Fundamental groups of some special quadric arrangements

In this work we obtain presentations of fundamental groups of the complements of three families of quadric arrangements in $\mathbb{P}^2$. The first arrangement is a union of $n$ quadrics, which are tangent to each other at two common points. The second arrangement is composed of $n$ quadrics which are tangent to each other at one common point. The third arrangement is composed of $n$ quadrics, $n-1$ of them are tangent to the $n$'th one and each one of the $n-1$ quadrics is transversal to the other $n-2$ ones.Comment: 21 pages, 12 main figures, appears in Revista Mathematica

The Hardness of Finding Linear Ranking Functions for Lasso Programs

Finding whether a linear-constraint loop has a linear ranking function is an important key to understanding the loop behavior, proving its termination and establishing iteration bounds. If no preconditions are provided, the decision problem is known to be in coNP when variables range over the integers and in PTIME for the rational numbers, or real numbers. Here we show that deciding whether a linear-constraint loop with a precondition, specifically with partially-specified input, has a linear ranking function is EXPSPACE-hard over the integers, and PSPACE-hard over the rationals. The precise complexity of these decision problems is yet unknown. The EXPSPACE lower bound is derived from the reachability problem for Petri nets (equivalently, Vector Addition Systems), and possibly indicates an even stronger lower bound (subject to open problems in VAS theory). The lower bound for the rationals follows from a novel simulation of Boolean programs. Lower bounds are also given for the problem of deciding if a linear ranking-function supported by a particular form of inductive invariant exists. For loops over integers, the problem is PSPACE-hard for convex polyhedral invariants and EXPSPACE-hard for downward-closed sets of natural numbers as invariants.Comment: In Proceedings GandALF 2014, arXiv:1408.5560. I thank the organizers of the Dagstuhl Seminar 14141, "Reachability Problems for Infinite-State Systems", for the opportunity to present an early draft of this wor

On Decidable Growth-Rate Properties of Imperative Programs

In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple "core" programming language - an imperative language with bounded loops, and arithmetics limited to addition and multiplication - it was possible to decide precisely whether a program had certain growth-rate properties, namely polynomial (or linear) bounds on computed values, or on the running time. This work emphasized the role of the core language in mitigating the notorious undecidability of program properties, so that one deals with decidable problems. A natural and intriguing problem was whether more elements can be added to the core language, improving its utility, while keeping the growth-rate properties decidable. In particular, the method presented could not handle a command that resets a variable to zero. This paper shows how to handle resets. The analysis is given in a logical style (proof rules), and its complexity is shown to be PSPACE-complete (in contrast, without resets, the problem was PTIME). The analysis algorithm evolved from the previous solution in an interesting way: focus was shifted from proving a bound to disproving it, and the algorithm works top-down rather than bottom-up

A Comment on Budach's Mouse-in-an-Octant Problem

Budach's Mouse-in-an-Octant Problem (attributed to Lothar Budach in a 1980 article by van Emde Boas and Karpinski) concerns the behaviour of a very simple finite-state machine ("the mouse") moving on the integer two-dimensional grid. Its decidability is apparently still open. This note sketches a proof that an extended version of the problem (a super-mouse) is undecidable.Comment: 3 pages, 2 bibliographic reference

Tight polynomial worst-case bounds for loop programs

In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple programming language - representing non-deterministic imperative programs with bounded loops, and arithmetics limited to addition and multiplication - it is possible to decide precisely whether a program has certain growth-rate properties, in particular whether a computed value, or the program's running time, has a polynomial growth rate. A natural and intriguing problem was to move from answering the decision problem to giving a quantitative result, namely, a tight polynomial upper bound. This paper shows how to obtain asymptotically-tight, multivariate, disjunctive polynomial bounds for this class of programs. This is a complete solution: whenever a polynomial bound exists it will be found. A pleasant surprise is that the algorithm is quite simple; but it relies on some subtle reasoning. An important ingredient in the proof is the forest factorization theorem, a strong structural result on homomorphisms into a finite monoid

Ranking Functions for Size-Change Termination II

Size-Change Termination is an increasingly-popular technique for verifying program termination. These termination proofs are deduced from an abstract representation of the program in the form of "size-change graphs". We present algorithms that, for certain classes of size-change graphs, deduce a global ranking function: an expression that ranks program states, and decreases on every transition. A ranking function serves as a witness for a termination proof, and is therefore interesting for program certification. The particular form of the ranking expressions that represent SCT termination proofs sheds light on the scope of the proof method. The complexity of the expressions is also interesting, both practicaly and theoretically. While deducing ranking functions from size-change graphs has already been shown possible, the constructions in this paper are simpler and more transparent than previously known. They improve the upper bound on the size of the ranking expression from triply exponential down to singly exponential (for certain classes of instances). We claim that this result is, in some sense, optimal. To this end, we introduce a framework for lower bounds on the complexity of ranking expressions and prove exponential lower bounds.Comment: 29 pages

A Dynamical Study of Galaxies in the Hickson Compact Groups

In order to investigate dynamical properties of spiral galaxies in the Hickson compact groups (HCGs), we present rotation curves of 30 galaxies in 20 HCGs. We found as follows. 1) There is not significant relation between dynamical peculiarity and morphological peculiarity in HCG spirals. 2) There is no significant relation between the dynamical properties and the frequency distribution of nuclear activities in HCG spirals. 3) There are no significant correlations between the dynamical properties of HCG spirals and any group properties (i.e., the size, the velocity dispersion, the galaxy number density, and the crossing time). 4) Asymmetric and peculiar rotation curves are more frequently seen in the HCG spirals than in field spirals and in cluster ones. However, this tendency is more obviously seen in late-type HCG spirals. These results suggest that the dynamical properties of HCG spirals do not strongly correlate with the morphology, the nuclear activity, and the group properties. Our results also suggest that more frequent galaxy collisions occur in the HCGs than in the field and in the clusters.Comment: 24 pages test (aasms4 LaTeX), 50 page tables (aasms4 LaTeX), and 16 Postscript figures, Accepted for The Astronomical Journa

Case studies of NOAA 6/TIROS N data impact on numerical weather forecasts

The impact of satellite temperatures from systems which predate the launching of the third generation of vertical sounding instruments aboard TIROS N (13 Oct 1978) and NOAA 6 (27 June 1979) is reported. The first evaluation of soundings from TIROS N found that oceanic, cloudy retrievals over NH mid latitudes show a cold bias in winter. It is confirmed for both satellite systems using a larger data base. It is shown that RMS differences between retrievals and colocated radiosonde observations within the swath 30-60N during the 1979-80 winter were generally 2-3K in clear air and higher for cloudy columns. A positive impact of TIROS N temperatures on the analysis of synoptic weather systems is shown. Analyses prepared from only satellite temperatures seemed to give a better definition to weather systems' thermal structure than that provided by corresponding NMC analyses without satellite data. The results of a set of 14 numerical forecast experiments performed with the PE model of the Israel Meteorological Service (IMS) are summarized; these were designed to test the impact of TIROS N and NOAA 6 temperatures within the IMS analysis and forecast cycle. The satellite data coverage over the NH, the mean area/period S1 and RMS verification scores and the spatial distribution of SAT versus NO SAT forecast differences are discussed and it is concluded that positive forecast impact occurs over ocean areas where the extra data improve the specification which is otherwise available from conventional observations. The forecast impact for three cases from the same set of experiments was examined and it is found that satellite temperatures, observed over the Atlantic Ocean contribute to better forecasts over Iceland and central Europe although a worse result was verified over Spain. It is also shown that the better scores of a forecast based also on satellite data and verified over North America actually represent a mixed impact on the forecast synoptic patterns. A superior 48 hr 500 mb forecast over the western US due to the better initial specification afforded by satellite observed temperatures over the North Pacific Ocean is shown