The complexity of the Multiple Pattern Matching Problem for random strings

We generalise a multiple string pattern matching algorithm, recently proposed by Fredriksson and Grabowski [J. Discr. Alg. 7, 2009], to deal with arbitrary dictionaries on an alphabet of size $s$. If $r_m$ is the number of words of length $m$ in the dictionary, and $\phi(r) = \max_m \ln(s\, m\, r_m)/m$, the complexity rate for the string characters to be read by this algorithm is at most $\kappa_{{}_\textrm{UB}}\, \phi(r)$ for some constant $\kappa_{{}_\textrm{UB}}$. On the other side, we generalise the classical lower bound of Yao [SIAM J. Comput. 8, 1979], for the problem with a single pattern, to deal with arbitrary dictionaries, and determine it to be at least $\kappa_{{}_\textrm{LB}}\, \phi(r)$. This proves the optimality of the algorithm, improving and correcting previous claims.Comment: 25 pages, 4 figure

The merger remnant NGC 3610 and its globular cluster system: a large-scale study

We present a photometric study of the prototype merger remnant NGC 3610 and its globular cluster (GC) system, based on new GEMINI/GMOS and ACS/HST archival images. Thanks to the large FOV of our GMOS data, larger than previous studies, we are able to detect a `classical' bimodal GC colour distribution, corresponding to metal-poor and metal-rich GCs, at intermediate radii and a small subsample of likely young clusters of intermediate colours, mainly located in the outskirts. The extent of the whole GC system is settled as about 40 kpc. The GC population is quite poor, about 500 +/- 110 members, that corresponds to a low total specific frequency S_N ~ 0.8. The effective radii of a cluster sample are determined, including those of two spectroscopically confirmed young and metal-rich clusters, that are in the limit between GC and UCD sizes and brightness. The large-scale galaxy surface-brightness profile can be decomposed as an inner embedded disc and an outer spheroid, determining for both larger extents than earlier research (10 kpc and 30 kpc, respectively). We detect boxy isophotes, expected in merger remnants, and show a wealth of fine-structure in the surface-brightness distribution with unprecedented detail, coincident with the outer spheroid. The lack of symmetry in the galaxy colour map adds a new piece of evidence to the recent merger scenario of NGC 3610.Comment: 14 pages, 16 figures. Accepted for publication in MNRA

On the genericity of Whitehead minimality

We show that a finitely generated subgroup of a free group, chosen uniformly at random, is strictly Whitehead minimal with overwhelming probability. Whitehead minimality is one of the key elements of the solution of the orbit problem in free groups. The proofs strongly rely on combinatorial tools, notably those of analytic combinatorics. The result we prove actually depends implicitly on the choice of a distribution on finitely generated subgroups, and we establish it for the two distributions which appear in the literature on random subgroups

Rich and slim, but relatively short Explaining the halt in the secular trend in Japan

An almost complete halt in the secular trend in stature at a relatively low level is observed in Japan since the late 1980s with average height of around 171 cm for males and 158 cm for females at age 18. Unidentified characteristics in the Japanese genetic pool or in the nutritional intake do not provide a convincing explanation. Japan is unique among OECD countries in combining contrasted health outcomes: a stagnation of height suggests a decline in biological well-being, but this picture is not consistent with high life expectancy and extremely low prevalence of infant mortality, overweight/obesity, and other pathologies. Individual data that could allow investigating the influence of socio-economic and other environmental conditions are unavailable. As a second best, we take advantage of the regional variance in average height and other indicators across the 47 Japanese prefectures and use data covering the period 1950-2005. A positive and significant influence of income and housing conditions on height is identified but the effect is fading. Caloric restraint of pregnant women, and the decrease in sleeping time observed since the 1980s appear as possible explanatory variables of the halt in the secular trend and a symptom of a decline in well-being. Public health policy implications are considered.height, income, housing, sleep, sexual dimorphism, Japan

Generic properties of subgroups of free groups and finite presentations

Asymptotic properties of finitely generated subgroups of free groups, and of finite group presentations, can be considered in several fashions, depending on the way these objects are represented and on the distribution assumed on these representations: here we assume that they are represented by tuples of reduced words (generators of a subgroup) or of cyclically reduced words (relators). Classical models consider fixed size tuples of words (e.g. the few-generator model) or exponential size tuples (e.g. Gromov's density model), and they usually consider that equal length words are equally likely. We generalize both the few-generator and the density models with probabilistic schemes that also allow variability in the size of tuples and non-uniform distributions on words of a given length.Our first results rely on a relatively mild prefix-heaviness hypothesis on the distributions, which states essentially that the probability of a word decreases exponentially fast as its length grows. Under this hypothesis, we generalize several classical results: exponentially generically a randomly chosen tuple is a basis of the subgroup it generates, this subgroup is malnormal and the tuple satisfies a small cancellation property, even for exponential size tuples. In the special case of the uniform distribution on words of a given length, we give a phase transition theorem for the central tree property, a combinatorial property closely linked to the fact that a tuple freely generates a subgroup. We then further refine our results when the distribution is specified by a Markovian scheme, and in particular we give a phase transition theorem which generalizes the classical results on the densities up to which a tuple of cyclically reduced words chosen uniformly at random exponentially generically satisfies a small cancellation property, and beyond which it presents a trivial group

CCD photometry in the region of NGC 6994: the remains of an old open cluster

We present the results of BV(RI)_KC CCD photometry down to V=21 mag in the region of NGC 6994. To our knowledge, no photometry has previously been reported for this object and we find evidences that it is a poor and sparse old open cluster, with a minimum angular diameter of 9 arcmin, i.e. larger than the 3 arcmin originally assigned to it. We obtain a color excess E(B-V) = 0.07 +/- 0.02 mag by means of the BVI_(C) technique. Based on the theoretical isochrones from VandenBergh (1985) that are in better agreement with our data, we estimate for this cluster a distance from the Sun of 620 pc (Vo-Mv = 9 +/- 0.25 mag) and an age lying within the range of 2 - 3 Gyr, adopting solar metallicity. Thus, the corresponding cluster's Galactocentric distance is 8.1 kpc and is placed at about 350 pc below the Galactic plane. According to this results, NGC 6994 belongs to the old open cluster population located in the outer disk and at large distances from the Galactic plane, and must have suffered significant individual dynamical evolution, resulting in mass segregation and evaporation of low mass stars.Comment: 10 pages including 11 figures and 1 table. Accepted for publication in Astronomy & Astrophysic

Footprints in the sand: What can globular clusters tell us about NGC 4753 past?

NGC 4753 is a bright (M_V approx -22.3) lenticular galaxy. It is a very interesting target to test different theories of formation of lenticular galaxies, due to its low-density environment and complex structure. We perform the first comprehensive study of NGC 4753 globular cluster system (GCS), using Gemini/GMOS and CTIO/MosaicII images. Our results indicate a rather poor GCS of approx 1000 members. Its azimuthal distribution follows the shape of the galaxy bulge. The GC colour distribution is peculiar, presenting an intermediate subpopulation in addition to blue and red ones. This intermediate subgroup can be explained by a single stellar population with an age of 1.5-3 Gyr and 0.5-1 Z_o. The GC specific frequency S_N = 1.3+/-0.15 is surprisingly low for a galaxy of its class. The GC luminosity function (GCLF) is also peculiar, with an excess of bright GCs compared to the expected gaussian distribution. The underlying galaxy body has significant substructure, with remnants of spiral arms, dust filaments, and isophote twisting. This, and the fact that NGC 4753 hosted two type Ia SNe, support the possibility that the intermediate GC subpopulation may have originated during a recent merger, 1-3 Gyr ago.Comment: 10 pages, 10 figures, accepted on MNRA

Intra Cluster Globular Clusters around NGC 1399 in Fornax?

We investigate whether the globular clusters (GCs) in the recently published sample of GCs in the Fornax cluster by Bergond and coworkers are indeed intra-cluster objects. We combine the catalogue of radial velocity measurements by Bergond et al. with our CTIO MOSAIC photometry in the Washington system and analyse the relation of metal-poor and metal-rich GCs with their host galaxies. The metal-rich GCs appear to be kinematically associated with their respective host galaxies. The vast majority of the metal-poor GCs found in between the galaxies of the Fornax cluster have velocities which are consistent with them being members of the very extended NGC 1399 GC system. We find that when the sample is restricted to the most accurate velocity measurements, the GC velocity dispersion profile can be described with a mass model derived for the NGC 1399 GC system within 80 kpc. We identify one ``vagrant'' GC whose radial velocity suggests that it is not bound to any galaxy unless its orbit has a very large apogalactic distance.Comment: 4 pages, 4 figures, accepted for publication as a Letter in A&

Certamen Homeri et Hesiodi: Introduction, Critical Edition and Commentary

This dissertation provides an up-to-date introduction to the Certamen Homeri et Hesiodi, a critical edition of the text, and the first commentary in English on it. The Certamen is an anonymous work composed around the second century AD. It gives an account of the lives of Homer and Hesiod and of their poetic contest by re-elaborating biographical anecdotes attested from the sixth century BC onwards. As a biographical work that draws on older texts and oral traditions which developed over hundreds of years, it yields unique insights into the reception of early Greek Epic in the course of classical antiquity. This thesis begins with an introduction to the tradition of the contest between Homer and Hesiod that collects and discusses the extant ancient accounts of that story. It argues that all versions are equally authoritative in principle, for they testify to different acts of reception of the poets in different contexts. The thesis then offers an up-to-date analysis of the manuscript witnesses of the Certamen and of their contribution to our understanding of the textual tradition of this text, and shows that the ancient biographies of the poets form a corpus that is naturally open to variation. The Edition provides a text that accounts for such an open tradition. The line-by-line Commentary offers a systematic analysis of both general and specific issues related to the text: this is a necessary and urgent task, not least because the Certamen is a stratified text, bringing together traditions of very different provenance, which can only be assessed and interpreted through a process of close reading. The ultimate aim of the thesis is to show how the story of the contest between Homer and Hesiod provides crucial insights into the processes of reception and canonisation of early hexameter epic from the archaic period to late antiquity