In this paper, we introduce the notion of infinity branches as well as approaching curves. We present some properties which allow us to obtain an algorithm that compares the behavior of two implicitly defined algebraic plane curves at the infinity. As an important result, we prove that if two plane algebraic curves have the same asymptotic behavior, the Hausdorff distance between them is finite.

Herschel/PACS observations of 29 local (Ultra-)Luminous Infrared Galaxies, including both starburst and AGNdominated sources as diagnosed in the mid-infrared/optical, show that the equivalent width of the absorbing OH 65 µm Π 3/2 J = 9/2 − 7/2 line (W eq (OH65)) with lower level energy E low ≈ 300 K, is anticorrelated with the [C ii]158 µm line to far-infrared luminosity ratio, and correlated with the far-infrared luminosity per unit gas mass and with the 60-to-100 µm far-infrared color. While all sources are in the active L IR /M H2 > 50 L ⊙ /M ⊙ mode as derived from previous CO line studies, the OH65 absorption shows a bimodal distribution with a discontinuity at L FIR /M H2 ≈ 100 L ⊙ /M ⊙ . In the most buried sources, OH65 probes material partially responsible for the silicate 9.7 µm absorption. Combined with observations of the OH 71 µm Π 1/2 J = 7/2 − 5/2 doublet (E low ≈ 415 K), radiative transfer models characterized by the equivalent dust temperature, T dust , and the continuum optical depth at 100 µm, τ 100 , indicate that strong [C ii]158 µm deficits are associated with far-IR thick (τ 100 0.7, N H 10 24 cm −2 ), warm (T dust 60 K) structures where the OH 65 µm absorption is produced, most likely in circumnuclear disks/tori/cocoons. With their high L FIR /M H2 ratios and columns, the presence of these structures is expected to give rise to strong [C ii] deficits. W eq (OH65) probes the fraction of infrared luminosity arising from these compact/warm environments, which is 30 − 50% in sources with high W eq (OH65). Sources with high W eq (OH65) have surface densities of both L IR and M H2 higher than inferred from the half-light (CO or UV/optical) radius, tracing coherent structures that represent the most buried/active stage of (circum)nuclear starburst-AGN co-evolution.

Approximate aggregation techniques consist of introducing certain approximations that allow one to reduce a complex system involving many coupled variables obtaining a simpler "aggregated system" governed by a few "macrovariables". Moreover, they give results that allow one to extract information about the complex original system in terms of the behavior of the reduced one. Often, the feature that allows one to carry out such a reduction is the presence of different time scales.In this work we deal with the approximate aggregation of a model for a population subjected to demographic stochasticity and whose dynamics is controlled by two processes with different time scales. There are no restrictions on the slow process while the fast process is supposed to be conservative of the total number of individuals. The incorporation of the effects of demographic stochasticity in the dynamics of the population makes both the fast and the slow processes being modelled by two multi-type Galton-Watson branching processes. We present a multi-type global model that incorporates the combined effect of the fast and slow processes and develop a method that takes advantage of the difference of time scales to reduce the model obtaining an "aggregated" simpler system. We show that, given the separation of time scales between the two processes is high enough, we can obtain relevant information about the behavior of the multi-type global model through the study of this simple aggregated system.

We develop a method for computing all the generalized asymptotes of a real plane algebraic curve C over C implicitly defined by an irreducible polynomial f (x, y) ∈ R[x, y]. The approach is based on the notion of perfect curve introduced from the concepts and results presented in [2].

In this paper, we generalize the results presented in [4] for the case of real algebraic space curves. More precisely, given an algebraic space curve C implicitly defined, we show how to compute the generalized asymptotes. In addition, we show how to deal with this problem for the case of a given curve C parametrically defined. The approaches are based on the notion of approaching curves introduced in [5].