Wrapping the cube and other polyhedra

An infinite series of twofold, two-way weavings of the cube, corresponding to 'wrappings', or double covers of the cube, is described with the aid of the two-parameter Goldberg- Coxeter construction. The strands of all such wrappings correspond to the central circuits (CCs) of octahedrites (four-regular polyhedral graphs with square and triangular faces), which for the cube necessarily have octahedral symmetry. Removing the symmetry constraint leads to wrappings of other eight-vertex convex polyhedra. Moreover, wrappings of convex polyhedra with fewer vertices can be generated by generalizing from octahedrites to i-hedrites, which additionally include digonal faces. When the strands of a wrapping correspond to the CCs of a four-regular graph that includes faces of size greater than 4, non-convex 'crinkled' wrappings are generated. The various generalizations have implications for activities as diverse as the construction of woven-closed baskets and the manufacture of advanced composite components of complex geometry. © 2012 The Royal Society.

Sensor graphs for guaranteed cooperative localization performance

A group of mobile robots can localize cooperatively, using relative position and absolute orientation measurements, fused through an extended Kalman filter (ekf). The topology of the graph of relative measurements is known to affect the steady-state value of the position error covariance matrix. Classes of sensor graphs are identified, for which tight bounds for the trace of the covariance matrix can be obtained based on the algebraic properties of the underlying relative measurement graph. The string and the star graph topologies are considered, and the explicit form of the eigenvalues of error covariance matrix is given. More general sensor graph topologies are considered as combinations of the string and star topologies, when additional edges are added. It is demonstrated how the addition of edges increases the trace of the steady-state value of the position error covariance matrix, and the theoretical predictions are verified through simulation analysis.

Random function priors for exchangeable arrays with applications to graphs and relational data

A fundamental problem in the analysis of structured relational data like graphs, networks, databases, and matrices is to extract a summary of the common structure underlying relations between individual entities. Relational data are typically encoded in the form of arrays; invariance to the ordering of rows and columns corresponds to exchangeable arrays. Results in probability theory due to Aldous, Hoover and Kallenberg show that exchangeable arrays can be represented in terms of a random measurable function which constitutes the natural model parameter in a Bayesian model. We obtain a flexible yet simple Bayesian nonparametric model by placing a Gaussian process prior on the parameter function. Efficient inference utilises elliptical slice sampling combined with a random sparse approximation to the Gaussian process. We demonstrate applications of the model to network data and clarify its relation to models in the literature, several of which emerge as special cases.

Model reductions for inference: generality of pairwise, binary, and planar factor graphs.

We offer a solution to the problem of efficiently translating algorithms between different types of discrete statistical model. We investigate the expressive power of three classes of model-those with binary variables, with pairwise factors, and with planar topology-as well as their four intersections. We formalize a notion of "simple reduction" for the problem of inferring marginal probabilities and consider whether it is possible to "simply reduce" marginal inference from general discrete factor graphs to factor graphs in each of these seven subclasses. We characterize the reducibility of each class, showing in particular that the class of binary pairwise factor graphs is able to simply reduce only positive models. We also exhibit a continuous "spectral reduction" based on polynomial interpolation, which overcomes this limitation. Experiments assess the performance of standard approximate inference algorithms on the outputs of our reductions.

Research on multi-degree-of-freedom neurons with weighted graphs

In this paper, we redefine the sample points set in the feature space from the point of view of weighted graph and propose a new covering model - Multi-Degree-of-Freedorn Neurons (MDFN). Base on this model, we describe a geometric learning algorithm with 3-degree-of-freedom neurons. It identifies the sample points secs topological character in the feature space, which is different from the traditional "separation" method. Experiment results demonstrates the general superiority of this algorithm over the traditional PCA+NN algorithm in terms of efficiency and accuracy.

Research on multi-degree-of-freedom neurons with weighted graphs

In this paper, we redefine the sample points set in the feature space from the point of view of weighted graph and propose a new covering model - Multi-Degree-of-Freedorn Neurons (MDFN). Base on this model, we describe a geometric learning algorithm with 3-degree-of-freedom neurons. It identifies the sample points secs topological character in the feature space, which is different from the traditional "separation" method. Experiment results demonstrates the general superiority of this algorithm over the traditional PCA+NN algorithm in terms of efficiency and accuracy.

InGaAsP-InP Square Microlasers With a Vertex Output Waveguide

InGaAsP-InP square microlasers with a vertex output waveguide are fabricated by planar processes, and the etched sidewalls of the lasers are confined by insulating layer SiO2 and p-electrode TiAu metals. For a square microlaser with a side length of 30 mu m and a 2-mu m-wide output waveguide, a continuous-wave threshold current is 26 mA at room temperature and output power is 0.72 mW at 86 mA. The mode interval of 21 and 7.4 nm is observed for the microlasers with the side length of 10 and 30 mu m, respectively. Finite-difference time-domain (FDTD) simulations indicate that the lasing modes have incident angles of about 45 degrees at the boundaries of the resonator. In addition, square resonators surrounded by air, SiO2-Ti-Au, and SiO2-Au are compared by FDTD simulations.

A Constant-Factor Approximation Algorithm for Embedding Unweighted Graphs into Trees

We present a constant-factor approximation algorithm for computing an embedding of the shortest path metric of an unweighted graph into a tree, that minimizes the multiplicative distortion.