Chunky and Equal-Spaced Polynomial Multiplication

Finding the product of two polynomials is an essential and basic problem in computer algebra. While most previous results have focused on the worst-case complexity, we instead employ the technique of adaptive analysis to give an improvement in many "easy" cases. We present two adaptive measures and methods for polynomial multiplication, and also show how to effectively combine them to gain both advantages. One useful feature of these algorithms is that they essentially provide a gradient between existing "sparse" and "dense" methods. We prove that these approaches provide significant improvements in many cases but in the worst case are still comparable to the fastest existing algorithms.Comment: 23 Pages, pdflatex, accepted to Journal of Symbolic Computation (JSC

An intelligent, free-flying robot

The ground based demonstration of the extensive extravehicular activity (EVA) Retriever, a voice-supervised, intelligent, free flying robot, is designed to evaluate the capability to retrieve objects (astronauts, equipment, and tools) which have accidentally separated from the Space Station. The major objective of the EVA Retriever Project is to design, develop, and evaluate an integrated robotic hardware and on-board software system which autonomously: (1) performs system activation and check-out; (2) searches for and acquires the target; (3) plans and executes a rendezvous while continuously tracking the target; (4) avoids stationary and moving obstacles; (5) reaches for and grapples the target; (6) returns to transfer the object; and (7) returns to base

Detecting lacunary perfect powers and computing their roots

We consider solutions to the equation f = h^r for polynomials f and h and integer r > 1. Given a polynomial f in the lacunary (also called sparse or super-sparse) representation, we first show how to determine if f can be written as h^r and, if so, to find such an r. This is a Monte Carlo randomized algorithm whose cost is polynomial in the number of non-zero terms of f and in log(deg f), i.e., polynomial in the size of the lacunary representation, and it works over GF(q)[x] (for large characteristic) as well as Q[x]. We also give two deterministic algorithms to compute the perfect root h given f and r. The first is output-sensitive (based on the sparsity of h) and works only over Q[x]. A sparsity-sensitive Newton iteration forms the basis for the second approach to computing h, which is extremely efficient and works over both GF(q)[x] (for large characteristic) and Q[x], but depends on a number-theoretic conjecture. Work of Erdos, Schinzel, Zannier, and others suggests that both of these algorithms are unconditionally polynomial-time in the lacunary size of the input polynomial f. Finally, we demonstrate the efficiency of the randomized detection algorithm and the latter perfect root computation algorithm with an implementation in the C++ library NTL.Comment: to appear in Journal of Symbolic Computation (JSC), 201

Vision technology/algorithms for space robotics applications

The thrust of automation and robotics for space applications has been proposed for increased productivity, improved reliability, increased flexibility, higher safety, and for the performance of automating time-consuming tasks, increasing productivity/performance of crew-accomplished tasks, and performing tasks beyond the capability of the crew. This paper provides a review of efforts currently in progress in the area of robotic vision. Both systems and algorithms are discussed. The evolution of future vision/sensing is projected to include the fusion of multisensors ranging from microwave to optical with multimode capability to include position, attitude, recognition, and motion parameters. The key feature of the overall system design will be small size and weight, fast signal processing, robust algorithms, and accurate parameter determination. These aspects of vision/sensing are also discussed

Certification of Bounds of Non-linear Functions: the Templates Method

The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.Comment: 16 pages, 3 figures, 2 table

Concurrent processing simulation of the space station

The development of a new capability for the time-domain simulation of multibody dynamic systems and its application to the study of a large angle rotational maneuvers of the Space Station is described. The effort was divided into three sequential tasks, which required significant advancements of the state-of-the art to accomplish. These were: (1) the development of an explicit mathematical model via symbol manipulation of a flexible, multibody dynamic system; (2) the development of a methodology for balancing the computational load of an explicit mathematical model for concurrent processing; and (3) the implementation and successful simulation of the above on a prototype Custom Architectured Parallel Processing System (CAPPS) containing eight processors. The throughput rate achieved by the CAPPS operating at only 70 percent efficiency, was 3.9 times greater than that obtained sequentially by the IBM 3090 supercomputer simulating the same problem. More significantly, analysis of the results leads to the conclusion that the relative cost effectiveness of concurrent vs. sequential digital computation will grow substantially as the computational load is increased. This is a welcomed development in an era when very complex and cumbersome mathematical models of large space vehicles must be used as substitutes for full scale testing which has become impractical