Número de Milnor associado a curvas reduzidas

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Topological invariants of isolated complete intersection curve singularities

In this paper we present some formulae for topological invariants of projective complete intersection curves with isolated singularities in terms of the Milnor number, the Euler characteristic and the topological genus. We also present some conditions, involving the Milnor number and the degree of the curve, for the irreducibility of complete intersection curves.

Finite determinacy and Whitney equisingularity of map germs from C(n) to C(2n-1)

We show that a holomorphic map germ f : (C(n), 0) -> (C(2n-1), 0) is finitely determined if and only if the double point scheme D(f) is a reduced curve. If n >= 3, we have that mu(D(2)(f)) = 2 mu(D(2)(f)/S(2))+C(f)-1, where D(2)(f) is the lifting of the double point curve in (C(n) x C(n), 0), mu(X) denotes the Milnor number of X and C(f) is the number of cross-caps that appear in a stable deformation of f. Moreover, we consider an unfolding F(t, x) = (t, f(t)(x)) of f and show that if F is mu-constant, then it is excellent in the sense of Gaffney. Finally, we find a minimal set of invariants whose constancy in the family f(t) is equivalent to the Whitney equisingularity of F. We also give an example of an unfolding which is topologically trivial, but it is not Whitney equisingular.

Multiplicity of Boardman strata and deformations of map germs

We define algebraically for each map germ f: Kn, 0→ Kp, 0 and for each Boardman symbol i = (il, . . ., ik) a number ci(f) which is script A sign-invariant. If f is finitely determined, this number is the generalization of the Milnor number of f when p = 1, the number of cusps of f when n = p = 2, or the number of cross caps when n = 2, p = 3. We study some properties of this number and prove that, in some particular cases, this number can be interpreted geometrically as the number of Σi points that appear in a generic deformation of f. In the last part, we compute this number in the case that the map germ is a projection and give some applications to catastrophe map germs.

Double point curves for corank 2 map germs from C-2 to C-3

We characterize finite determinacy of map germs f : (C-2, 0) -> (C-3, 0) in terms of the Milnor number mu(D(f)) of the double point curve D(f) in (C-2, 0) and we provide an explicit description of the double point scheme in terms of elementary symmetric functions. Also we prove that the Whitney equisingularity of 1-parameter families of map germs f(t) : (C-2, 0) -> (C-3, 0) is equivalent to the constancy of both mu(D(f(t))) and mu(f(t)(C-2)boolean AND H) with respect to t, where H subset of C-3 is a generic plane. (C) 2011 Elsevier B.V. All rights reserved.

Lê–Greuel type formula for the Euler obstruction and applications

The Euler obstruction of a function f can be viewed as a generalization of the Milnor number for functions defined on singular spaces. In this work, using the Euler obstruction of a function, we establish several Lê–Greuel type formulas for germs f:(X,0)→(C,0) and g:(X,0)→(C,0). We give applications when g is a generic linear form and when f and g have isolated singularities.

Watson-Crick pairing, the Heisenberg group and Milnor invariants

We study the secondary structure of RNA determined by Watson-Crick pairing without pseudo-knots using Milnor invariants of links. We focus on the first non-trivial invariant, which we call the Heisenber invariant. The Heisenberg invariant, which is an integer, can be interpreted in terms of the Heisenberg group as well as in terms of lattice paths. We show that the Heisenberg invariant gives a lower bound on the number of unpaired bases in an RNA secondary structure. We also show that the Heisenberg invariant can predict allosteric structures for RNA. Namely, if the Heisenberg invariant is large, then there are widely separated local maxima (i.e., allosteric structures) for the number of Watson-Crick pairs found.

On the Milnor fibre and L numbers of semi-weighted homogeneous arrangements

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

On the Milnor fibre and L numbers of semi-weighted homogeneous arrangements

Inthispaperwestudygermsofpolynomialsformedbytheproductofsemi-weighted homogeneous polynomials of the same type, which we call semi-weighted homogeneous arrangements. It is shown how the L numbers of such polynomials are computed using only their weights and degree of homogeneity. A key point of the main theorem is to find the number called polar ratio of this polynomial class. An important consequence is the description of the Euler characteristic of the Milnor fibre of such arrangements only depending on their weights and degree of homogeneity. The constancy of the L numbers in families formed by such arrangements is shown, with the deformed terms having weighted degree greater than the weighted degree of the initial germ. Moreover, using the results of Massey applied to families of function germs, we obtain the constancy of the homology of the Milnor fibre in this family of semi-weighted homogeneous arrangements.

Smoothing a Discrete Hazard Function for the Number of Patients Colonized with Methicillin-Resistance Staphylococcus Aureus in an Intensive Care Unit

A new method for estimating the time to colonization of Methicillin-resistant Staphylococcus Aureus (MRSA) patients is developed in this paper. The time to colonization of MRSA is modelled using a Bayesian smoothing approach for the hazard function. There are two prior models discussed in this paper: the first difference prior and the second difference prior. The second difference prior model gives smoother estimates of the hazard functions and, when applied to data from an intensive care unit (ICU), clearly shows increasing hazard up to day 13, then a decreasing hazard. The results clearly demonstrate that the hazard is not constant and provide a useful quantification of the effect of length of stay on the risk of MRSA colonization which provides useful insight.

Particle number and mass emission factors from combustion of wood samples collected from Queensland forest

Knowledge of particle emission characteristics associated with forest fires and in general, biomass burning, is becoming increasingly important due to the impact of these emissions on human health. Of particular importance is developing a better understanding of the size distribution of particles generated from forest combustion under different environmental conditions, as well as provision of emission factors for different particle size ranges. This study was aimed at quantifying particle emission factors from four types of wood found in South East Queensland forests: Spotted Gum (Corymbia citriodora), Red Gum (Eucalypt tereticornis), Blood Gum (Eucalypt intermedia), and Iron bark (Eucalypt decorticans); under controlled laboratory conditions. The experimental set up included a modified commercial stove connected to a dilution system designed for the conditions of the study. Measurements of particle number size distribution and concentration resulting from the burning of woods with a relatively homogenous moisture content (in the range of 15 to 26 %) and for different rates of burning were performed using a TSI Scanning Mobility Particle Sizer (SMPS) in the size range from 10 to 600 nm and a TSI Dust Trak for PM2.5. The results of the study in terms of the relationship between particle number size distribution and different condition of burning for different species show that particle number emission factors and PM2.5 mass emission factors depend on the type of wood and the burning rate; fast burning or slow burning. The average particle number emission factors for fast burning conditions are in the range of 3.3 x 1015 to 5.7 x 1015 particles/kg, and for PM2.5 are in the range of 139 to 217 mg/kg.

Long-term trends in fine particle number concentrations in the urban atmosphere of Brisbane : the relevance of traffic emissions and new particle formation

The measurement of submicrometre (< 1.0 m) and ultrafine particles (diameter < 0.1 m) number concentration have attracted attention since the last decade because the potential health impacts associated with exposure to these particles can be more significant than those due to exposure to larger particles. At present, ultrafine particles are not regularly monitored and they are yet to be incorporated into air quality monitoring programs. As a result, very few studies have analysed their long-term and spatial variations in ultrafine particle concentration, and none have been in Australia. To address this gap in scientific knowledge, the aim of this research was to investigate the long-term trends and seasonal variations in particle number concentrations in Brisbane, Australia. Data collected over a five-year period were analysed using weighted regression models. Monthly mean concentrations in the morning (6:00-10:00) and the afternoon (16:00-19:00) were plotted against time in months, using the monthly variance as the weights. During the five-year period, submicrometre and ultrafine particle concentrations increased in the morning by 105.7% and 81.5% respectively whereas in the afternoon there was no significant trend. The morning concentrations were associated with fresh traffic emissions and the afternoon concentrations with the background. The statistical tests applied to the seasonal models, on the other hand, indicated that there was no seasonal component. The spatial variation in size distribution in a large urban area was investigated using particle number size distribution data collected at nine different locations during different campaigns. The size distributions were represented by the modal structures and cumulative size distributions. Particle number peaked at around 30 nm, except at an isolated site dominated by diesel trucks, where the particle number peaked at around 60 nm. It was found that ultrafine particles contributed to 82%-90% of the total particle number. At the sites dominated by petrol vehicles, nanoparticles (< 50 nm) contributed 60%-70% of the total particle number, and at the site dominated by diesel trucks they contributed 50%. Although the sampling campaigns took place during different seasons and were of varying duration these variations did not have an effect on the particle size distributions. The results suggested that the distributions were rather affected by differences in traffic composition and distance to the road. To investigate the occurrence of nucleation events, that is, secondary particle formation from gaseous precursors, particle size distribution data collected over a 13 month period during 5 different campaigns were analysed. The study area was a complex urban environment influenced by anthropogenic and natural sources. The study introduced a new application of time series differencing for the identification of nucleation events. To evaluate the conditions favourable to nucleation, the meteorological conditions and gaseous concentrations prior to and during nucleation events were recorded. Gaseous concentrations did not exhibit a clear pattern of change in concentration. It was also found that nucleation was associated with sea breeze and long-range transport. The implications of this finding are that whilst vehicles are the most important source of ultrafine particles, sea breeze and aged gaseous emissions play a more important role in secondary particle formation in the study area.

Year 6 students' idiosyncratic notions of unitising, reunitising and regrouping decimal number places

Having flexible notions of the unit (e.g., 26 ones can be thought of as 2.6 tens, 1 ten 16 ones, 260 tenths, etc.) should be a major focus of elementary mathematics education. However, often these powerful notions are relegated to computations where the major emphasis is on "getting the right answer" thus procedural knowledge rather than conceptual knowledge becomes the primary focus. This paper reports on 22 high-performing students' reunitising processes ascertained from individual interviews on tasks requiring unitising, reunitising and regrouping; errors were categorised to depict particular thinking strategies. The results show that, even for high-performing students, regrouping is a cognitively complex task. This paper analyses this complexity and draws inferences for teaching.