Showing papers on "Singularity published in 2021"
TL;DR: In this article, it was shown that the K-polystability of Q-Fano varieties is equivalent to the equivariant K polystability, that is, to check K-Polystability it is sufficient to check special test configurations which are equivariant under the torus action.
Abstract: We prove two new results on the K-polystability of Q-Fano varieties based on purely algebro-geometric arguments. The first one says that any K-semistable log Fano cone has a special degeneration to a uniquely determined K-polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to complete the proof of Donaldson-Sun's Conjecture which says that the metric tangent cone of any close point appearing on a Gromov-Hausdorff limit of Kahler-Einstein Fano manifolds only depends on the algebraic structure of the singularity. The second result says that for any log Fano variety with a torus action, the K-polystability is equivalent to the equivariant K-polystability, that is, to check K-polystability, it is sufficient to check special test configurations which are equivariant under the torus action.
77 citations
08 Feb 2021
TL;DR: In this paper, the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero were studied and two fundamental theorems of fractional calculus were proved.
Abstract: In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First, the Sonine kernels and their important special classes and particular cases are discussed. In particular, we introduce a class of the Sonine kernels that possess an integrable singularity of power function type at the point zero. For the general fractional integrals and derivatives with the Sonine kernels from this class, two fundamental theorems of fractional calculus are proved. Then, we construct the n-fold general fractional integrals and derivatives and study their properties.
65 citations
TL;DR: With the proposed control scheme, no piecewise continuous functions are required any more in the controller design to avoid the singularity, and the fixed-time stability of the entire closed-loop system in the reaching phase and sliding phase is analyzed with a rigorous theoretical proof.
Abstract: In this article, a neural-network-based adaptive fixed-time control scheme is proposed for the attitude tracking of uncertain rigid spacecrafts. A novel singularity-free fixed-time switching function is presented with the directly nonsingular property, and by introducing an auxiliary function to complete the switching function in the controller design process, the potential singularity problem caused by the inverse of the error-related matrix could be avoided. Then, an adaptive neural controller is developed to guarantee that the attitude tracking error and angular velocity error can both converge into the neighborhood of the equilibrium within a fixed time. With the proposed control scheme, no piecewise continuous functions are required any more in the controller design to avoid the singularity, and the fixed-time stability of the entire closed-loop system in the reaching phase and sliding phase is analyzed with a rigorous theoretical proof. Comparative simulations are given to show the effectiveness and superiority of the proposed scheme.
58 citations
TL;DR: In this paper, an improved method for disconnection and re-connection of a six-DOF manipulator with simple geometry is proposed. But the method is not suitable for the case of the wrist joint variables.
53 citations
TL;DR: In this paper, the existence and uniqueness of solutions to complex Monge-Ampere equations with prescribed singularity type were proved for the case of big cohomology classes.
Abstract: Let $$(X,\omega )$$
be a compact Kahler manifold. We prove the existence and uniqueness of solutions to complex Monge–Ampere equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is dropped, and we work with general model type singularities. We state and prove our theorems in the context of big cohomology classes, however our results are new in the Kahler case as well. As an application we confirm a conjecture by Boucksom–Eyssidieux–Guedj–Zeriahi concerning log-concavity of the volume of closed positive (1, 1)-currents. Finally, we show that log-concavity of the volume in complex geometry corresponds to the Brunn–Minkowski inequality in convex geometry, pointing out a dictionary between our relative pluripotential theory and P-relative convex geometry. Applications related to stability and existence of csck metrics are treated elsewhere.
47 citations
TL;DR: In this article, the authors studied the extremization equations in the semiclassical region far from the singularity and found that the quantum extremal surfaces always bend in the direction away from singularity.
Abstract: We study aspects of entanglement and extremal surfaces in various families of spacetimes exhibiting cosmological, Big-Crunch, singularities, in particular isotropic AdS Kasner. The classical extremal surface dips into the bulk radial and time directions. Explicitly analysing the extremization equations in the semiclassical region far from the singularity, we find the surface bends in the direction away from the singularity. In the 2-dim cosmologies obtained by dimensional reduction of these and other singularities, we have studied quantum extremal surfaces by extremizing the generalized entropy. The resulting extremization shows the quantum extremal surfaces to always be driven to the semiclassical region far from the singularity. We give some comments and speculations on our analysis.
44 citations
TL;DR: The regularized method of moments uses the origin intensity factor technique which is free of mesh and integration to deal with the singularity at origin of the basis function and can reduce the computational time by half, while the stability and accuracy stay about the same.
43 citations
39 citations
38 citations
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TL;DR: In this paper, the authors review recent progress in the study of flat band systems, especially focusing on the fundamental physics related to the singularity of the flat band's Bloch wave functions.
Abstract: We review recent progresses in the study of flat band systems, especially focusing on the fundamental physics related to the singularity of the flat band’s Bloch wave functions. We first explain th...
38 citations
TL;DR: The authors derived loop quantum gravity corrections to the Raychaudhuri equation in the interior of a Schwarzschild black hole and near the classical singularity, and showed that the resulting effective equation implies defocusing of geodesics due to the appearance of repulsive terms.
Abstract: We derive loop quantum gravity corrections to the Raychaudhuri equation in the interior of a Schwarzschild black hole and near the classical singularity. We show that the resulting effective equation implies defocusing of geodesics due to the appearance of repulsive terms. This prevents the formation of conjugate points, renders the singularity theorems inapplicable, and leads to the resolution of the singularity for this spacetime.
TL;DR: In this paper, the authors argue that the proper time from the event horizon to the singularity can be extracted from the thermal expectation values of certain operators outside the horizon, by varying the mass of the field.
Abstract: We argue that the proper time from the event horizon to the black hole singularity can be extracted from the thermal expectation values of certain operators outside the horizon. This works for fields which couple to higher-curvature terms, so that they can decay into two gravitons. To extract this proper time, it is necessary to vary the mass of the field.
TL;DR: In this paper, the authors proved sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and showed the asymptotic radial symmetry of the solutions near the singularity.
TL;DR: In this article, a complete list of all envelope patterns and a varied of dynamical properties of patterns, which include solitons, singularity, periodicity and double periodicity are presented.
TL;DR: In this article, a high-order compact alternating direction implicit scheme is considered to solve the two-dimensional time-fractional integro-differential equation with weak singularity near the initial time.
TL;DR: In this article, it was shown that the geometry near the singularity takes a universal Kasner form when the kinetic term of the scalar hair dominates, while novel behaviors different from the Kasner-form are uncovered when scalar potential become important to the background.
Abstract: We establish a no inner-horizon theorem for black holes with charged scalar hairs. Considering a general gravitational theory with a charged scalar field, we prove that there exists no inner Cauchy horizon for both spherical and planar black holes with non-trivial scalar hair. The hairy black holes approach to a spacelike singularity at late interior time. This result is independent of the form of scalar potentials as well as the asymptotic boundary of spacetimes. We prove that the geometry near the singularity takes a universal Kasner form when the kinetic term of the scalar hair dominates, while novel behaviors different from the Kasner form are uncovered when the scalar potential become important to the background. For the hyperbolic horizon case, we show that hairy black hole can only has at most one inner horizon, and a concrete example with an inner horizon is presented. All these features are also valid for the Einstein gravity coupled with neutral scalars.
TL;DR: In this article, it was shown that negative precession and shadow can exist simultaneously in deformed Kerr naked singularity spacetime. But the authors did not consider the case of a singularity with a shadow.
Abstract: It is now known that the shadow is not only the property of a black hole, it can also be cast by other compact objects like naked singularities. However, there exist some novel features of the shadow of the naked singularities which are elaborately discussed in some recent articles. In the earlier literature, it is also shown that a naked singularity may admit negative precession of bound timelike orbits which cannot be seen in Schwarzschild and Kerr black hole spacetimes. This distinguishable behavior of timelike bound orbit in the presence of the naked singularity along with the novel features of the shadow may be useful to distinguish between a black hole and a naked singularity observationally. However, in this paper, it is shown that deformed Kerr spacetime can allow negative precession of bound timelike orbits, when the central singularity of that spacetime is naked. We also show that negative precession and shadow both can exist simultaneously in deformed Kerr naked singularity spacetime. Therefore, any observational evidence of negative precession of bound orbits, along with the central shadow may indicate the presence of a deformed Kerr naked singularity.
TL;DR: A high-order method based on orthogonal spline collocation (OSC) method is formulated for the solution of the fourth-order subdiffusion problem on the rectangle domain in 2D with sides parallel to the coordinate axes, whose solutions display a typical weak singularity at the initial time.
Abstract: A high-order method based on orthogonal spline collocation (OSC) method is formulated for the solution of the fourth-order subdiffusion problem on the rectangle domain in 2D with sides parallel to the coordinate axes, whose solutions display a typical weak singularity at the initial time. By introducing an auxiliary variable v = Δ u , the fourth-order problem is reduced into a couple of second-order system. The L1 scheme on graded mesh is considered for the Caputo fractional derivatives of order α ∈ ( 0 , 1 ) by inserting more grid points near the initial time. By virtue of some properties, such as complementary discrete convolution kernel and discrete fractional Gronwall inequality, we establish unconditional stability and convergence for the original unknown u and auxiliary variable v . Some numerical experiments are provided to further verify our theoretical analysis.
TL;DR: In this paper, the authors analyzed the dynamics associated with spherically symmetric, regular black holes taking the full backreaction between the infalling matter and geometry into account, and identified the crucial features taming the growth of the mass function and diminishing the curvature singularity at the Cauchy horizon.
Abstract: Non-singular black hole geometries typically come with two spacetime horizons: an (outer) event horizon and an (inner) Cauchy horizon. This nurtures the speculation that they may be subject to a mass-inflation effect which renders the Cauchy horizon unstable. We analyze the dynamics associated with spherically symmetric, regular black holes taking the full backreaction between the infalling matter and geometry into account. On this basis, we identify the crucial features taming the growth of the mass function and diminishing the curvature singularity at the Cauchy horizon. It is demonstrated explicitly that the regular black hole solutions proposed by Hayward and obtained from Asymptotic Safety satisfy these properties.
TL;DR: In this paper, a singularity preserving regularization method for fractional derivatives of order α ∈ (0, 1 ) is proposed. And the convergence analysis is carried out and the optimal error estimates are obtained.
TL;DR: In this paper, a class of nonlocal viscous Cahn-Hilliard equations with Neumann boundary conditions for the chemical potential was considered and well-posedness for the nonlocal equation was proved in a suitable variational sense.
Abstract: We consider a class of nonlocal viscous Cahn–Hilliard equations with Neumann boundary conditions for the chemical potential. The double-well potential is allowed to be singular (e.g. of logarithmic type), while the singularity of the convolution kernel does not fall in any available existence theory under Neumann boundary conditions. We prove well-posedness for the nonlocal equation in a suitable variational sense. Secondly, we show that the solutions to the nonlocal equation converge to the corresponding solutions to the local equation, as the convolution kernels approximate a Dirac delta. The asymptotic behaviour is analyzed by means of monotone analysis and Gamma convergence results, both when the limiting local Cahn–Hilliard equation is of viscous type and of pure type.
29 Jan 2021
TL;DR: In this paper, a Poincare algebra of conserved charges associated with the dynamics of the interior of a black hole is revealed, and the existence of this symmetry provides a powerful criterion to discriminate between different regularization and quantization schemes.
Abstract: We reveal an $\mathfrak{iso}(2,1)$ Poincare algebra of conserved charges associated with the dynamics of the interior of black holes The action of these Noether charges integrates to a symmetry of the gravitational system under the Poincare group ISO$(2,1)$, which allows to describe the evolution of the geometry inside the black hole in terms of geodesics and horocycles of AdS${}_2$ At the Lagrangian level, this symmetry corresponds to Mobius transformations of the proper time together with translations Remarkably, this is a physical symmetry changing the state of the system, which also naturally forms a subgroup of the much larger $\textrm{BMS}_{3}=\textrm{Diff}(S^1)\ltimes\textrm{Vect}(S^1)$ group, where $S^1$ is the compactified time axis It is intriguing to discover this structure for the black hole interior, and this hints at a fundamental role of BMS symmetry for black hole physics The existence of this symmetry provides a powerful criterion to discriminate between different regularization and quantization schemes Following loop quantum cosmology, we identify a regularized set of variables and Hamiltonian for the black hole interior, which allows to resolve the singularity in a black-to-white hole transition while preserving the Poincare symmetry on phase space This unravels new aspects of symmetry for black holes, and opens the way towards a rigorous group quantization of the interior
TL;DR: While for general unstructured differential-algebraic systems the characterization of these distances are partially open problems, it is shown that for dissipative Hamiltonian systems and related matrix polynomials there exist explicit characterizations that can be implemented numerically.
TL;DR: In this article, the lattice results for the form factors of the quenched three-gluon vertex of QCD, in two special kinematic configurations that depend on a single momentum scale, are presented.
TL;DR: In this article, the authors combine singularity theory and Clifford algebra to study singular ruled surfaces, and make the singular rules transform into the dual singular curves on the dual unit sphere.
Abstract: Singular ruled surface is an interesting research object and is the breakthrough point of exploring new problems. However, because of singularity, it’s difficult to study the properties of singular ruled surfaces. In this paper, we combine singularity theory and Clifford algebra to study singular ruled surfaces. We take advantage of the dual number of Clifford algebra to make the singular ruled surfaces transform into the dual singular curves on the dual unit sphere. By using the research method on the singular curves, we give the definition of the dual evolute of the dual front in the dual unit sphere, we further provide the k-th dual evolute of the dual front. Moreover, we consider the ruled surface corresponding to the dual evolute and k-th dual evolute and provide the developable conditions of these ruled surfaces.
TL;DR: The main purpose of this work is to develop a spectrally accurate collocation method for solving weakly singular integral equations of the second kind with nonsmooth solutions in high dimensions based on a multivariate Jacobi approximation in the frequency space.
Abstract: The main purpose of this work is to develop a spectrally accurate collocation method for solving weakly singular integral equations of the second kind with nonsmooth solutions in high dimensions. The proposed spectral collocation method is based on a multivariate Jacobi approximation in the frequency space. The essential idea is to adopt a smoothing transformation for the spectral collocation method to circumvent the curse of singularity at the beginning of time. As such, the singularity of the numerical approximation can be tailored to that of the singular solutions. A rigorous convergence analysis is provided and confirmed by numerical tests with nonsmooth solutions in two dimensions. The results in this paper seem to be the first spectral approach with a theoretical justification for high-dimensional nonlinear weakly singular Volterra type equations with nonsmooth solutions.
TL;DR: In this article, the authors extend the notion of the Nieh-Yan invariant to generic metric-affine geometries, where both torsion and nonmetricity are taken into account.
Abstract: We extend the notion of the Nieh-Yan invariant to generic metric-affine geometries, where both torsion and nonmetricity are taken into account. Notably, we show that the properties of projective invariance and topologicity can be independently accommodated by a suitable choice of the parameters featuring this new Nieh-Yan term. We then consider a special class of modified theories of gravity able to promote the Immirzi parameter to a dynamical scalar field coupled to the Nieh-Yan form, and we discuss in more detail the dynamics of the effective scalar tensor theory stemming from such a revised theoretical framework. We focus, in particular, on cosmological Bianchi I models and we derive classical solutions where the initial singularity is safely removed in favor of a big bounce, which is ultimately driven by the nonminimal coupling with the Immirzi field. These solutions, moreover, turn out to be characterized by finite time singularities, but we show that such critical points do not spoil the geodesic completeness and wave regularity of these spacetimes.
TL;DR: In this article, the authors compute thermal 2-point correlation functions in the black brane background dual to 4d CFT's at finite temperature for operators of large scaling dimension and find a formula that matches the expected structure of the OPE.
Abstract: We compute thermal 2-point correlation functions in the black brane $AdS_5$ background dual to 4d CFT's at finite temperature for operators of large scaling dimension. We find a formula that matches the expected structure of the OPE. It exhibits an exponentiation property, whose origin we explain. We also compute the first correction to the two-point function due to graviton emission, which encodes the proper time from the event horizon to the black hole singularity.
TL;DR: In this article, a new nonlinear parabolic equation on a Riemann surface is studied, which arises as a reduction of the Anomaly flow on a fibration, and a criterion for long-time existence for this flow is given.
Abstract: We initiate the study of a new nonlinear parabolic equation on a Riemann surface. The evolution equation arises as a reduction of the Anomaly flow on a fibration. We obtain a criterion for long-time existence for this flow, and give a range of initial data where a singularity forms in finite time, as well as a range of initial data where the solution exists for all time. A geometric interpretation of these results is given in terms of the Anomaly flow on a Calabi-Yau threefold.
TL;DR: In this paper, the Petrov-Galerkin spectral method is adopted to deal with the initial singularity in the temporal direction in which the first kind Jacobi poly-fractonomials are utilized as temporal trial functions and the second kind JF as temporal test functions.