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

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

Extented ionized gas emission and kinematics of the compact group galaxies in HCG 16: Signatures of mergers

We report on kinematic observations of Ha emission line from four late-type galaxies of Hickson Compact Group 16 (H16a,b,c and d) obtained with a scanning Fabry-Perot interferometer and samplings of 16 km/s and 1". The velocity fields show kinematic peculiarities for three of the four galaxies: H16b, c and d. Misalignments between the kinematic and photometric axes of gas and stellar components (H16b,c,d), double gas systems (H16c) and severe warping of the kinematic major axis (H16b and c) were some of the peculiarities detected. We conclude that major merger events have taken place in at least two of the galaxies group. H16c and d, based on their significant kinematic peculiarities, their double nuclei and high infrared luminosities. Their Ha gas content is strongly spatially concentred - H16d contains a peculiar bar-like structure confined to the inner $\sim$ 1 h^-1 kpc region. These observations are in agreement with predictions of simulations, namely that the gas flows towards the galaxy nucleus during mergers, forms bars and fuel the central activity. Galaxy H16b, and Sb galaxy, also presents some of the kinematic evidences for past accretion events. Its gas content, however, is very spare, limiting our ability to find other kinematic merging indicators, if they are present. We find that isolated mergers, i.e., they show an anormorphous morphology and no signs of tidal tails. Tidal arms and tails formed during the mergers may have been stripped by the group potential (Barnes & Hernquist 1992) ar alternatively they may have never been formed. Our observations suggest that HCG 16 may be a young compact group in formation throught the merging of close-by objects in a dense environment.Comment: Accepted for publication in ApJ. 35 pages, 13 figures. tar file gzipped and uuencode

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