Generic Singularities Research Articles

Stability of local transitivity of a generic control system on a surface with boundary

A classification of generic singularities of local transitivity of smooth control systems on surfaces with boundary is obtained. The stability of these singularities and of the entire set of points with identical properties of local transitivity with respect to small perturbations of a generic system is proved.

Invariant identification of naked singularities in spherically symmetric spacetimes

The study of generic naked singularities and their implications for the cosmic censorship conjecture is still an open issue in the framework of general relativity. One of the obstacles can be traced to the procedures for identifying naked singularities. Usually, the methods applied are not only model and coordinate dependent, but they very often rely in some strong assumptions on the degree of differentiability of the physical magnitudes of the model (such as the mass, density, etc) in the singularity. In this paper, we present a coordinate independent framework for identifying naked singularities based on invariants which is also devoid of strong differentiability requirements. The approach is intended to analyse whole families of models and to provide general results related to the cosmic censorship conjecture. Moreover, since the framework has a strict geometrical nature it can be used with alternative theories of gravitation as long as they assume the existence of a Lorentzian manifold. We exemplify its strength by applying it to the study of the collapse of radiation in radiative coordinates and the collapse of dust in comoving coordinates.

Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity

We consider a (possibly) nonlinear interface problem in 2D and 3D, which is solved by use of various adaptive FEM-BEM coupling strategies, namely the Johnson–Nédélec coupling, the Bielak–MacCamy coupling, and Costabel’s symmetric coupling. We provide a framework to prove that the continuous as well as the discrete Galerkin solutions of these coupling methods additionally solve an appropriate operator equation with a Lipschitz continuous and strongly monotone operator. Therefore, the original coupling formulations are well-defined, and the Galerkin solutions are quasi-optimal in the sense of a Céa-type lemma. For the respective Galerkin discretizations with lowest-order polynomials, we provide reliable residual-based error estimators. Together with an estimator reduction property, we prove convergence of the adaptive FEM-BEM coupling methods. A key point for the proof of the estimator reduction are novel inverse-type estimates for the involved boundary integral operators which are advertized. Numerical experiments conclude the work and compare performance and effectivity of the three adaptive coupling procedures in the presence of generic singularities.

Generic singularities in solutions of the shallow water equations

Generic singularities in solutions of the shallow water equations

Propagations from a space curve in three space with indicatrix a surface

Generic singularities of rays emanating from a space curve in R 3 in all directions with the rate determined by an indicatrix (independent of the point in R 3 ) defined by a surface are classified. Similarly rays emanating from surface defined by an indicatrix given by a curve are also considered. Some applications to control theory are indicated. In this paper we solve two problems on the classification of local geometrical singularities that are related to control theory. We use some techniques from the singularity theory of caustics and wave fronts to study singularities of exponential mappings in a class of control problems which correspond to special integrable Hamiltonian systems with straight lines as extremals. The first problem concerns a control system on a three-dimensional ane space with points q 2 R 3 . We identify the tangent space TqR 3 with R 3 itself. At each point q we choose an indicatrix Iq of admissible velocities _ q = @q @ of motion which we assume is independent of the point q itself. Assume that this set is parametrised locally by a regular mapping (x;y)7! r2(x;y) whose image is a surface M. We shall now write M in place of Iq. An admissible motion is a smooth curve ( )2 R 3 , parametrised by a segment of the (ane) time axis , such that the velocity at each point _ belongs to the set of admissible velocities M. Let qb( ) be the trajectory of an admissible motion of an initial point b2 N, issuing at = 0 from an initial set N, where N is a space curve which is a submanifold in R 3 .

Singularities of Tangent Varieties to Curves and Surfaces

It is given the dieomorphism classification on generic singularities of tangent varieties to curves with arbitrary codimension in a projective space. The generic classifica- tions are performed in terms of certain geometric structures and dierential systems on flag manifolds, via several techniques in dierentiable algebra. It is provided also the generic dif- feomorphism classification of singularities on tangent varieties to contact-integral curves in the standard contact projective space. Moreover we give basic results on the classification of singularities of tangent varieties to generic surfaces and Legendre surfaces.

Generic singular configurations of linkages

We study the topological and differentiable singularities of the configuration space C ( Γ ) of a mechanical linkage Γ in R d , defining an inductive sufficient condition to determine when a configuration is singular. We show that this condition holds for generic singularities, provide a mechanical interpretation, and give an example of a type of mechanism for which this criterion identifies all singularities.

Betti Numbers of Transversal Monomial Ideals

In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of t-minors of a generic pluri-circulant matrix is a transversal monomial ideal. Using a Gröbner basis for this ideal, it is shown that the initial ideal of a generic pluri-circulant matrix is a stable monomial ideal when the matrix has two square blocks. By means of the Eliahou-Kervaire resolution for stable monomial ideals, the Betti numbers of this initial ideal is computed and it is proved that for some significant values of t, this ideal has the same Betti numbers as the corresponding transversal monomial ideal. The ideals treated in this paper naturally arise in the study of generic singularities of algebraic varieties.

$${\mathcal{F}}$$ -stability of self-similar solutions to harmonic map heat flow

Inspired by the work of Colding and Minicozzi II: ”Generic mean curvature flow I: generic singularities”, we explore the notion of generic singularities for the harmonic map heat flow. We introduce \({\mathcal{F}}\) -functional and entropy for maps from Euclidean spaces. The critical points of the \({\mathcal{F}}\) -functional are exactly the weakly self-similar solutions to the harmonic map heat flow. We define the notion of \({\mathcal{F}}\) -stability for weakly self-similar solutions. The \({\mathcal{F}}\) -stability can be characterized by the semi-positive definiteness of the Jacobi operator acting on a subspace of variation fields.

Singularities of the Minkowski set and affine equidistants for a curve and a surface

Generic singularities of envelopes of families of chords and bifurcations of affine equidistants defined by a pair of a curve and a surface in R 3 are classified. The chords join pairs of points of the curve and the surface such that the tangent line to the curve is parallel to the tangent plane to the surface. The classification contains singularities of stable Lagrange and Legendre projections, boundary singularities and some less known classes appearing at the points of the surface and the curve themselves.

Generic bifurcations of framed curves in a space form and their envelopes

The generic singularities and bifurcations are classified for one-parameter families of curves with frames in the space forms E n + 1 , S n + 1 , H n + 1 . Two kinds of frames are considered; adapted frames and osculating frames. In particular, we give the classification results on the singularities of envelopes associated to framed curves. The associated envelopes and their singularities are classified. characterised in term of geometric invariants of framed curves. We apply to the global problem of framed curves and to the extension problem of surfaces with boundaries in three space. generalising the results obtained in Ishikawa (2010) [12].

ONE-DIMENSIONAL LIE FOLIATIONS WITH GENERIC SINGULARITIES IN COMPLEX DIMENSION THREE

AbstractWe prove that a germ of a one-dimensional holomorphic foliation with a generic singularity in dimension two or three that exhibits a Lie group transverse structure in the complement of some codimension one analytic subset is logarithmic, that is, given by a system of closed meromorphic one-forms with simple poles. In the global context, we prove that a foliation by curves in a three-dimensional complex manifold with generic singularities and a Lie group transverse structure off a codimension one analytic subset is logarithmic; that is, it is given by a system of closed meromorphic one-forms with simple poles.

Poincaré–Bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields

We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface, and prove a Poincaré–Bendixson theorem describing recurrence properties and ω-limit sets of geodesics for a meromorphic connection on P 1 ( C ) . We then show how to associate to a homogeneous vector field Q in C n a rank 1 singular holomorphic foliation F of P n − 1 ( C ) and a (partial) meromorphic connection ∇ o along F so that integral curves of Q are described by the geodesic flow of ∇ o along the leaves of F , which are Riemann surfaces. The combination of these results yields powerful tools for a detailed study of the dynamics of homogeneous vector fields. For instance, in dimension two we obtain a description of recurrence properties of integral curves of Q, and of the behavior of the geodesic flow in a neighborhood of a singularity, classifying the possible singularities both from a formal point of view and (for generic singularities) from a holomorphic point of view. We also get examples of unexpected new phenomena, we put in a coherent context scattered results previously known, and we obtain (as far as we know for the first time) a complete description of the dynamics in a full neighborhood of the origin for a substantial class of holomorphic maps tangent to the identity. Finally, as an example of application of our methods we study in detail the dynamics of quadratic homogeneous vector fields in C 2 .

On the similarity of Hamiltonian and reversible vector fields in 4D

We study the existence of formal conjugacies between reversible vector fields and Hamiltonian vector fields in 4D around a generic singularity. We construct conjugacies for a generic class of reversible vector fields. We also show that reversible vector fields are formally orbitally equivalent to polynomial decoupled Hamiltonian vector fields. The main tool we employ is the normal formtheory.

Generic bifurcations of low codimension of planar Filippov Systems

In this article some qualitative and geometric aspects of non-smooth dynamical systems theory are discussed. The main aim of this article is to develop a systematic method for studying local (and global) bifurcations in non-smooth dynamical systems. Our results deal with the classification and characterization of generic codimension-2 singularities of planar Filippov Systems as well as the presentation of the bifurcation diagrams and some dynamical consequences.

WORMHOLES AND CHILD UNIVERSES

Evidence to the case that classical gravitation provides the clue to make sense out of quantum gravity is presented. The key observation is the existence in classical gravitation of child universe solutions or "almost" solutions, "almost" because of some singularity problems. The difficulties of these child universe solutions that are due to their generic singularity problems will be very likely be cured by quantum effects, just like for example "almost" instanton solutions are made relevant in gauge theories with the breaking of conformal invariance. Some well-motivated modifcations of general relativity where these singularity problems are absent even at the classical level are discussed. High energy density excitations, responsible for UV divergences in quantum field theories, including quantum gravity, are likely to be the source of child universes which carry them out of the original space–time. This decoupling could prevent these high UV excitations from having any influence on physical amplitudes. Child universe production could therefore be responsible for UV regularization in quantum field theories which take into account semiclassically gravitational effects. Child universe production in the last stages of black hole evaporation, the prediction of absence of trans-Planckian primordial perturbations, connection to the minimum length hypothesis, and in particular the connection to the maximal curvature hypothesis are discussed. Some discussion of superexcited states in the case these states such as Kaluza–Klein excitations are carried out. Finally, the possibility of obtaining "string like" effects from the wormholes associated with the child universes is discussed.

Examining gravitational collapse with test scalar fields

Numerical simulations are performed of a test scalar field in a spacetime undergoing gravitational collapse. The behavior of the scalar field near the singularity is examined and implications for generic singularities are discussed. In particular, our example is the first confirmation of the Belinskii, Khalatnikov and Lifschitz (BKL) conjecture for an asymptotically flat spacetime.

Singularity Spectrum of Generic α-Hölder Regular Functions After Time Subordination

A question of Yves Meyer motivated the research concerning “time” subordinations of real functions. Denote by \({\mathcal {B}}_{1}^{\alpha}\) the metric space of functions with Lipschitz constant 1 defined on [0,1], equipped with the complete metric defined via the supremum norm. Given a function \(g\in {\mathcal {B}}_{1}^{\alpha}\) one obtains a time subordination of g simply by considering the composite function Z=g○f, where f∈ℳ:={f:f(0)=0, f(1)=1 and f is a continuous nondecreasing function on [0,1]}. The metric space \(\mathcal {E}^{\alpha}=\mathcal {M}\times {\mathcal {B}}_{1}^{\alpha}\) equipped with the product supremum metric is a complete metric space. In this paper for all α∈[0,1) multifractal properties of g○f are investigated for a generic (typical) element (f,g)∈ℰ α . In particular we determine the generic Holder singularity spectrum of g○f.

Resonance scattering and singularities of the scattering function

Recent studies of transport phenomena with complex potentials are explained by generic square root singularities of spectrum and eigenfunctions of non-Hermitian Hamiltonians. Using a two channel problem we demonstrate that such singularities produce a significant effect upon the pole behaviour of the scattering matrix, and more significantly upon the associated residues. This mechanism explains why by proper choice of the system parameters the resonance cross section is increased drastically in one channel and suppressed in the other channel.

Convex hulls of surfaces with boundaries and corners and singularities of transitivity zone in ℝ3

Generic singularities of the boundary of the local transitivity set of a control system on two- and three-dimensional manifolds are classified. The indicatrices of the system are assumed to be given by generic equations and inequalities.