Finite Singularities Research Articles

Zero Level Perturbation of a Certain Third-Order Linear Solvable ODE with an Irregular Singularity at the Origin of Poincaré Rank 1

We study an irregular singularity of Poincare rank 1 at the origin of a certain third-order linear solvable homogeneous ODE. We perturb the equation by introducing a small parameter $\varepsilon \in ({\mathbb R}_{+},0)$ (e < 1), which causes the splitting of the irregular singularity into two finite Fuchsian singularities. We show that when the solutions of the perturbed equation contain logarithmic terms, the Stokes matrices of the initial equation are limits of the part of the monodromy matrices around the finite resonant Fuchsian singularities of the perturbed equation.

Global C∞ integrability of quartic–linear polynomial differential systems

ABSTRACTThe quartic–linear polynomial differential systems having at least one finite singularity are affine equivalent to systems of the formwhere P and Q are coprime, Pi are homogeneous polynomials of degree i and (otherwise it is cubic–linear) and Q(x, y) is either x or y. In this paper, we classify all the quartic–linear systems with Q(x, y) = y which have a global C∞ first integral. We use the local characterization of first integrals, partition of unity in and smoothness of first integrals in canonical regions.

Hausdorff measure of escaping sets on certain meromorphic functions

We consider transcendental meromorphic function for which the set of finite singularities of its inverse is bounded. Bergweiler and Kotus gave bounds for the Hausdorff dimension of the escaping sets if the function has no logarithmic singularities over ∞, the multiplicities of poles are bounded and the order is finite. We study the case of infinite order and find gauge functions for which the Hausdorff measure of escaping sets is zero or ∞.

Finite conformal quantum gravity and spacetime singularities

We show that a class of finite quantum non-local gravitational theories is conformally invariant at classical as well as at quantum level. This is actually a range of conformal anomaly-free theories in the spontaneously broken phase of the Weyl symmetry. At classical level we show how the Weyl conformal invariance is able to tame all the spacetime singularities that plague not only Einstein gravity, but also local and weakly non-local higher derivative theories. The latter statement is proved by a singularity theorem that applies to a large class of weakly non-local theories. Therefore, we are entitled to look for a solution of the spacetime singularity puzzle in a missed symmetry of nature, namely the Weyl conformal symmetry. Following the seminal paper by Narlikar and Kembhavi, we provide an explicit construction of singularity-free black hole exact solutions in a class of conformally invariant theories.

On the splash and splat singularities for the one-phase inhomogeneous Muskat Problem

In this paper, we study finite time splash and splat singularities formation for the interface of one fluid in a porous media with two different permeabilities. We prove that the smoothness of the interface breaks down in finite time into a splash singularity but this is not going to happen into a splat singularity.

One Existence Theorem for non-CSC Extremal Kähler Metrics with Conical Singularities on $S^2$

We often call an extremal Kähler metric with finite singularities on a compact Riemann surface an HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric. In this paper we consider the following question: if we give $N$ points $p_1, \ldots, p_N$ on $S^2$ and $N$ positive real numbers $2\pi \alpha_1, \ldots, 2\pi \alpha_N$ with $\alpha_n \neq 1$, $n = 1, \ldots, N$, what condition can guarantee the existence of a non-CSC HCMU metric which has conical singularities $p_1, \ldots, p_N$ with singular angles $2\pi \alpha_1, \ldots, 2\pi \alpha_N$ respectively. We prove that if there are at least $N-2$ integers in $\alpha_1, \ldots, \alpha_N$ then there exists one non-CSC HCMU metric on $S^2$ satisfying the condition stated above no matter where the given points are.

Full groups of Cuntz–Krieger algebras and Higman–Thompson groups

In this paper, we will study representations of the continuous full group \Gamma_A of a one-sided topological Markov shift (X_A,\sigma_A) for an irreducible matrix A with entries in \{0,1\} as a generalization of Higman–Thompson groups V_N, 1 &lt; N \in {\mathbb{N}} . We will show that the group \Gamma_A can be represented as a group \Gamma_A^{\operatorname{tab}} of matrices, called A -adic tables, with entries in admissible words of the shift space X_A , and a group \Gamma_A^{\operatorname{PL}} of right continuous piecewise linear functions, called A -adic PL functions, on [0,1] with finite singularities.

Generalized Lagrangian mean curvature flows in almost Calabi–Yau manifolds

In this paper, we study the generalized Lagrangian mean curvature flow in almost Einstein manifold proposed by T. Behrndt. We show that the singularity of this flow is characterized by the second fundamental form. We also show that the rescaled flow at a singularity converges to a finite union of Special Lagrangian cones for generalized Lagrangian mean curvature flow with zero-Maslov class in almost Calabi–Yau manifold. As a corollary, there is no finite time Type-I singularity for such a flow.

Finite time future singularities in the interacting dark sector

We construct a piecewise model that gives a physical viable realization of finite-time future singularity for a spatially flat Friedmann-Robertson-Walker universe within the interacting dark matter--dark energy framework, with the latter one in the form of a variable vacuum energy. The scale factor solutions provided by the model are accommodated in several branches defined in four regions delimited by the scale factor and the effective energy density. A branch starts from a big bang singularity and describes an expanding matter-dominated universe until the sudden future singularity occurs. Then, an expanding branch emerges from a past singularity, reaches a maximum, reverses its expansion and possibly collapses into itself while another expanding branch emerges from the latter singularity and has a stable de Sitter phase which is intrinsically stable. We obtain a different piecewise scale factor which describes a contracting de Sitter universe in the distant past until the finite-time future singularity happens. It emerges and continues in a contracting phase, bounces at the minimum, reverses, and enters into a stable de Sitter phase without a dramatic final. Also, we explore the aforesaid cosmic scenarios by focusing on the leading contributions of some physical quantities near the sudden future singularity and applying the geometric Tipler and Kr\'olak criteria in order to inspect the behavior of timelike geodesic curves around such singularity.

Scalar field collapse with an exponential potential

An analogue of the Oppenheimer-Synder collapsing model is treated analytically, where the matter source is a scalar field with an exponential potential. An exact solution is derived followed by matching to a suitable exterior geometry, and an analysis of the visibility of the singularity. In some situations, the collapse indeed leads to a finite time curvature singularity, which is always hidden from the exterior by an apparent horizon.

Ferromagnetic model on the Apollonian packing.

This work investigates the influence of geometrical features of the Apollonian packing (AP) on the behavior of magnetic models. The proposed model differs from previous investigations on the Apollonian network (AN), where the magnetic coupling constants depend only on the properties of the network structure defined by the packing, but not on quantitative aspects of its geometry. In opposition to the exact scale invariance observed in the AN, the circle's sizes in the AP are scaled by different factors when one goes from one generation to the next, requiring a different approach for the evaluation of the model's properties. If the nearest-neighbors coupling constants are defined by J_{i,j}∼1/(r_{i}+r_{j})^{α}, where r_{i} indicates the radius of the circle i containing the node i, the results for the correlation length ξ indicate that the model's behavior depend on α. In the thermodynamic limit, the uniform model (α=0) is characterized by ξ→∞ for all T>0. Our results indicate that, on increasing α, the system changes to an uncorrelated pattern, with finite ξ at all T>0, at a value α_{c}≃0.743. For any fixed value of α, no finite temperature singularity in the specific heat is observed, indicating that changes in the magnetic ordering occur only when α is changed. This is corroborated by the results for the magnetization and magnetic susceptibility.

Cosmic anisotropic doomsday in Bianchi type I universes

In this paper we study finite time future singularities in anisotropic Bianchi type I models. It is shown that there exist future singularities similar to Big Rip ones (which appear in the framework of phantom Friedmann-Robertson-Walker cosmologies). Specifically, in an ellipsoidal anisotropic scenario or in a fully anisotropic scenario, the three directional and average scale factors may diverge at a finite future time, together with energy densities and anisotropic pressures. We call these singularities “Anisotropic Big Rip Singularities.” We show that there also exist Bianchi type I models filled with matter, where one or two directional scale factors may diverge. Another type of future anisotropic singularities is shown to be present in vacuum cosmologies, i.e., Kasner spacetimes. These singularities are induced by the shear scalar, which also blows up at a finite time. We call such a singularity “Vacuum Rip.” In this case one directional scale factor blows up, while the other two and average scale factors tend to zero.

The Klein–Gordon–Fock equation in the curved spacetime of the Kerr–Newman (anti) de Sitter black hole

Exact solutions of the Klein–Gordon–Fock (KGF) general relativistic equation that describe the dynamics of a massive, electrically charged scalar particle in the curved spacetime geometry of an electrically charged, rotating Kerr–Newman–(anti) de Sitter black hole are investigated. In the general case of a rotating, charged, cosmological black hole the solution of the KGF equation with the method of separation of variables results in Fuchsian differential equations for the radial and angular parts which for most of the parameter space contain more than three finite singularities and thereby generalise the Heun differential equations. For particular values of the physical parameters (i.e. mass of the scalar particle) these Fuchsian equations reduce to the case of the Heun equation and the closed form analytic solutions we derive are expressed in terms of Heun functions. For other values of the parameters some of the extra singular points are false singular points. We derive the conditions on the coefficients of the generalised Fuchsian equation such that a singular point is a false point. In such a case the exact solution of the Fuchsian equation can in principle be simplified and expressed in terms of Heun functions. This is the generalisation of the case of a Heun equation with a false singular point in which the exact solution of Heun’s differential equation is expressed in terms of Gauß hypergeometric function. We also derive the exact solutions of the radial and angular equations for a charged massive scalar particle in the Kerr–Newman spacetime. The analytic solutions are expressed in terms of confluent Heun functions. Moreover, we derived the constraints on the parameters of the theory such that the solution simplifies and expressed in terms of confluent Kummer hypergeometric functions. We also investigate the radial solutions in the KN case in the regions near the event horizon and far from the black hole. Finally, we construct several expansions of the solutions of the Heun equation in terms of generalised hypergeometric functions of Lauricella–Appell.

Some New Facts Concerning the Delta Neutral Case of Fox’s H Function

In this paper, we find several new properties of a class of Fox’s H functions which we call delta neutral. In particular, we find an expansion in the neighborhood of the finite non-zero singularity and give new Mellin transform formulas under a special restriction on parameters. The last result is applied to prove a conjecture regarding the representing measure for gamma ratio in Bernstein’s theorem. Furthermore, we find the weak limit of measures expressed in terms of the H function which furnishes a regularization method for integrals containing the delta neutral and zero-balanced cases of Fox’s H function. We apply this result to extend a recently discovered integral equation to the zero-balanced case. In the last section of the paper, we consider a reduced form of this integral equation for Meijer’s G function. This leads to certain expansions believed to be new even in the case of the Gauss hypergeometric function.

Curved branes with regular support

We study spacetime singularities in a general five-dimensional braneworld with curved branes satisfying four-dimensional maximal symmetry. The bulk is supported by an analog of perfect fluid with the time replaced by the extra coordinate. We show that contrary to the existence of finite distance singularities from the brane location in any solution with flat (Minkowski) branes, in the case of curved branes there are singularity-free solutions for a range of equations of state compatible with the null energy condition.

Singular multicontact structures

We describe the automorphisms of a singular multicontact structure, that is a generalisation of the Martinet distribution. Such a structure is interpreted as a para-CR structure on a hypersurface M of a direct product space R+2×R−2. We introduce the notion of a finite type singularity analogous to CR geometry and, along the way, we prove extension results for para-CR functions and mappings on embedded para-CR manifolds into the ambient space.

Classification of Global Phase Portrait of Planar Quintic Quasi-Homogeneous Coprime Polynomial Systems

This paper is devoted to the complete classification of global phase portraits (short for GPP) of quasi-homogeneous but non-homogeneous coprime planar quintic polynomial differential systems (short for QCQS). We firstly study the canonical forms of QCQS. It is shown that these canonical forms can have 0, 1, 2, 4 parameters. Then, we investigate the global topological structures of all canonical forms, by using the quasi-homogeneous blow-up technique for the finite singularities and the Poincare–Lyapunov compactification for the infinite singularities. We finally perform a topological classification for the set of GPP.

Slowly rotating black holes in Einstein-æther theory

We study slowly rotating, asymptotically flat black holes in Einstein-aether theory and show that solutions that are free from naked finite area singularities form a two-parameter family. These parameters can be thought of as the mass and angular momentum of the black hole, while there are no independent aether charges. We also show that the aether has non-vanishing vorticity throughout the spacetime, as a result of which there is no hypersurface that resembles the universal horizon found in static, spherically symmetric solutions. Moreover, for experimentally viable choices of the coupling constants, the frame-dragging potential of our solutions only shows percent-level deviations from the corresponding quantities in General Relativity and Horava gravity. Finally, we uncover and discuss several subtleties in the correspondence between Einstein-aether theory and Horava gravity solutions in the $c_\omega\to\infty$ limit.

Cosmological evolution with the cubic order field derivative coupling

We investigate cosmological evolution in the scalar–tensor theory with the field derivative coupling to the double-dual of the Riemann tensor (the cubic-type theory). The theory can be seen as the straightforward extension of the scalar–tensor with the quadratic order field derivative coupling to the Einstein tensor (the quadratic-type theory). Both the field derivative couplings to the Einstein tensor and the double-dual of the Riemann tensor have been argued in terms of the successful realization of the self-tuning of the cosmological constant within the Horndeski theory. Assuming the constant potential given by the sum of the cosmological constant and the quantum vacuum energy, the shift symmetry for the scalar field and no matter fields, in the spatially-flat Friedmann–Lemaitre–Robertson–Walker spacetime, we can reduce the set of the field equations to the first-order ordinary nonlinear differential equation for the Hubble parameter, showing the existence of the self-tuned and runaway de Sitter solutions, in addition to the standard de Sitter solutions in general relativity and the finite Hubble singularities which can be reached within the finite time. We then argue the possible cosmological evolution in terms of the values of the effective cosmological constant, the kinetic coupling constants and the initial Hubble parameter. Although the behavior of the universe around each of the de Sitter solutions as well as the finite time singularities is very similar in both theories, we find that the crucial difference appears in terms of no bounce or turnaround behavior across the vanishing Hubble parameter as well as no limitation for the range of the Hubble parameter in the cubic-type theory.

Anisotropic cosmological models with bulk viscosity and particle creation in Saez–Ballester theory of gravitation

The paper deals with the study of particle creation and bulk viscosity in the evolution of spatially homogeneous and anisotropic Bianchi type-V cosmological models in the framework of Saez–Ballester theory of gravitation. Particle creation and bulk viscosity are considered as separate irreversible processes. The energy–momentum tensor is modified to accommodate the viscous pressure and creation pressure which is associated with the creation of matter out of gravitational field. A special law of variation of Hubble parameter is applied to obtain exact solutions of field equations in two types of cosmologies, one with power-law expansion and the other with exponential expansion. Cosmological model with power-law expansion has a Big-Bang singularity at time t = 0, whereas the model with exponential expansion has no finite singularity. We study bulk viscosity and particle creation in each model in four different cases. The bulk viscosity coefficient is obtained for full causal, Eckart’s and truncated theories. All physical parameters are calculated and thoroughly discussed in both models.