Strong Singularity Research Articles

Disentanglement as a strong cosmic censor

If entanglement builds spacetime, then conversely, disentanglement ought to destroy spacetime. From the quantum null energy condition and quantum focusing conjecture, we obtain disentanglement criteria which necessitate infinite energies and strong spacetime singularities. We apply our results to the strong cosmic censorship proposal, where singularities at the Cauchy horizons in black holes are desirable. Using our disentanglement criteria and without resorting to any detailed calculations, we provide an exceedingly general and physically transparent discussion of strong cosmic censorship in semiclassical black holes. We argue that strong cosmic censorship is enforced in asymptotically flat and de Sitter black holes by disentanglement and describe how similar disentanglement might be avoided in some anti-de Sitter cases.

Superconvergence results for hypersingular integral equation of first kind by Chebyshev spectral projection methods

In this article, we propose Chebyshev spectral projection methods to solve the hypersingular integral equation of first kind. The presence of strong singularity in Hadamard sense in the first part of the integral equation makes it challenging to get superconvergence results. To overcome this, we transform the first kind hypersingular integral equation into a second kind integral equation. This is achieved by defining a bounded inverse of the hypersingular integral operator in some suitable Hilbert space. Using iterated Chebyshev spectral Galerkin method on the equivalent second kind integral equation, we obtain improved convergence of O(N−2r), where N is the highest degree of Chebyshev polynomials employed in the approximation space and r is the smoothness of the solution. Further, using commutativity of projection operator and inverse of the hypersingular integral operator, we are able to obtain superconvergence of O(N−3r) and O(N−4r), by Chebyshev spectral multi-Galerkin method (CSMGM) and iterated CSMGM, respectively. Finally, numerical examples are presented to verify our theoretical results.

Three Weak Solutions for a Critical Non-Local Problem with Strong Singularity in High Dimension

Three Weak Solutions for a Critical Non-Local Problem with Strong Singularity in High Dimension

Inverse scattering for repulsive potential and strong singular interactions

In a previous work of 2014 on a quantum system governed by the repulsive Hamiltonian, the author proved uniqueness for short-range interactions described by a scattering operator consisting of regular and singular parts. In this paper, the singular part is assumed to have much stronger singularities and the same uniqueness theorem is proved. By applying the time-dependent method invented by Enss and Weder [J. Math. Phys. 36(8), 3902–3921 (1995)], the high-velocity limit for a wider class of the scattering operator with stronger singularities also uniquely determines the interactions of a multi-dimensional system.

Tipler naked singularities in N dimensions

Abstract A spacetime singularity, identified by the existence of incomplete causal geodesics in the spacetime, is called a (Tipler) strong curvature singularity if the volume form acting on independent Jacobi fields along causal geodesics vanishes in the approach of the singularity. It is called naked if at least one of these causal geodesics is past incomplete. Here, we study the formation of strong curvature naked singularities arising from spherically symmetric gravitational collapse of general type-I matter fields in an arbitrarily finite number of dimensions. In the spirit of Joshi and Dwivedi (1993 Phys. Rev. D 47 5357), and Goswami and Joshi (2007 Phys. Rev. D 76 084026), beginning with regular initial data, we derive two distinct (but not mutually exclusive) conditions, which we call the positive root condition (PRC) and the simple positive root condition (SPRC), that serve as necessary and sufficient conditions, respectively, for the existence of naked singularities. In doing so, we generalize the results of both the aforementioned works. We further restrict the PRC and the SPRC by imposing the curvature growth condition (CGC) of Clarke and Krolak (1985 J. Geom. Phys. 2(2) 127) on all causal curves that satisfy the causal convergence condition. The CGC then gives a sufficient condition ensuring that the naked singularities implying the PRC and implied by the SPRC, are of strong curvature type and hence correspond to the inextendibility of the spacetime. Using the CGC, we extend the results of Mosani et. al. (2020 Phys. Rev. D 101 044052) (that hold for dimension N=4) to the case N=5, showing that strong curvature naked singularities can occur in this case. However, for the case N≥6, we show that past-incomplete causal curves that identify naked singularities do not satisfy the CGC. These results shed light on the validity of the cosmic censorship conjectures in arbitrary dimensions.

Positive periodic solution of second-order difference equation with a repulsive singularity

A second-order difference equation with periodic boundary value conditions is studied in this article, the nonlinear term may be singular. A few conditions are given to guarantee the existence of positive periodic solution by fixed-point theorem in cones. Our results can be applied to both superlinearity and semilinearity in the case that the difference equation has no singularity, and can be applied to both strong singularity and weak singularity in the case that the difference equation has a repulsive singularity.

A Singular Tempered Sub-Diffusion Fractional Model Involving a Non-Symmetrically Quasi-Homogeneous Operator

In this paper, we focus on the existence of positive solutions for a singular tempered sub-diffusion fractional model involving a quasi-homogeneous nonlinear operator. By using the spectrum theory and computing the fixed point index, some new sufficient conditions for the existence of positive solutions are derived. It is worth pointing out that the nonlinearity of the equation contains a tempered fractional sub-diffusion term, and is allowed to possess strong singularities in time and space variables. In particular, the quasi-homogeneous operator is a nonlinear and non-symmetrical operator.

Singular space-times with bounded algebraic curvature scalars

We show that the absence of unbounded algebraic curvature invariants constructed from polynomials of the Riemann tensor cannot guarantee the absence of strong singularities. As a consequence, it is not sufficient to rely solely on the analysis of such scalars to assess the regularity of a given space-time. This conclusion follows from the analysis of incomplete geodesics within the internal region of asymmetric wormholes supported by scalar matter which arise in two distinct metric-affine gravity theories. These wormholes have bounded algebraic curvature scalars everywhere, which highlights that their finiteness does not prevent the emergence of pathologies (singularities) in the geodesic structure of space-time. By analyzing the tidal forces in the internal wormhole region, we find that the angular components are unbounded along incomplete radial time-like geodesics. The strength of the singularity is determined by the evolution of Jacobi fields along such geodesics, finding that it is of strong type, as volume elements are torn apart as the singularity is approached. Lastly, and for completeness, we consider the wormhole of the quadratic Palatini theory and present an analysis of the tidal forces in the entire space-time.

Sharp regularization effect for the non-cutoff Boltzmann equation with hard potentials

For the Maxwellian molecules or hard potentials case, we verify the smoothing effect for the spatially inhomogeneous Boltzmann equation without angular cutoff. Given initial data with low regularity, we prove solutions at any positive time are analytic for strong angular singularity, and in the Gevrey class with optimal index for mild angular singularity. To overcome the degeneracy in the spatial variable, a family of well-chosen vector fields with time-dependent coefficients will play a crucial role, and the sharp regularization effect of weak solutions relies on a quantitative estimate on directional derivatives in these vector fields.

Constant sign and nodal solutions for singular Gierer–Meinhardt-type system

We establish the existence of three solutions for singular semilinear elliptic system, two of which are of opposite constant sign. Under a strong singularity effect, the third solution is nodal with synchronous sign components. The approach combines sub-supersolutions method and Leray–Schauder topological degree involving perturbation argument.

Nonautonomous double‐phase equations with strong singularity and concave perturbation

AbstractWe consider a nonlinear Dirichlet problem driven by a nonautonomous double‐phase differential operator and with a reaction consisting of a “strongly” singular term plus a concave perturbation. Using the Nehari method, we show the existence of a bounded strictly positive solution.

Research on Singularity in Calculation of Induced Electric Field in Pointed Conductors

In the electric field environment of thunderstorm, the tip of a grounded object at a high place is more likely to produce corona discharge and finally cause lightning flash. Therefore, the study of the surface electric field of the grounded object tip in the electric field environment has important theoretical value and practical significance for the analysis of the characteristics of the head-on discharge. Due to the strong singularity of the tip structure, the traditional numerical analysis method cannot accurately calculate its surface electric field. Therefore, the traditional literature usually chamfers the conductor tip, but in lightning physics, researchers are most concerned about the maximum field strength in the local area. In order to study the electric field distribution on the tip conductor surface, a semi-analytical boundary element method is proposed to calculate the tip structure directly. Firstly, the analytical formula of the electric field on the tip conductor surface and the boundary element semi-analytical common method are derived, and the tip conductor model is established to calculate the tip electric field excited by the external electric field using the analytical solution and the semi-analytical method. The correctness of the semi-analytical method is verified by calculating the L2 loss function of the two, and the influence of the tip angle on the semi-analytical calculation results is studied. The results show that for the three-dimensional cone model, the L2 loss function is 10-5, which meets the calculation accuracy. The special treatment of the semi-analytical method at the strong singular integral makes the calculation accuracy much higher than that of the Gaussian integral, where the L2 loss function is greatly reduced. Further, based on the change of tip angle, as the conductor tip becomes sharper, the greater the electric field distortion is, and the greater the L2 loss function is. The research results provide a method reference for the calculation of the electric field on the tip conductor surface.

Transverse free vibration analysis of thin sectorial plates by the weak form quadrature element method

The weak form quadrature element method is applied to free vibration analysis of thin sectorial plates with arbitrary vertex angles and boundary conditions. To tackle the strong stress singularity around the vertex, analytical displacement descriptions are introduced into the inner sectorial subdomain, while the outer annular subdomain is modeled by a single weak form quadrature thin plate element. The continuity on the interface between the two subdomains is enforced afterwards. Eventually, a generalized eigenvalue formulation is established after introducing Hamilton’s principle. The first six non-dimensional frequency parameters for various vertex angles and boundary conditions are obtained and compared with available results. Several typical free vibration modes are plotted. The accuracy, convergence rate, and computational cost of the present formulation are discussed at length.

On the evolution of the hierarchy of shock waves in a two-dimensional isobaric medium

In the proposed paper, the process of propagation of shock waves in two-dimensional media without its own pressure drop is studied. The model of such media is a system of equations of gas dynamics, where formally the pressure is assumed to be zero. From the point of view of the theory of systems of conservation laws, the system of equations under consideration is in some sense degenerate, and, consequently, the corresponding generalized solutions have strong singularities (evolving shock waves with density in the form of delta functions on manifolds of different dimensions). We will denote this property as the evolution of the hierarchy of strong singularities or the evolution of the hierarchy of shock waves. In the paper, in the two-dimensional case, the existence of such an interaction of strong singularities with density delta function along curves in the space $\mathbb{R}^2$ is proved, at which a density concentration occurs at a point, that is, a hierarchy of shock waves arises. The properties of such dynamics of strong singularities are described. The results obtained provide a starting point for moving on to a much more interesting multidimensional case in the future.

A unified hp-HDG framework for Friedrichs' PDE systems

This work proposes a unified hp-adaptivity framework for hybridized discontinuous Galerkin (HDG) method for a large class of partial differential equations (PDEs) of Friedrichs' type. In particular, we present unified hp-HDG formulations for abstract one-field and two-field structures and prove their well-posedness. In order to handle non-conforming interfaces we simply take advantage of HDG built-in mortar structures. With split-type mortars and the approximation space of trace, a numerical flux can be derived via Godunov approach and be naturally employed without any additional treatment. As a consequence, the proposed formulations are parameter-free. We perform several numerical experiments for time-independent and linear PDEs including elliptic, hyperbolic, and mixed-type to verify the proposed unified hp-formulations and demonstrate the effectiveness of hp-adaptation. Two adaptivity criteria are considered: one is based on a simple and fast error indicator, while the other is rigorous but more expensive using an adjoint-based error estimate. The numerical results show that these two approaches are comparable in terms of convergence rate even for problems with strong gradients, discontinuities, and singularities.

Visualizing near-coexistence of massless Dirac electrons and ultra-massive saddle point electrons

Strong singularities in the electronic density of states amplify correlation effects and play a key role in determining the ordering instabilities in various materials. Recently high order van Hove singularities (VHSs) with diverging power-law scaling have been classified in single-band electron models. We show that the 110 surface of Bismuth exhibits high order VHS with an usually high density of states divergence \sim (E)^{-0.7}∼(E)−0.7. Detailed mapping of the surface band structure using scanning tunneling microscopy and spectroscopy combined with first-principles calculations show that this singularity occurs in close proximity to Dirac bands located at the center of the surface Brillouin zone. The enhanced power-law divergence is shown to originate from the anisotropic flattening of the Dirac band just above the Dirac node. Such near-coexistence of massless Dirac electrons and ultra-massive saddle points enables to study the interplay of high order VHS and Dirac fermions.

Existence of solution for a class of integro-differential sublinear problems with strong singularity

Existence of solution for a class of integro-differential sublinear problems with strong singularity

Active learning-based domain adaptive localized polynomial chaos expansion

The paper presents a novel methodology to build surrogate models of complicated functions by an active learning-based sequential decomposition of the input random space and construction of localized polynomial chaos expansions, referred to as domain adaptive localized polynomial chaos expansion (DAL-PCE). The approach utilizes sequential decomposition of the input random space into smaller sub-domains approximated by low-order polynomial expansions. This allows the approximation of functions with strong nonlinearities, discontinuities, and/or singularities that often appear in dynamical systems. Decomposition of the input random space and local approximations alleviates the Gibbs phenomenon for these types of problems and confines error to a very small vicinity near the non-linearity. The global behavior of the surrogate model is therefore significantly better than existing methods, as shown in numerical examples, including an engineering dynamical system exhibiting discontinuous response. The whole process is driven by an active learning routine that uses the recently proposed Θ criterion to assess local variance contributions (Novák et al., 2021). The proposed approach balances both exploitation of the surrogate model and exploration of the input random space and thus leads to efficient and accurate approximation of the original mathematical model. The numerical results show the superiority of the DAL-PCE in comparison to (i) a single global polynomial chaos expansion and (ii) the recently proposed stochastic spectral embedding (SSE) method (Marelli et al., 2021) developed as an accurate surrogate model and which is based on a similar domain decomposition process. This method represents a general framework upon which further extensions and refinements can be based and which can be combined with any technique for non-intrusive polynomial chaos expansion construction.

Boundary blow-up solutions to singular k-Hessian equations with gradient terms: Sufficient and necessary conditions and asymptotic behavior

Let Ω be a smooth, bounded and strictly (k−1)-convex domain in RN(N≥2). Assume that b and f are smooth positive functions and b may be singular near ∂Ω. We provide sufficient and necessary conditions on b, f and q for the existence of k-convex solutions to the boundary blow-up k-Hessian problemSk(D2u)=b(x)f(u)|∇u|q for x∈Ω,u(x)→+∞ as dist(x,∂Ω)→0. The asymptotic behavior of such solutions is also investigated. Moreover, when b has strong singularity near ∂Ω, we study the existence and nonexistence of boundary blow-up solutions.

Kirchhoff equations with strong singularity in closed manifolds

In this paper, we study a class of Kirchhoff-type problems involving strong singular nonlinearities on compact Riemannian manifolds. With the help of variational method and Nehari method, we prove that this problem has a unique positive solution.