Using the distribution of cells by dimension in a cylindrical algebraic decomposition

We investigate the distribution of cells by dimension in cylindrical algebraic decompositions (CADs). We find that they follow a standard distribution which seems largely independent of the underlying problem or CAD algorithm used. Rather, the distribution is inherent to the cylindrical structure and determined mostly by the number of variables. This insight is then combined with an algorithm that produces only full-dimensional cells to give an accurate method of predicting the number of cells in a complete CAD. Since constructing only full-dimensional cells is relatively inexpensive (involving no costly algebraic number calculations) this leads to heuristics for helping with various questions of problem formulation for CAD, such as choosing an optimal variable ordering. Our experiments demonstrate that this approach can be highly effective.Comment: 8 page

Program Verification in the presence of complex numbers, functions with branch cuts etc

In considering the reliability of numerical programs, it is normal to "limit our study to the semantics dealing with numerical precision" (Martel, 2005). On the other hand, there is a great deal of work on the reliability of programs that essentially ignores the numerics. The thesis of this paper is that there is a class of problems that fall between these two, which could be described as "does the low-level arithmetic implement the high-level mathematics". Many of these problems arise because mathematics, particularly the mathematics of the complex numbers, is more difficult than expected: for example the complex function log is not continuous, writing down a program to compute an inverse function is more complicated than just solving an equation, and many algebraic simplification rules are not universally valid. The good news is that these problems are theoretically capable of being solved, and are practically close to being solved, but not yet solved, in several real-world examples. However, there is still a long way to go before implementations match the theoretical possibilities

A "Piano Movers" Problem Reformulated

It has long been known that cylindrical algebraic decompositions (CADs) can in theory be used for robot motion planning. However, in practice even the simplest examples can be too complicated to tackle. We consider in detail a "Piano Mover's Problem" which considers moving an infinitesimally thin piano (or ladder) through a right-angled corridor. Producing a CAD for the original formulation of this problem is still infeasible after 25 years of improvements in both CAD theory and computer hardware. We review some alternative formulations in the literature which use differing levels of geometric analysis before input to a CAD algorithm. Simpler formulations allow CAD to easily address the question of the existence of a path. We provide a new formulation for which both a CAD can be constructed and from which an actual path could be determined if one exists, and analyse the CADs produced using this approach for variations of the problem. This emphasises the importance of the precise formulation of such problems for CAD. We analyse the formulations and their CADs considering a variety of heuristics and general criteria, leading to conclusions about tackling other problems of this form.Comment: 8 pages. Copyright IEEE 201

Choosing a variable ordering for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition

Cylindrical algebraic decomposition (CAD) is a key tool for solving problems in real algebraic geometry and beyond. In recent years a new approach has been developed, where regular chains technology is used to first build a decomposition in complex space. We consider the latest variant of this which builds the complex decomposition incrementally by polynomial and produces CADs on whose cells a sequence of formulae are truth-invariant. Like all CAD algorithms the user must provide a variable ordering which can have a profound impact on the tractability of a problem. We evaluate existing heuristics to help with the choice for this algorithm, suggest improvements and then derive a new heuristic more closely aligned with the mechanics of the new algorithm

PLIT: An alignment-free computational tool for identification of long non-coding RNAs in plant transcriptomic datasets

Long non-coding RNAs (lncRNAs) are a class of non-coding RNAs which play a significant role in several biological processes. RNA-seq based transcriptome sequencing has been extensively used for identification of lncRNAs. However, accurate identification of lncRNAs in RNA-seq datasets is crucial for exploring their characteristic functions in the genome as most coding potential computation (CPC) tools fail to accurately identify them in transcriptomic data. Well-known CPC tools such as CPC2, lncScore, CPAT are primarily designed for prediction of lncRNAs based on the GENCODE, NONCODE and CANTATAdb databases. The prediction accuracy of these tools often drops when tested on transcriptomic datasets. This leads to higher false positive results and inaccuracy in the function annotation process. In this study, we present a novel tool, PLIT, for the identification of lncRNAs in plants RNA-seq datasets. PLIT implements a feature selection method based on L1 regularization and iterative Random Forests (iRF) classification for selection of optimal features. Based on sequence and codon-bias features, it classifies the RNA-seq derived FASTA sequences into coding or long non-coding transcripts. Using L1 regularization, 31 optimal features were obtained based on lncRNA and protein-coding transcripts from 8 plant species. The performance of the tool was evaluated on 7 plant RNA-seq datasets using 10-fold cross-validation. The analysis exhibited superior accuracy when evaluated against currently available state-of-the-art CPC tools

Design, development and test of shuttle/Centaur G-prime cryogenic tankage thermal protection systems

The thermal protection systems for the shuttle/Centaur would have had to provide fail-safe thermal protection during prelaunch, launch ascent, and on-orbit operations as well as during potential abort. The thermal protection systems selected used a helium-purged polyimide foam beneath three rediation shields for the liquid-hydrogen tank and radiation shields only for the liquid-oxygen tank (three shields on the tank sidewall and four on the aft bulkhead). A double-walled vacuum bulkhead separated the two tanks. The liquid-hydrogen tank had one 0.75-in-thick layer of foam on the forward bulkhead and two layers on the larger area sidewall. Full scale tests of the flight vehicle in a simulated shuttle cargo bay that was purged with gaseous nitrogen gave total prelaunch heating rates of 88,500 Btu/hr and 44,000 Btu/hr for the liquid-hydrogen and -oxygen tanks, respectively. Calorimeter tests on a representative sample of the liquid-hydrogen tank sidewall thermal protection system indicated that the measured unit heating rate would rapidly decrease from the prelaunch rate of approx 100 Btu/hr/sq ft to a desired rate of less than 1.3 Btu/hr/sq ft once on orbit

A comparison of three heuristics to choose the variable ordering for CAD

Cylindrical algebraic decomposition (CAD) is a key tool for problems in real algebraic geometry and beyond. When using CAD there is often a choice over the variable ordering to use, with some problems infeasible in one ordering but simple in another. Here we discuss a recent experiment comparing three heuristics for making this choice on thousands of examples