Locked and Unlocked Chains of Planar Shapes

We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the families of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the distance from each edge along the path along the boundary of the slender adornment to each hinge should be monotone. In contrast, we show that isosceles triangles with any desired apex angle less than 90 degrees admit locked chains, which is precisely the threshold beyond which the inward-normal property no longer holds.Comment: 23 pages, 25 figures, Latex; full journal version with all proof details. (Fixed crash-induced bugs in the abstract.

Generic Global Rigidity in Complex and Pseudo-Euclidean Spaces

In this paper we study the property of generic global rigidity for frameworks of graphs embedded in d-dimensional complex space and in a d-dimensional pseudo-Euclidean space R$^{2}$ with a metric of indefinite signature). We show that a graph is generically globally rigid in Euclidean space iff it is generically globally rigid in a complex or pseudo-Euclidean space. We also establish that global rigidity is always a generic property of a graph in complex space, and give a sufficient condition for it to be a generic property in a pseudo-Euclidean space. Extensions to hyperbolic space are also discussed.Engineering and Applied Science

Cyfip1 Haploinsufficiency Does Not Alter GABAA Receptor δ-Subunit Expression and Tonic Inhibition in Dentate Gyrus PV+ Interneurons and Granule Cells

Copy number variation (CNV) at chromosomal region 15q11.2 is linked to increased risk of neurodevelopmental disorders including autism and schizophrenia. A significant gene at this locus is cytoplasmic fragile X mental retardation protein (FMRP) interacting protein 1 (CYFIP1). CYFIP1 protein interacts with FMRP, whose monogenic absence causes fragile X syndrome (FXS). Fmrp knock-out has been shown to reduce tonic GABAergic inhibition by interacting with the δ-subunit of the GABAA receptor (GABAAR). Using in situ hybridization (ISH), qPCR, Western blotting techniques, and patch clamp electrophysiology in brain slices from a Cyfip1 haploinsufficient mouse, we examined δ-subunit mediated tonic inhibition in the dentate gyrus (DG). In wild-type (WT) mice, DG granule cells (DGGCs) responded to the δ-subunit-selective agonist THIP with significantly increased tonic currents. In heterozygous mice, no significant difference was observed in THIP-evoked currents in DGGCs. Phasic GABAergic inhibition in DGGC was also unaltered with no difference in properties of spontaneous IPSCs (sIPSCs). Additionally, we demonstrate that DG granule cell layer (GCL) parvalbumin-positive interneurons (PV+-INs) have functional δ-subunit-mediated tonic GABAergic currents which, unlike DGGC, are also modulated by the α1-selective drug zolpidem. Similar to DGGC, both IPSCs and THIP-evoked currents in PV+-INs were not different between Cyfip1 heterozygous and WT mice. Supporting our electrophysiological data, we found no significant change in hippocampal δ-subunit mRNA expression or protein level and no change in α1/α4-subunit mRNA expression. Thus, Cyfip1 haploinsufficiency, mimicking human 15q11.2 microdeletion syndrome, does not alter hippocampal phasic or tonic GABAergic inhibition, substantially differing from the Fmrp knock-out mouse model

Characterizing the universal rigidity of generic frameworks

A framework is a graph and a map from its vertices to E^d (for some d). A framework is universally rigid if any framework in any dimension with the same graph and edge lengths is a Euclidean image of it. We show that a generic universally rigid framework has a positive semi-definite stress matrix of maximal rank. Connelly showed that the existence of such a positive semi-definite stress matrix is sufficient for universal rigidity, so this provides a characterization of universal rigidity for generic frameworks. We also extend our argument to give a new result on the genericity of strict complementarity in semidefinite programming.Comment: 18 pages, v2: updates throughout; v3: published versio

Micro-Electromechanical Instrument and Systems Development at the Charles Stark Draper Laboratory

Several generations of micromechanical gyros and accelerometers have been developed at Draper. Current design effort centers on tuning-fork gyro design and pendulous accelerometer configurations. Over 200 gyros of different generations have been packaged and tested. These units have successfully performed across a temperature range of -40 to 85 degrees C, and have survived 30,000-g shock tests along all axes. Draper is currently under contract to develop an integrated micro-mechanical inertial sensor assembly (MMISA) and global positioning system (GPS) receiver configuration. The ultimate projections for size, weight, and power for an MMISA, after electronic design of the application specific integrated circuit (ASIC ) is completed, are 2 x 2 x 0.5 cm, 5 gm, and less than 1 W, respectively. This paper describes the fabrication process, the current gyro and accelerometer designs, and system configurations

Localizability of Wireless Sensor Networks: Beyond Wheel Extension

A network is called localizable if the positions of all the nodes of the network can be computed uniquely. If a network is localizable and embedded in plane with generic configuration, the positions of the nodes may be computed uniquely in finite time. Therefore, identifying localizable networks is an important function. If the complete information about the network is available at a single place, localizability can be tested in polynomial time. In a distributed environment, networks with trilateration orderings (popular in real applications) and wheel extensions (a specific class of localizable networks) embedded in plane can be identified by existing techniques. We propose a distributed technique which efficiently identifies a larger class of localizable networks. This class covers both trilateration and wheel extensions. In reality, exact distance is almost impossible or costly. The proposed algorithm based only on connectivity information. It requires no distance information