creasing instabilities, which occur on the surface of elastomers under compression. Plane-strain finite element (FE) analysis is performed by combining a nonlinear perturbation approach to find two bifurcated solutions for the flat surface in a metastable state. The first solution is the stable deformation path for crease evolution, and the second solution is the previously unknown unstable deformation path from wrinkling to creasing. The two deformation paths are not independent, and they are connected at the critical strain for creasing εC. The resulting deformation at εC acts as a precursive deformation to cause creasing. Although the unstable path is initiated by the wrinkling instability as a bifurcation, the creasing instability (i.e., εC) is interpreted as the singular point at which the unstable path switches to the stable path. A chain reaction model is developed to describe the mechanism and deformation of the wrinkle–crease transformation, and it shows good agreement with the FE prediction within an inevitable limitation of FEs.

Global existence and partial regularity for the Landau-Lifshitz-Gilbert equation with helicity

In this paper, the Landau-Lifshitz-Gilbert equation with helicity is considered. In R3 or a bounded regular domain Ω of R3, we establish the global existence of a weak solution. In Rn, a global existence criterion and uniqueness of the smooth solution are given. In R1, the local smooth solution is indeed global with large initial data. In R2, we prove the existence of a global weak solution, which is smooth with the exception of at most finite singular points.

Gaussian maps for singular curves on Enriques surfaces

Gaussian maps for singular curves on Enriques surfaces

Topological defects and boundary states in four-dimensional topological insulator

We show the connection between the second Chern number and topological defects, in a (4+1)-dimensional time-reversal invariant Dirac lattice model. It is discovered that two types of topological defects, the five-dimensional (5D) and four-dimensional (4D) point defects arise from the singular points of wave functions together with the geometric meaning of the second Chern number. We demonstrated that the 5D point defects appear at the band crossing positions with a topological transition, leading to a jump of the second Chern number. The 4D point defects exist in an insulating bulk, whose topological charges can give the evaluations of the second Chern number of energy bands. Finally, we discussed the possible structures of the boundary states in the light of the realization way of the 4D model. Our theory provides not only a new perspective to grasp the second Chern number, but also a simple approach to derive its values without calculating any integrals.

The very singular solution for the Anisotropic Fast Diffusion Equation and its consequences

We construct the Very Singular Solution (VSS) for the Anisotropic Fast Diffusion Equation (AFDE) in the suitably good exponent range. VSS is a solution that, starting from an infinite mass located at one point as initial datum, evolves as an admissible solution of the corresponding equation everywhere away from the point singularity. It is expected to represent important properties of the fundamental solutions when the initial mass is very big. Here we work in the whole Euclidean space.In this setting we show how the diffusion process distributes mass from the initial infinite singularity with different rates along the different space directions. Indeed, and up to constant factors, there is a simple partition formula for the anisotropic mass expansion, given approximately as the minimum of separate 1-D VSS solutions. This striking fact is a consequence of the improved scaling properties of the special solution, and it has strong consequences.If we consider the family of fundamental solutions for different masses, we prove that they all share the same universal tail behaviour (i.e., for large |x|) as the VSS. Namely, their tail is asymptotically convergent to the unique VSS tail. This means that the VSS partition formula holds also for the fundamental solutions at spatial infinity. With the help of this analysis we study the behaviour of the class of nonnegative finite-mass solutions of the Anisotropic FDE, and prove the Global Harnack Principle (GHP) and the Asymptotic Convergence in Relative Error (ACRE) under a natural assumption on the decay of the initial tail.

Analysis of long transients and detection of early warning signals of extinction in a class of predator-prey models exhibiting bistable behavior.

In this paper, we develop a method of analyzing long transient dynamics in a class of predator-prey models with two species of predators competing explicitly for their common prey, where the prey evolves on a faster timescale than the predators. In a parameter regime near a singular zero-Hopf bifurcation of the coexistence equilibrium state, we assume that the system under study exhibits bistability between a periodic attractor that bifurcates from the singular Hopf point and another attractor, which could be a periodic attractor or a point attractor, such that the invariant manifolds of the coexistence equilibrium point play central roles in organizing the dynamics. To find whether a solution that starts in a vicinity of the coexistence equilibrium approaches the periodic attractor or the other attractor, we reduce the equations to a suitable normal form, and examine the basin boundary near the singular Hopf point. A key component of our study includes an analysis of the long transient dynamics, characterized by their rapid oscillations with a slow variation in amplitude, by applying a moving average technique. We obtain a set of necessary and sufficient conditions on the initial values of a solution near the coexistence equilibrium to determine whether it lies in the basin of attraction of the periodic attractor. As a result of our analysis, we devise a method of identifying early warning signals, significantly in advance, of a future crisis that could lead to extinction of one of the predators. The analysis is applied to the predator-prey model considered in Sadhu (Discrete Contin Dyn Syst B 26:5251-5279, 2021) and we find that our theory is in good agreement with the numerical simulations carried out for this model.

The Shell of the Cosmos

The Big Bang theory, widely accepted by cosmologists, explains the origin, expansion, and eventual fate of the universe. It was further developed by inflation theory, which solves issues like the horizon and flatness problems. This theory states that the Cosmic Microwave Background (CMB), a remnant of the Big Bang, permeates the entire universe. Cosmologists believe these photons were emitted during the recombination epoch, approximately 380,000 years after the Big Bang when the cosmos had a temperature of around 3,000 Kelvin. Additionally, the Lambda-Cold Dark Matter (ΛCDM) model addresses questions arising from observations of the cosmos, such as the existence of light elements like hydrogen, helium, lithium, and the anisotropy in the CMB, and eventually, the continuous expansion of space. In this model, dark energy, represented as the cosmological constant lambda (Λ), exerts negative pressure on empty space, counteracting the effects of gravity with a repulsive force. While the Standard Model of Particle Physics (SMPP) posits that the universe's matter originated from fundamental particles (quarks and leptons), it cannot explain the origin of these particles and how they came into existence, because general relativity and the SMPP cannot be integrated to explain matter production at the singularity point. T-Consciousness Cosmology introduces the ‘Spherical Cosmos Model’ (SCM) to answer questions about the cause of the explosion, expansion, and shape of the universe, the nature of ordinary matter and energy, dark matter and energy, the fate of the universe, the reason for the high density of objects in the depths of space, etc. In this model, the spherical cosmos has a shell called the ‘Shell of the Cosmos,’ made of Taheri Absolute Matter (TAM) that not only isolates it but also produces dark matter and dark energy, ordinary matter and ordinary energy, and finally space mesh from the inner surface to the inside of the cosmos since the birth of the universe. The Shell of this isolated sphere is currently expanding at a speed faster than the speed of light. In this perspective, dark matter and energy are the same space mesh that have been compressed to varying degrees. Dark energy, which is constantly being released from the Shell into the cosmos, unlike the standard model of cosmology, is one of the factors in the expansion of the isolated cosmos by creating positive pressure in it. Also in this model, the recombination epoch will always be located spherically at a certain distance from the Shell until the ultimate stage of Rebound. In other words, not only is the origin of the CMB not related to the past of the cosmos, but it also exists now, and given the vastness of the sphere and the position of the Earth within this sphere, we detect it in the microwave wavelength with a delay of several billion years. Therefore, the observed distant galaxies that have been attributed to the early epochs of the cosmos in the Big Bang model are currently being created by the Shell according to the Spherical Cosmos Model, and we are surrounded by particles and objects that are constantly being produced.

EXPONENTIAL ASYMPTOTICS USING NUMERICAL RATIONAL APPROXIMATION IN LINEAR DIFFERENTIAL EQUATIONS

Abstract Singularly perturbed ordinary differential equations often exhibit Stokes’ phenomenon, which describes the appearance and disappearance of oscillating exponentially small terms across curves in the complex plane known as Stokes lines. These curves originate at singular points in the leading-order solution to the differential equation. In many important problems, it is impossible to obtain a closed-form expression for these leading-order solutions, and it is therefore challenging to locate these singular points. We present evidence that the analytic leading-order solution of a linear differential equation can be replaced with a numerical rational approximation using the adaptive Antoulas–Anderson (AAA) method. Despite such an approximation having completely different singularity types and locations, we show that the subsequent exponential asymptotic analysis accurately predicts the exponentially small behaviour present in the solution. For sufficiently small values of the asymptotic parameter, this approach breaks down; however, the range of validity may be extended by increasing the number of poles in the rational approximation. We present a related nonlinear problem and discuss the challenges that arise due to nonlinear effects. Overall, our approach allows for the study of exponentially small asymptotic effects without requiring an exact analytic form for the leading-order solution; this permits exponential asymptotic methods to be used in a much wider range of applications.

On singularity and normality of regular nilpotent Hessenberg varieties

Regular nilpotent Hessenberg varieties form an important family of subvarieties of the flag variety, which are often singular and sometimes not normal varieties. Like Schubert varieties, they contain distinguished points called permutation flags. In this paper, we give a combinatorial characterization for a permutation flag of a regular nilpotent Hessenberg variety to be a singular point. We also apply this result to characterize regular nilpotent Hessenberg varieties which are normal algebraic varieties.

Large Péclet number forced convection from a circular wire in a uniform stream: hybrid approximations at small Reynolds numbers

Abstract We consider heat or mass transport from a circular cylinder under a uniform crossflow at small Reynolds numbers, $\mathrm{Re}\ll 1$ . This problem has been thwarted in the past by limitations inherent in the classical analyses of the singular flow problem, which have used asymptotic expansions in inverse powers of $\log \mathrm{Re}$ . We here make use of the hybrid approximation of Kropinski, Ward & Keller [(1995) SIAM J. Appl. Math.55, 1484], based upon a robust asymptotic expansion in powers of $\mathrm{Re}$ . In that approximation, the “inner” streamfunction is provided by the product of a pre-factor $S$ , a slowly varying function of $\mathrm{Re}$ , with a $\mathrm{Re}$ -independent “canonical” solution of a simple mathematical form. The pre-factor, in turn, is determined as an implicit function of $\log \mathrm{Re}$ via asymptotic matching with a numerical solution of the nonlinear single-scaled “outer” problem, where the cylinder appears as a point singularity. We exploit the hybrid approximation to analyse the transport problem in the limit of large Péclet number, $\mathrm{Pe}\gg 1$ . In that limit, transport is restricted to a narrow boundary layer about the cylinder surface – a province contained within the inner region of the flow problem. With $S$ appearing as a parameter, a similarity solution is readily constructed for the boundary-layer problem. It provides the Nusselt number as $0.5799(S\,\mathrm{Pe})^{1/3}$ . This asymptotic prediction is in remarkably close agreement with that of the numerical solution of the exact problem [Dennis, Hudson & Smith (1968) Phys. Fluids11, 933] even for moderate $\mathrm{Re}$ -values.

A Singular Tempered Sub-Diffusion Fractional Equation with Changing-Sign Perturbation

In this paper, we establish some new results on the existence of positive solutions for a singular tempered sub-diffusion fractional equation involving a changing-sign perturbation and a lower-order sub-diffusion term of the unknown function. By employing multiple transformations, we transform the changing-sign singular perturbation problem to a positive problem, then establish some sufficient conditions for the existence of positive solutions of the problem. The asymptotic properties of solutions are also derived. In deriving the results, we only require that the singular perturbation term satisfies the Carathéodory condition, which means that the disturbance influence is significant and may even achieve negative infinity near some time singular points.