The two components of the SO(3)-character space of the fundamental group of a closed surface of genus 2

We use geometric techniques to explicitly find the topological structure of the space of SO(3)-representations of the fundamental group of a closed surface of genus 2 quotient by the conjugation action by SO(3). There are two components of the space. We will describe the topology of both components and describe the corresponding SU(2)-character spaces by parametrizing them by spherical triangles. There is the sixteen to one branch-covering for each component, and the branch locus is a union of 2-spheres or 2-tori. Along the way, we also describe the topology of both spaces. We will later relate this result to future work into higher-genus cases and the SL(3,R)-representations

Geometric Satake, Springer correspondence, and small representations II

For a split reductive group scheme $G$ over a commutative ring $k$ with Weyl group $W$, there is an important functor $Rep(G,k) \to Rep(W,k)$ defined by taking the zero weight space. We prove that the restriction of this functor to the subcategory of small representations has an alternative geometric description, in terms of the affine Grassmannian and the nilpotent cone of the Langlands dual group to $G$. The translation from representation theory to geometry is via the Satake equivalence and the Springer correspondence. This generalizes the result for the $k=\mathbb{C}$ case proved by the first two authors, and also provides a better explanation than in that earlier paper, since the current proof is uniform across all types.Comment: Version 4: minor revisions; 73 page

Generalized event structures and probabilities

For the classical mind, quantum mechanics is boggling enough; nevertheless more bizarre behavior could be imagined, thereby concentrating on propositional structures (empirical logics) that transcend the quantum domain. One can also consistently suppose predictions and probabilities which are neither classical nor quantum, but which are subject to subclassicality; that is, the additivity of probabilities for mutually exclusive, co-measurable observables, as formalized by admissibility rules and frame functions.Comment: 10 pages, 10 figures in Information and Complexity, World Scientific Series in Information Studies: Volume 6, ed. by Mark Burgin and Cristian S Calude, Chapter 11, pp. 276-300, (World Scientific, Singapore, 2016

Integrated sensors for robotic laser welding

A welding head is under development with integrated sensory systems for robotic laser welding applications. Robotic laser welding requires sensory systems that are capable to accurately guide the welding head over a seam in three-dimensional space and provide information about the welding process as well as the quality of the welding result. In this paper the focus is on seam tracking. It is difficult to measure three-dimensional parameters of a ream during a robotic laser welding task, especially when sharp corners are present. The proposed sensory system is capable to provide the three dimensional parameters of a seam in one measurement and guide robots over sharp corners

A New 3D Tool for Planning Plastic Surgery

Face plastic surgery (PS) plays a major role in today medicine. Both for reconstructive and cosmetic surgery, achieving harmony of facial features is an important, if not the major goal. Several systems have been proposed for presenting to patient and surgeon possible outcomes of the surgical procedure. In this paper, we present a new 3D system able to automatically suggest, for selected facial features as nose, chin, etc, shapes that aesthetically match the patient's face. The basic idea is suggesting shape changes aimed to approach similar but more harmonious faces. To this goal, our system compares the 3D scan of the patient with a database of scans of harmonious faces, excluding the feature to be corrected. Then, the corresponding features of the k most similar harmonious faces, as well as their average, are suitably pasted onto the patient's face, producing k+1 aesthetically effective surgery simulations. The system has been fully implemented and tested. To demonstrate the system, a 3D database of harmonious faces has been collected and a number of PS treatments have been simulated. The ratings of the outcomes of the simulations, provided by panels of human judges, show that the system and the underlying idea are effectiv

Sensor integration for robotic laser welding processes

The use of robotic laser welding is increasing among industrial applications, because of its ability to weld objects in three dimensions. Robotic laser welding involves three sub-processes: seam detection and tracking, welding process control, and weld seam inspection. Usually, for each sub-process, a separate sensory system is required. The use of separate sensory systems leads to heavy and bulky tools, in contrast to compact and light sensory systems that are needed to reach sufficient accuracy and accessibility. In the solution presented in this paper all three subprocesses are integrated in one compact multipurpose welding head. This multi-purpose tool is under development and consists of a laser welding head, with integrated sensors for seam detection and inspection, while also carrying interfaces for process control. It can provide the relative position of the tool and the work piece in three-dimensional space. Additionally, it can cope with the occurrence of sharp corners along a three-dimensional weld path, which are difficult to detect and weld with conventional equipment due to measurement errors and robot dynamics. In this paper the process of seam detection will be mainly elaborated

On deformations of pasting diagrams, II

We continue the development of the infinitesimal deformation theory of pasting diagrams of k-linear categories begun in TAC, Vol 22, #2. In that article the standard result that all obstructions are cocycles was established only for the elementary, composition-free parts of pasting diagrams. In the present work we give a proof for pasting diagrams in general. As tools we use the method developed by Shrestha of simultaneously representing formulas for obstructions, along with the corresponding cocycle and cobounding conditions by suitably labeled polygons, giving a rigorous exposition of the previously heuristic method; and deformations of pasting diagrams in which some cells are required to be deformed trivially

Iso-array rewriting P systems with context-free iso-array rules

A new computing model called P system is a highly distributed and parallel theoretical model, which is proposed in the area of membrane computing. Ceterchi et al. initially proposed array rewriting P systems by extending the notion of string rewriting P systems to arrays (2003). A theoretical model for picture generation using context-free iso-array grammar rules and puzzle iso-array grammar rules are introduced by Kalyani et al. (2004, 2006). Also iso-array rewriting P systems for iso-picture languages have been studied by Annadurai et al. (2008). In this paper we consider the context-free iso-array rules and context-free puzzle iso-array rules in iso-array rewriting P systems and examine the generative powers of these P systems

Two-Dimensional Critical Percolation: The Full Scaling Limit

We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.Comment: 45 pages, 12 figures. This is a revised version of math.PR/0504036 without the appendice