Kinds Of Singularities Research Articles

Scattering singularities in Fano-resonant multilayer grating structure with phase-change materials

In non-Hermitian systems, the singularities refer to poles or zeros of the scattering and transfer matrices. Here, we investigate the kinds of singularities in a non-parity-time-symmetric multilayer structure which mainly consists of double layers of silver grating and one layer of phase-change material Ge2Sb2Te5 (GST). When the GST is in its crystalline phase, a unidirectional zero-reflection singularity, i.e., exceptional point (EP) is achieved with the aid of Fano resonance. Interestingly, a coherent perfect absorber (CPA)-laser singularity adjacent to the EP is observed in the Fano peak. Meanwhile, a few CPA singularities exist at other wavelengths. By switching the GST from its crystalline to amorphous phase, the CPA-laser singularity will be absent, and the CPA singularity can be switched to a laser singularity. The scattering spectra near the singularities are sensitive to a tiny perturbation of the gain material and the environmental refractive index. The study is beneficial for making switchable active optical implementations.

The $L^{p}$-boundedness of wave operators for fourth order Schrödinger operators on $\R^{4}$

We prove that the wave operators of the scattering theory for the fourth order Schrö­dinger operator \Delta^{2} + V(x) on {\mathbb{R}}^{4} are bounded in L^{p}({\mathbb{R}}^{4}) for the set of p ’s of (1,\infty) depending on the kind of spectral singularities of H at zero which can be described by the space of bounded solutions of (\Delta^{2} + V(x))u(x)=0 .

Coulomb and Higgs phases of G2-manifolds

Ricci flat manifolds of special holonomy are a rich framework as models of the extra dimensions in string/M-theory. At special points in vacuum moduli space, special kinds of singularities occur and demand a physical interpretation. In this paper we show that the topologically distinct G2-holonomy manifolds arising from desingularisations of codimension four orbifold singularities due to Joyce and Karigiannis correspond physically to Coulomb and Higgs phases of four dimensional gauge theories. The results suggest generalisations of the Joyce-Karigiannis construction to arbitrary ADE-singularities and higher order twists which we explore in detail in explicitly solvable local models. These models allow us to derive an isomorphism between moduli spaces of Ricci flat metrics on these non-compact G2-manifolds and flat ADE-connections on compact flat 3-manifolds which we establish explicitly for SU(n).

Parameter agnostic stacked wavelet transformer for detecting singularities

Machine learning algorithms especially deep neural networks have seen tremendous growth in their real-world deployment. While these algorithms have to yield high performances for the task at hand, it is of equal importance that they are robust against the different kinds of singularities or adversarial anomalies. Researchers have generally addressed singularity detection independently for each anomaly, however, for the algorithms to be effective in the real world, the singularity detection algorithm must be able to handle a wide variety of attacks whether trained individually on them or not seen at the time of training. With this objective, in this paper, we propose a unique end-to-end transformation domain network to detect a variety of well-known attacks. The proposed architecture utilizes a combination of two different wavelet transformations to simultaneously learn low-level and high-level image features in a deep stacked layer fashion. The proposed method is generalized to handle a broad set of singularities of several computer vision algorithms whether operating in the object recognition domain or biometrics recognition. We showcase the results on a wide range of singularity points including (i) adversarial perturbations which are learned using deep learning networks, (ii) physical presentation attacks on face recognition which aim to produce fake data for the sensor for acquisition, and identification, (iii) synthetic images, and (iv) digital retouching of the images. Even with a zero-day or open-world attack setting, the results of the proposed algorithm show improvement over state-of-the-art results. The proposed architecture is agnostic to parameters, computationally efficient to provide a sustainable detector to resource-constrained institutions, provides robustness to attacks, and supports green computing.

Quantum thermometry with a dissipative quantum Rabi system

Dissipative quantum Rabi system, a finite-component system composed of a single two-level atom interacting with an optical cavity field mode, exhibits a quantum phase transition, which can be exploited to greatly enhance the estimation precision of unitary parameters (frequency and coupling strength). Here, using the quantum Langevin equation, standard mean field theory and adiabatic elimination, we investigate the quantum thermometry of a thermal bath surrounding the atom with quantum optical probes. With the increase in coupling strength between the atom and the cavity field, two kinds of singularities can be observed. One type of singularity is the critical point (CP) of phase transition from the normal to superradiant phase. The other type of singularity is the exceptional point (EP) in the anti-parity-time (anti-\(\mathcal{P}\mathcal{T}\)) symmetrical cavity field. In many previous jobs, the optimal measurement precision can appear around EPs. However, we show that the optimal measurement precision occurs at the CP, instead of the EP. And the direct photon detection represents an excellent proxy for the optimal measurement near the CP. In the case where the thermal bath to be tested is independent of the extra thermal bath interacting with the cavity field, the estimation precision of the temperature always increases with the coupling strength. Oppositely, if the thermal bath to be tested is in equilibrium with the extra thermal bath interacting with the cavity field, noises that suppress the information of the temperature will be introduced when increasing the coupling strength unless it is close to the CP.

A dynamic model of adaptive consumers with endogenous unimodal preferences

We consider a discrete-time dynamic model to describe the repeated choices of adaptive consumers that at each time period adjust their request for a given good according to the discrepancies between the observed price and the fair price perceived on the basis of their utility function. Moreover, consumers’ preferences are endogenously modified on the basis of past consumption experience. The model considered is derived from the one proposed in D’Orlando and Rodano (2006), with a different assumption about the way the utility function changes according to the past consumption. In fact, in D’Orlando and Rodano (2006) the consumption preferences increase whenever past consumption increases, whereas in the model proposed in this paper a saturation effect is introduced, so that the same assumption holds for low and moderate past consumption, whereas current consumption decreases if the quantity consumed in the previous time period was too high. This leads to a unimodal preference function instead of an increasing one, which implies that the two-dimensional map, whose iteration represents the time evolution of the consumers’ choices, is transformed from an invertible map to a noninvertible one. Hence different global dynamic properties are obtained that influence the structure of the attractors and basins of attraction. These global dynamic features, described by the method of critical curves, interact with the property that the map is also characterized by the presence of a denominator that can vanish, giving rise to different kinds of singularities denoted as focal points and prefocal curves in Bischi et al. (1999), Bischi et al. (2003), Bischi et al. (2005), that strongly influence the structure of the basins of attraction. We describe the structure of the basins of attraction, and the contact bifurcations that change the qualitative properties of their global structure, by using geometric and numerical methods guided by the combined study of critical and prefocal curves.

Black bounces, wormholes, and partly phantom scalar fields

Simpson and Visser recently proposed a phenomenological way to avoid some kinds of space-time singularities by replacing a parameter whose zero value corresponds to a singularity (say, $r$) with the manifestly nonzero expression $r(u) = \sqrt{u^2 + b^2}$, where $u$ is a new coordinate, and $b =$ \const $>0$. This trick, generically leading to a regular minimum of $r$ beyond a black hole horizon (called a "black bounce"), may hopefully mimic some expected results of quantum gravity, and was previously applied to regularize the Schwarzschild, Reissner-Nordstr\"om, Kerr and some other metrics. In this paper it is applied to regularize the Fisher solution with a massless canonical scalar field in general relativity (resulting in a traversable wormhole) and a family of static, spherically symmetric dilatonic black holes (resulting in regular black holes and wormholes). These new regular metrics represent exact solutions of general relativity with a sum of stress-energy tensors of a scalar field with a nonzero self-interaction potential and a magnetic field in the framework of nonlinear electrodynamics with a Lagrangian function $L(F)$, $F = F_{\mu\nu} F^{\mu\nu}$. A novel feature in the present study is that the scalar fields involved have "trapped ghost" properties, that is, are phantom in a strong-field region and canonical outside it, with a smooth transition between the regions. It is also shown that any static, spherically symmetric metric can be obtained as an exact solution to the Einstein equations with the stress-energy tensor of the above field combination.

Note on solutions of scattering equations

In the CHY-frame for the amplitudes, there are two kinds of singularities we need to deal with. The first one is the pole singularities when the kinematics is not general, such that some of $S_A\to 0$. The second one is the collapse of locations of points after solving scattering equations (i.e., the singular solutions). These two types of singularities are tightly related to each other, but the exact mapping is not well understood. In this paper, we have initiated the systematic study of the mapping. We have demonstrated the different mapping patterns using three typical situations, i.e., the factorization limit, the soft limit and the forward limit.

A high-order BEM for acoustic problems in a subsonic uniform flow

The present work is to develop a boundary element method (BEM) with curved surface elements for 3D acoustic problems in a subsonic uniform flow. For exterior sound propagation problems in moving flows, the convected conventional boundary integral equation (CBIE) suffers from the irregular frequency issue which becomes more severe in the presence of moving flows. In this paper, a hybrid convected boundary integral formula based on the Burton-Miller method is implemented to avoid the fictitious eigenfrequency problem. However, the hypersingular integral involved in the convected normal derivative boundary integral equation (NDBIE) is one of the crucial troublesomeness in the implementation of BEM with curved continuous elements for 3D acoustic problems. Moreover, the presence of the Mach number further complicates the evaluation of the singular integrals. To resolve this dilemma, an effective method based on polar coordinate transformation (PCT) is developed to compute the singular integrals. By mapping physical coordinates to the intrinsic local coordinates, this method can effectually remove various kinds of singularities appearing in the CBIE and NDBIE. Additionally, a complete scheme on the high-order convected BEM including the scheme of collocation points is also constructed for acoustic problems in a subsonic uniform flow. Several numerical examples are elaborately designed to verify the accuracy of the developed method.

High-order quadrature on multi-component domains implicitly defined by multivariate polynomials

A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly defined geometry as the graph of an implicitly defined, multi-valued height function, and applies a dimension reduction approach needing only one-dimensional quadrature. In particular, we explore the use of Gauss-Legendre and tanh-sinh methods and demonstrate that the quadrature algorithm inherits their high-order convergence rates. Under the action of h-refinement with q fixed, the quadrature schemes yield an order of accuracy of 2q, where q is the one-dimensional node count; numerical experiments demonstrate up to 22nd order. Under the action of q-refinement with the geometry fixed, the convergence is approximately exponential, i.e., doubling q approximately doubles the number of accurate digits of the computed integral. Complex geometry is automatically handled by the algorithm, including, e.g., multi-component domains, tunnels, and junctions arising from multiple polynomial level sets, as well as self-intersections, cusps, and other kinds of singularities. A variety of numerical experiments demonstrates the quadrature algorithm on two- and three-dimensional problems, including: randomly generated geometry involving multiple high-curvature pieces; challenging examples involving high degree singularities such as cusps; adaptation to simplex constraint cells in addition to hyperrectangular constraint cells; and boolean operations to compute integrals on overlapping domains.

Cylindrical Gravitational Wave: Source and Resonance

Gravitational waves are regarded as linear waves in the weak field approximation, which ignores the spacetime singularity. In this paper, we analyze singularities in exact gravitational wave solutions. We provide an exact general solution of the gravitational wave with cylindrical symmetry. The general solution includes some known cylindrical wave solutions as special cases. We show that there are two kinds of singularities in the cylindrical gravitational wave solution. The first kind of singularity corresponds to a singular source. The second kind of singularity corresponds to a resonance between different gravitational waves. When two gravitational waves coexist, the interference term in the source may vanish in the sense of time averaging.

Crossing symmetry in the planar limit

Crossing symmetry asserts that particles are indistinguishable from antiparticles traveling back in time. In quantum field theory, this statement translates to the long-standing conjecture that probabilities for observing the two scenarios in a scattering experiment are described by one and the same function. Why could we expect it to be true? In this work we examine this question in a simplified setup and take steps towards illuminating a possible physical interpretation of crossing symmetry. To be more concrete, we consider planar scattering amplitudes involving any number of particles with arbitrary spins and masses to all loop orders in perturbation theory. We show that by deformations of the external momenta one can smoothly interpolate between pairs of crossing channels without encountering singularities or violating mass-shell conditions and momentum conservation. The analytic continuation can be realized using two types of moves. The first one makes use of an $iϵ$ prescription for avoiding singularities near the physical kinematics and allows us to adjust the momenta of the external particles relative to one another within their light cones. The second, more violent, step involves a rotation of subsets of particle momenta via their complexified light cones from the future to the past and vice versa. We show that any singularity along such a deformation would have to correspond to two beams of particles scattering off each other. For planar Feynman diagrams, these kinds of singularities are absent because of the particular flow of energies through their propagators. We prescribe a five-step sequence of such moves that combined together proves crossing symmetry for planar scattering amplitudes in perturbation theory, paving a way towards settling this question for more general scattering processes in quantum field theories.

Singularity Analysis and Representation of 6DOF Parallel Robot Using Natural Coordinates

Singularity research is carried out. The problem, which is about six-dimensional parameters of position and orientation can not realize three-dimensional visualization for 6DOF parallel robot, has been solved. Firstly, according to the structural characteristics of the 6DOF parallel robot with the planar platform, the position and orientation of the mobile platform are described, respectively, and the six equations of forward kinematics are established by choosing the natural coordinates of three representative points as parameters. Then, the singularities of the 6DOF parallel robot with a planar platform are divided into input singularity and output singularity. Aiming at the output singularity, in combination with six constraint equations among the position vectors of three representative points, an analytical algorithm is proposed to express the coupling singularity of position and orientation and the analytical expression is derived. In further research, three kinds of output singularities are found, the spatial distribution of the output singular trajectory is determined, and a unified three-dimensional fully visualized description of six-dimensional coupling variables is realized for the first time. The problems of finding the singular orientation at a given position or the singular position at a given orientation are solved. The analysis of the singularity lays a solid foundation for the description of the three-dimensional complete visualization of a six-dimensional singularity-free workspace based on forward kinematics. What is more, it has great significance for both trajectory planning and control design of the parallel robot.

Propagation of conductive crack along interface of piezoelectric/piezomagnetic bimaterials

This paper investigates the fracture characteristics of a Yoffe conductive crack moving along the interface of piezoelectric (PE)/piezomagnetic (PM) bimaterials. By assuming that the tangential electric- and magnetic-fields along the crack surface are zero and that the speed of the moving crack is lower than the minimum shear wave speed of the bimaterial system, the considered problem can be transformed into a Riemann–Hilbert boundary value problem of vector form. Then, the singularity parameters are exactly solved for different speed regions. In contrast to the anti-plane moving crack model including impermeable and permeable crack-face assumptions along the interface of magnetoelectroelastic (MEE) bimaterials studied before, which shows inverse square-root singularity, three novel kinds of singularities are found as the speed of the moving crack is varied for the present PE/PM interface model, which can be defined as δ1,2 = − 1/2 ± ie (Case 1), δ1,2 = − 1 ± ie (Case 2) and δ1,2 = − 1/2 ± κ (Case 3), and the third parameter δ3 = − 1/2 always holds true for all three cases. Two bimaterial combinations, i.e., BaTiO3/CoFe2O4 and BaTiO3/Terfenol-D, are numerically examined. Different from the piezoelectric case, Case 3 does not appear for BaTiO3/CoFe2O4 bimaterial combination. Above all, the singularity parameters significantly depend on the speed of the moving crack and the material properties of bimaterial systems.

Explosive Parametric Instability of the Free Surface of a Liquid Metal in a Radio Frequency Electric Field

The possibility of the appearance of microirregularities on the surface of a liquid metal under the action of a radio frequency electric field is investigated as applied to the conditions of the compact linear collider accelerating structures. These microirregularities are formed as a result of the development of parametric instability caused by the resonant interaction between the oscillating field and surface electrocapillary waves. It is shown that if the amplitude of the boundary oscillations becomes large enough, the dynamics of the system begins to be governed by nonlinear effects. In this case, it should be expected that the instability will become explosive and the boundary of the melt will cease to be smooth in a finite time: various kinds of singularities (cuspidal points, drops, and jets) will form. This will lead to a sharp increase in the emissive ability of the electrode surface and, as a consequence, to failure of the electric strength of the system.

A well‐conditioned integral equation for electromagnetic scattering from composite inhomogeneous bi‐anisotropic material and closed perfect electric conductor objects

A well-conditioned volume-surface integral equation, called the volume integral equation-combined field integral equation, is applied to analyse electromagnetic (EM) scattering from arbitrarily shaped three-dimensional composite objects comprising both inhomogeneous bi-anisotropic material and closed perfect electric conductors (PECs). The equivalent surface and volume currents are respectively expanded using the commonly used RWG and SWG basis functions, while a matrix equation is derived by the method of moments. Because the magnetic field integral equation is involved in modelling the surface electric current, and the constitutive parameters are all tensors, some new kinds of singularities are encountered and properly handled in the filling process of the impedance matrix. Several numerical results of EM scattering from composite bi-anisotropy and closed PEC objects are shown to illustrate the accuracy and efficiency of the proposed scheme. The validity of the continuity condition of electric flux enforced on the bi-anisotropy-PEC interfaces, which can be used to eliminate the volumetric electric unknowns, is also verified.

A Beale-Kato-Majda criterion for three dimensional compressible viscous non-isentropic magnetohydrodynamic flows without heat-conductivity

In this paper, we prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth (strong) solutions for the three dimensional (3D) compressible non-isentropic magnetohydrodynamic (MHD) equations with zero heat-conductivity. Therefore, if the maximum norm of the deformation tensor of velocity gradients remains bounded, it is not possible for other kinds of singularities. Our results are same as Beale-Kato-Majda type criterion for compressible viscous barotropic flows (Huang et al., 2011 [17]), and do not depend on further sophistication of the non-isentropic MHD model, it is independent of the pressure and magnetic field. Furthermore, this extends the corresponding Zhong's results (Zhong, 2019 [46]), and removes the viscous coefficients restriction condition 3μ>λ and the boundedness of pressure. As a byproduct, the same results also hold for compressible non-isentropic Navier-Stokes equations without heat-conductivity.

Periodic wave solutions of non-Newtonian filtration equations with nonlinear sources and singularities

This paper is concerned with a kind of non-Newtonian filtration equation with nonlinear sources and singularities. Based on the mountain pass theorem and variational methods, a sufficient criterion for the new results on the periodic wave solutions has been provided. Here not only the structure is more general and practical than the existing works but the conditions imposed are concise. Consequently, compared with the previous results on the singular equations and non-Newtonian filtration equations, the results we established are more generalized and some previous ones can been complemented and improved. Finally, the effectiveness of the established results are validated via two numerical examples and simulations.

Ontology of the Multitude and Heterarchy of the Common

This paper explores ontology of multitude, reflecting the General Intellect theory by A. Negri, M. Lazzarato, P. Virno, M. Pasquinelli and others. General Intellect is used as a synonym of the cognitive capacity of society, that may either liberate it from capitalism or be exploited by the capitalistic organisation of society. In this paper, General Intellect is analysed as a property of a social connection structure, hereinafter referred to as heterarchy. The connection structure heterarchy forms different kinds of singularities, i.e. aggregations consisting of statistical repetitions of relations and individual egos creating values through their goal-setting and other intellectual activities. The article argues that though General Intellect may denote capacity for the self-organization of society to a certain extent, it is difficult to identify with the only particular institutional organisation or political regime. General Intellect manifests itself in any type of social structuring through self-organising processes

Dispersive blow-up for solutions of the Zakharov-Kuznetsov equation

The main purpose here is the study of dispersive blow-up for solutions of the Zakharov-Kuznetsov equation. Dispersive blow-up refers to point singularities due to the focusing of short or long waves. We will construct initial data such that solutions of the linear problem present this kind of singularities. Then we show that the corresponding solutions of the nonlinear problem present dispersive blow-up inherited from the linear component part of the equation. Similar results are obtained for the generalized Zakharov-Kuznetsov equation.