Generic Singularities Research Articles

Generic singularities of relay systems

Generic singularities of relay systems

Generic singularities for 2D pressureless flows

Generic singularities for 2D pressureless flows

On Generic Singularities of Solutions to the 1D Gas Flow Equations: Chaplygin and Bechert–Stanyukovich Cases

On Generic Singularities of Solutions to the 1D Gas Flow Equations: Chaplygin and Bechert–Stanyukovich Cases

Stability Enhancement by Zero‐Point Spin Fluctuations: A Quantum Perspective on Bloch Point Topological Singularities

AbstractBloch points represent singularities within magnetic materials. From a macroscopic viewpoint, their cores are points where the magnetization vector is undefined, resulting in unique topological characteristics that influence the magnetic behavior of their hosts. The picture is very different at the microscopic level, where quantum effects enter the scene. The spin variables' quantum dynamics effect on the BP's stability is revealed. Zero‐point fluctuations, intrinsic fluctuations within the quantum mechanical ground state originating from the uncertainty principle, play a fundamental role. It is found that quantum fluctuations bloom in the vicinity of the singularity, thereby reducing the effective magnetic moment in its neighborhood. This increases the overall stability of the BP. These methods also allow for a characterization of the magnonic eigenmodes surrounding and bound to the singularity. The latter leads to predict on quite general grounds several features of the magnonic spectra, its degeneration structure, and its splitting response under a magnetic field. The last result is coherent with the association of a magnetic moment to the orbital angular momentum of the magnons. This approach allows integration with multiscale algorithms to provide a realistic description of generic topological singularities.

Quantum chaos of the BKL scenario

The Belinski-Khalatnikov-Lifshitz (BKL) scenario concerns the existence of generic singularity of general relativity. Evolving towards that singularity, spacetime enters a chaotic phase. We consider a model of the BKL scenario to get insight into the corresponding quantum dynamics. The integral quantization of the BKL model leads to quantum evolution devoid of singularity. The quantum fluctuations seem to be unable to suppress the classical chaos. These interesting features of quantum dynamics result from the never vanishing variance of considered quantum dynamics. We suggest that these results generalize to a quantum model (to be constructed) of the original BKL scenario.

Tubular surface generated by a curve lying on a regular surface and its characterizations

<abstract><p>In this research, we have constructed and studied special tubular surfaces in Euclidean 3-space $ \mathbb{R}^{3} $. We examined the singularities and geometrical properties of these surfaces. We achieved some significant results for these surfaces via Darboux frame. Also, we have proposed a few geometric invariants that illustrate the geometric characteristics of these surfaces, such as tubular Weingarten surfaces, using the traditional methods of differential geometry. Additionally, taking advantage of the singularity theory, we have given the classification of generic singularities of these surfaces. At last, we have presented some computational examples as an instance of use to validate our theoretical findings.</p></abstract>

Integrability of SLE via conformal welding of random surfaces

AbstractWe demonstrate how to obtain integrability results for the Schramm‐Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating‐of‐trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact formula for the law of a conformal derivative of a classical variant of SLE called . Our proof is built on two connections between SLE, LCFT, and mating‐of‐trees. Firstly, LCFT and mating‐of‐trees provide equivalent but complementary methods to describe natural random surfaces in LQG. Using a novel tool that we call the uniform embedding of an LQG surface, we extend earlier equivalence results by allowing fewer marked points and more generic singularities. Secondly, the conformal welding of these random surfaces produces SLE curves as their interfaces. In particular, we rely on the conformal welding results proved in our companion paper Ang, Holden and Sun (2023). Our paper is an essential part of a program proving integrability results for SLE, LCFT, and mating‐of‐trees based on these two connections.

The Łojasiewicz exponent of nondegenerate surface singularities

AbstractLet f be an isolated singularity at the origin of $\mathbb {C}^n$ . One of many invariants that can be associated with f is its Łojasiewicz exponent $\mathcal {L}_0 (f)$ , which measures, to some extent, the topology of f. We give, for generic surface singularities f, an effective formula for $\mathcal {L}_0 (f)$ in terms of the Newton polyhedron of f. This is a realization of one of Arnold’s postulates.

Generic features of a polymer quantum black hole

Non-singular black holes models can be described by modified classical equations motivated by loop quantum gravity. We investigate what happens when the sine function typically used in the modification is replaced by an arbitrary bounded function, a generalization meant to study the effect of ambiguities such as the choice of representation of the holonomy. A number of features can be determined without committing to a specific choice of functions. We find generic singularity resolution. The presence and number of horizons is determined by global features of the function regularizing the angular components of the connection, and the presence and number of bounces by global features of the function regularizing the time component. The trapping or anti-trapping nature of regions inside horizons depends on the relative location with respect to eventual bounces. We use these results to comment on some of the ambiguities of polymer black hole models.

Local entropy and generic multiplicity one singularities of mean curvature flow of surfaces

In this paper we prove that the generic singularities of mean curvature flow of closed embedded surfaces in $\mathbb{R}^3$ modelled by closed self-shrinkers with multiplicity has multiplicity one. Together with the previous result by Colding–Minicozzi in [CM12], we conclude that the only generic singularity of mean curvature flow of closed embedded surfaces in $\mathbb{R}^3$ modelled by closed self-shrinkers is a multiplicity one sphere. We also construct particular perturbations of the flow to avoid those singularities with multiplicity higher than one. Our result partially addresses the well-known multiplicity one conjecture by Ilmanen.

Quantum dynamics corresponding to the chaotic BKL scenario

We quantize the solution to the Belinski– Khalatnikov–Lifshitz (BKL) scenario using the integral quantization method. Quantization smears the gravitational singularity, preventing its localization in the configuration space. The latter is defined in terms of spatial and temporal coordinates, which are treated on the same footing that enables the respective covariance of general relativity. The relative quantum perturbations grow as the system evolves towards the gravitational singularity. The quantum randomness amplifies the deterministic classical chaos of the BKL scenario. Additionally, our results suggest that the generic singularity of general relativity can be avoided at a quantum level, giving support to the expectation that quantum gravity has a good chance of being a regular theory.

Singularities for Timelike Developable Surfaces in Minkowski 3-Space

In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give the classification of generic singularities of these developable surfaces. Furthermore, an example of application is given to illustrate the applications of the results.

Pseudo‐spherical evolutes of lightlike loci on mixed type surfaces in Minkowski 3‐space

A surface with nonempty timelike, lightlike, and spacelike points in Minkowski 3‐space is a mixed type surface. The mixed type surface has a signature‐changing metric, and its lightlike points can be seen as singularities of such metric. In this paper, we study singular properties of pseudo‐spherical evolutes of lightlike loci on mixed type surfaces. We classify the generic singularities of pseudo‐spherical evolutes by the singularity theory. These singularities and the contact between lightlike loci and model submanifolds are closely associated. We also show the Legendrian duality among pseudo‐spherical evolutes, lightlike loci, and Darboux vectors. Moreover, the singularities of pseudo‐spherical evolutes are studied from the viewpoint of opening map‐germ. Finally, we give an example to demonstrate the theoretical results.

Approximation of mean curvature flow with generic singularities by smooth flows with surgery

We construct smooth mean curvature flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We conclude by improving the entropy bound on the low-entropy Schoenflies conjecture.

Inheritance of Generic Singularities of Solutions of a Linear Wave Equation by Solutions of Isoentropic Gas Motion Equations

Inheritance of Generic Singularities of Solutions of a Linear Wave Equation by Solutions of Isoentropic Gas Motion Equations

The Poynting Vector Field Generic Singularities in Resonant Scattering of Plane Linearly Polarized Electromagnetic Waves by Subwavelength Particles.

We present the results of a study of the Poynting vector field generic singularities at the resonant light scattering of a plane monochromatic linearly polarized electromagnetic wave by a subwavelength particle. We reveal the impact of the problem symmetry, the spatial dimension, and the energy conservation law on the properties of the singularities. We show that, in the cases when the problem symmetry results in the existence of an invariant plane for the Poynting vector field lines, a formation of a standing wave in the immediate vicinity of a singularity gives rise to a saddle-type singular point. All other types of singularities are associated with vanishing at the singular points, either (i) magnetic field, for the polarization plane parallel to the invariant plane, or (ii) electric field, at the perpendicular orientation of the polarization plane. We also show that in the case of two-dimensional problems (scattering by a cylinder), the energy conservation law restricts the types of possible singularities only to saddles and centers in the non-dissipative media and to saddles, foci, and nodes in dissipative. Finally, we show that dissipation affects the (i)-type singularities much stronger than the (ii)-type. The same conclusions are valid for the imaginary part of the Poynting vector in problems where the latter is regarded as a complex quantity. The singular points associated with the formation of standing waves are different for real and imaginary parts of this complex vector field, while all other singularities are common. We illustrate the general discussion by analyzing singularities at light scattering by a subwavelength Germanium cylinder with the actual dispersion of its refractive index.

Generic naked singularities in Vaidya spacetimes

We investigate the occurrence of naked singularities, local and global, in the incoming Vaidya spacetimes with zero initial mass. While it is well-known that these spacetimes admit locally and globally naked singularities, we demonstrate that globally naked singularities are significantly more common than is stressed in the literature, being generic in a natural topology on this collection of spacetimes. A heuristic consequence of the results is that the slow accumulation of mass is both necessary and sufficient for both types of naked singularity in these spacetimes. We demonstrate that the naked singularity, as long as it exists, always has divergent curvature (Kretschmann scalar) associated to it along emerging null curves, regardless of the form of the mass function. In particular, a consequence is that the curvature strength of the singularity cannot be smoothed away.

Uniqueness and generic regularity of global weak conservative solutions to the Constantin-Lannes equation

ABSTRACT This paper is devoted to the uniqueness of global-in-time conservative solutions and generic regularity for the shallow water waves of moderate amplitude equation (Constantin-Lannes equation). The Constantin-Lannes equation possible development of singularities in finite time, and beyond the occurrence of wave breaking, it exists global conservative solutions. In the present paper, we will prove the uniqueness of global-in-time conservative solutions for the Constantin-Lannes equation with general initial data by analyzing the evolution of the quantities u and along each characteristic. Moreover, we consider that piecewise smooth solutions with only generic singularities are dense in the whole solution set by Thom's transversality Lemma.

Generic optical singularities and beam-field phenomena due to general paraxial beam reflection at a plane dielectric interface

A single paraxial beam reflection at a plane dielectric interface, configured appropriately, can lead to the formation of a polarization singularity in the inhomogeneously polarized output beam field for any central angle of incidence. In this paper we derive the necessary condition to realize this effect. We explore the phase singularity characteristics associated with this polarization-singular field and explore the dynamics of the singularities due to controlled variations of the input polarization. The simulation-generated exact field information lead to the exploration of the unique Goos-H\"anchen, Imbert-Fedorov, and spin shifts of the optical-singular fields and the anticipation of an exact mathematical characterization of spin-orbit interaction phenomena involved therein. The formation of a phase singularity independent of a polarization singularity is explained subsequently. Interrelating these seemingly unconnected beam-field phenomena and generic optical singularities can lead to a significant and fundamental understanding of the inhomogeneously polarized beam field; additionally, our singularity generation method can find potential application in experimental characterization of the involved dielectric media.

Generation of Stokes singularities using polarization lateral shear interferometer.

Lateral shear interferometer, being a self-referenced interferometer, has proven to be an important tool in scalar optics. Here we employ a vectorial counterpart - polarization lateral shear interferometer, in which the two interfering beams apart from being derived from the test wavefront, are in orthogonal states of polarization. Therefore when the test wavefront has spatially varying phase gradient across the beam cross-section, the resulting shearogram produces polarization fringes instead of intensity fringes. Further, the shearogram becomes inhomogeneously polarized. This polarization lateral shear interferometer may have potential uses in metrology, but in this article we demonstrate the ability of the interferometer in the generation of all Stokes singularities in the single beam by launching a phase singular beam into it. It is found that a vortex dipole is formed along with other generic Stokes singularities. Experimental observations support the results and they are discussed in the article.