Interior Singularities Research Articles

Torus-like solutions for the Landau-de Gennes model. Part II: Topology of [formula omitted]-equivariant minimizers

We study energy minimization of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in a three-dimensional axisymmetric domain and in a restricted class of S1-equivariant (i.e., axially symmetric) configurations. We assume smooth and nonvanishing S1-equivariant (e.g. homeotropic) Dirichlet boundary condition and a physically relevant norm constraint (the Lyuksyutov constraint) in the interior. Relying on our previous results in the nonsymmetric setting [16], we prove partial regularity of minimizers away from a possible finite set of interior singularities located on the symmetry axis. For a suitable class of domains and boundary data, we show that for smooth minimizers (torus solutions) the level sets of the signed biaxiality are generically finite unions of tori of revolution. Concerning nonsmooth minimizers (split solutions), we characterize their asymptotic behavior around any singular point in terms of explicit S1-equivariant harmonic maps into S4, whence the generic level sets of the signed biaxiality contains invariant topological spheres. Finally, in the model case of a nematic droplet, we provide existence of torus solutions, at least when the boundary data are suitable uniaxial deformations of the radial anchoring, and existence of split solutions for boundary data which are suitable linearly full harmonic spheres.

A dialog on the fate of information in black hole evaporation

We present two alternative perspectives for the resolution of Hawking's information puzzle in black hole evaporation. The two views are deeply contrasting, yet they share several common aspects. One of them is the central role played by the existence of the interior singularity (whose physical relevance is implied by the singularity theorems of Penrose) that we expect to be replaced by a region described by a more fundamental quantum gravity formulation. Both views rely on the notion that the standard effective quantum field theoretic perspective would require some deep modifications. In this respect, both of our scenarios are deeply influenced by ideas that Penrose has advocated at various times and, thus, serves to illustrate the lasting influence that his deep thinking on these and related matters continues to have on the modern thinking about fundamental aspects of both quantum theory and gravitation. Despite that, there is of course no claim that Penrose would agree with any of the concrete proposals that will be discussed here.

Optimal rates of convergence and error localization of Gegenbauer projections

Abstract Motivated by comparing the convergence behavior of Gegenbauer projections and best approximations, we study the optimal rate of convergence for Gegenbauer projections in the maximum norm. We show that the rate of convergence of Gegenbauer projections is the same as that of best approximations under conditions of the underlying function being either analytic on and within an ellipse and $\lambda \leqslant 0$ or differentiable and $\lambda \leqslant 1$, where $\lambda $ is the parameter in Gegenbauer projections. If the underlying function is analytic and $\lambda>0$ or differentiable and $\lambda>1$, then the rate of convergence of Gegenbauer projections is slower than that of best approximations by factors of $n^{\lambda }$ and $n^{\lambda -1}$, respectively. An exceptional case is functions with endpoint singularities, for which Gegenbauer projections and best approximations converge at the same rate for all $\lambda>-1/2$. For functions with interior or endpoint singularities, we provide a theoretical explanation for the error localization phenomenon of Gegenbauer projections and for why the accuracy of Gegenbauer projections is better than that of best approximations except in small neighborhoods of the critical points. Our analysis provides fundamentally new insight into the power of Gegenbauer approximations and related spectral methods.

Tamed spaces – Dirichlet spaces with distribution-valued Ricci bounds

We develop the theory of tamed spaces which are Dirichlet spaces with distribution-valued lower bounds on the Ricci curvature and investigate these from an Eulerian point of view. To this end we analyze in detail singular perturbations of Dirichlet form by a broad class of distributions. The distributional Ricci bound is then formulated in terms of an integrated version of the Bochner inequality using the perturbed energy form and generalizing the well-known Bakry-Émery curvature-dimension condition. Among other things we show the equivalence of distributional Ricci bounds to gradient estimates for the heat semigroup in terms of the Feynman-Kac semigroup induced by the taming distribution as well as consequences in terms of functional inequalities. We give many examples of tamed spaces including in particular Riemannian manifolds with interior singularities and singular boundary behavior.

Nonsingular black holes from charged dust collapse: A concrete mechanism to evade interior singularities in general relativity

In this paper, we examine the gravitational collapse of a nonrelativistic charged perfect fluid interacting with a dark energy component. Given a simple factor for the energy transfer, we obtain a nonsingular interior solution which naturally matches the Reissner–Nordström–de Sitter exterior geometry. We also show that the interacting parameter is proportional to the overall charge of the final black hole thus formed. For the case of quasi-extremal configurations, we propose a statistical model for the entropy of the collapsed matter. This entropy extends Bekenstein’s geometrical entropy by an additive constant proportional to the area of the extremal black hole.

Cut-enhanced PolyCube-maps for feature-aware all-hex meshing

Volumetric PolyCube-Map-based methods offer automatic ways to construct all-hexahedral meshes for closed 3D polyhedral domains, but their meshing quality is limited by the lack of interior singularities and feature alignment. In the presented work, we propose cut-enhanced PolyCube-Maps , to introduce essential interior singularities and preserve most input features. Our main idea is simple and intuitive: by inserting proper parameterization seams into the initial PolyCube-Map via novel PolyCube cutting operations, the mapping distortion can be reduced significantly. The cut-enhanced PolyCube-Map computation includes feature-aware PolyCube-Map construction and cut-enhanced PolyCube deformation. The former aims to preserve input feature edges during the initial PolyCube-Map construction. The latter introduces seams into the volumetric PolyCube shape by cutting it through selective PolyCube edges and deforms the modified PolyCube under the seamless constraints to compute a low-distortion PolyCube-Map. The hexahedral mesh induced by the final PolyCube-Map can be further enhanced by our mesh improvement algorithm. We validate the efficacy of our method on a collection of more than one hundred CAD models and demonstrate its advantages over other automatic all-hex meshing methods and padding strategies. The limitations of cut-enhanced PolyCube-Maps are also discussed thoroughly.

Optimal decay rates on the asymptotics of orthogonal polynomial expansions for functions of limited regularities

In this paper, new and optimal asymptotics on the decay of the coefficients for functions of limited regularity expanded in terms of Jacobi and Gegenbauer polynomial series are presented. For a class of functions with interior singularities, the decay of the coefficient is of the same asymptotic order for arbitrary $$\alpha ,\,\beta >-1$$, which confirms that the decay of the coefficients in the Jacobi polynomial series without normalization is a factor of $$ \sqrt{n}$$ slower compared with the Chebyshev expansion. While for functions with boundary singularities, the decay depends on $$\alpha $$ and $$\beta $$ with $$\alpha ,\,\beta >-1$$. For Gegenbauer expansion, it is related to the parameter $$\lambda $$ whatever f with interior or boundary singularities. All of these asymptotic analysis are optimal. Moreover, under the optimal asymptotic analysis, it derives that the truncated spectral expansions with some specific parameters can achieve the optimal convergence rates, i.e., the same as the best polynomial approximation in the sense of absolute maximum error norm. Numerical examples illustrate the perfect coincidence with the estimates.

RETRACTED ARTICLE: A New Viewpoint on Control Algorithms for Anthropomorphic Robotic Arms

We investigate the singularity for the anthropomorphic robotic arm and develop an algorithm to handle the calculation. It is demonstrated that although the limit characteristics and the interior characteristics of these robotic arms can be distinguished through an examination of their unique properties, it is still not possible to obtain decent kinematic decoupling properties for modern automated arms. In light of this, the methodology of our calculation is changed to deal with these two cases. The control contribution is altered to again allow bending of the robotic arm from the solitary straight stance. The interior singularities are taken into consideration in the calculation to control against invalid spatial movement of the automated arm. By incorporating fuzzy control theorem, we demonstrate that modeling system enables the robotic arm to move within solitary areas and return back to certain locations, so that the usable workspace is expanded. The viability of the proposed calculation method for reproduction and use in ongoing control tests is proven.

Manypoint Boundary Value Problems for Elliptic Differential-Operator Equations with Interior Singularities

In this work, a elliptic differential-operator equation together with boundary-transmission conditions is considered on two disjoint intervals. The boundary-transmission conditions may contain finite number interior points and abstract linear operators. We establish such important properties as coerciveness and Fredholmness of the problem under consideration. Moreover, we prove that the system of root functions forms an Abel basis in the corresponding Hilbert space.

Optimal error estimates for Chebyshev approximations of functions with limited regularity in fractional Sobolev-type spaces

In this paper, we introduce a new theoretical framework built upon fractional Sobolev-type spaces involving Riemann-Liouville (RL) fractional integrals/derivatives, which is naturally arisen from exact representations of Chebyshev expansion coefficients, for optimal error estimates of Chebyshev approximations to functions with limited regularity. The essential pieces of the puzzle for the error analysis include (i) fractional integration by parts (under the weakest possible conditions), and (ii) generalised Gegenbauer functions of fractional degree (GGF-Fs): a new family of special functions with notable fractional calculus properties. Under this framework, we are able to estimate the optimal decay rate of Chebyshev expansion coefficients for a large class of functions with interior and endpoint singularities, which are deemed suboptimal or complicated to characterize in existing literature. We can then derive optimal error estimates for spectral expansions and the related Chebyshev interpolation and quadrature measured in various norms, and also improve the available results in usual Sobolev spaces of integer regularity exponentials in several senses. As a by-product, this study results in some analytically perspicuous formulas particularly on GGF-Fs, which are potentially useful in spectral algorithms. The idea and analysis techniques can be extended to general Jacobi spectral approximations.

Interior singularity analysis for a 2(3HUS+S) parallel manipulator with descending matrix rank method

Singularity analysis is one of the basic problems for parallel manipulators. When a manipulator moves in a singular configuration, the motion and transmission performance are poor. In certain serious cases, the normal operation could be damaged. Based on the topology structure and kinematics analysis of a 2(3HUS+S) parallel manipulator, the Jacobian matrices were established. Then, the singular locus surface was obtained by numerical simulation. In addition, the relationship between the motion path curve and the singular locus surface was analyzed. In this study, α, β, and γ are the attitude angles that describe the motion of moving platforms. There is a nonsingular attitude space in singular locus surfaces, and the singular locus surface is a single surface in a small attitude angle range. The nonsingular attitude space increases as the absolute value of γ increases, and singularity could be avoided when γ is large. Furthermore, the motion path curve passes through the singular locus surface two times, and the two intersection points are consistent with the positions where the motion dexterity is equal to zero. This study provides new insights on the singularity analysis of parallel manipulators, particularly for the structure parameter optimization of the nonsingular attitude space.

Existence of positive weak solutions for fractional Lane–Emden equations with prescribed singular sets

In this paper, we consider the problem of the existence of positive weak solutions of $$\begin{aligned}{\left\{ \begin{array}{ll} (-\Delta )^s{u}=u^p&{}\text {in}~\Omega \\ u=0&{}\text {on}~{\mathbb R}^n \backslash \Omega \end{array}\right. }\\\end{aligned}$$ having prescribed isolated interior singularities. We prove that if $$\frac{n}{n-2s}<p<p_1$$ for some critical exponent $$p_1$$ defined in the introduction which is related to the stability of the singular solution $$u_s$$ , and if S is a closed subset of $$\Omega $$ , then there are infinitely many positive weak solutions with S as its singular set. We also show the existence of solutions to the fractional Yamabe problem with singular set to be the whole space $${\mathbb R}^n$$ . These results are the extension of Chen and Lin’s result (Chen and Lin in J Geom Anal 9(2):221–246, 1999) to the fractional case.

Development of a new 3-DOF parallel manipulator for minimally invasive surgery.

This article proposes a novel dexterous endoscopic parallel manipulator for minimally invasive surgery. The proposed manipulator has 3 degrees of freedom (3-DOF), which consist of two rotational DOFs and one translational DOF (2R1T DOFs). The manipulator consists of 3 limbs exhibiting identical kinematic structure. Each limb contains an active prismatic joint followed by 2 consecutive passive universal joints. The proposed manipulator has a unique arrangement of its joints' axes. This unique arrangement permits large bending angles, ±90° in any direction, and a workspace almost free from interior singularities. These advantages allow the proposed manipulator to outperforms existing surgical manipulators. However, this unique arrangement makes the analysis of the robot extremely difficult. Therefore, a geometrical/analytical approach is used to facilitate its singularity analysis. Construction of the virtual prototype is accomplished using ADAMS software to validate the proposed manipulator and its bending capability. A closed-form solution for inverse kinematics is obtained analytically. Also, the forward kinematics solution is obtained numerically. Moreover, evaluation of the workspace is achieved using motion/force transmissibility indices. A practical experiment has been performed using a scaling technique and PID controller. The experimental results show the feasibility of the teleoperated surgical system using the proposed parallel manipulator as the slave.

MINIMIZATION PRINCIPLE AND GENERALIZED FOURIER SERIES FOR DISCONTINUOUS STURM-LIOUVILLE SYSTEMS IN DIRECT SUM SPACES

By modifing the Green's function method we study certain spectral aspects of discontinuous Sturm-Liouville problems with interior singularities. Firstly, we define four eigen-solutions and construct the Green's function in terms of them. Based on the Green's function we establish the uniform convergeness of generalized Fourier series as eigenfunction expansion in the direct sum of Lebesgue spaces L2 where the usual inner product replaced by new inner product. Finally, we extend and generalize such important spectral properties as Parseval equation, Rayleigh quotient and Rayleigh-Ritz formula (minimization principle) for the considered problem.

The Hardy–Schrödinger operator with interior singularity: the remaining cases

We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem $L_\gamma u-\lambda u=\frac{u^{2^*(s)-1}}{|x|^s}$ on a smooth bounded domain $\Omega$ in $\mathbb{R}^n$ ($n\geq 3$) having the singularity $0$ in its interior. Here $\gamma 0$ or if $\gamma > 0$, i.e., in {\it the truly singular case}, we show that in low dimensions, a solution is guaranteed by the positivity of the ``Hardy-singular internal of $\Omega$, a notion that we introduce herein. On the other hand, and just like the case wnen $\gamma=s=0$ studied by Brezis-Nirenberg and completed by Druet, $n=3$ is the critical dimension, and the classical positive mass theorem is sufficient for the {\it merely singular case}, that is when $s=0$, $\gamma \leq 0$.

Spectral Geometry of the Steklov Problem on Orbifolds

We consider how the geometry and topology of a compact $n$-dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in terms of only boundary data. As a consequence, we prove that the Steklov spectrum detects the presence and number of orbifold singularities on the boundary of an orbisurface and it detects the number each of smooth and singular boundary components. Moreover, we find that the Steklov spectrum also determines the lengths of the boundary components modulo an equivalence relation, and we show by examples that this result is the best possible. We construct various examples of Steklov isospectral Riemannian orbifolds which demonstrate that these two-dimensional results do not extend to higher dimensions. In addition, we give two-dimensional examples which show that the Steklov spectrum does \emph{not} detect the presence of interior singularities nor does it determine the orbifold Euler characteristic. In fact, a flat disk is Steklov isospectral to a cone. In another direction, we obtain upper bounds on the Steklov eigenvalues of a Riemannian orbifold in terms of the isoperimetric ratio and a conformal invariant. We generalize results of B. Colbois, A. El Soufi and A. Girouard, and the fourth author to the orbifold setting; in the process, we gain a sharpness result on these bounds that was not evident in the manifold setting. In dimension two, our eigenvalue bounds are solely in terms of the orbifold Euler characteristic and the number each of smooth and singular boundary components.

Minimization problem on the Hardy–Sobolev inequality

We study minimization problems on Hardy–Sobolev type inequality. We consider the case where singularity is in interior of bounded domain \(\Omega \subset \mathbb {R}^N\). The attainability of best constants for Hardy–Sobolev type inequalities with boundary singularities have been studied so far, for example Ghoussoub and Kang (Ann Inst Henri Poincare Anal Non Lineaire 21(6):767–793, 2004), Ghoussoub and Robert (IMRP 21867:1–85, 2006), Ghoussoub and Robert (Trans Am Math Soc 361(9):4843–4870, 2009) etc.... According to their results, the mean curvature of \(\partial \Omega \) at singularity affects the attainability of the best constants. In contrast with boundary singularity case, in interior singularity case it is well known that the best Hardy–Sobolev constant $$\begin{aligned} \mu _s(\Omega ):=\left\{ \int _\Omega |\nabla u|^2 dx \Bigg | u \in H_0^1(\Omega ),\ \int _\Omega \frac{|u|^{2^*(s)}}{|x|^s}dx = 1 \right\} \end{aligned}$$is never achieved for all bounded domain \(\Omega \). We can see that the position of singularity on domain is related to the existence of minimizer. In this paper, we consider the attainability of the best constant for the embedding \(H^1(\Omega ) \hookrightarrow L^{2^*(s)}(\Omega ,|x|^{-s}dx)\) for bounded domain \(\Omega \) with \(0 \in \Omega \). In this problem, scaling invariance doesn’t hold and we can not obtain information of singularity like mean curvature.

Singular solutions of a superlinear parabolic equation with homogeneous Neumann boundary conditions

This paper discusses the existence of singular solutions of a superlinear parabolic partial differential equation on bounded domains with homogeneous Neumann boundary conditions. It is shown that in some parameter range, there exists a solution with a spatial singularity whose position and strength depend on time. Both interior and boundary singularities are studied.

Spectral Analysis of Legendre Type Fourth Order Operators on the Real Axis

We describe the continuous spectrum of Legendre type fourth order operators with interior singularities determined by their deficiency numbers.

The Dirichlet problem with prescribed interior singularities

In this paper we solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in Rn. The main results apply, in particular, to subequations with a Riesz characteristic p≥2. It is shown that, without requiring uniform ellipticity, the Dirichlet problem can be solved uniquely for arbitrary continuous boundary data with singularities asymptotic to the Riesz kernel ΘjKp(x−xj) whereKp(x)={−1|x|p−2for2<p<∞,log⁡|x|ifp=2. at any prescribed finite set of points {x1,...,xk} in the domain and any finite set of positive real numbers Θ1,...,Θk. This sharpens a previous result of the authors concerning the discreteness of high-density sets of subsolutions.Uniqueness and existence results are also established for finite-type singularities such as Θj|x−xj|2−p for 1≤p<2.The main results apply similarly with prescribed singularities asymptotic to the fundamental solutions of Armstrong–Sirakov–Smart (in the uniformly elliptic case).