Automatic Deduction in Dynamic Geometry using Sage

We present a symbolic tool that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction. The main prototype has been developed as two different worksheets for the open source computer algebra system Sage, corresponding to two different ways of coding a geometric construction. In one worksheet, diagrams constructed with the open source dynamic geometry system GeoGebra are accepted. In this worksheet, Groebner bases are used to either compute the equation of a geometric locus in the case of a locus construction or to determine the truth of a general geometric statement included in the GeoGebra construction as a boolean variable. In the second worksheet, locus constructions coded using the common file format for dynamic geometry developed by the Intergeo project are accepted for computation. The prototype and several examples are provided for testing. Moreover, a third Sage worksheet is presented in which a novel algorithm to eliminate extraneous parts in symbolically computed loci has been implemented. The algorithm, based on a recent work on the Groebner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Detailed examples are discussed.Comment: In Proceedings THedu'11, arXiv:1202.453

A Unified Approach for Representing Structurally-Complex Models in SBML Level 3

The aim of this document is to explore a unified approach to handling several of the proposed extensions to the SBML Level 3 Core specification. The approach is illustrated with reference to Simile, a modelling environment which appears to have most of the capabilities of the various SBML Level 3 package proposals which deal with model structure. Simile (http://www.simulistics.com) is a visual modelling environment for continuous systems modelling which includes the ability to handle complex disaggregation of model structure, by allowing the modeller to specify classes of object and the relationships between them.

The note is organised around the 6 packages listed on the SBML Level 3 Proposals web page (http://sbml.org/Community/Wiki/SBML_Level_3_Proposals) which deal with model structure, namely comp, arrays, spatial, geom, dyn and multi. For each one, I consider how the requirements which motivated the package can be handled using Simile's unified approach. Although Simile has a declarative model-representation language (in both Prolog and XML syntax), I use Simile diagrams and equation syntax throughout, since this is more compact and readable than large chunks of XML.

The conclusion is that Simile can indeed meet most of the requirements of these various packages, using a generic set of constructs - basically, the multiple-instance submodel, the concept of a relationship (association) between submodels, and array variables. This suggests the possibility of having a single SBML Level 3 extension package similar to the Simile data model, rather than a series of separate packages. Such an approach has a number of potential advantages and disadvantages compared with having the current set of discrete packages: these are discussed in this paper

Sequential importance sampling for multiway tables

We describe an algorithm for the sequential sampling of entries in multiway contingency tables with given constraints. The algorithm can be used for computations in exact conditional inference. To justify the algorithm, a theory relates sampling values at each step to properties of the associated toric ideal using computational commutative algebra. In particular, the property of interval cell counts at each step is related to exponents on lead indeterminates of a lexicographic Gr\"{o}bner basis. Also, the approximation of integer programming by linear programming for sampling is related to initial terms of a toric ideal. We apply the algorithm to examples of contingency tables which appear in the social and medical sciences. The numerical results demonstrate that the theory is applicable and that the algorithm performs well.Comment: Published at http://dx.doi.org/10.1214/009053605000000822 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Relative fixed-width stopping rules for Markov chain Monte Carlo simulations

Markov chain Monte Carlo (MCMC) simulations are commonly employed for estimating features of a target distribution, particularly for Bayesian inference. A fundamental challenge is determining when these simulations should stop. We consider a sequential stopping rule that terminates the simulation when the width of a confidence interval is sufficiently small relative to the size of the target parameter. Specifically, we propose relative magnitude and relative standard deviation stopping rules in the context of MCMC. In each setting, we develop sufficient conditions for asymptotic validity, that is conditions to ensure the simulation will terminate with probability one and the resulting confidence intervals will have the proper coverage probability. Our results are applicable in a wide variety of MCMC estimation settings, such as expectation, quantile, or simultaneous multivariate estimation. Finally, we investigate the finite sample properties through a variety of examples and provide some recommendations to practitioners.Comment: 24 page

Multirobot heterogeneous control considering secondary objectives

Cooperative robotics has considered tasks that are executed frequently, maintaining the shape and orientation of robotic systems when they fulfill a common objective, without taking advantage of the redundancy that the robotic group could present. This paper presents a proposal for controlling a group of terrestrial robots with heterogeneous characteristics, considering primary and secondary tasks thus that the group complies with the following of a path while modifying its shape and orientation at any time. The development of the proposal is achieved through the use of controllers based on linear algebra, propounding a low computational cost and high scalability algorithm. Likewise, the stability of the controller is analyzed to know the required features that have to be met by the control constants, that is, the correct values. Finally, experimental results are shown with di erent configurations and heterogeneous robots, where the graphics corroborate the expected operation of the proposalThis research was funded by Corporación Ecuatoriana para el Desarrollo de la Investigación y Academia–CEDI

An efficient multi-core implementation of a novel HSS-structured multifrontal solver using randomized sampling

We present a sparse linear system solver that is based on a multifrontal variant of Gaussian elimination, and exploits low-rank approximation of the resulting dense frontal matrices. We use hierarchically semiseparable (HSS) matrices, which have low-rank off-diagonal blocks, to approximate the frontal matrices. For HSS matrix construction, a randomized sampling algorithm is used together with interpolative decompositions. The combination of the randomized compression with a fast ULV HSS factorization leads to a solver with lower computational complexity than the standard multifrontal method for many applications, resulting in speedups up to 7 fold for problems in our test suite. The implementation targets many-core systems by using task parallelism with dynamic runtime scheduling. Numerical experiments show performance improvements over state-of-the-art sparse direct solvers. The implementation achieves high performance and good scalability on a range of modern shared memory parallel systems, including the Intel Xeon Phi (MIC). The code is part of a software package called STRUMPACK -- STRUctured Matrices PACKage, which also has a distributed memory component for dense rank-structured matrices