Properties of active galaxies at the extreme of Eigenvector 1

Context. Eigenvector 1 (EV1) is the formal parameter which allows the introduction of some order in the properties of the unobscured type 1 active galaxies. Aims. We aim to understand the nature of this parameter by analyzing the most extreme examples of quasars with the highest possible values of the corresponding eigenvalues RFe. Methods. We selected the appropriate sources from the Sloan Digital Sky Survey (SDSS) and performed detailed modeling, including various templates for the Fe II pseudo-continuum and the starlight contribution to the spectrum. Results. Out of 27 sources with RFe larger than 1.3 and with the measurement errors smaller than 20% selected from the SDSS quasar catalog, only six sources were confirmed to have a high value of RFe, defined as being above 1.3. All other sources have an RFe of approximately 1. Three of the high RFe objects have a very narrow Hβ line, below 2100 km s−1 but three sources have broad lines, above 4500 km s−1, that do not seem to form a uniform group, differing considerably in black hole mass and Eddington ratio; they simply have a very similar EW([OIII]5007) line. Therefore, the interpretation of the EV1 remains an open issue.

EXTENSION OF FUNCTORS FOR ALGEBRAS OF FORMAL DEFORMATION

Suppose we are given complex manifoldsXandYtogether with substacks$\mathcal{S}$and$\mathcal{S}'$of modules over algebras of formal deformation$\mathcal{A}$onXand$\mathcal{A}'$onY, respectively. Also, suppose we are given a functor Φ from the category of open subsets ofXto the category of open subsets ofYtogether with a functorFof prestacks from$\mathcal{S}$to$\mathcal{S}'\circ\Phi$. Then we give conditions for the existence of a canonical functor, extension ofFto the category of coherent$\mathcal{A}$-modules such that the cohomology associated to the action of the formal parameter$\hbar$takes values in$\mathcal{S}$. We give an explicit construction and prove that when the initial functorFis exact on each open subset, so is its extension. Our construction permits to extend the functors of inverse image, Fourier transform, specialisation and micro-localisation, nearby and vanishing cycles in the framework of$\mathcal{D}[[\hbar]]$-modules. We also obtain the Cauchy–Kowalewskaia–Kashiwara theorem in the non-characteristic case as well as comparison theorems for regular holonomic$\mathcal{D}[[\hbar]]$-modules and a coherency criterion for proper direct images of good$\mathcal{D}[[\hbar]]$-modules.

Diagrammatic logic applied to a parameterisation process

This paper provides an abstract definition of a class of logics, called diagrammatic logics, together with a definition of morphisms and 2-morphisms between them. The definition of the 2-category of diagrammatic logics relies on category theory, mainly on adjunction, categories of fractions and limit sketches. This framework is applied to the formalisation of a parameterisation process. This process, which consists of adding a formal parameter to some operations in a given specification, is presented as a morphism of logics. Then the parameter passing process for recovering a model of the given specification from a model of the parameterised specification and an actual parameter is shown to be a 2-morphism of logics.

DEFORMATIONS AND INVERSION FORMULAS FOR FORMAL AUTOMORPHISMS IN NONCOMMUTATIVE VARIABLES

Let z = (z1, z2,…, zn) be noncommutative free variables and t a formal parameter which commutes with z. Let k be any unital integral domain of any characteristic and Ft(z) = z - Ht(z) with Ht(z) ∈ k[[t]]〈〈z〉〉×n and the order o(Ht(z))≥ 2. Note that Ft(z) can be viewed as a deformation of the formal map F(z):= z - Ht=1(z) when it makes sense (for example, when Ht(z) ∈ k[t]〈〈z〉〉×n). The inverse map Gt(z) of Ft(z) can always be written as Gt(z) = z+Mt(z) with Mt(z) ∈ k[[t]]〈〈z〉〉×n and o(Mt(z)) ≥ 2. In this paper, we first derive the PDEs satisfied by Mt(z) and u(Ft), u(Gt) ∈ k[[t]]〈〈z〉〉 with u(z) ∈ k〈〈z〉〉 in the general case as well as in the special case when Ht(z) = tH(z) for some H(z) ∈ k〈〈z〉〉×n. We also show that the elements above are actually characterized by certain Cauchy problems of these PDEs. Secondly, we apply the derived PDEs to prove a recurrent inversion formula for formal maps in noncommutative variables. Finally, for the case char. k = 0, we derive an expansion inversion formula by the planar binary rooted trees.