Types Of Singularities Research Articles

Singularity Analysis for the Perspective-Four and Five-Line Problems

This paper deals with image-based visual servoing and pose estimation by observing four and five lines. Our main interest is to determine the relative configurations of the camera and the observed lines that lead to problems in control and stability. Since it is equivalent to finding the singularities of the corresponding Jacobian matrix, we use tools from computational algebraic geometry to seek configurations such that all of its minors vanish simultaneously. By choosing a suitable basis for this matrix, we revisit the problem in the case of three lines to show that one type of the singularities is when the camera lies on the hyperboloid of one sheet uniquely defined by the lines. This result is further exploited to prove that the one-dimensional singularities, if any, in the case of n lines appear when the camera lies on the transversals to the observed lines. Thus, by forcing the transversals to be complex, we can avoid the aforementioned type of singularities in the case of four lines although the algebra shows that there can always be up to 10 inevitable singular locations of the camera for the other type of singularity. For five lines, we find out that there are no singularities in the generic case. The singularities are also characterized for four and five lines with orthogonality and parallelism constraints. Furthermore, a visual servoing library is used to conduct some simulated experiments to substantiate the theoretical results. As expected, we observe problems in control in the vicinity of a singularity as well as increased errors in pose estimation.

Quantization of open strings in time-dependent black holes

In this paper, the open string is quantized in a time-dependent black hole background. By time-dependent black hole, one means a Vaydia geometry in the adiabatic approximation, which in [Formula: see text] dimensions is a BTZ black hole with a time-dependent mass. The worldsheet two-point function is derived and it is shown to have the same type of singularity as the flat space one. However, the equal times two-point function depends on the particular Cauchy surface where the worldsheet fields are defined. Finite temperature effects are incorporated through the Liouville–von Neumann approach to nonequilibrium thermodynamics.

Dynamic mode III stress intensity factors of multiple axisymmetric interfacial cracks in an FGM coated orthotropic layer

In the present study, mode III dynamic stress intensity factors (DSIFs) for the multiple axisymmetric interfacial cracks in an orthotropic layer with FGM orthotropic coating under transient torsional loading are formulated. It is assumed that the mass density and the shear modulus of the FGM orthotropic coating vary exponentially and power-law form along the thickness of the layer. At first, the solution for Somigliana-type dynamic rotational ring dislocation in the layer and its coating is obtained by using the Laplace and Hankel transforms. Then, the dislocation solution is used to derive a set of singular integral equations for a system of coaxial axisymmetric interfacial cracks, including penny-shaped and annular cracks. The integral equations are of Cauchy singular type, which are solved by Erdogan’s collocation method for dislocation density on the surface of interfacial crack and the results are used to determine DSIFs for axisymmetric interfacial cracks. Finally, several examples are provided to study the effects of the non-homogeneity constant, orthotropy parameter and thickness of FGM coating as well as the interaction of multiple interfacial cracks on the DSIFs.

Kinematic singularities of Feynman integrals and principal A-determinants

We consider the analytic properties of Feynman integrals from the perspective of general A-discriminants and mathcal{A} -hypergeometric functions introduced by Gelfand, Kapranov and Zelevinsky (GKZ). This enables us, to give a clear and mathematically rigorous description of the singular locus, also known as Landau variety, via principal A-determinants. We also comprise a description of the various second type singularities. Moreover, by the Horn-Kapranov-parametrization we give a very efficient way to calculate a parametrization of Landau varieties. We furthermore present a new approach to study the sheet structure of multivalued Feynman integrals by the use of coamoebas.

Multiple soft radiation at one-loop order and the emission of a soft quark\u2013antiquark pair

We consider the radiation of two or more soft partons in QCD hard-scattering at one-loop order. The corresponding scattering amplitude is singular, and the singular behaviour is controlled by a process-independent soft current. Using regularization in d=4 - 2epsilon space-time dimensions, we explicitly evaluate the ultraviolet and infrared divergent (epsilon -pole) terms of the one-loop soft current for emission of an arbitrary number of soft partons in a generic hard-scattering process. Then we consider the specific case of soft quark–antiquark (q{{bar{q}}}) emission and we compute the one-loop current by including the finite terms. We find that the one-loop soft-q{{bar{q}}} current exhibits a new type of transverse-momentum singularity, which has a quantum (absorptive) origin and a purely non-abelian character. At the squared amplitude (cross section) level, this transverse-momentum singularity produces contributions to multijet production processes in hadron collisions. The one-loop squared current also leads to charge asymmetry terms, which are a distinctive features of soft-q{{bar{q}}} radiation. We also extend these results to the cases of QED and mixed QCDtimes QED radiative corrections for soft fermion–antifermion emission.

Did the Universe experience a pressure non-crushing type cosmological singularity in the recent past?

In the recent literature it has been shown that the H 0 tension may be eliminated if an abrupt physics transition changed the Cepheid parameters in the near past of the Universe, nearly 70–150 Myrs ago. In this letter we stress the possibility that this abrupt transition was caused by the smooth passage of our Universe through a pressure finite-time cosmological singularity. Being a non-crushing type singularity the pressure singularity can leave its imprints in the Universe, since it occurs globally and literally everywhere. We discuss how this scenario could easily be realized by F(R) gravity, with the strong energy conditions being satisfied without the need for a scalar field or specific matter fluids. We also stress the fact that the pressure singularity can affect the effective gravitational constant of F(R) gravity. Moreover, we stress the fact that pressure singularities can disrupt the trajectories of bound objects in the Universe, which is also pointed out in the literature, even in the context of general relativity. We also show numerically in a general relativistic framework that elliptic trajectories are distorted and changed to different elliptic trajectories when the Universe passes through the pressure singularity. Such a disruption of the trajectories could have tidal effects on the surface of the Earth, for example on sea waters and oceans, regarding the distortion of Moon's elliptic trajectory. Accordingly, the distortion of Earth's trajectory around the Sun could have affected climatologically the Earth 70–150 Myrs ago.

Unveiling quantum entanglement and correlation of sub-Ohmic and Ohmic baths for quantum phase transitions in dissipative systems

By employing the spin-boson model in a dense limit of environmental modes, quantum entanglement and correlation of sub-Ohmic and Ohmic baths for dissipative quantum phase transitions are numerically investigated based on the variational principle. With several measures borrowed from quantum information theory, three different types of singularities are found for the first-order, second-order, and Kosterlitz-Thouless phase transitions, respectively, and the values of transition points and critical exponents are accurately determined. Besides, the scaling form of the quantum discord in the Ohmic case is identified, quite different from that in the sub-Ohmic regime. In a two-spin model, two distinct behaviors of the quantum discord are uncovered: one is related to the quantum entanglement between two spins and the other is decided by the correlation function in the position space rather than the entanglement.

Singularities and soft-Big Bang in a viscous ΛCDM model

In this paper we explore the different types of singularities that arise in the $\Lambda$CDM model when dissipative processes are considered, in the framework of the Eckart's theory. In particular, we study the late-time behavior of $\Lambda$CDM model with viscous cold dark matter (CDM) and an early-time viscous radiation domination era with cosmological constant (CC). The fluids are described by the barotropic equation of state (EoS) $p=(\gamma-1)\rho$, where $p$ is the equilibrium pressure of the fluid, $\rho$ their energy density, and $\gamma$ is the barotropic index. We explore two particular cases for the bulk viscosity $\xi$, a constant bulk viscosity $\xi=\xi_{0}$, and a bulk viscosity proportional to the energy density of the fluid $\xi=\xi_{0}\rho$. Due to some previous investigations that have explored to describe the behavior of the universe with a negative CC, we extend our analysis to this case. We found that future singularities like Big-Rip are allowed but without having a phantom EoS associated to the DE fluid. Big-Crunch singularities also appears when a negative CC is present, but also de Sitter and even Big-Rip types are allowed due to the negative pressure of the viscosity, which opens the possibility of an accelerated expansion in AdS cosmologies. We also discuss a very particular solution without Big Bang singularity that arises in the early-time radiation dominant era of our model known as Soft-Big Bang.

The closures of test configurations and algebraic singularity types

Given a Kähler manifold X with an ample line bundle L, we consider the metric space of finite energy geodesic rays associated to the Chern class c1(L). We characterize rays that can be approximated by ample test configurations. At the same time, we also characterize the closure of algebraic singularity types among all singularity types of quasi-plurisubharmonic functions, pointing out the very close relationship between these two seemingly unrelated problems.By Bonavero's holomorphic Morse inequalities, the arithmetic and non-pluripolar volumes of algebraic singularity types coincide. We show that in general the arithmetic volume dominates the non-pluripolar one, and equality holds exactly on the closure of algebraic singularity types. Analogously, we give an estimate for the Monge–Ampère energy of a general finite energy ray in terms of the arithmetic volumes along its Legendre transform. Equality holds exactly for rays approximable by test configurations.Various other cohomological and potential theoretic characterizations are given in both settings. As applications, we give a concrete formula for the non-Archimedean Monge–Ampère energy in terms of asymptotic expansion, and show the continuity of the projection map from L1 rays to non-Archimedean rays.

On the Timelike Sweeping Surfaces and Singularities in Minkowski 3-Space E 1 3

The Bishop frame or rotation minimizing frame (RMF) is an alternative approach to define a moving frame that is well defined even when the curve has vanished second derivative, and it has been widely used in the areas of computer graphics, engineering, and biology. The main aim of this paper is using the RMF for classification of singularity type of timelike sweeping surface and Bishop spherical Darboux image which is mightily concerning a unit speed spacelike curve with timelike binormal vector in E 1 3 .

Reflections on the matter of 3DN=1vacua and localSpin(7)compactifications

We use Higgs bundles to study the 3d $\mathcal{N} = 1$ vacua obtained from M-theory compactified on a local $Spin(7)$ space given as a four-manifold $M_4$ of ADE singularities with further generic enhancements in the singularity type along one-dimensional subspaces. There can be strong quantum corrections to the massless degrees of freedom in the low energy effective field theory, but topologically robust quantities such as "parity" anomalies are still calculable. We show how geometric reflections of the compactification space descend to 3d reflections and discrete symmetries. The "parity" anomalies of the effective field theory descend from topological data of the compactification. The geometric perspective also allows us to track various perturbative and non-perturbative corrections to the 3d effective field theory. We also provide some explicit constructions of well-known 3d theories, including those which arise as edge modes of 4d topological insulators, and 3d $\mathcal{N} = 1$ analogs of grand unified theories. An additional result of our analysis is that we are able to track the spectrum of extended objects and their transformations under higher-form symmetries.

Extremal $1/2$ Calabi–Yau $3$-folds and six-dimensional F-theory applications

We discuss a method for classifying the singularity types of 1/2 Calabi-Yau 3-folds, a family of rational elliptic 3-folds introduced in a previous study in relation to various U(1) factors in 6D F-theory models. A projective dual pair of del Pezzo manifolds recently studied by Mukai is used to analyze the singularity types. In particular, we studied the maximal rank seven singularity types of 1/2 Calabi-Yau 3-folds. The structures of the singular fibers are analyzed using blow-ups. Double covers of the 1/2 Calabi-Yau 3-folds yield elliptic Calabi-Yau 3-folds and applications to six-dimensional $N = 1$ F-theory on the Calabi-Yau 3-folds are also discussed. The deduced singular fibers have applications in studying the gauge groups formed in 6D F-theory compactifications. The blow-up methods used to analyze the singular fibers and sections utilized in this research might have applications in studying the U(1) factors and hypermultiplets charged under U(1) in 6D F-theory.

Positivity of direct images with a Poincaré type twist

Abstract We consider a holomorphic family $f:\mathscr {X} \to S$ of compact complex manifolds and a line bundle $\mathscr {L}\to \mathscr {X}$ . Given that $\mathscr {L}^{-1}$ carries a singular hermitian metric that has Poincaré type singularities along a relative snc divisor $\mathscr {D}$ , the direct image $f_*(K_{\mathscr {X}/S}\otimes \mathscr {D} \otimes \mathscr {L})$ carries a smooth hermitian metric. If $\mathscr {L}$ is relatively positive, we give an explicit formula for its curvature. The result applies to families of log-canonically polarized pairs. Moreover, we show that it improves the general positivity result of Berndtsson-Păun in a special situation of a big line bundle.

Interpolation of a regular subspace complementing the span of a radially singular function

We analyze the interpolation of the sum of a subspace, consisting of regular functions, with the span of a function with $r^{\alpha }$-type singularity. In particular, we determine all interpolation parameters, for which the interpolation space of the sub

On the fifth Whitney cone of a complex analytic curve

We present an explicit procedure to determine the $C_5$-cone of a reduced complex analytic curve $X \subset \mathbb{C}^n$ at a singular point $0 \in X$, which is known to be a union of a finite number of planes. We also give upper bounds on the number of these planes. The procedure comes out with a collections of integers that we call auxiliary multiplicities and we prove they determine the Lipschitz type of complex curve singularities. We finish presenting an example which shows that the number of irreducible components of the $C_5$-cone of a curve is not a bi-Lipschitz invariant.

On the traveling wave solutions of pulse propagation in monomode fiber via the extended Kudryashov’s approach

In this research paper, we tackle with the solitary wave solutions to the pulse prop?agation in monomode optic fiber by defining non-linear Schrodinger equation with higher order. We applied the extended Kudryashov?s method with Bernoulli-Riccati equation and successfully gained soliton solutions and their contour, 2-D and 3-D graphical representations, such as dark, singular, periodic and kink type solutions. We also discussed the obtained results in the related section.

Detailed Study of Stellar Evolution and Stellar Gravitational Collapse Leading to The Formation of Black Holes

In this paper we talk about the formation mechanism of the stellar gravitational singularities. To begin with we talk about the formation and evolution of stars and understand the Hertzsprung Russell diagram that teaches us how to classify stars. With the help of the diagram, we categorize the stars based of their physical parameters such as color, temperature, and mass. We then talk about the death cycle of different mass stars and what comes after when their fuels have been exhausted. Stars below the Chandrashekhar limit form a white dwarf at the end of their lives, while stars above the limit form a neutron star of a singularity. Further to find which of the heavy mass stars forms a singularity we look at the Tolman-Oppenheimer-Volkoff limit that states stars with mass above the limit will form singularities. The types of singularities formed depend on the solution of the general theory of relativity given by Schwarzschild, Kerr, Kerr-Newman, and Reissner-Nordström. Each of the theory aspect of the four solutions has been described to give a better understanding of the structure of the singularity formed. The paper also explains theories such as Wormholes and time travel in brief to try and explain what can replace the singularity.

Asymptotic stability of spiky steady states for a singular chemotaxis model with signal-suppressed motility.

We study the nonlinear stability of spiky solutions to a chemotaxis model of consumption type with singular signal-suppressed motility in the half space. We show that, when the no-flux boundary condition for the bacteria density and the nonhomogeneous Dirichlet boundary condition for the nutrient are prescribed, this chemotaxis model admits a unique smooth spiky steady state, and it is nonlinearly stable under appropriate perturbations. The challenge of the problem is that there are two types of singularities involved in the model: one is the logarithmic singularity of the sensitive function; and the other is the inverse square singularity of the motility. We employ a Cole-Hopf transformation to relegate the former singularity to a nonlocality that can be resolved by the method of anti-derivative. To deal with the latter singularity, we construct an approximate system that retains a key structure of the original singular system in the local theory, and develop a new strategy, which combines a weighted elliptic estimate and the weighted energy estimate, to establish a priori estimate in the global theory.

Analytic torsion for log-Enriques surfaces and Borcherds product

Abstract We introduce a holomorphic torsion invariant of log-Enriques surfaces of index two with cyclic quotient singularities of type $\frac {1}{4}(1,1)$ . The moduli space of such log-Enriques surfaces with k singular points is a modular variety of orthogonal type associated with a unimodular lattice of signature $(2,10-k)$ . We prove that the invariant, viewed as a function of the modular variety, is given by the Petersson norm of an explicit Borcherds product. We note that this torsion invariant is essentially the BCOV invariant in the complex dimension $2$ . As a consequence, the BCOV invariant in this case is not a birational invariant, unlike the Calabi-Yau case.

A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions

Laplace’s equation is one of the important equations in studying applied mathematics and engineering problems including the study of temperature distribution of steady-state heat conduction or the concentration distribution of steady-state diffusion problems. In this study, the analytical method has been applied to solve the Laplace's equation in a two-dimensional domain. For the specified Neumann or Dirichlet boundary conditions, the analytical solution of temperature distribution in the quarter-plane can be found by several methods including the Fourier transform method, similarity method, and the method of Green’s function with images. For different boundary conditions, the solution of temperature distribution of the Laplace’s equation will be in a totally different form. Nevertheless, the merit of this research is that the solutions of steady-state temperature distribution in the quarter plane with Neumann and Dirichlet boundary conditions are unified under the singular similarity solution with source type singularity. With the typical benchmarked examples for finding the temperature distribution by the numerical integral method, it is shown that Gibbs phenomenon behaves at a jump discontinuity, where serious oscillation result was found especially near the singular points of the boundary. In addition, the temperature distribution in the domain can be easily calculated without oscillation phenomenon near the singular points from the similarity solutions.