Every normal complex surface singularity with Q-homology sphere link has a universal abelian cover. It has been conjectured by Neumann and Wahl that the universal abelian cover of a rational or minimally elliptic singularity is a complete intersection singularity defined by a system of “splice diagram equations”. In this paper we introduce a Neumann-Wahl system, which is an analogue of the syst...

The Ricci curvature at an isolated singularity of an immersed hypersurface exhibits a local behaviour echoing a global property of the Ricci curvature on a complete hypersurface in euclidean space (i.e., the Efimov theorem [12], that sup Ric ≥ 0). This local behaviour takes the form of a near universal bound on the decay of the Ricci curvature at a simple singularity (eg. a cone singularity) in...

We explicitly analyze the trajectories of learning near singularities in hierarchical networks, such as multilayer perceptrons and radial basis function networks, which include permutation symmetry of hidden nodes, and show their general properties. Such symmetry induces singularities in their parameter space, where the Fisher information matrix degenerates and odd learning behaviors, especiall...

Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of microelectromechanical systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a ...

The kinematic and algorithmic singularities in controlling robotic manipulators have been investigated intensively because they are not predictable or difficult to avoid. The problem with handling these singularities is an unnecessary performance reduction in non-singular region and the difficulty in performance tuning. In this paper, we propose a method of avoiding kinematic and algorithmic si...

I first recount Raychaudhuri’s deep involvement with the singularity problem in general relativity. I then argue that precisely the same situation has arisen today in loop quantum cosmology as obtained when Raychaudhuri discovered his celebrated equation. We thus need a new analogue of the Raychaudhuri equation in quantum gravity. PACS numbers: 04.20.Jb, 04.2.Cv, 98.80 Dr 1 Singularity and AKR ...

The topology of a normal surface singularity does not determine the analytical invariants of its equisingularity class, but recent partial results indicated that this should be true under two restrictions, a topological one, that the link of the singularity is a rational homology sphere, and an analytical one, that the singularity is Q-Gorenstein. Neumann and Wahl conjectured that the singulari...

As we all know, singularity is commonly encountered in parallel mechanisms. It is shown that the behavior of singularity of parallel mechanisms could be more complicated than that of serial ones. However, it is not very clear whether singularity will bring problems to kinematics, dynamics or other characteristics and what result will be caused when parallel mechanisms fall into the neighborhood...

We analyze here the issue of local versus the global visibility of a singularity that forms in gravitational collapse of a dust cloud, which has important implications for the weak and strong versions of the cosmic censorship hypothesis. We find conditions as to when a singularity will be only locally naked, rather than being globally visible, thus preseving the weak censorship hypothesis. The ...

It is well known that singular configurations are an inherent nature of mechanical systems and have important influences on their properties. Therefore, these special configurations of the system are usually avoided in design and application of mechanisms. For simple 2-DOF and 3-DOF planar manipulators, the singularity loci have been obtained in [1, 2]. For spherical 3DOF parallel manipulators ...